View source for Integrability of trigonometric series

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Given a [[Trigonometric series|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|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|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 } | &lt; \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 [[#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

\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 $\widehat{L^1}$ were considered:

1) The so-called Boas–Telyakovskii space $\operatorname{ bt}$ (see, e.g., [[#References|[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 $a _{p}$ [[#References|[a7]]] for $1 &lt; p &lt; \infty$, $1 / p + 1 / p ^ { \prime } = 1$:

\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 } &lt; \infty . \end{equation*}

3) The Sidon–Telyakovskii space [[#References|[a17]]]:

\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]]]).

5) The amalgam space [[#References|[a1]]], [[#References|[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 } &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]]]:

\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|Dirichlet kernel]] of order $k$.

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]]]),

\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]]]).

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

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

====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 $L^1$-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>