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[edit]

[a1] |  B. Aubertin, J.J.F. Fournier, "Integrability theorems for trigonometric series" Studia Math. , 107 (1993) pp. 33–59

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[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