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Given a [[Trigonometric series|trigonometric series]]
 
Given a [[Trigonometric series|trigonometric 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/i/i130/i130040/i1300401.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
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\begin{equation} \tag{a1} \frac { a _ { 0 } } { 2 } + \sum _ { k = 1 } ^ { \infty } ( a _ { k } \operatorname { cos } k x + b _ { k } \operatorname { sin } k x ), \end{equation}
  
the problem of its integrability asks under which assumptions on its coefficients this series is the [[Fourier series|Fourier series]] of an integrable function (i.e., belonging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130040/i1300402.png" />). Frequently, the series
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the problem of its integrability asks under which assumptions on its coefficients this series is the [[Fourier series|Fourier series]] of an integrable function (i.e., belonging to $\widehat{L^1}$). Frequently, 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/i/i130/i130040/i1300403.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
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\begin{equation} \tag{a2} \frac { a_0 } { 2 } + \sum _ { k = 1 } ^ { \infty } a _ { k } \operatorname { cos } k x \end{equation}
  
 
and
 
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/i/i130/i130040/i1300404.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>
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\begin{equation} \tag{a3} \sum _ { k = 1 } ^ { \infty } b _ { k } \operatorname { sin } k x \end{equation}
  
 
are investigated separately, since there is a difference in their behaviour, and usually integrability of (a3) requires additional assumptions. Of course, one may also consider trigonometric series in complex form.
 
are investigated separately, since there is a difference in their behaviour, and usually integrability of (a3) requires additional assumptions. Of course, one may also consider trigonometric series in complex form.
  
There exists no convenient description of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130040/i1300405.png" /> in terms of a given sequence alone. Hence, subspaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130040/i1300406.png" /> are studied. In view of the Riemann–Lebesgue lemma (cf. [[Fourier series|Fourier series]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130040/i1300407.png" /> is a subspace of the space of null sequences, while the space of sequences of bounded variation
+
There exists no convenient description of $\widehat{L^1}$ in terms of a given sequence alone. Hence, subspaces of $\widehat{L^1}$ are studied. In view of the Riemann–Lebesgue lemma (cf. [[Fourier series|Fourier series]]) $\widehat{L^1}$ is a subspace of the space of null sequences, while the space of sequences of bounded variation
  
<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/i/i130/i130040/i1300408.png" /></td> </tr></table>
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\begin{equation*}  \operatorname{bv} = \left\{ d = \{ d _ { k } \} : \| d \| _ {  \operatorname{bv} } = \sum _ { k = 0 } ^ { \infty } | \Delta d _ { k } | &lt; \infty \right\} \end{equation*}
  
is not a subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130040/i1300409.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130040/i13004010.png" />. Having a null sequence of bounded variation as its Fourier coefficients, the series (a2) converges for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130040/i13004011.png" />, while (a3) converges everywhere.
+
is not a subspace of $\widehat{L^1}$. Here $\Delta d_k = d_k - d_{k + 1}$. Having a null sequence of bounded variation as its Fourier coefficients, the series (a2) converges for every $x \neq 0 ( \operatorname { mod } 2 \pi )$, while (a3) converges everywhere.
  
In 1913, W.H. Young [[#References|[a20]]] proved that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130040/i13004012.png" /> is a convex null sequence, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130040/i13004013.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130040/i13004014.png" />, then (a1) is the Fourier series of an integrable function (cf. also [[Trigonometric series|Trigonometric series]]). In 1923, A.N. Kolmogorov [[#References|[a12]]] extended this result to the class of quasi-convex sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130040/i13004015.png" />, namely, those satisfying
+
In 1913, W.H. Young [[#References|[a20]]] proved that if $\{ a _ { k } \}$ is a convex null sequence, that is, $\Delta ^ { 2 } a _ { k } = \Delta ( \Delta a _ { k } ) \geq 0$ for $k = 0,1 , \ldots$, then (a1) is the Fourier series of an integrable function (cf. also [[Trigonometric series|Trigonometric series]]). In 1923, A.N. Kolmogorov [[#References|[a12]]] extended this result to the class of quasi-convex sequences $\{ a _ { k } \}$, namely, those satisfying
  
