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''of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f0410901.png" /> in a system of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f0410902.png" /> which are orthonormal on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f0410903.png" />''
+
''of a function $f$ in a system of functions $\{\phi_n\}$ which are orthonormal on an interval $(a,b)$.''
 +
 
 +
{{MSC|42A|42B}}
 +
{{TEX|done}}
 +
$\newcommand{\abs}[1]{\left|#1\right|}$
  
 
The series
 
The series
 
+
$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f0410904.png" /></td> </tr></table>
+
\sum_{k=0}^\infty c_k \phi_k
 
+
$$
 
whose coefficients are determined by
 
whose coefficients are determined by
 +
\begin{equation}
 +
\label{eq1}
 +
c_k = \int_a^b f(x)\phi_k(x) \rd x, \quad k=1,2,\ldots
 +
\end{equation}
 +
These  coefficients are called the Fourier coefficients of $f$. In general it  is assumed that $f$ is square integrable on $(a,b) $. For many systems  $\{\phi_k\}$ this requirement can be relaxed by replacing it by another  which ensures the existence of all the integrals in \ref{eq1}.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f0410905.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
The Fourier series in the trigonometric system is defined for every function $f$ that is integrable on $(0,2\pi)$. It is the series
 
+
\begin{equation}
These coefficients are called the Fourier coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f0410906.png" />. In general it is assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f0410907.png" /> is square integrable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f0410908.png" />. For many systems <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f0410909.png" /> this requirement can be relaxed by replacing it by another which ensures the existence of all the integrals in (1).
+
\label{eq2}
 
+
\frac{a_0}{2} + \sum_{k=1}^\infty \bigl( a_k \cos kx + b_k \sin kx \bigr)
The Fourier series in the trigonometric system is defined for every function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109010.png" /> that is integrable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109011.png" />. It is the series
+
\end{equation}
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109012.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
 
 
 
 
with coefficients
 
with coefficients
 +
\begin{equation}
 +
\label{eq3}
 +
a_k = \frac{1}{\pi}\int_0^{2\pi} f(x)\cos kx \rd x, \quad
 +
b_k = \frac{1}{\pi}\int_0^{2\pi} f(x)\sin kx \rd x.
 +
\end{equation}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109013.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
Fourier series for functions in several variables are constructed analogously. A further generalization leads to Fourier coefficients and Fourier series for elements of a Hilbert space.
 
 
Fourier series for functions in several variables are constructed analogously. A further generalization leads to Fourier coefficients and Fourier series for elements of a Hilbert space.
 
 
 
The theory of Fourier series in the trigonometric system has been most thoroughly developed, and these were the first examples of Fourier series. If one has in mind Fourier series in the trigonometric system, it is usual to talk simply of Fourier series, without indicating the system by which they are constructed.
 
 
 
Fourier series form a considerable part of the theory of [[Trigonometric series|trigonometric series]]. Fourier series first appeared in the papers of J. Fourier (1807) devoted to an investigation of the problems of heat conduction. He suggested representing a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109014.png" /> given on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109015.png" /> by the trigonometric series (2) with coefficients determined by (3). Such a choice of coefficients is natural from many points of view. For example, if the series (2) converges uniformly to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109016.png" />, then term-by-term integration leads to the expressions for the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109018.png" /> given in (3). These formulas had been obtained already by L. Euler (1777) by term-by-term integration.
 
  
Using (3) the Fourier series (2) can be constructed for every function that is integrable over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109019.png" />. Integrability of the function can be understood in various senses, for example integrability according to Riemann or Lebesgue. Depending on this, one speaks of Fourier–Riemann series, Fourier–Lebesgue series, etc. The concepts of the Riemann and the Lebesgue integral themselves arose to a considerable extent in connection with research on Fourier series. The modern presentation of the theory of Fourier series was developed after the construction of the Lebesgue integral, and since then it has developed mainly as the theory of Fourier–Lebesgue series. Below it is assumed that the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109020.png" /> has period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109021.png" /> and is Lebesgue integrable over the period.
+
The theory of Fourier series in the trigonometric system has been most thoroughly developed, and these were the first examples of Fourier series. If one has in mind  Fourier series in the trigonometric system, it is usual to talk simply  of Fourier series, without indicating the system by which they are  constructed.
  
In the theory of Fourier series one studies the relation between the properties of functions and the properties of their Fourier series; in particular, one investigates questions on the representation of functions by Fourier series.
+
Fourier series form a considerable part of  the theory of [[Trigonometric series|trigonometric series]]. Fourier series first appeared in the papers of J.&nbsp;Fourier (1807)  devoted to an investigation of the problems of heat conduction. He  suggested representing a function $f$ given on $(0,2\pi)$ by the trigonometric series \ref{eq2} with coefficients determined by  \ref{eq3}. Such a choice of coefficients is natural from many points of  view. For example, if the series \ref{eq2} converges uniformly to $f$, then term-by-term integration leads to the expressions for the coefficients $a_k$ and $b_k$ given in \ref{eq3}.  These formulas had  been obtained already by L.&nbsp;Euler (1777) by term-by-term  integration.
  
The proof of a minimum property of the partial sums of Fourier series goes back to the work of F. Bessel (1828): Given an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109022.png" />, then among all the trigonometric polynomials of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109024.png" />,
+
Using \ref{eq3} the Fourier series  \ref{eq2} can be constructed for every function that is integrable over  $[0,2\pi]$. Integrability of the function can be understood in various  senses, for example integrability according to Riemann or Lebesgue.  Depending on this, one speaks of Fourier–Riemann series,  Fourier–Lebesgue series, etc. The concepts of the Riemann and the  Lebesgue integral themselves arose to a considerable extent in  connection with research on Fourier series.  The modern presentation of the theory of Fourier series was developed after the construction of the  Lebesgue integral, and since then it has developed mainly as the theory  of Fourier–Lebesgue series. Below it is assumed that the function $f$  has period $2\pi$ and is Lebesgue integrable over the period.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109025.png" /></td> </tr></table>
+
In  the theory of Fourier series one studies the relation between the  properties of functions and the properties of their Fourier series; in  particular, one investigates questions on the representation of  functions by Fourier series.
  
 +
The proof of a minimum  property of the partial sums of Fourier series goes back to the work of  F.&nbsp;Bessel (1828): Given an $f\in L_2$, then among all the  trigonometric polynomials of order $n$,
 +
$$
 +
t_n(x) = A_0 + \sum_{k=0}^n \bigl( A_k \cos kx + B_k \sin kx \bigr),
 +
$$
 
the smallest value of the integral
 
the smallest value of the integral
 
+
$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109026.png" /></td> </tr></table>
+
\frac{1}{\pi}\int_0^{2\pi} \bigl( f(x) - t_n(x) \bigr)^2 \rd x
 
+
$$
is attained for the partial sum of the Fourier series (2) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109027.png" />:
+
is attained for the partial sum of the Fourier series \ref{eq2} of $f$:
 
+
$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109028.png" /></td> </tr></table>
+
s_n(f,x) = \frac{a_0}{2} + \sum_{k=1}^n \bigl( a_k \cos kx + b_k \sin kx \bigr).
 
+
$$
 
This smallest value is equal to
 
This smallest value is equal to
 
+
$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109029.png" /></td> </tr></table>
+
\frac{1}{\pi} \int_0^{2\pi} f^2(x) \rd x -  
 
+
\left( \frac{a_0^2}{2} + \sum_{k=1}^n \bigl( a_k^2 + b_k^2 \bigr) \right).
 +
$$
 
This implies the [[Bessel inequality|Bessel inequality]]
 
This implies the [[Bessel inequality|Bessel inequality]]
 +
$$
 +
\frac{a_0^2}{2} + \sum_{k=1}^\infty \bigl( a_k^2 + b_k^2 \bigr) \leq
 +
\frac{1}{\pi} \int_0^{2\pi} f^2(x) \rd x,
 +
$$
 +
which is satisfied for every function $f$ in $L_2$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109030.png" /></td> </tr></table>
+
The system of trigonometric functions is a closed system (cf. [[Closed system of elements (functions)]]), that is, if $f \in L_2$, then the [[Parseval equality]]
 
+
$$
which is satisfied for every function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109031.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109032.png" />.
+
\frac{a_0^2}{2} + \sum_{k=1}^\infty \bigl( a_k^2 + b_k^2 \bigr) =
 
+
\frac{1}{\pi} \int_0^{2\pi} f^2(x) \rd x
The system of trigonometric functions is a closed system (cf. [[Closed system of elements (functions)|Closed system of elements (functions)]]), that is, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109033.png" />, then the [[Parseval equality|Parseval equality]]
+
$$
 
+
is valid, where $a_k$, $b_k$ are the Fourier coefficients of $f$. In particular, for functions $f$ in $L_2$ the series
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109034.png" /></td> </tr></table>
+
\begin{equation}
 
+
\label{eq4}
is valid, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109035.png" /> are the Fourier coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109036.png" />. In particular, for functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109037.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109038.png" /> the series
+
\frac{a_0^2}{2} + \sum_{k=1}^\infty \bigl( a_k^2 + b_k^2 \bigr)
 
+
\end{equation}
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109039.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
is convergent. The converse assertion also holds: If for a system of numbers $a_k$, $b_k$ the series \ref{eq4} converges, then these numbers are the Fourier coefficients of a certain function $f \in L_2$  (F.&nbsp;Riesz and E.&nbsp;Fischer, 1907).
 
 
is convergent. The converse assertion also holds: If for a system of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109040.png" /> the series (4) converges, then these numbers are the Fourier coefficients of a certain function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109041.png" /> (F. Riesz and E. Fischer, 1907).
 
