Abstract
A theorem of Fejér states that if a periodic function F is of bounded variation on the closed interval [0, 2π], then the nth partial sum of its formally differentiated Fourier series divided by n converges to π-1[F(x+0)-F(x-0)] at each point x. The generalization of this theorem for Fourier-Stieltjes series of (nonperiodic) functions of bounded variation is also well known.
The aim of the present article is to extend these results to the (m, n)th rectangular partial sum of double Fourier or Fourier-Stieltjes series of a function F(x, y) of bounded variation over the closed square [0, 2π]×[0, 2π] in the sense of Hardy and Krause. As corollaries, we also obtain the following results:
-
(i)
The terms of the Fourier or Fourier-Stieltjes series of F(x, y) determine the atoms of the (periodic) Borel measure induced by (an appropriate extension of) F.
-
(ii)
In the case of periodic functions F(x, y) of bounded variation, the class of double Fourier-Stieltjes series coincides with the class of series that can be obtained from their Fourier series by a formal termwise differentiation with respect to both x and y.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adams, C.R. and Clarkson, J.A. (1934). Properties of functionsf(x, y) of bounded variation,Trans. Am. Math. Soc.,36, 711–730.
Clarkson, J.A. and Adams, C.R. (1933). On definitions of bounded variation for functions of two variables,Trans. Am. Math. Soc.,35, 824–854.
Fejér, L. (1913). Über die Bestimmung des Sprunges der Funktion aus ihrer Fourierreihe,J. Reine Angew. Math.,142, 165–188.
Hardy, G.H. (1906). On double Fourier series,Quart. J. Math.,37, 53–79.
Hobson, E.W. (1927).The Theory of Functions of a Real Variable and the Theory of Fourier's Series, 3rd ed., Vol. 1, Cambridge University Press.
Móricz, F. Order of magnitude of double Fourier coefficients of functions of bounded variation,Acta Sci. Math., (Szeged), (submitted).
Riesz, F. and Sz-Nogy, B. (1990).Functional Analysis, Dover Publications, Inc., New York.
Zygmund, A. (1959).Trigonometric Series, Vol. 1, Cambridge University Press.
Author information
Authors and Affiliations
Additional information
Communicated by Tom Körner
Rights and permissions
About this article
Cite this article
Móricz, F. Extension of a theorem of Fejér to double Fourier-Stieltjes series. The Journal of Fourier Analysis and Applications 7, 601–614 (2001). https://doi.org/10.1007/BF02513078
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02513078