<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/i/i130/i130040/i13004016.png" /></td> </tr></table>
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\begin{equation*} \sum _ { k = 0 } ^ { \infty } ( k + 1 ) \left| \Delta ^ { 2 } \alpha _ { k } \right| &lt; \infty. \end{equation*}
  
Such a sequence is the difference of two convex sequences. In 1956, R.P. Boas generalized all previous results [[#References|[a6]]]. Subsequently, more general subspaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130040/i13004017.png" /> were considered:
+
Such a sequence is the difference of two convex sequences. In 1956, R.P. Boas generalized all previous results [[#References|[a6]]]. Subsequently, more general subspaces of $\widehat{L^1}$ were considered:
  
1) The so-called Boas–Telyakovskii space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130040/i13004018.png" /> (see, e.g., [[#References|[a16]]]):
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1) The so-called Boas–Telyakovskii space $\operatorname{ bt}$ (see, e.g., [[#References|[a16]]]):
  
<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/i/i130/i130040/i13004019.png" /></td> </tr></table>
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\begin{equation*} \| d \| _ { b t } = \| d \| _ { \operatorname {bv} } + \sum _ { n = 2 } ^ { \infty } \left| \sum _ { k = 1 } ^ { n / 2 } \frac { \Delta d _ { n - k } - \Delta d _ { n + k }  } { k }\right|. \end{equation*}
  
2) The Fomin space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130040/i13004020.png" /> [[#References|[a7]]] for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130040/i13004021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130040/i13004022.png" />:
+
2) The Fomin space $a _{p}$ [[#References|[a7]]] for $1 &lt; p &lt; \infty$, $1 / p + 1 / p ^ { \prime } = 1$:
  
<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/i/i130/i130040/i13004023.png" /></td> </tr></table>
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\begin{equation*} \| d \| _ { a _ { p } } = \sum _ { n = 0 } ^ { \infty } 2 ^ { n / p ^ { \prime } } \left\{ \sum _ { k = 2 ^ { n } } ^ { 2 ^ { n + 1 } - 1 } | \Delta d _ { k } | ^ { p } \right\} ^ { 1 / p } &lt; \infty . \end{equation*}
  
 
3) The Sidon–Telyakovskii space [[#References|[a17]]]:
 
3) The Sidon–Telyakovskii space [[#References|[a17]]]:
  
<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/i/i130/i130040/i13004024.png" /></td> </tr></table>
+
\begin{equation*} A _ { k } \downarrow 0 ( k \rightarrow \infty ) , \sum _ { k = 0 } ^ { \infty } A _ { k } &lt; \infty , | \Delta d _ { k } | &lt; A _ { k }. \end{equation*}
  
 
4) The Buntinas–Tanovic–Miller spaces (see, e.g., [[#References|[a2]]]).
 
4) The Buntinas–Tanovic–Miller spaces (see, e.g., [[#References|[a2]]]).
Line 41: Line 49:
 
5) The amalgam space [[#References|[a1]]], [[#References|[a3]]]:
 
5) The amalgam space [[#References|[a1]]], [[#References|[a3]]]:
  
<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/i/i130/i130040/i13004025.png" /></td> </tr></table>
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\begin{equation*} \sum _ { n = 0 } ^ { \infty } \left\{ \sum _ { m = 1 } ^ { \infty } \left[ \sum _ { k = m 2 ^ { n } } ^ { ( m + 1 ) 2 ^ { n } - 1 } | \Delta d _ { k } | \right] ^ { 2 } \right\} ^ { 1 / 2 } &lt; \infty. \end{equation*}
  