 
 
The Fourier coefficients of any integrable function tend to zero. This statement is called the Riemann–Lebesgue theorem. B. Riemann proved it for Fourier–Riemann series and e dimension','../l/l057830.htm','Lebesgue function','../l/l057840.htm','Lebesgue inequality','../l/l057850.htm','Lebesgue integral','../l/l057860.htm','Lebesgue measure','../l/l057870.htm','Lebesgue summation method','../l/l057940.htm','Lebesgue theorem','../l/l057950.htm','Measure','../m/m063240.htm','Metric space','../m/m063680.htm','Metric theory of functions','../m/m063700.htm','Orthogonal series','../o/o070370.htm','Perron method','../p/p072370.htm','Potential theory','../p/p074140.htm','Regular boundary point','../r/r080680.htm','Singular integral','../s/s085570.htm','Suslin theorem','../s/s091480.htm','Urysohn–Brouwer lemma','../u/u095860.htm','Vitali variation','../v/v096790.htm')" style="background-color:yellow;">H. Lebesgue for Fourier–Lebesgue series.
 
 
 
If the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109042.png" /> is absolutely continuous, then the Fourier series for the derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109043.png" /> can be obtained by term-by-term differentation of the Fourier series for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109044.png" />. This implies that if the derivative of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109045.png" /> of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109046.png" /> is absolutely continuous, then the estimates
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109047.png" /></td> </tr></table>
 
 
 
are valid for the Fourier coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109048.png" />.
 
  
The first convergence criterion for Fourier series was obtained by P.G.L. Dirichlet in 1829. His result (the [[Dirichlet theorem|Dirichlet theorem]]) can be formulated as follows: If a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109049.png" /> has a finite number of maxima and minima over the period and is everywhere continuous, except at a finite number of points where it may have discontinuities of the first kind, then the Fourier series of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109050.png" /> converges for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109051.png" />, and, moreover, at points of continuity it converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109052.png" /> and at points of discontinuity it converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109053.png" />. Subsequently, this assertion was extended to arbitrary functions of bounded variation (C. Jordan, 1881).
+
The Fourier coefficients of any integrable function tend to zero. This  statement is called the Riemann–Lebesgue theorem. B. Riemann proved it for Fourier–Riemann series and H.&nbsp;Lebesgue for Fourier–Lebesgue  series.
  
According to the localization principle proved by Riemann (1853), the convergence or divergence of the Fourier series of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109054.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109055.png" />, and the value of the sum when it converges, depends only on the behaviour of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109056.png" /> in an arbitrarily small neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109057.png" />.
+
If the function $f$ is absolutely continuous, then the Fourier series for the derivative $f'$ can be obtained by  term-by-term differentation of the Fourier series for $f$. This implies  that if the derivative of order $r \geq 0$ of a function $f$ is  absolutely continuous, then the estimates
 +
$$
 +
a_k, b_k = o\bigl( k^{-(r+1)} \bigr), \quad k \rightarrow \infty,
 +
$$
 +
are valid for the Fourier coefficients of $f$.
  
Many different convergence criteria for Fourier series at a point are known. R. Lipschitz (1864) established that the Fourier series of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109058.png" /> converges at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109059.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109060.png" /> is satisfied for all sufficiently small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109061.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109062.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109063.png" /> are certain positive constants (the Lipschitz criterion). The [[Dini criterion|Dini criterion]] is more general: The Fourier series of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109064.png" /> converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109065.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109066.png" /> if the integral
+
The  first convergence criterion for Fourier series was obtained by  P.G.L.&nbsp; Dirichlet in 1829. His result (the [[Dirichlet  theorem]]) can be formulated as follows: If a function $f$ has a finite  number of maxima and minima over the period and is everywhere  continuous, except at a finite number of points where it may have  discontinuities of the first kind, then the Fourier series of $fconverges for all $x$, and, moreover, at points of continuity it  converges to $f(x)$ and at points of discontinuity it converges to $\bigl( f(x+0) + f(x-0) \bigr)/2$. Subsequently, this assertion was  extended to arbitrary functions of bounded variation  (C.&nbsp;Jordan, 1881).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109067.png" /></td> </tr></table>
+
According to the  localization principle proved by Riemann (1853), the convergence or  divergence of the Fourier series of a function $f$ at a point $x$, and  the value of the sum when it converges, depends only on the behaviour of  $f$ in an arbitrarily small neighbourhood of $x$.
  
converges, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109068.png" />. The value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109069.png" /> is usually taken for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109070.png" />. For example, if the Fourier series of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109071.png" /> converges at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109072.png" /> where this function is continuous, then the sum of the series is necessarily equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109073.png" />.
+
Many  different convergence criteria for Fourier series at a point are known.  R.&nbsp;Lipschitz (1864) established that the Fourier series of a  function $f$ converges at a point $x$ if $\abs{f(x+h) - f(x)} \leq M  \abs{h}^\alpha$ is satisfied for all sufficiently small $h$, where $M$  and $\alpha$ are certain positive constants (the Lipschitz criterion).  The [[Dini criterion]] is more general: The Fourier series of a function  $f$ converges to $S$ at a point $x$ if the integral
 +
$$
 +
\int_0^\pi \abs{\phi_x(t)} \frac{\mathrm{d}t}{t}
 +
$$
 +
converges,  where $\phi_x(t) = f(x+t) + f(x-t) - 2S$. The value $f(x)$ is usually taken for $S$. For example, if the Fourier series of $f$ converges at a point $x$ where this function is continuous, then the sum of the series is necessarily equal to $f(x)$.
  
 
Lebesgue (1905) proved that if
 
Lebesgue (1905) proved that if
 +
$$
 +
\int_0^h \abs{\phi_x(t)} \rd t = o(h), \quad
 +
\int_h^\pi \abs{\phi_x(t+h) - \phi_x(t)} \frac{\mathrm{d}t}{t} = o(1),
 +
$$
 +
as  $h \rightarrow 0$, then the Fourier series of $f$ converges to $S$ at  $x$. This [[Lebesgue criterion]] is stronger than all those given above  and stronger than the [[De la Vallée-Poussin criterion]] and the [[Young  criterion]].  But verifying it is usually difficult.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109074.png" /></td> </tr></table>
+
A convergence criterion of another type is given by the Hardy–Littlewood theorem (1932): The Fourier series of a function $f$ converges at a point $x$ if the following conditions are satisfied:
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109075.png" /></td> </tr></table>
 
 
 
as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109076.png" />, then the Fourier series of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109077.png" /> converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109078.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109079.png" />. This [[Lebesgue criterion|Lebesgue criterion]] is stronger than all those given above and stronger than the [[De la Vallée-Poussin criterion|de la Vallée-Poussin criterion]] and the [[Young criterion|Young criterion]]. But verifying it is usually difficult.
 
