Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-26T07:24:36.407Z Has data issue: false hasContentIssue false
  • English
  • Français

Pointwise Convergence of Trigonometric Series

Part of: Harmonic analysis in one variable

Published online by Cambridge University Press:  09 April 2009

Chang-Pao Chen
Chang-Pao Chen
Affiliation:
Department of MathematicsStanford UniversityStanford, California 94305, U.S.A.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We establish two results in the pointwise convergence problem of a trigonometric series for some nonnegative integer m. These results not only generalize Hardy's theorem, the Jordan test theorem and Fatou's theorem, but also complement the results on pointwise convergence of those Fourier series associated with known L1-convergence classes. A similar result is also established for the case that , where {ln} satisfies certain conditions.

Keywords

pointwise convergence of a trigonometric seriesFourier series.

MSC classification

Secondary: 42A20: Convergence and absolute convergence of Fourier and trigonometric series 42A32: Trigonometric series of special types (positive coefficients, monotonic coefficients, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 43 , Issue 3 , December 1987 , pp. 291 - 300
Copyright
Copyright © Australian Mathematical Society 1987

References

[1]Bojanic, R. and Stanojević, č. V., ‘A class of l 1-convergence’, Trans. Amer. Math. Soc. 269 (1982), 677–683.Google Scholar
[2]Chen, C.-P., ‘l1-convergence of Fourier series,‘ J. Austral. Math. Soc. (Series A) 41 (1986), 376–390.Google Scholar
[3]Edwards, R. E., Fourier Series. A Modern Introduction, Vols. 1 and 2 (Holt, Rinehart and Winston, New York, 1967).Google Scholar
[4]Karamata, J., Teorija i praksa Stieltjes-ova integrala (Srpska Akademija Nauka, Beograd, 1949).Google Scholar
[5]Katznelson, Y., An Introduction to Harmonic Analysis (John Wiley and Sons, New York, 1968)Google Scholar
[6]Kolmogorov, A. N., ‘Sur l'ordre de grandeur des coefficients de la série de Fourier-Lebesgue’, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. (1923), 83–86.Google Scholar
[7]Stanojević, Ĉ. V., ‘Classes of L 1-convergence of Fourier and Fourier-Stieltjes series,‘ Proc. Amer. Math. Soc. 82 (1981), 209–215.Google Scholar
[8]Titchmarsh, E. C., Theory of Functions (2nd ed., Oxford University Press, 1958).Google Scholar
[9]Zygmund, A., Trigonometric Series (Cambridge Univ. Press, 1959).Google Scholar