 
A classical way to prove such results is by using Sidon-type inequalities (see, e.g., [[#References|[a8]]]), a typical example of which is the one obtained by S. Sidon [[#References|[a15]]]:
 
A classical way to prove such results is by using Sidon-type inequalities (see, e.g., [[#References|[a8]]]), a typical example of which is the one obtained by S. Sidon [[#References|[a15]]]:
  
<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/i/i130/i130040/i13004026.png" /></td> </tr></table>
+
\begin{equation*} ( N + 1 ) ^ { - 1 } \left\| \sum _ { k = 0 } ^ { N } c _ { k } D _ { k } \right\| _ { L^{1} } \leq \operatorname { max } _ { 0 \leq k \leq N } | c  _ { k } |, \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130040/i13004027.png" /> is the [[Dirichlet kernel|Dirichlet kernel]] of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130040/i13004028.png" />.
+
where $D _ { k }$ is the [[Dirichlet kernel|Dirichlet kernel]] of order $k$.
  
In [[#References|[a13]]] a new approach to these problems was suggested. First, a locally absolutely continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130040/i13004029.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130040/i13004030.png" /> is considered such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130040/i13004031.png" /> (cf. also [[Absolute continuity|Absolute continuity]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130040/i13004032.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130040/i13004033.png" /> is a subspace of the space of functions of bounded variation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130040/i13004034.png" /> and is a generalization of a known space of sequences; e.g., 1)–3) above. Then the asymptotic behaviour of the Fourier transform of a function from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130040/i13004035.png" /> is investigated. Using the following result from [[#References|[a19]]] (an earlier version for functions with compact support can be found in [[#References|[a5]]]),
+
In [[#References|[a13]]] a new approach to these problems was suggested. First, a locally absolutely continuous function $f$ on $[ 0 , \infty )$ is considered such that $\operatorname { lim } _ { x \rightarrow \infty } f ( x ) = 0$ (cf. also [[Absolute continuity|Absolute continuity]]) and $f \in X$, where $X$ is a subspace of the space of functions of bounded variation $\operatorname{BV}$ and is a generalization of a known space of sequences; e.g., 1)–3) above. Then the asymptotic behaviour of the Fourier transform of a function from $X$ is investigated. Using the following result from [[#References|[a19]]] (an earlier version for functions with compact support can be found in [[#References|[a5]]]),
  
<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/i/i130/i130040/i13004036.png" /></td> </tr></table>
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\begin{equation*} \operatorname { sup } _ { 0 &lt; | y | &lt; \pi } \left| \int _ { - \infty } ^ { \infty } \varphi ( x ) e ^ { - i y x } d x - \sum _ { - \infty } ^ { \infty } \varphi ( k ) e ^ { - i k x } \right| \leq C \| \varphi \| _ { \operatorname{BV} }, \end{equation*}
  
 
one obtains even stronger results than those known earlier (for early results, see [[#References|[a18]]], [[#References|[a19]]]).
 