 
 
A convergence criterion of another type is given by the Hardy–Littlewood theorem (1932): The Fourier series of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109080.png" /> converges at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109081.png" /> if the following conditions are satisfied:
 
  
 
1)
 
1)
 
+
$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109082.png" /></td> </tr></table>
+
f(x+h) - f(x) = o\left( \ln^{-1}\frac{1}{\abs{h}} \right),
 
+
$$
as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109083.png" />; and
+
as $h \rightarrow 0$; and
  
 
2) the estimates
 
2) the estimates
 +
$$
 +
a_k, b_k = O\bigl( k^{-\delta} \bigr), \quad \delta > 0,
 +
$$
 +
are valid for the Fourier coefficients of $f$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109084.png" /></td> </tr></table>
+
Besides  convergence criteria for Fourier series at a point, criteria for  uniform convergence have been studied also. Let a function $f$ have  period $2\pi$ and be continuous. Then its Fourier series converges  uniformly to it on the whole real line if the [[Continuity, modulus  of|modulus of continuity]] $\omega(f,\delta)$ of $f$ satisfies the  condition
 +
$$
 +
\text{$\omega(f,\delta) \ln \delta \rightarrow 0$ as $\delta \rightarrow 0$}
 +
$$
 +
(the [[Dini–Lipschitz criterion]]) or if $f$ has bounded variation (the [[Jordan criterion]]).
  
are valid for the Fourier coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109085.png" />.
+
From  this one can obtain criteria for uniform convergence of Fourier series  on a certain interval if the localization principle for uniform  convergence is used. The latter is formulated as follows. If two  functions are equal on an interval $[a,b]$, then on each strictly  interior interval $[a+\epsilon,b-\epsilon]$, $\epsilon > 0$, either  the Fourier series of these functions are both uniformly convergent or  neither is uniformly convergent. In other words, the uniform convergence  of the Fourier series of a function $f$ on an interval depends only on  the behaviour of $f$ in an arbitrarily small extension of this interval.
  
Besides convergence criteria for Fourier series at a point, criteria for uniform convergence have been studied also. Let a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109086.png" /> have period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109087.png" /> and be continuous. Then its Fourier series converges uniformly to it on the whole real line if the modulus of continuity (cf. [[Continuity, modulus of|Continuity, modulus of]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109088.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109089.png" /> satisfies the condition
+
P.  du Bois-Reymond (1876) established that the continuity of a function at a certain point does not guarantee that its Fourier series converges at  this point. Later it was proved that the Fourier series of a continuous  function may diverge on an everywhere-dense set of measure zero that is  of the second category.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109090.png" /></td> </tr></table>
+
If nothing is assumed about  the function except that it is integrable, then its Fourier series may  turn out to be divergent almost-everywhere, or even everywhere. The  first examples of such functions were constructed by  A.N.&nbsp;Kolmogorov (1923, 1926). Later it was shown that this may  be true both for the Fourier series of the function itself and for the  function conjugate to it.
  
(the [[Dini–Lipschitz criterion|Dini–Lipschitz criterion]]) or if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109091.png" /> has bounded variation (the [[Jordan criterion|Jordan criterion]]).
+
As early as 1915,  N.N.&nbsp;Luzin made the conjecture that the Fourier series of every  $L_2$-function converges almost-everywhere. For a long time only  partial results were obtained in this direction. The general form of the  problem turned out to be very difficult and it was only in 1966 that  L.&nbsp;Carleson proved the validity of this conjecture (see  [[Carleson theorem]]). The Fourier series of $L_p$-functions when  $p>1$ also converge almost-everywhere. Kolmogorov's example shows  that it is impossible to strengthen this result any further in terms of  the spaces $L_p$.
  
From this one can obtain criteria for uniform convergence of Fourier series on a certain interval if the localization principle for uniform convergence is used. The latter is formulated as follows. If two functions are equal on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109092.png" />, then on each strictly interior interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109093.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109094.png" />, either the Fourier series of these functions are both uniformly convergent or neither is uniformly convergent. In other words, the uniform convergence of the Fourier series of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109095.png" /> on an interval depends only on the behaviour of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109096.png" /> in an arbitrarily small extension of this interval.
+
Since the partial sums of a Fourier  series do not always converge, one also considers the [[Summation of  Fourier series|summation of Fourier series]] by some average of the  partial sums and uses this to represent the function. One of the  simplest examples are the [[Fejér sum|Fejér sums]], which are the  arithmetical means of the partial sums $s_k(f,x)$ of the Fourier series:
 +
$$
 +
\sigma_n(f,x) = \frac{1}{n+1}\sum_{k=0}^n s_k(f,x).
 +
$$
 +
For  every integrable function $f$ the sums $\sigma_n(f,x)$ converge to  $f(x)$ almost-everywhere and, moreover, converge at every point where  $f$ is continuous; if $f$ is continuous everywhere, then they converge  uniformly.
  
P. du Bois-Reymond (1876) established that the continuity of a function at a certain point does not guarantee that its Fourier series converges at this point. Later it was proved that the Fourier series of a continuous function may diverge on an everywhere-dense set of measure zero that is of the second category.
+
According to the [[Denjoy–Luzin theorem]],  if the trigonometric series \ref{eq2} at every $x$ converges absolutely  on a set of positive measure, then the series
 +
\begin{equation}
 +
\label{eq5}
 +
\sum_k \bigl( \abs{a_k} + \abs{b_k} \bigr)
 +
\end{equation}
 +
converges,  and hence the series \ref{eq2} converges absolutely for all $x$. Thus,  the absolute convergence of \ref{eq2} is equivalent to convergence of \ref{eq5}.
  
If nothing is assumed about the function except that it is integrable, then its Fourier series may turn out to be divergent almost-everywhere, or even everywhere. The first examples of such functions were constructed by A.N. Kolmogorov (1923, 1926). Later it was shown that this may be true both for the Fourier series of the function itself and for the function conjugate to it.
+
S.N.&nbsp;Bernstein  [S.N.&nbsp;Bernshtein] (1934) proved that if the modulus of  continuity $\omega(f,\delta)$ of a function $f$ satisfies
 +
$$
 +
\sum_{n=1}^\infty \frac{1}{\sqrt{n}}\omega\bigl(f,1/n\bigr) < \infty,
 +
$$
 +
then the Fourier series of $f$ converges absolutely. It is impossible to weaken this condition: If $\omega(\delta)$ is a modulus of continuity of  function type [Remark - what does this mean?] such that the series
 +
$$
 +
\sum_{n=1}^\infty \frac{1}{\sqrt{n}} \omega\bigl(1/n\bigr)
 +
$$
 +
diverges, then a function $f$ can be found with modulus of continuity satisfying  $\omega(f,\delta)\leq\omega(\delta)$ and whose Fourier series does not  converge absolutely.
  
As early as 1915, N.N. Luzin made the conjecture that the Fourier series of every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109098.png" />-function converges almost-everywhere. For a long time only partial results were obtained in this direction. The general form of the problem turned out to be very difficult and it was only in 1966 that L. Carleson proved the validity of this conjecture (see [[Carleson theorem|Carleson theorem]]). The Fourier series of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109099.png" />-functions when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090100.png" /> also converge almost-everywhere. Kolmogorov's example shows that it is impossible to strengthen this result any further in terms of the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090101.png" />.
+
In particular, the Fourier series of functions satisfying a [[Lipschitz condition]] of order $\alpha > 1/2$ converge absolutely. When $\alpha=1/2$, absolute convergence need  not hold (Bernshtein, 1914).
  
Since the partial sums of a Fourier series do not always converge, one also considers the [[Summation of Fourier series|summation of Fourier series]] by some average of the partial sums and uses this to represent the function. One of the simplest examples are the Fejér sums (cf. [[Fejér sum|Fejér sum]]), which are the arithmetical means of the partial sums <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090102.png" /> of the Fourier series:
+
If $f$ is a function of bounded variation and if its modulus of continuity satisfies
 +
\begin{equation}
 +
\label{eq6}
 +
\sum_{n=1}^\infty \frac{1}{n} \sqrt{\omega\bigl(f,1/n\bigr)} < \infty,
 +
\end{equation}
 +
then the Fourier series of $f$ converges absolutely (see {{Cite|Sa}}). Condition \ref{eq6} cannot be weakened (see {{Cite|Bo}}).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090103.png" /></td> </tr></table>
+
In contrast to the above, the following theorem gives a criterion for the absolute convergence for an individual function. A necessary and sufficient condition for the absolute convergence of the Fourier series of a function $f$ is that the series
 
+
$$
For every integrable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090104.png" /> the sums <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090105.png" /> converge to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090106.png" /> almost-everywhere and, moreover, converge at every point where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090107.png" /> is continuous; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090108.png" /> is continuous everywhere, then they converge uniformly.
+
\sum_{n=1}^\infty \frac{1}{\sqrt{n}} e_n(f)
 
+
$$
According to the [[Denjoy–Luzin theorem|Denjoy–Luzin theorem]], if the trigonometric series (2) at every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090109.png" /> converges absolutely on a set of positive measure, then the series
+
converges, where $e_n(f)$ is the [[Best approximation|best approximation]] to $fin the metric of $L_2$ by trigonometric polynomials containing $n$  harmonics (see {{Cite|St}}).
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090110.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>
 
 
 
converges, and hence the series (2) converges absolutely for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090111.png" />. Thus, the absolute convergence of (2) is equivalent to convergence of (5).
 
