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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" />''
| + | Post $\TeX$ remarks |
| − | | + | * There was a load of gibberish in two places in the original imported text (looked like a part of the index had been cut-and-pasted accidentally). By looking at the google version of EoM I determined where it started and ended, and so could remove it cleanly. |
| − | The series
| + | * In several places I have merged the explicit reference and the link, so "<nowiki>sometimes called the A of B (see [[B, A of]])</nowiki>" becomes "<nowiki>sometimes called the [[B, A of|A of B]])</nowiki>". |
| − | | + | * The text contains the phrase "a modulus of continuity of function type" which is a google-whack -- according to google it occurs only in this document. I guess it is a literal translation of Russian terminology, find what it is called in the West! |
| − | <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>
| + | --[[User:Jjg|Jjg]] 00:10, 26 April 2012 (CEST) |
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| − | whose coefficients are determined by
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| − | <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>
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| − | 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).
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| − | 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
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| − | <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>
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| − | with coefficients
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| − | <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>
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| − | 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.
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| − | 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.
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| − | 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.
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| − | 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.
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| − | 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. | |
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| − | 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" />,
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| − | <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> | |
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| − | the smallest value of the integral | |
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| − | <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>
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| − | 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" />:
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| − | <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>
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| − | This smallest value is equal to
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| − | <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>
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| − | This implies the [[Bessel inequality|Bessel inequality]]
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| − | <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>
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| − | 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" />.
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| − | 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]]
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| − | <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>
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| − | 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
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| − | <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>
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| − | 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).
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| − | 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.
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| − | 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
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| − | <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>
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| − | are valid for the Fourier coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109048.png" />.
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| − | 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).
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| − | 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" />.
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| − | 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
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| − | <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>
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| − | 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" />.
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| − | Lebesgue (1905) proved that if
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| − | <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>
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| − | <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>
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| − | 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.
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| − | 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: | |
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| − | 1)
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| − | <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>
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| − | as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109083.png" />; and
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| − | 2) the estimates
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| − | <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>
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| − | are valid for the Fourier coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f04109085.png" />.
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| − | 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
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| − | <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>
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| − | (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]]).
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| − | | |
| − | 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.
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| − | | |
| − | 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.
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| − | | |
| − | 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.
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| − | | |
| − | 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" />.
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| − | | |
| − | 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:
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| − | | |
| − | <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>
| |
| − | | |
| − | 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.
| |
| − | | |
| − | 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
| |
| − | | |
| − | <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>
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| − | | |
| − | 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>
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| − | | |
| − | 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>
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| − | | |
| − | 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).
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| − | | |
| − | 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>
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| − | | |
| − | 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]]]).
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| − | | |
| − | 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
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| − | | |
| − | <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>
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| − | | |
| − | 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]]]).
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| − | | |
| − | The series (2) can be considered as the real part of the power series
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| − | | |
| − | <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>
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| − | | |
| − | The imaginary part
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| − | | |
| − | <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>
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| − | | |
| − | is called the series conjugate to the series (2).
| |
| − | | |
| − | 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
| |
| − | | |
| − | <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>
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| − | | |
| − | 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).
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| − | | |
| − | 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.
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| − | | |
| − | 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]]]).
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| − | | |
| − | 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.
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| − | | |
| − | 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
| |
| − | | |
| − | <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>
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| − | | |
| − | where the summation is over all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041090/f041090154.png" /> and
| |
| − | | |
| − | <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>
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| − | | |
| − | 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.
| |
| − | | |
| − | Various definitions of the partial sums of the series (8) are possible; for example, partial sums over rectangles
| |
| − | | |
| − | <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>
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| − | | |
| − | and over circles
| |
| − | | |
| − | <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>
| |
| − | | |
| − | 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" />.
| |
| − | | |
| − | The circular partial sums (9) are not so suitable for representing functions as are their Riesz means
| |
| − | | |
| − | <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]]]).
| |
| − | | |
| − | 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====
| |
| − | <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="to
| |
| − | p"> 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>
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| − | | |
| − | | |
| − | | |
| − | ====Comments====
| |
| − | 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====
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| − | <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>
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