one obtains even stronger results than those known earlier (for early results, see [[#References|[a18]]], [[#References|[a19]]]).
Line 60: Line 68:
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Aubertin,  J.J.F. Fournier,  "Integrability theorems for trigonometric series"  ''Studia Math.'' , '''107'''  (1993)  pp. 33–59</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Buntinas,  N. Tanović-Miller,  "Integrability classes and summability"  ''Israel Math. Conf. Proc.'' , '''4'''  (1991)  pp. 75–88</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Buntinas,  N. Tanović-Miller,  "New integrability and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130040/i13004037.png" />-convergence classes for even trigonometric series II"  J. Szabados (ed.)  K Tandori (ed.) , ''Approximation Theory'' , ''Colloq. Math. Soc. János Bolyai'' , '''58''' , North-Holland  (1991)  pp. 103–125</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  L. Bausov,  "On linear methods for the summation of Fourier series"  ''Mat. Sb.'' , '''68'''  (1965)  pp. 313–327  (In Russian)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  E. Belinskii,  "On asymptotic behavior of integral norms of trigonometric polynomials" , ''Metric Questions of the Theory of Functions and Mappings'' , '''6''' , Nauk. Dumka, Kiev  (1975)  pp. 15–24  (In Russian)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  R.P. Boas,  "Absolute convergence and integrability of trigonometric series"  ''J. Rat. Mech. Anal.'' , '''5'''  (1956)  pp. 621–632</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  G.A. Fomin,  "A class of trigonometric series"  ''Math. Notes'' , '''23'''  (1978)  pp. 117–123  ''Mat. Zametki'' , '''23'''  (1978)  pp. 213–222</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  S. Fridli,  "Integrability and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130040/i13004038.png" /> convergence of trigonometric and Walsh series"  ''Ann. Univ. Sci. Budapest, Sect. Comput.'' , '''16'''  (1996)  pp. 149–172</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  M. Ganzburg,  "Best approximation of functions like <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130040/i13004039.png" />"  ''J. Approx. Th.'' , '''92'''  (1998)  pp. 379–410</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  M. Ganzburg,  E. Liflyand,  "Estimates of best approximation and Fourier transforms in integral metrics"  ''J. Approx. Th.'' , '''83'''  (1995)  pp. 347–370</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  D.V. Giang,  F. Móricz,  "Multipliers of Fourier transforms and series on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130040/i13004040.png" />"  ''Archiv Math.'' , '''62'''  (1994)  pp. 230–238</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  A.N. Kolmogorov,  "Sur l'ordre de grandeur des coefficients de la série de Fourier–Lebesgue"  ''Bull. Acad. Polon.''  (1923)  pp. 83–86</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  E.R. Liflyand,  "On asymptotics of Fourier transform for functions of certain classes"  ''Anal. Math.'' , '''19''' :  2  (1993)  pp. 151–168</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  E.R. Liflyand,  "A family of function spaces and multipliers"  ''Israel Math. Conf. Proc.'' , '''13'''  (1999)  pp. 141–149</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  S. Sidon,  "Hinreichende Bedingungen für den Fourier–Charakter einer trigonometrischen Reihe"  ''J. London Math. Soc.'' , '''14'''  (1939)  pp. 158–160</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  S.A. Telyakovskii,  "An estimate, useful in problems of approximation theory, of the norm of a function by means of its Fourier coefficients"  ''Proc. Steklov Inst. Math.'' , '''109'''  (1971)  pp. 73–109  (In Russian)</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top">  S.A. Telyakovskii,  "Concerning a sufficient condition of Sidon for the integrability of trigonometric series"  ''Math. Notes'' , '''14'''  (1973)  pp. 742–748  ''Mat. Zametki'' , '''14'''  (1973)  pp. 317–328</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top">  R.M. Trigub,  "On integral norms of polynomials"  ''Math. USSR Sb.'' , '''30'''  (1976)  pp. 279–295  ''Mat. Sb.'' , '''101 (143)'''  (1976)  pp. 315–333</TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top">  R.M. Trigub,  "Multipliers of Fourier series and approximation of functions by polynomials in spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130040/i13004041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130040/i13004042.png" />"  ''Soviet Math. Dokl.'' , '''39''' :  3  (1989)  pp. 494–498  ''Dokl. Akad. Nauk