 
 
e theorem','../l/l057530.htm','Lebesgue constants','../l/l057800.htm','Limit theorems','../l/l058920.htm','Lyapunov theorem','../l/l061200.htm','Markov–Bernstein-type inequalities','../m/m110060.htm','Orthogonal polynomials','../o/o070340.htm')" style="background-color:yellow;">S.N. Bernstein [S.N. Bernshtein] (1934) proved that if the modulus of continuity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090112.png" /> of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090113.png" /> satisfies
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090114.png" /></td> </tr></table>
 
 
 
then the Fourier series of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090115.png" /> converges absolutely. It is impossible to weaken this condition: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090116.png" /> is a modulus of continuity of function type such that the series
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090117.png" /></td> </tr></table>
 
 
 
diverges, then a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090118.png" /> can be found with modulus of continuity satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090119.png" /> and whose Fourier series does not converge absolutely.
 
 
 
In particular, the Fourier series of functions satisfying a [[Lipschitz condition|Lipschitz condition]] of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090120.png" /> converge absolutely. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090121.png" />, absolute convergence need not hold (Bernshtein, 1914).
 
 
 
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090122.png" /> is a function of bounded variation and if its modulus of continuity satisfies
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090123.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>
 
 
 
then the Fourier series of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090124.png" /> converges absolutely (see [[#References|[9]]]). Condition (6) cannot be weakened (see [[#References|[10]]]).
 
 
 
In contrast to the above, the following theorem gives a criterion for the absolute convergence for an individual function. A necessary and sufficient condition for the absolute convergence of the Fourier series of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090125.png" /> is that the series
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090126.png" /></td> </tr></table>
 
 
 
converges, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090127.png" /> is the [[Best approximation|best approximation]] to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090128.png" /> in the metric of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090129.png" /> by trigonometric polynomials containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090130.png" /> harmonics (see [[#References|[11]]]).
 
 
 
The series (2) can be considered as the real part of the power series
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090131.png" /></td> </tr></table>
 
  
 +
The series \ref{eq2} can be considered as the real part of the power series
 +
$$
 +
\frac{a_0}{2} + \sum_{k=1}^\infty \bigl( a_k - i b_k \bigr) e^{ikx}.
 +
$$
 
The imaginary part
 
The imaginary part
 +
\begin{equation}
 +
\label{eq7}
 +
\sum_{k=1}^\infty \bigl( -b_k \cos kx + a_k \sin kx \bigr)
 +
\end{equation}
 +
is called the series conjugate to the series \ref{eq2}.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090132.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7)</td></tr></table>
+
Let $f\in L_2$ and let \ref{eq2} be its Fourier series. Then for almost-all $x$ the function
 
+
$$
is called the series conjugate to the series (2).
+
\tilde{f}(x) = \lim_{\epsilon \rightarrow +0}
 
+
\frac{1}{\pi} \int_\epsilon^\pi \frac{f(x-t) - f(x+t)}{2 \tan t/2} \rd t
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090133.png" /> and let (2) be its Fourier series. Then for almost-all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090134.png" /> the function
+
$$
 
+
exists (I.I.&nbsp;Privalov, 1919). The function $\tilde{f}$ is called the conjugate function to $f$; it need not be integrable. However, if $\tilde{f} \in L_1$, then the Fourier series of $\tilde{f}$ is the series \ref{eq7} (V.I.&nbsp;Smirnov, 1928).
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090135.png" /></td> </tr></table>
 
 
 
exists (I.I. Privalov, 1919). The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090136.png" /> is called the conjugate function to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090137.png" />; it need not be integrable. However, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090138.png" />, then the Fourier series of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090139.png" /> is the series (7) (V.I. Smirnov, 1928).
 
 
 
In many cases one can deduce some property or other of the conjugate series (7) from the properties of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090140.png" /> or its Fourier series (2), for example, convergence in the metric of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090141.png" />, convergence or summability at a point, or almost-everywhere, etc.
 
 
 
Properties of Fourier series under special assumptions on their coefficients have also been studied. For example, [[Lacunary trigonometric series|lacunary trigonometric series]], when the only non-zero coefficients are those indexed by numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090142.png" /> forming a [[Lacunary sequence|lacunary sequence]], that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090143.png" />. Another example of special series are series with monotone coefficients.
 
  
All that has been said above concerns Fourier series of the form (2). For Fourier series in a rearranged trigonometric system certain properties of the Fourier series in the trigonometric system, taken in the usual order, do not hold. For example, there is a continuous function such that its Fourier series after a certain rearrangement diverges almost-everywhere (see [[#References|[12]]]–[[#References|[15]]]).
+
In many  cases one can deduce some property or other of the conjugate series \ref{eq7} from the properties of the function $f$ or its Fourier series \ref{eq2}, for example, convergence in the metric of $L_p$, convergence  or summability at a point, or almost-everywhere, etc.
  
The theory of Fourier series for functions in several variables (multiple Fourier series) has been developed to a lesser extent. Some of the multi-dimensional results are analogous to the one-dimensional results. But there are crucial differences.
+
Properties  of Fourier series under special assumptions on their coefficients have  also been studied. For example, [[Lacunary trigonometric series|lacunary  trigonometric series]], when the only non-zero coefficients are those  indexed by numbers $n_m$ forming a [[Lacunary sequence|lacunary  sequence]], that is, $n_{m+1}/n_m \geq \lambda > 1$. Another example  of special series are series with monotone coefficients.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090144.png" /> be a point of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090145.png" />-dimensional space, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090146.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090147.png" />-dimensional vector with integer coordinates and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090148.png" />. For a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090149.png" /> with period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090150.png" /> in each variable and Lebesgue integrable over the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090151.png" />-dimensional cube <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090152.png" />, the Fourier series in the trigonometric system is
+
All  that has been said above concerns Fourier series of the form \ref{eq2}. For Fourier series in a rearranged trigonometric system certain  properties of the Fourier series in the trigonometric system, taken in  the usual order, do not hold. For example, there is a continuous  function such that its Fourier series after a certain rearrangement  diverges almost-everywhere (see {{Cite|KoMe}}, {{Cite|Za}},  {{Cite|Ul2}}, {{Cite|Ol}});
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090153.png" /></td> <td valign="top" style="width:5%;text-align:right;">(8)</td></tr></table>
+
The theory of Fourier  series for functions in several variables (multiple Fourier series) has  been developed to a lesser extent. Some of the multi-dimensional results  are analogous to the one-dimensional results. But there are crucial  differences.
  
where the summation is over all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090154.png" /> and
+
Let $x = (x_1,\ldots,x_N)$ be a point of  the $N$-dimensional space, let $k=(k_1,\ldots,k_N)$ be an  $N$-dimensional vector with integer coordinates and let  $(k,x)=k_1x_1+\cdots+k_Nx_N$. For a function $f(x)$ with period $2\pi$  in each variable and Lebesgue integrable over the $N$-dimensional cube  $[0,2\pi]^N$, the Fourier series in the trigonometric system is
 
+
\begin{equation}
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090155.png" /></td> </tr></table>
+
\label{eq8}
 
+
\sum_k c_k e^{i(k,x)},
are the Fourier coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090156.png" />. The Fourier series (8) is written in complex form. Writing it in trigonometric form as a series in the products of multiple cosines and sines is rather more clumsy.
+
\end{equation}
 
+
where the summation is over all $k$ and
Various definitions of the partial sums of the series (8) are possible; for example, partial sums over rectangles
+
$$
 
+
c_k = \frac{1}{(2\pi)^N} \int_0^{2\pi} \cdots \int_0^{2\pi} f(x) e^{-i(k,x)} \rd x
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090157.png" /></td> </tr></table>
+
$$
 +
are the Fourier coefficients of $f$. The Fourier series \ref{eq8} is written in complex form. Writing it in trigonometric form as a series in the products of multiple cosines and sines is rather more clumsy.
  
