Hostname: page-component-7f64f4797f-42qgm Total loading time: 0 Render date: 2025-11-04T13:57:21.408Z Has data issue: false hasContentIssue false
  • English
  • Français

A Note on Convergence of Fourier Series of a Function on Wiener's Class Vp

Published online by Cambridge University Press:  20 November 2018

Rafat N. Siddiqi
Rafat N. Siddiqi*
Affiliation:
Department of Physics-Mathematics, Université de Moncton Moncton, N.B., Canada Summer research Institute of Canadian Mathematical Congress Dalhousie University Halifax, N.S., Canada
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.

Let f be a 2π-periodic function defined on [0, 2π]. We set

where suprema have been taken with respect to all partitions P:a = t0<t1<t2<…<tn<=b of any segment [a, b] contained in [0, 2π]. We call the pth total variation of f on [a, b]. If we denote pth total variaiton of f on [0, 2π] by Vp (f), then we can define Wiener's class simply by

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 20 , Issue 2 , 01 June 1977 , pp. 243 - 247
Copyright
Copyright © Canadian Mathematical Society 1977

References

1. Bari, N., A treatise on trigonometric series, Vol. 1, Oxford, Pergamon Press, 1964.Google Scholar
2. Rudin, W., Fourier Analysis on groups, New York, Interscience Publishers, 1962.Google Scholar
3. Siddiqi, R. N., The order of Fourier coefficients of a function of higher variation, Proc. Japan Acad., 48 (1972), 569-572.Google Scholar
4. Siddiqi, R. N., Some properties of Fourier-Stieltjes coefficients of a function of Wiener's class Vp , Bull. Math. De Roumanie, Tome 16 (64), nr. 1 (1972), 105-112.Google Scholar
5. Wiener, N., The quadratic variation of a function and its Fourier coefficients, Massachusetts J. Math. 3 (1924), 72-94.Google Scholar
6. Zygmund, A., Trigonometric series, Vol. 1, Cambridge, 1959.Google Scholar