SSSR'' , '''306'''  (1989)  pp. 292–296</TD></TR><TR><TD valign="top">[a20]</TD> <TD valign="top">  W.H. Young,  "On the Fourier series of bounded functions"  ''Proc. London Math. Soc.'' , '''12''' :  2  (1913)  pp. 41–70</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top">  B. Aubertin,  J.J.F. Fournier,  "Integrability theorems for trigonometric series"  ''Studia Math.'' , '''107'''  (1993)  pp. 33–59</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  M. Buntinas,  N. Tanović-Miller,  "Integrability classes and summability"  ''Israel Math. Conf. Proc.'' , '''4'''  (1991)  pp. 75–88</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  M. Buntinas,  N. Tanović-Miller,  "New integrability and $L^1$-convergence classes for even trigonometric series II"  J. Szabados (ed.)  K Tandori (ed.) , ''Approximation Theory'' , ''Colloq. Math. Soc. János Bolyai'' , '''58''' , North-Holland  (1991)  pp. 103–125</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  L. Bausov,  "On linear methods for the summation of Fourier series"  ''Mat. Sb.'' , '''68'''  (1965)  pp. 313–327  (In Russian)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  E. Belinskii,  "On asymptotic behavior of integral norms of trigonometric polynomials" , ''Metric Questions of the Theory of Functions and Mappings'' , '''6''' , Nauk. Dumka, Kiev  (1975)  pp. 15–24  (In Russian)</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  R.P. Boas,  "Absolute convergence and integrability of trigonometric series"  ''J. Rat. Mech. Anal.'' , '''5'''  (1956)  pp. 621–632</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  G.A. Fomin,  "A class of trigonometric series"  ''Math. Notes'' , '''23'''  (1978)  pp. 117–123  ''Mat. Zametki'' , '''23'''  (1978)  pp. 213–222</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  S. Fridli,  "Integrability and $L^1$ convergence of trigonometric and Walsh series"  ''Ann. Univ. Sci. Budapest, Sect. Comput.'' , '''16'''  (1996)  pp. 149–172</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  M. Ganzburg,  "Best approximation of functions like $| x | ^ { \lambda } \operatorname { exp } ( - A | x | ^ { - \alpha } )$"  ''J. Approx. Th.'' , '''92'''  (1998)  pp. 379–410</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  M. Ganzburg,  E. Liflyand,  "Estimates of best approximation and Fourier transforms in integral metrics"  ''J. Approx. Th.'' , '''83'''  (1995)  pp. 347–370</td></tr><tr><td valign="top">[a11]</td> <td valign="top">  D.V. Giang,  F. Móricz,  "Multipliers of Fourier transforms and series on $L^1$" ''Archiv Math.'' , '''62'''  (1994)  pp. 230–238</td></tr><tr><td valign="top">[a12]</td> <td valign="top">  A.N. Kolmogorov,  "Sur l'ordre de grandeur des coefficients de la série de Fourier–Lebesgue"  ''Bull. Acad. Polon.''  (1923)  pp. 83–86</td></tr><tr><td valign="top">[a13]</td> <td valign="top">  E.R. Liflyand,  "On asymptotics of Fourier transform for functions of certain classes"  ''Anal. Math.'' , '''19''' :  2  (1993)  pp. 151–168</td></tr><tr><td valign="top">[a14]</td> <td valign="top">  E.R. Liflyand,  "A family of function spaces and multipliers"  ''Israel Math. Conf. Proc.'' , '''13'''  (1999)  pp. 141–149</td></tr><tr><td valign="top">[a15]</td> <td valign="top">  S. Sidon,  "Hinreichende Bedingungen für den Fourier–Charakter einer trigonometrischen Reihe"  ''J. London Math. Soc.'' , '''14'''  (1939)  pp. 158–160</td></tr><tr><td valign="top">[a16]</td> <td valign="top">  S.A. Telyakovskii,  "An estimate, useful in problems of approximation theory, of the norm of a function by means of its Fourier coefficients"  ''Proc. Steklov Inst. Math.'' , '''109'''  (1971)  pp. 73–109  (In Russian)</td></tr><tr><td valign="top">[a17]</td> <td valign="top">  S.A. Telyakovskii,  "Concerning a sufficient condition of Sidon for the integrability of trigonometric series"  ''Math. Notes'' , '''14'''  (1973)  pp. 742–748  ''Mat. Zametki'' , '''14'''  (1973)  pp. 317–328</td></tr><tr><td valign="top">[a18]</td> <td valign="top">  R.M. Trigub,  "On integral norms of polynomials"  ''Math. USSR Sb.'' , '''30'''  (1976)  pp. 279–295  ''Mat. Sb.'' , '''101 (143)'''  (1976)  pp. 315–333</td></tr><tr><td valign="top">[a19]</td> <td valign="top">  R.M. Trigub,  "Multipliers of Fourier series and approximation of functions by polynomials in spaces $C$ and $L$"  ''Soviet Math. Dokl.'' , '''39''' :  3  (1989)  pp. 494–498  ''Dokl. Akad. Nauk SSSR'' , '''306'''  (1989)  pp. 292–296</td></tr><tr><td valign="top">[a20]</td> <td valign="top">  W.H. Young,  "On the Fourier series of bounded functions"  ''Proc. London Math. Soc.'' , '''12''' :  2  (1913)  pp. 41–70</td></tr></table>