 +
Various definitions of the partial sums of the series \ref{eq8} are possible; for example, partial sums over rectangles
 +
$$
 +
\sum_{\abs{k_1} \leq n_1} \cdots \sum_{\abs{k_1} \leq n_1} c_k e^{i(k,x)},
 +
$$
 
and over circles
 
and over circles
 +
\begin{equation}
 +
\label{eq9}
 +
\sum_{\abs{k} \leq n} c_k e^{i(k,x)},
 +
\end{equation}
 +
where $n$ is the radius and $\abs{k} = \sqrt{k_1^2 + \cdots + k_N^2}$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090158.png" /></td> <td valign="top" style="width:5%;text-align:right;">(9)</td></tr></table>
+
The circular partial sums \ref{eq9} are not so suitable for representing functions as are their Riesz means
 
+
$$
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090159.png" /> is the radius and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090160.png" />.
+
\sum_{\abs{k} \leq n} \bigl(1 - \abs{k}/n\bigr)^\alpha c_k e^{i(k,x)}.  
 
+
$$
The circular partial sums (9) are not so suitable for representing functions as are their Riesz means
+
For Riesz means of order $\alpha \geq (N-1)/2$ of Fourier series of $L_2$-functions the localization principle is valid; this is not so for smaller $\alpha$ (S.&nbsp;Bochner, 1936). The Riesz means of circular partial sums of critical order $\alpha = (N-1)/2$ play an essential role also in other questions about Fourier series of functions in several variables.
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090161.png" /></td> </tr></table>
 
 
 
For Riesz means of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090162.png" /> of Fourier series of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090163.png" />-functions the localization principle is valid; this is not so for smaller <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090164.png" /> (S. Bochner, 1936). The Riesz means of circular partial sums of critical order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090165.png" /> play an essential role also in other questions about Fourier series of functions in several variables.
 
  
There is a continuous function in two variables with a Fourier series that does not converge over rectangles at any interior point of the square <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090166.png" /> (see [[#References|[16]]]).
+
There is a continuous function in two variables with a Fourier series that does not converge over rectangles at any interior point of the square $[0,2\pi]^N$ (see {{Cite|Fe}}).
  
Certain results about Fourier series in the trigonometric system can be generalized considerably; for example, they can be carried over in a corresponding way to the spectral decompositions corresponding to self-adjoint elliptic differential operators.
+
Certain results about Fourier series in the trigonometric system can be generalized considerably; for example, they can be carried over in a corresponding way to the spectral decompositions corresponding to self-adjoint elliptic differential operators.
  
====References====
+
====References====  
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> N.K. [N.K. Bari] Bary,   "A treatise on trigonometric series" , Pergamon  (1964) (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A. Zygmund,  "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1988)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"G.H. Hardy,   W.W. Rogosinsky,   "Fourier series" , Cambridge Univ. Press (1965)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> N.N. Luzin,   "The integral and trigonometric series" , Moscow-Leningrad (1951) (In Russian)  (Thesis; also: Collected Works, Vol. 1, Moscow, 1953, pp. 48–212)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"H. Lebesgue,   "Leçons sur les séries trigonométriques" , Gauthier-Villars (1906)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> A.B. Paplauskas,   "Trigonometric series from Euler to Lebesgue" , Moscow  (1966) (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> P.L. Ul'yanov,   "Solved and unsolved problems in the theory of trigonometric and orthogonal series"  ''Russian Math. Surveys'' , '''19''' :  1  (1964) pp. 1–62  ''Uspekhi Mat. Nauk'' , '''19''' :  1  (1964) pp. 3–69</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> Sh.A. Alimov,  V.A. Il'in,  E.M. Nikishin,   "Convergence problems of multiple trigonometric series and spectral decomposition. I" ''Russian Math. Surveys'' , '''31''' :  6 (1976) pp. 29–86  ''Uspekhi Mat. Nauk'' , '''31''' :  6  (1976) pp. 28–83</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"R. Salem,   "On a theorem of Zygmund" ''Duke Math. J.'' , '''10''' (1943) pp. 23–31</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  S.V. Bochkarev,  "On a problem of Zygmund"  ''Math. USSR Izv.'' , '''7''' : 3  (1973)  pp. 629–637  ''Izv. Akad. Nauk SSSR'' , '''37'''  (1973)  pp. 630–638</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  S.B. Stechkin,   "On absolute convergence of orthogonal series"  ''Dokl. Akad. Nauk SSSR'' , '''102''' (1955) pp. 37–40 (In Russian)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> A.N. [A.N. Kolmogorov] Kolmogoroff,  D.E. [D.E. Menshov] Menschoff,   "Sur la convergence des séries de fonctions orthogonales" ''Math. Z.'' , '''26'''  (1927) pp. 432–441</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  Z. Zahorski,  "Une série de Fourier permutée d'une fonction de classe <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090167.png" /> divergente partout"  ''C.R. Acad. Sci. Paris'' , '''251''' (1960) pp. 501–503</TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top"P.L. Ul'yanov,   "Divergent Fourier series" ''Russian Math. Surveys'' , '''16''' : 3 (1961) pp. 1–75 ''Uspekhi Mat. Nauk'' , '''16''' : 3 (1961) pp. 61–142</TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top"> A.M. Olevskii,   "Divergent Fourier series for continuous functions" ''Soviet Math. Dokl.'' , '''2'''  (1961)  pp. 1382–1386  ''Dokl. Akad. Nauk SSSR'' , '''141''' (1961) pp. 28–31</TD></TR><TR><TD valign="top">[16]</TD> <TD valign="top">  C. Fefferman,   "On the divergence of multiple Fourier series" ''Bull. Amer. Math. Soc.'' , '''77''' (1971) pp. 191–195</TD></TR></table>
+
{|
 +
|-
 +
|valign="top"|{{Ref|AlIlNi}}||valign="top"|  Sh.A.  Alimov, V.A. Il'in, E.M. Nikishin, "Convergence problems of  multiple trigonometric series and spectral decomposition. I" ''Russian  Math.  Surveys'', '''31''' : 6 (1976) pp. 29–86 ''Uspekhi Mat. Nauk'',  '''31''' : 6 (1976) pp. 28–83
 +
|-
 +
|valign="top"|{{Ref|Ba}}||valign="top"| N.K. [N.K.   Bari] Bary, "A treatise on trigonometric series", Pergamon  (1964) (Translated from Russian)
 +
|-
 +
|valign="top"|{{Ref|Bo}}||valign="top"| S.V. Bochkarev, "On a problem of Zygmund" ''Math.  USSR Izv.'', '''7'''  : 3 (1973) pp. 629–637 ''Izv. Akad. Nauk SSSR'', '''37''' (1973) pp.   630–638
 +
|-
 +
|valign="top"|{{Ref|Fe}}||valign="top"| C.  Fefferman, "On the divergence of multiple Fourier series" ''Bull. Amer.  Math.  Soc.'', '''77''' (1971) pp. 191–195
 +
|-
 +
|valign="top"|{{Ref|HaRo}}||valign="top"| G.H. Hardy, W.W. Rogosinsky, "Fourier series", Cambridge Univ. Press (1965)
 +
|-
 +
|valign="top"|{{Ref|KoMe}}||valign="top"| A.N. [A.N. Kolmogorov] Kolmogoroff, D.E. [D.E. Menshov] Menschoff, "Sur la convergence des séries de fonctions orthogonales" ''Math. Z.'''''26''' (1927) pp. 432–441
 +
|-
 +
|valign="top"|{{Ref|Le}}||valign="top"| H. Lebesgue, "Leçons sur les séries trigonométriques", Gauthier-Villars (1906)
 +
|-
 +
|valign="top"|{{Ref|Lu}}||valign="top"| N.N. Luzin, "The integral and trigonometric series", Moscow-Leningrad (1951) (In Russian) (Thesis; also: Collected Works, Vol. 1, Moscow,  1953, pp.  48–212)
 +
|-
 +
|valign="top"|{{Ref|Ol}}||valign="top"| A.M. Olevskii, "Divergent Fourier series for continuous functions"  ''Soviet Math. Dokl.'', '''2''' (1961) pp. 1382–1386 ''Dokl. Akad. Nauk SSSR'', '''141''' (1961) pp. 28–31
 +
|-
 +
|valign="top"|{{Ref|Pa}}||valign="top"| A.B. Paplauskas, "Trigonometric series from Euler to Lebesgue", Moscow (1966) (In Russian)
 +
|-
 +
|valign="top"|{{Ref|Sa}}||valign="top"| R. Salem, "On a theorem of Zygmund" ''Duke Math. J.'', '''10''' (1943) pp. 23–31
 +
|-
 +
|valign="top"|{{Ref|St}}||valign="to p"| S.B. Stechkin, "On absolute convergence of orthogonal series"  ''Dokl. Akad. Nauk SSSR'', '''102''' (1955) pp. 37–40 (In Russian)
 +
|-
 +
|valign="top"|{{Ref|Ul}}||valign="top"| P.L. Ul'yanov, "Solved and unsolved problems in the theory of  trigonometric and orthogonal series" ''Russian Math. Surveys'', '''19'''  : 1 (1964) pp. 1–62 ''Uspekhi Mat. Nauk'', '''19''' : 1 (1964) pp.   3–69
 +
|-
 +
|valign="top"|{{Ref|Ul2}}||valign="top"| P.L. Ul'yanov, "Divergent Fourier series" ''Russian Math. Surveys'', '''16''' : 3 (1961) pp. 1–75 ''Uspekhi Mat. Nauk'', '''16''' : 3 (1961) pp.   61–142
 +
|-
 +
|valign="top"|{{Ref|Za}}||valign="top"| Z. Zahorski, "Une série de Fourier permutée d'une fonction de classe $L_2$  divergente partout" ''C.R. Acad. Sci. Paris'', '''251''' (1960) pp. 501–503
 +
|-
 +
|valign="top"|{{Ref|Zy}}||valign="top"| A. Zygmund, "Trigonometric series", '''1–2''', Cambridge Univ. Press (1988)
 +
|-
 +
|}
  
