Vol. 80, No. 1, 1979

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ISSN: 0030-8730
The Lebesgue constants for (f,dn)-summability

Richard Arthur Shoop

Vol. 80 (1979), No. 1, 255–263
Abstract

It is well-known that the Fourier series of a continuous periodic function need not be pointwise convergent. This fact is a consequence of the unboundedness of the Lebesgue constants, which are the norms of the partial sum operators. It is equally-known that the Fourier series of a continuous function is uniformly (C,1)-summable to the value of the function. Thus, the question naturally arises as to which summability matrices are effective in the limitation of Fourier series of continuous functions. In this paper we consider a very general class of matrices, the (f,dn) means, and show that their Lebesgue constants are unbounded. An interesting corollary is that the Fourier series of a continuous periodic function need not be everywhere almost convergent.

Mathematical Subject Classification 2000
Primary: 42A24
Milestones
Received: 1 April 1976
Revised: 26 July 1978
Published: 1 January 1979
Authors
Richard Arthur Shoop