Since the trigonometric Fourier series of an integrable function does not necessarily converge to the function in the mean, several additional conditions have been devised to guarantee the convergence. For instance, sufficient conditions can be constructed by using the Fourier coefficients or the integral modulus of the corresponding function. In this paper we give a Hardy-Karamata type Tauberian condition on the Fourier coefficients and prove that it implies the convergence of the Fourier series in integral norm, almost everywhere, and if the function itself is in the real Hardy space, then also in the Hardy norm. We also compare it to the previously known conditions.
@article{bwmeta1.element.bwnjournal-article-smv125i2p161bwm,
author = {S. Fridli},
title = {On the $L\_1$-convergence of Fourier series},
journal = {Studia Mathematica},
volume = {122},
year = {1997},
pages = {161-174},
zbl = {0883.42005},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv125i2p161bwm}
}
Fridli, S. On the $L_1$-convergence of Fourier series. Studia Mathematica, Tome 122 (1997) pp. 161-174. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv125i2p161bwm/
[00000] [1] B. Aubertin and J. J. F. Fournier, Integrability theorems for trigonometric series, Studia Math. 107 (1993), 33-59. | Zbl 0809.42001
[00001] [2] R. Bojanic and Č. Stanojević, A class of convergence, Trans. Amer. Math. Soc. 269 (1982), 677-683.
[00002] [3] W. O. Bray and Č. Stanojević, Tauberian -convergence classes of Fourier series II, Math. Ann. 269 (1984), 469-486. | Zbl 0535.42007
[00003] [4] C. P. Chen, -convergence of Fourier series, J. Austral. Math. Soc. Ser. A 41 (1986), 376-390. | Zbl 0642.42005
[00004] [5] G. A. Fomin, A class of trigonometric series, Mat. Zametki 23 (1978), 213-222 (in Russian).
[00005] [6] S. Fridli, An inverse Sidon type inequality, Studia Math. 105 (1993), 283-308. | Zbl 0811.42001
[00006] [7] D. E. Grow and Č. V. Stanojević, Convergence and the Fourier character of trigonometric transforms with slowly varying convergence moduli, Math. Ann. 302 (1995), 433-472. | Zbl 0827.42003
[00007] [8] B. S. Kashin and A. A. Saakyan, Orthogonal Series, Nauka, Moscow, 1984 (in Russian). | Zbl 0632.42017
[00008] [9] A. N. Kolmogorov, Sur l'ordre de grandeur des coefficients de la série de Fourier-Lebesgue, Bull. Internat. Acad. Polon. Sci. Lettres Sér. (A) Sci. Math. 1923, 83-86.
[00009] [10] M. A. Krasnosel'skiǐ and Ya. B. Rutickiĭ, Convex Functions and Orlicz Spaces, Noordhoff, Groningen, 1961.
[00010] [11] F. Schipp, Sidon-type inequalities, in: Approximation Theory, Lecture Notes in Pure and Appl. Math. 138, Marcel Dekker, New York, 1992, 421-436 . | Zbl 0801.42024
[00011] [12] F. Schipp, W. R. Wade and P. Simon (with assistance from J. Pál), Walsh Series, Adam Hilger, Bristol, 1990. | Zbl 0727.42017
[00012] [13] S. Sidon, Hinreichende Bedingungen für den Fourier-Charakter einer trigonometrischen Reihe, J. London Math. Soc. 14 (1939), 158-160. | Zbl 65.0255.02
[00013] [14] Č. V. Stanojević, Tauberian conditions for the -convergence of Fourier series, Trans. Amer. Math. Soc. 271 (1982), 234-244.
[00014] [15] Č. V. Stanojević, Structure of Fourier and Fourier-Stieltjes coefficients of series with slowly varying convergence moduli, Bull. Amer. Math. Soc. 19 (1988), 283-286. | Zbl 0663.42008
[00015] [16] Č. V. Stanojević and V. B. Stanojević, Generalizations of the Sidon-Telyakovskiĭ theorem, Proc. Amer. Math. Soc. 101 (1987), 679-684. | Zbl 0647.42007
[00016] [17] N. Tanović-Miller, On integrability and convergence of cosine series, Boll. Un. Mat. Ital. B (7) 4 (1990), 499-516. | Zbl 0725.42007
[00017] [18] S. A. Telyakovskiǐ, On a sufficient condition of Sidon for the integrability of trigonometric series, Mat. Zametki 14 (1973), 317-328 (in Russian).
[00018] [19] W. H. Young, On the Fourier series of bounded functions, Proc. London Math. Soc. (2) 12 (1913), 41-70. | Zbl 44.0300.03
[00019] [20] A. Zygmund, Trigonometric Series, University Press, Cambridge, 1959. | Zbl 0085.05601