 +
====Comments====
  
 +
A  closed system is also called a [[Complete system of functions|complete  system]]. Instead of Riemann–Lebesgue theorem one often uses  Riemann–Lebesgue lemma.
  
====Comments====
+
For multiple Fourier series see, e.g., Chapt. 7 of {{Cite|StWe}}.
A closed system is also called a complete system (cf. [[Complete system of functions|Complete system of functions]]). Instead of Riemann–Lebesgue theorem one often uses Riemann–Lebesgue lemma.
 
  
For multiple Fourier series see, e.g., Chapt. 7 of [[#References|[a5]]].
+
====References====
  
====References====
+
{|
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"R.E. Edwards,   "Fourier series. A modern introduction" , '''1–2''' , Springer (1979–1982)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"J.-P. Kahane,   "Séries de Fourier absolument convergentes" , Springer (1970)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"Y. Katznelson,   "An introduction to harmonic analysis" , Wiley (1968)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H. Dym,  H.P. McKean,  "Fourier series and integrals" , Acad. Press  (1972)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> E.M. Stein,   G. Weiss,   "Introduction to Fourier analysis on Euclidean spaces" , Princeton Univ. Press (1971)</TD></TR></table>
+
|-
 +
|valign="top"|{{Ref|DyMc}}||valign="top"| H. Dym, H.P.  McKean, "Fourier series and integrals", Acad. Press (1972)
 +
|-
 +
|valign="top"|{{Ref|Ed}}||valign="top"| R.E. Edwards, "Fourier series. A modern introduction", '''1–2''', Springer (1979–1982)
 +
|-
 +
|valign="top"|{{Ref|Ka}}||valign="top"| J.-P. Kahane, "Séries de Fourier absolument convergentes", Springer (1970)
 +
|-
 +
|valign="top"|{{Ref|Ka2}}||valign="top"| Y. Katznelson, "An introduction to harmonic analysis", Wiley (1968)
 +
|-
 +
|valign="top"|{{Ref|StWe}}||valign="top"| E.M. Stein, G. Weiss, "Introduction to Fourier analysis on Euclidean spaces", Princeton Univ. Press (1971)
 +
|-
 +
|}

Revision as of 21:58, 25 April 2012

of a function $f$ in a system of functions $\{\phi_n\}$ which are orthonormal on an interval $(a,b)$.

2020 Mathematics Subject Classification: Primary: 42A Secondary: 42B [MSN][ZBL] $\newcommand{\abs}[1]{\left|#1\right|}$

The series $$ \sum_{k=0}^\infty c_k \phi_k $$ whose coefficients are determined by \begin{equation} \label{eq1} c_k = \int_a^b f(x)\phi_k(x) \rd x, \quad k=1,2,\ldots \end{equation} These coefficients are called the Fourier coefficients of $f$. In general it is assumed that $f$ is square integrable on $(a,b) $. For many systems $\{\phi_k\}$ this requirement can be relaxed by replacing it by another which ensures the existence of all the integrals in \ref{eq1}.

The Fourier series in the trigonometric system is defined for every function $f$ that is integrable on $(0,2\pi)$. It is the series \begin{equation} \label{eq2} \frac{a_0}{2} + \sum_{k=1}^\infty \bigl( a_k \cos kx + b_k \sin kx \bigr) \end{equation} with coefficients \begin{equation} \label{eq3} a_k = \frac{1}{\pi}\int_0^{2\pi} f(x)\cos kx \rd x, \quad b_k = \frac{1}{\pi}\int_0^{2\pi} f(x)\sin kx \rd x. \end{equation}

Fourier series for functions in several variables are constructed analogously. A further generalization leads to Fourier coefficients and Fourier series for elements of a Hilbert space.

The theory of Fourier series in the trigonometric system has been most thoroughly developed, and these were the first examples of Fourier series. If one has in mind Fourier series in the trigonometric system, it is usual to talk simply of Fourier series, without indicating the system by which they are constructed.

Fourier series form a considerable part of the theory of trigonometric series. Fourier series first appeared in the papers of J. Fourier (1807) devoted to an investigation of the problems of heat conduction. He suggested representing a function $f$ given on $(0,2\pi)$ by the trigonometric series \ref{eq2} with coefficients determined by \ref{eq3}. Such a choice of coefficients is natural from many points of view. For example, if the series \ref{eq2} converges uniformly to $f$, then term-by-term integration leads to the expressions for the coefficients $a_k$ and $b_k$ given in \ref{eq3}. These formulas had been obtained already by L. Euler (1777) by term-by-term integration.

Using \ref{eq3} the Fourier series \ref{eq2} can be constructed for every function that is integrable over $[0,2\pi]$. Integrability of the function can be understood in various senses, for example integrability according to Riemann or Lebesgue. Depending on this, one speaks of Fourier–Riemann series, Fourier–Lebesgue series, etc. The concepts of the Riemann and the Lebesgue integral themselves arose to a considerable extent in connection with research on Fourier series. The modern presentation of the theory of Fourier series was developed after the construction of the Lebesgue integral, and since then it has developed mainly as the theory of Fourier–Lebesgue series. Below it is assumed that the function $f$ has period $2\pi$ and is Lebesgue integrable over the period.

In the theory of Fourier series one studies the relation between the properties of functions and the properties of their Fourier series; in particular, one investigates questions on the representation of functions by Fourier series.

The proof of a minimum property of the partial sums of Fourier series goes back to the work of F. Bessel (1828): Given an $f\in L_2$, then among all the trigonometric polynomials of order $n$, $$ t_n(x) = A_0 + \sum_{k=0}^n \bigl( A_k \cos kx + B_k \sin kx \bigr), $$ the smallest value of the integral $$ \frac{1}{\pi}\int_0^{2\pi} \bigl( f(x) - t_n(x) \bigr)^2 \rd x $$ is attained for the partial sum of the Fourier series \ref{eq2} of $f$: $$ s_n(f,x) = \frac{a_0}{2} + \sum_{k=1}^n \bigl( a_k \cos kx + b_k \sin kx \bigr). $$ This smallest value is equal to $$ \frac{1}{\pi} \int_0^{2\pi} f^2(x) \rd x - \left( \frac{a_0^2}{2} + \sum_{k=1}^n \bigl( a_k^2 + b_k^2 \bigr) \right). $$ This implies the Bessel inequality $$ \frac{a_0^2}{2} + \sum_{k=1}^\infty \bigl( a_k^2 + b_k^2 \bigr) \leq \frac{1}{\pi} \int_0^{2\pi} f^2(x) \rd x, $$ which is satisfied for every function $f$ in $L_2$.

