On the L 1 -convergence of Fourier series

S. Fridli

Studia Mathematica (1997)

  • Volume: 125, Issue: 2, page 161-174
  • ISSN: 0039-3223

Abstract

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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.

How to cite

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Fridli, S.. "On the $L_1$-convergence of Fourier series." Studia Mathematica 125.2 (1997): 161-174. <http://eudml.org/doc/216429>.

@article{Fridli1997,
abstract = {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.},
author = {Fridli, S.},
journal = {Studia Mathematica},
keywords = {Fourier series; $L_1$-convergence; a.e. convergence; Hardy-Karamata type Tauberian condition; Fourier coefficients; convergence; integral norm; Hardy norm; lacunary Fourier series},
language = {eng},
number = {2},
pages = {161-174},
title = {On the $L_1$-convergence of Fourier series},
url = {http://eudml.org/doc/216429},
volume = {125},
year = {1997},
}

TY - JOUR
AU - Fridli, S.
TI - On the $L_1$-convergence of Fourier series
JO - Studia Mathematica
PY - 1997
VL - 125
IS - 2
SP - 161
EP - 174
AB - 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.
LA - eng
KW - Fourier series; $L_1$-convergence; a.e. convergence; Hardy-Karamata type Tauberian condition; Fourier coefficients; convergence; integral norm; Hardy norm; lacunary Fourier series
UR - http://eudml.org/doc/216429
ER -

References

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  1. [1] B. Aubertin and J. J. F. Fournier, Integrability theorems for trigonometric series, Studia Math. 107 (1993), 33-59. Zbl0809.42001
  2. [2] R. Bojanic and Č. Stanojević, A class of L 1 convergence, Trans. Amer. Math. Soc. 269 (1982), 677-683. 
  3. [3] W. O. Bray and Č. Stanojević, Tauberian L 1 -convergence classes of Fourier series II, Math. Ann. 269 (1984), 469-486. Zbl0535.42007
  4. [4] C. P. Chen, L 1 -convergence of Fourier series, J. Austral. Math. Soc. Ser. A 41 (1986), 376-390. Zbl0642.42005
  5. [5] G. A. Fomin, A class of trigonometric series, Mat. Zametki 23 (1978), 213-222 (in Russian). 
  6. [6] S. Fridli, An inverse Sidon type inequality, Studia Math. 105 (1993), 283-308. Zbl0811.42001
  7. [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. Zbl0827.42003
  8. [8] B. S. Kashin and A. A. Saakyan, Orthogonal Series, Nauka, Moscow, 1984 (in Russian). Zbl0632.42017
  9. [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. 
  10. [10] M. A. Krasnosel'skiǐ and Ya. B. Rutickiĭ, Convex Functions and Orlicz Spaces, Noordhoff, Groningen, 1961. 
  11. [11] F. Schipp, Sidon-type inequalities, in: Approximation Theory, Lecture Notes in Pure and Appl. Math. 138, Marcel Dekker, New York, 1992, 421-436 . Zbl0801.42024
  12. [12] F. Schipp, W. R. Wade and P. Simon (with assistance from J. Pál), Walsh Series, Adam Hilger, Bristol, 1990. Zbl0727.42017
  13. [13] S. Sidon, Hinreichende Bedingungen für den Fourier-Charakter einer trigonometrischen Reihe, J. London Math. Soc. 14 (1939), 158-160. Zbl65.0255.02
  14. [14] Č. V. Stanojević, Tauberian conditions for the L 1 -convergence of Fourier series, Trans. Amer. Math. Soc. 271 (1982), 234-244. 
  15. [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. Zbl0663.42008
  16. [16] Č. V. Stanojević and V. B. Stanojević, Generalizations of the Sidon-Telyakovskiĭ theorem, Proc. Amer. Math. Soc. 101 (1987), 679-684. Zbl0647.42007
  17. [17] N. Tanović-Miller, On integrability and L 1 convergence of cosine series, Boll. Un. Mat. Ital. B (7) 4 (1990), 499-516. Zbl0725.42007
  18. [18] S. A. Telyakovskiǐ, On a sufficient condition of Sidon for the integrability of trigonometric series, Mat. Zametki 14 (1973), 317-328 (in Russian). 
  19. [19] W. H. Young, On the Fourier series of bounded functions, Proc. London Math. Soc. (2) 12 (1913), 41-70. Zbl44.0300.03
  20. [20] A. Zygmund, Trigonometric Series, University Press, Cambridge, 1959. Zbl0085.05601

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