# zbMATH — the first resource for mathematics

Convergence and the Fourier character of trigonometric transforms with slowly varying convergence moduli. (English) Zbl 0827.42003
For $$p> 1$$, the convergence modulus of the trigonometric transform $$\sum_{|n|< \infty} c(n) e^{int}$$ is defined in Math. Ann. 279, 103-115 (1987; Zbl 0625.42003) by the second author as $$k^p_n(c)= \sum_{|k|\leq n} |k|^{p- 1}|\Delta c(k)|^p$$. The convergence modulus for $$c= \widehat f$$, $$f\in L^1$$, is a Tauberian device whose restriction in growth: (i) recovers $$L^1$$-convergence of the Fourier series to $$f$$, (ii) recovers a.e. convergence of the Fourier series to $$f$$ and (iii) determines the structure of the Fourier coefficients.
Later [the second author, Proc. Third Annual Meeting Int. Workshop Anal. Appl., Inst. Math., Novi Sad 1-5 (1991)] an equivalent form of the convergence modulus is generalized to $$n^{{1\over p}}|\sigma_n(c)- S_n(c)|_p$$, and results analogous to (i), (ii) and (iii) are proved.
In this study, the authors develop a theory of convergence and integrability of the trigonometric transforms satisfying ${1\over n} \sum_{|k|\leq n} |k|^p|\Delta c(k)|^p= O(1),\quad n> 1, \quad p\in (1, 2],\tag{i}$
$n^{{1\over p}}|\sigma_n(c)- S_n(c)|_p= O(1),\quad n\to \infty,\quad p>1.\tag{ii}$ The results obtained follow a natural direction of inquiry and significantly extend previous results.

##### MSC:
 42A20 Convergence and absolute convergence of Fourier and trigonometric series
Full Text:
##### References:
  L. Carleson, On convergence and growth of partial sums of Fourier series, Acta Math.116 (1966), 135–157 · Zbl 0144.06402  W.O. Bray and Č.V. Stanojević, TauberianL 1-covergence classes of Fourier series I, Trans. Amer. Math. Soc.275 (1983), 59–69  W.O. Bray and Č.V. Stanojević, TauberianL 1-covergence classes of Fourier series II, Math. Ann.269 (1984), 469–486 · Zbl 0545.42005  W. Rudin, Trigonometric series with gaps, J. Math. Mech.9 (1960), 203–207 · Zbl 0091.05802  G.A. Fomin, A class of trigonometric series, Math. Notes23 (1978), 117–124  D. Grow and V.B. Stanojević, Representations of Fourier coefficients and TauberianL 1-covergence classes, J. Math. Anal. Appl.,160 (1992), 47–50 · Zbl 0737.42003  R.A. Hunt, On the covergence of Fourier series, Orthogonal Expansions and Their Continuous Analogues, Southern Illinois University Press, Carbondale, 1968, 235–255, D.T. Haimo, ed  M.A. Kolmogorov, Sur l’ordre de grandeur des coefficients de la série de Fourier-Lebesgue, Bulletin International de l’Academie Polonaise des Sciences et des Lettres, Classe de Sciences Mathematiques et Naturelles (1923), 83–86  W. Rudin, Fourier Analysis on Groups, Interscience Publishers, New York, 1962 · Zbl 0107.09603  S. Sidon, Hinreichende Bedingungen fur den Fourier Charakter einer Trigonometrischen Reihe, J. London Math. Soc. (2)14 (1939), 158–160 · Zbl 0021.40301  Č.V. Stanojević, Tauberian conditions for theL 1-convergence of Fourier series, Trans. Amer. Math. Soc.271 (1982), 237–244  Č.V. Stanojević, 0-Regularly varying convergence moduli of Fourier and Fourier-Stieltjes series, Math. Ann.279 (1987), 103–115 · Zbl 0607.42005  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  Č.V. Stanojević, The Fourier character of series with slowly varying convergence moduli, Proceedings of the Third Annual Meeting of the International Workshop in Analysis and its Applications, Publications de l’Institut Mathématique, Nouvelle série tome48 (62), 1990, 91–95, S. Aljančić and Č.V. Stanojević, editors  Č.V. Stanojević, Characterizations of Tauberian equiconvergence classes of trigonometric transforms, Proceedings of the Fourth Annual Meeting of the International Workshop in Analysis and its Applications, Institute of Mathematics, Novi Sad, 1991, 1–5, Č.V. Stanojević and O. Hadžić, editors  Č.V. Stanojević and V.B. Stanojević, Generalizations of the Sidon-Telyakovskii theorem, Proc. Amer. Math. Soc.101 (1987), 679–684 · Zbl 0647.42007  E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970 · Zbl 0207.13501  R.E. Edwards, Fourier Series: A Modern Introduction, Holt, Rinehart and Winston, Inc., New York, vol. II, 1967 · Zbl 0152.25902  S.A. Telyakovskii, On a sufficient condition of Sidon for the integrability of trigonometric series, Math. Notes14 (1973), 742–748  W.H. Young, On the Fourier series of bounded functions, Proc. London Math. Soc.12 (1913), 41–70 · JFM 44.0300.03  A. Zygmund, Trigonometric Series (second edition), Cambridge University Press, New York, vol. I, 1977 · Zbl 0367.42001  A. Zygmund, op. cit., vol. II, 1977
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.