The system of trigonometric functions is a closed system (cf. Closed system of elements (functions)), that is, if $f \in L_2$, then the Parseval equality $$ \frac{a_0^2}{2} + \sum_{k=1}^\infty \bigl( a_k^2 + b_k^2 \bigr) = \frac{1}{\pi} \int_0^{2\pi} f^2(x) \rd x $$ is valid, where $a_k$, $b_k$ are the Fourier coefficients of $f$. In particular, for functions $f$ in $L_2$ the series \begin{equation} \label{eq4} \frac{a_0^2}{2} + \sum_{k=1}^\infty \bigl( a_k^2 + b_k^2 \bigr) \end{equation} is convergent. The converse assertion also holds: If for a system of numbers $a_k$, $b_k$ the series \ref{eq4} converges, then these numbers are the Fourier coefficients of a certain function $f \in L_2$ (F. Riesz and E. Fischer, 1907).

The Fourier coefficients of any integrable function tend to zero. This statement is called the Riemann–Lebesgue theorem. B. Riemann proved it for Fourier–Riemann series and H. Lebesgue for Fourier–Lebesgue series.

If the function $f$ is absolutely continuous, then the Fourier series for the derivative $f'$ can be obtained by term-by-term differentation of the Fourier series for $f$. This implies that if the derivative of order $r \geq 0$ of a function $f$ is absolutely continuous, then the estimates $$ a_k, b_k = o\bigl( k^{-(r+1)} \bigr), \quad k \rightarrow \infty, $$ are valid for the Fourier coefficients of $f$.

The first convergence criterion for Fourier series was obtained by P.G.L.  Dirichlet in 1829. His result (the Dirichlet theorem) can be formulated as follows: If a function $f$ has a finite number of maxima and minima over the period and is everywhere continuous, except at a finite number of points where it may have discontinuities of the first kind, then the Fourier series of $f$ converges for all $x$, and, moreover, at points of continuity it converges to $f(x)$ and at points of discontinuity it converges to $\bigl( f(x+0) + f(x-0) \bigr)/2$. Subsequently, this assertion was extended to arbitrary functions of bounded variation (C. Jordan, 1881).

According to the localization principle proved by Riemann (1853), the convergence or divergence of the Fourier series of a function $f$ at a point $x$, and the value of the sum when it converges, depends only on the behaviour of $f$ in an arbitrarily small neighbourhood of $x$.

Many different convergence criteria for Fourier series at a point are known. R. Lipschitz (1864) established that the Fourier series of a function $f$ converges at a point $x$ if $\abs{f(x+h) - f(x)} \leq M \abs{h}^\alpha$ is satisfied for all sufficiently small $h$, where $M$ and $\alpha$ are certain positive constants (the Lipschitz criterion). The Dini criterion is more general: The Fourier series of a function $f$ converges to $S$ at a point $x$ if the integral $$ \int_0^\pi \abs{\phi_x(t)} \frac{\mathrm{d}t}{t} $$ converges, where $\phi_x(t) = f(x+t) + f(x-t) - 2S$. The value $f(x)$ is usually taken for $S$. For example, if the Fourier series of $f$ converges at a point $x$ where this function is continuous, then the sum of the series is necessarily equal to $f(x)$.

Lebesgue (1905) proved that if $$ \int_0^h \abs{\phi_x(t)} \rd t = o(h), \quad \int_h^\pi \abs{\phi_x(t+h) - \phi_x(t)} \frac{\mathrm{d}t}{t} = o(1), $$ as $h \rightarrow 0$, then the Fourier series of $f$ converges to $S$ at $x$. This Lebesgue criterion is stronger than all those given above and stronger than the De la Vallée-Poussin criterion and the Young criterion. But verifying it is usually difficult.

A convergence criterion of another type is given by the Hardy–Littlewood theorem (1932): The Fourier series of a function $f$ converges at a point $x$ if the following conditions are satisfied:

1) $$ f(x+h) - f(x) = o\left( \ln^{-1}\frac{1}{\abs{h}} \right), $$ as $h \rightarrow 0$; and

2) the estimates $$ a_k, b_k = O\bigl( k^{-\delta} \bigr), \quad \delta > 0, $$ are valid for the Fourier coefficients of $f$.

Besides convergence criteria for Fourier series at a point, criteria for uniform convergence have been studied also. Let a function $f$ have period $2\pi$ and be continuous. Then its Fourier series converges uniformly to it on the whole real line if the modulus of continuity $\omega(f,\delta)$ of $f$ satisfies the condition $$ \text{$\omega(f,\delta) \ln \delta \rightarrow 0$ as $\delta \rightarrow 0$} $$ (the Dini–Lipschitz criterion) or if $f$ has bounded variation (the Jordan criterion).

From this one can obtain criteria for uniform convergence of Fourier series on a certain interval if the localization principle for uniform convergence is used. The latter is formulated as follows. If two functions are equal on an interval $[a,b]$, then on each strictly interior interval $[a+\epsilon,b-\epsilon]$, $\epsilon > 0$, either the Fourier series of these functions are both uniformly convergent or neither is uniformly convergent. In other words, the uniform convergence of the Fourier series of a function $f$ on an interval depends only on the behaviour of $f$ in an arbitrarily small extension of this interval.

P. du Bois-Reymond (1876) established that the continuity of a function at a certain point does not guarantee that its Fourier series converges at this point. Later it was proved that the Fourier series of a continuous function may diverge on an everywhere-dense set of measure zero that is of the second category.

If nothing is assumed about the function except that it is integrable, then its Fourier series may turn out to be divergent almost-everywhere, or even everywhere. The first examples of such functions were constructed by A.N. Kolmogorov (1923, 1926). Later it was shown that this may be true both for the Fourier series of the function itself and for the function conjugate to it.

As early as 1915, N.N. Luzin made the conjecture that the Fourier series of every $L_2$-function converges almost-everywhere. For a long time only partial results were obtained in this direction. The general form of the problem turned out to be very difficult and it was only in 1966 that L. Carleson proved the validity of this conjecture (see Carleson theorem). The Fourier series of $L_p$-functions when $p>1$ also converge almost-everywhere. Kolmogorov's example shows that it is impossible to strengthen this result any further in terms of the spaces $L_p$.

Since the partial sums of a Fourier series do not always converge, one also considers the summation of Fourier series by some average of the partial sums and uses this to represent the function. One of the simplest examples are the Fejér sums, which are the arithmetical means of the partial sums $s_k(f,x)$ of the Fourier series: $$ \sigma_n(f,x) = \frac{1}{n+1}\sum_{k=0}^n s_k(f,x). $$ For every integrable function $f$ the sums $\sigma_n(f,x)$ converge to $f(x)$ almost-everywhere and, moreover, converge at every point where $f$ is continuous; if $f$ is continuous everywhere, then they converge uniformly.

According to the Denjoy–Luzin theorem, if the trigonometric series \ref{eq2} at every $x$ converges absolutely on a set of positive measure, then the series \begin{equation} \label{eq5} \sum_k \bigl( \abs{a_k} + \abs{b_k} \bigr) \end{equation} converges, and hence the series \ref{eq2} converges absolutely for all $x$. Thus, the absolute convergence of \ref{eq2} is equivalent to convergence of \ref{eq5}.

S.N. Bernstein [S.N. Bernshtein] (1934) proved that if the modulus of continuity $\omega(f,\delta)$ of a function $f$ satisfies $$ \sum_{n=1}^\infty \frac{1}{\sqrt{n}}\omega\bigl(f,1/n\bigr) < \infty, $$ then the Fourier series of $f$ converges absolutely. It is impossible to weaken this condition: If $\omega(\delta)$ is a modulus of continuity of function type [Remark - what does this mean?] such that the series $$ \sum_{n=1}^\infty \frac{1}{\sqrt{n}} \omega\bigl(1/n\bigr) $$ diverges, then a function $f$ can be found with modulus of continuity satisfying $\omega(f,\delta)\leq\omega(\delta)$ and whose Fourier series does not converge absolutely.

In particular, the Fourier series of functions satisfying a Lipschitz condition of order $\alpha > 1/2$ converge absolutely. When $\alpha=1/2$, absolute convergence need not hold (Bernshtein, 1914).

If $f$ is a function of bounded variation and if its modulus of continuity satisfies \begin{equation} \label{eq6} \sum_{n=1}^\infty \frac{1}{n} \sqrt{\omega\bigl(f,1/n\bigr)} < \infty, \end{equation} then the Fourier series of $f$ converges absolutely (see [Sa]). Condition \ref{eq6} cannot be weakened (see [Bo]).

In contrast to the above, the following theorem gives a criterion for the absolute convergence for an individual function. A necessary and sufficient condition for the absolute convergence of the Fourier series of a function $f$ is that the series $$ \sum_{n=1}^\infty \frac{1}{\sqrt{n}} e_n(f) $$ converges, where $e_n(f)$ is the best approximation to $f$ in the metric of $L_2$ by trigonometric polynomials containing $n$ harmonics (see [St]).