Revision as of 16:56, 1 July 2020

Given a trigonometric series

\begin{equation} \tag{a1} \frac { a _ { 0 } } { 2 } + \sum _ { k = 1 } ^ { \infty } ( a _ { k } \operatorname { cos } k x + b _ { k } \operatorname { sin } k x ), \end{equation}

the problem of its integrability asks under which assumptions on its coefficients this series is the Fourier series of an integrable function (i.e., belonging to $\widehat{L^1}$). Frequently, the series

\begin{equation} \tag{a2} \frac { a_0 } { 2 } + \sum _ { k = 1 } ^ { \infty } a _ { k } \operatorname { cos } k x \end{equation}

and

\begin{equation} \tag{a3} \sum _ { k = 1 } ^ { \infty } b _ { k } \operatorname { sin } k x \end{equation}

are investigated separately, since there is a difference in their behaviour, and usually integrability of (a3) requires additional assumptions. Of course, one may also consider trigonometric series in complex form.

There exists no convenient description of $\widehat{L^1}$ in terms of a given sequence alone. Hence, subspaces of $\widehat{L^1}$ are studied. In view of the Riemann–Lebesgue lemma (cf. Fourier series) $\widehat{L^1}$ is a subspace of the space of null sequences, while the space of sequences of bounded variation

\begin{equation*} \operatorname{bv} = \left\{ d = \{ d _ { k } \} : \| d \| _ { \operatorname{bv} } = \sum _ { k = 0 } ^ { \infty } | \Delta d _ { k } | < \infty \right\} \end{equation*}

is not a subspace of $\widehat{L^1}$. Here $\Delta d_k = d_k - d_{k + 1}$. Having a null sequence of bounded variation as its Fourier coefficients, the series (a2) converges for every $x \neq 0 ( \operatorname { mod } 2 \pi )$, while (a3) converges everywhere.

In 1913, W.H. Young [a20] proved that if $\{ a _ { k } \}$ is a convex null sequence, that is, $\Delta ^ { 2 } a _ { k } = \Delta ( \Delta a _ { k } ) \geq 0$ for $k = 0,1 , \ldots$, then (a1) is the Fourier series of an integrable function (cf. also Trigonometric series). In 1923, A.N. Kolmogorov [a12] extended this result to the class of quasi-convex sequences $\{ a _ { k } \}$, namely, those satisfying

\begin{equation*} \sum _ { k = 0 } ^ { \infty } ( k + 1 ) \left| \Delta ^ { 2 } \alpha _ { k } \right| < \infty. \end{equation*}

Such a sequence is the difference of two convex sequences. In 1956, R.P. Boas generalized all previous results [a6]. Subsequently, more general subspaces of $\widehat{L^1}$ were considered:

