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Journal of Mathematics
Volume 2013, Article ID 135245, 7 pages
http://dx.doi.org/10.1155/2013/135245
Research Article

Summation of Multiple Fourier Series in Matrix Weighted -Spaces

Department of Mathematical Sciences, Aalborg University, Fredrik Bajers Vej 7G, 9220 Aalborg East, Denmark

Received 28 January 2013; Accepted 16 February 2013

Academic Editor: Baoding Liu

Copyright © 2013 Morten Nielsen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper is concerned with rectangular summation of multiple Fourier series in matrix weighted -spaces. We introduce a product Muckenhoupt condition for matrix weights and prove that rectangular Fourier partial sums converge in the corresponding matrix weighted space , , if and only if the weight satisfies the product Muckenhoupt condition. The same result is shown to hold true for other summation methods such as Cesàro and summation with the Jackson kernel.