The series \ref{eq2} can be considered as the real part of the power series $$ \frac{a_0}{2} + \sum_{k=1}^\infty \bigl( a_k - i b_k \bigr) e^{ikx}. $$ The imaginary part \begin{equation} \label{eq7} \sum_{k=1}^\infty \bigl( -b_k \cos kx + a_k \sin kx \bigr) \end{equation} is called the series conjugate to the series \ref{eq2}.

Let $f\in L_2$ and let \ref{eq2} be its Fourier series. Then for almost-all $x$ the function $$ \tilde{f}(x) = \lim_{\epsilon \rightarrow +0} \frac{1}{\pi} \int_\epsilon^\pi \frac{f(x-t) - f(x+t)}{2 \tan t/2} \rd t $$ exists (I.I. Privalov, 1919). The function $\tilde{f}$ is called the conjugate function to $f$; it need not be integrable. However, if $\tilde{f} \in L_1$, then the Fourier series of $\tilde{f}$ is the series \ref{eq7} (V.I. Smirnov, 1928).

In many cases one can deduce some property or other of the conjugate series \ref{eq7} from the properties of the function $f$ or its Fourier series \ref{eq2}, for example, convergence in the metric of $L_p$, convergence or summability at a point, or almost-everywhere, etc.

Properties of Fourier series under special assumptions on their coefficients have also been studied. For example, lacunary trigonometric series, when the only non-zero coefficients are those indexed by numbers $n_m$ forming a lacunary sequence, that is, $n_{m+1}/n_m \geq \lambda > 1$. Another example of special series are series with monotone coefficients.

All that has been said above concerns Fourier series of the form \ref{eq2}. For Fourier series in a rearranged trigonometric system certain properties of the Fourier series in the trigonometric system, taken in the usual order, do not hold. For example, there is a continuous function such that its Fourier series after a certain rearrangement diverges almost-everywhere (see [KoMe], [Za], [Ul2], [Ol]);

The theory of Fourier series for functions in several variables (multiple Fourier series) has been developed to a lesser extent. Some of the multi-dimensional results are analogous to the one-dimensional results. But there are crucial differences.

Let $x = (x_1,\ldots,x_N)$ be a point of the $N$-dimensional space, let $k=(k_1,\ldots,k_N)$ be an $N$-dimensional vector with integer coordinates and let $(k,x)=k_1x_1+\cdots+k_Nx_N$. For a function $f(x)$ with period $2\pi$ in each variable and Lebesgue integrable over the $N$-dimensional cube $[0,2\pi]^N$, the Fourier series in the trigonometric system is \begin{equation} \label{eq8} \sum_k c_k e^{i(k,x)}, \end{equation} where the summation is over all $k$ and $$ c_k = \frac{1}{(2\pi)^N} \int_0^{2\pi} \cdots \int_0^{2\pi} f(x) e^{-i(k,x)} \rd x $$ are the Fourier coefficients of $f$. The Fourier series \ref{eq8} is written in complex form. Writing it in trigonometric form as a series in the products of multiple cosines and sines is rather more clumsy.

Various definitions of the partial sums of the series \ref{eq8} are possible; for example, partial sums over rectangles $$ \sum_{\abs{k_1} \leq n_1} \cdots \sum_{\abs{k_1} \leq n_1} c_k e^{i(k,x)}, $$ and over circles \begin{equation} \label{eq9} \sum_{\abs{k} \leq n} c_k e^{i(k,x)}, \end{equation} where $n$ is the radius and $\abs{k} = \sqrt{k_1^2 + \cdots + k_N^2}$.

The circular partial sums \ref{eq9} are not so suitable for representing functions as are their Riesz means $$ \sum_{\abs{k} \leq n} \bigl(1 - \abs{k}/n\bigr)^\alpha c_k e^{i(k,x)}. $$ For Riesz means of order $\alpha \geq (N-1)/2$ of Fourier series of $L_2$-functions the localization principle is valid; this is not so for smaller $\alpha$ (S. Bochner, 1936). The Riesz means of circular partial sums of critical order $\alpha = (N-1)/2$ play an essential role also in other questions about Fourier series of functions in several variables.

There is a continuous function in two variables with a Fourier series that does not converge over rectangles at any interior point of the square $[0,2\pi]^N$ (see [Fe]).

Certain results about Fourier series in the trigonometric system can be generalized considerably; for example, they can be carried over in a corresponding way to the spectral decompositions corresponding to self-adjoint elliptic differential operators.

References

[AlIlNi] Sh.A. Alimov, V.A. Il'in, E.M. Nikishin, "Convergence problems of multiple trigonometric series and spectral decomposition. I" Russian Math. Surveys, 31 : 6 (1976) pp. 29–86 Uspekhi Mat. Nauk, 31 : 6 (1976) pp. 28–83
[Ba] N.K. [N.K. Bari] Bary, "A treatise on trigonometric series", Pergamon (1964) (Translated from Russian)
[Bo] S.V. Bochkarev, "On a problem of Zygmund" Math. USSR Izv., 7 : 3 (1973) pp. 629–637 Izv. Akad. Nauk SSSR, 37 (1973) pp. 630–638
[Fe] C. Fefferman, "On the divergence of multiple Fourier series" Bull. Amer. Math. Soc., 77 (1971) pp. 191–195
[HaRo] G.H. Hardy, W.W. Rogosinsky, "Fourier series", Cambridge Univ. Press (1965)
[KoMe] A.N. [A.N. Kolmogorov] Kolmogoroff, D.E. [D.E. Menshov] Menschoff, "Sur la convergence des séries de fonctions orthogonales" Math. Z., 26 (1927) pp. 432–441
[Le] H. Lebesgue, "Leçons sur les séries trigonométriques", Gauthier-Villars (1906)
[Lu] N.N. Luzin, "The integral and trigonometric series", Moscow-Leningrad (1951) (In Russian) (Thesis; also: Collected Works, Vol. 1, Moscow, 1953, pp. 48–212)
[Ol] A.M. Olevskii, "Divergent Fourier series for continuous functions" Soviet Math. Dokl., 2 (1961) pp. 1382–1386 Dokl. Akad. Nauk SSSR, 141 (1961) pp. 28–31
[Pa] A.B. Paplauskas, "Trigonometric series from Euler to Lebesgue", Moscow (1966) (In Russian)
[Sa] R. Salem, "On a theorem of Zygmund" Duke Math. J., 10 (1943) pp. 23–31
[St] S.B. Stechkin, "On absolute convergence of orthogonal series" Dokl. Akad. Nauk SSSR, 102 (1955) pp. 37–40 (In Russian)
[Ul] P.L. Ul'yanov, "Solved and unsolved problems in the theory of trigonometric and orthogonal series" Russian Math. Surveys, 19 : 1 (1964) pp. 1–62 Uspekhi Mat. Nauk, 19 : 1 (1964) pp. 3–69
[Ul2] P.L. Ul'yanov, "Divergent Fourier series" Russian Math. Surveys, 16 : 3 (1961) pp. 1–75 Uspekhi Mat. Nauk, 16 : 3 (1961) pp. 61–142
[Za] Z. Zahorski, "Une série de Fourier permutée d'une fonction de classe $L_2$ divergente partout" C.R. Acad. Sci. Paris, 251 (1960) pp. 501–503
[Zy] A. Zygmund, "Trigonometric series", 1–2, Cambridge Univ. Press (1988)

Comments

A closed system is also called a complete system. Instead of Riemann–Lebesgue theorem one often uses Riemann–Lebesgue lemma.

For multiple Fourier series see, e.g., Chapt. 7 of [StWe].

References

[DyMc] H. Dym, H.P. McKean, "Fourier series and integrals", Acad. Press (1972)
[Ed] R.E. Edwards, "Fourier series. A modern introduction", 1–2, Springer (1979–1982)
[Ka] J.-P. Kahane, "Séries de Fourier absolument convergentes", Springer (1970)
[Ka2] Y. Katznelson, "An introduction to harmonic analysis", Wiley (1968)
[StWe] E.M. Stein, G. Weiss, "Introduction to Fourier analysis on Euclidean spaces", Princeton Univ. Press (1971)
How to Cite This Entry:
Fourier series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fourier_series&oldid=25470
This article was adapted from an original article by S.A. Telyakovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article