1) The so-called Boas–Telyakovskii space $\operatorname{ bt}$ (see, e.g., [a16]):

\begin{equation*} \| d \| _ { b t } = \| d \| _ { \operatorname {bv} } + \sum _ { n = 2 } ^ { \infty } \left| \sum _ { k = 1 } ^ { n / 2 } \frac { \Delta d _ { n - k } - \Delta d _ { n + k } } { k }\right|. \end{equation*}

2) The Fomin space $a _{p}$ [a7] for $1 < p < \infty$, $1 / p + 1 / p ^ { \prime } = 1$:

\begin{equation*} \| d \| _ { a _ { p } } = \sum _ { n = 0 } ^ { \infty } 2 ^ { n / p ^ { \prime } } \left\{ \sum _ { k = 2 ^ { n } } ^ { 2 ^ { n + 1 } - 1 } | \Delta d _ { k } | ^ { p } \right\} ^ { 1 / p } < \infty . \end{equation*}

3) The Sidon–Telyakovskii space [a17]:

\begin{equation*} A _ { k } \downarrow 0 ( k \rightarrow \infty ) , \sum _ { k = 0 } ^ { \infty } A _ { k } < \infty , | \Delta d _ { k } | < A _ { k }. \end{equation*}

4) The Buntinas–Tanovic–Miller spaces (see, e.g., [a2]).

5) The amalgam space [a1], [a3]:

\begin{equation*} \sum _ { n = 0 } ^ { \infty } \left\{ \sum _ { m = 1 } ^ { \infty } \left[ \sum _ { k = m 2 ^ { n } } ^ { ( m + 1 ) 2 ^ { n } - 1 } | \Delta d _ { k } | \right] ^ { 2 } \right\} ^ { 1 / 2 } < \infty. \end{equation*}

A classical way to prove such results is by using Sidon-type inequalities (see, e.g., [a8]), a typical example of which is the one obtained by S. Sidon [a15]:

\begin{equation*} ( N + 1 ) ^ { - 1 } \left\| \sum _ { k = 0 } ^ { N } c _ { k } D _ { k } \right\| _ { L^{1} } \leq \operatorname { max } _ { 0 \leq k \leq N } | c _ { k } |, \end{equation*}

where $D _ { k }$ is the Dirichlet kernel of order $k$.

In [a13] a new approach to these problems was suggested. First, a locally absolutely continuous function $f$ on $[ 0 , \infty )$ is considered such that $\operatorname { lim } _ { x \rightarrow \infty } f ( x ) = 0$ (cf. also Absolute continuity) and $f \in X$, where $X$ is a subspace of the space of functions of bounded variation $\operatorname{BV}$ and is a generalization of a known space of sequences; e.g., 1)–3) above. Then the asymptotic behaviour of the Fourier transform of a function from $X$ is investigated. Using the following result from [a19] (an earlier version for functions with compact support can be found in [a5]),

\begin{equation*} \operatorname { sup } _ { 0 < | y | < \pi } \left| \int _ { - \infty } ^ { \infty } \varphi ( x ) e ^ { - i y x } d x - \sum _ { - \infty } ^ { \infty } \varphi ( k ) e ^ { - i k x } \right| \leq C \| \varphi \| _ { \operatorname{BV} }, \end{equation*}

one obtains even stronger results than those known earlier (for early results, see [a18], [a19]).

Results on integrability of trigonometric series have numerous applications to approximation problems. The Lebesgue constants of linear means of Fourier series can be efficiently estimated in this way (see, e.g., [a16]). For applications to multiplier problems, see [a11] and [a14]. Other integrability conditions (see, e.g., [a4] and [a16]) were surprisingly applied to the approximation of infinitely differentiable functions in [a10] and [a9].

There exist various extensions of integrability conditions for trigonometric series to the multi-dimensional case (see, e.g., [a13]).

