Vol. 105, No. 1, 1983

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On the transformation of Fourier coefficients of certain classes of functions. II

Kenneth F. Andersen

Vol. 105 (1983), No. 1, 1–10
Abstract

If {aν}1 is the sequence of Fourier cosine coefficients of a function in the space Lp, 1 p < , a Theorem of Hardy states that the sequence of averages {j=1νaj)∕ν}1 arise as Fourier cosine coefficients of a function also in Lp. Analogous results for the sequence {Σj=νaj∕j}1 were obtained by Bellman. In this paper, sufficient conditions on the non-negative weight function ω(x) are given in order that the weighted Lebesgue space Lp(ω(x)dx) may replace the spaces Lp in the Theorems of Hardy and Bellman.

Mathematical Subject Classification 2000
Primary: 42A16
Secondary: 46E30
Milestones
Received: 19 August 1981
Published: 1 March 1983
Authors
Kenneth F. Andersen