Integrability of trigonometric series

Given a trigonometric series

(a1)

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 ). Frequently, the series

(a2)

and

(a3)

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 in terms of a given sequence alone. Hence, subspaces of are studied. In view of the Riemann–Lebesgue lemma (cf. Fourier series) is a subspace of the space of null sequences, while the space of sequences of bounded variation

is not a subspace of . Here . Having a null sequence of bounded variation as its Fourier coefficients, the series (a2) converges for every , while (a3) converges everywhere.

In 1913, W.H. Young [a20] proved that if is a convex null sequence, that is, for , 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 , namely, those satisfying

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 were considered:

1) The so-called Boas–Telyakovskii space (see, e.g., [a16]):

2) The Fomin space [a7] for , :

3) The Sidon–Telyakovskii space [a17]:

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

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

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]:

where is the Dirichlet kernel of order .

In [a13] a new approach to these problems was suggested. First, a locally absolutely continuous function on is considered such that (cf. also Absolute continuity) and , where is a subspace of the space of functions of bounded variation 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 is investigated. Using the following result from [a19] (an earlier version for functions with compact support can be found in [a5]),

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 -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 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 " 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 " 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 and " 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