References

[a1] B. Aubertin, J.J.F. Fournier, "Integrability theorems for trigonometric series" Studia Math. , 107 (1993) pp. 33–59
[a2] M. Buntinas, N. Tanović-Miller, "Integrability classes and summability" Israel Math. Conf. Proc. , 4 (1991) pp. 75–88
[a3] M. Buntinas, N. Tanović-Miller, "New integrability and $L^1$-convergence classes for even trigonometric series II" J. Szabados (ed.) K Tandori (ed.) , Approximation Theory , Colloq. Math. Soc. János Bolyai , 58 , North-Holland (1991) pp. 103–125
[a4] L. Bausov, "On linear methods for the summation of Fourier series" Mat. Sb. , 68 (1965) pp. 313–327 (In Russian)
[a5] E. Belinskii, "On asymptotic behavior of integral norms of trigonometric polynomials" , Metric Questions of the Theory of Functions and Mappings , 6 , Nauk. Dumka, Kiev (1975) pp. 15–24 (In Russian)
[a6] R.P. Boas, "Absolute convergence and integrability of trigonometric series" J. Rat. Mech. Anal. , 5 (1956) pp. 621–632
[a7] G.A. Fomin, "A class of trigonometric series" Math. Notes , 23 (1978) pp. 117–123 Mat. Zametki , 23 (1978) pp. 213–222
[a8] S. Fridli, "Integrability and $L^1$ convergence of trigonometric and Walsh series" Ann. Univ. Sci. Budapest, Sect. Comput. , 16 (1996) pp. 149–172
[a9] M. Ganzburg, "Best approximation of functions like $| x | ^ { \lambda } \operatorname { exp } ( - A | x | ^ { - \alpha } )$" J. Approx. Th. , 92 (1998) pp. 379–410
[a10] M. Ganzburg, E. Liflyand, "Estimates of best approximation and Fourier transforms in integral metrics" J. Approx. Th. , 83 (1995) pp. 347–370
[a11] D.V. Giang, F. Móricz, "Multipliers of Fourier transforms and series on $L^1$" Archiv Math. , 62 (1994) pp. 230–238
[a12] A.N. Kolmogorov, "Sur l'ordre de grandeur des coefficients de la série de Fourier–Lebesgue" Bull. Acad. Polon. (1923) pp. 83–86
[a13] E.R. Liflyand, "On asymptotics of Fourier transform for functions of certain classes" Anal. Math. , 19 : 2 (1993) pp. 151–168
[a14] E.R. Liflyand, "A family of function spaces and multipliers" Israel Math. Conf. Proc. , 13 (1999) pp. 141–149
[a15] S. Sidon, "Hinreichende Bedingungen für den Fourier–Charakter einer trigonometrischen Reihe" J. London Math. Soc. , 14 (1939) pp. 158–160
[a16] S.A. Telyakovskii, "An estimate, useful in problems of approximation theory, of the norm of a function by means of its Fourier coefficients" Proc. Steklov Inst. Math. , 109 (1971) pp. 73–109 (In Russian)
[a17] S.A. Telyakovskii, "Concerning a sufficient condition of Sidon for the integrability of trigonometric series" Math. Notes , 14 (1973) pp. 742–748 Mat. Zametki , 14 (1973) pp. 317–328
[a18] R.M. Trigub, "On integral norms of polynomials" Math. USSR Sb. , 30 (1976) pp. 279–295 Mat. Sb. , 101 (143) (1976) pp. 315–333
[a19] R.M. Trigub, "Multipliers of Fourier series and approximation of functions by polynomials in spaces $C$ and $L$" Soviet Math. Dokl. , 39 : 3 (1989) pp. 494–498 Dokl. Akad. Nauk SSSR , 306 (1989) pp. 292–296
[a20] W.H. Young, "On the Fourier series of bounded functions" Proc. London Math. Soc. , 12 : 2 (1913) pp. 41–70
How to Cite This Entry:
Integrability of trigonometric series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integrability_of_trigonometric_series&oldid=17312
This article was adapted from an original article by E.R. Liflyand (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article