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

Kenneth F. Andersen

Vol. 105 (1983), No. 1, 1–10

DOI: 10.2140/pjm.1983.105.1

Abstract

If {a_ν}₁^∞ is the sequence of Fourier cosine coefficients of a function in the space L^p, 1 ≤ p < ∞, a Theorem of Hardy states that the sequence of averages {(Σ_j=1^νa_j)∕ν}₁^∞ arise as Fourier cosine coefficients of a function also in L^p. Analogous results for the sequence {Σ_j=ν^∞a_j∕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 L^p(ω(x)dx) may replace the spaces L^p in the Theorems of Hardy and Bellman.

Mathematical Subject Classification 2000

Primary: 42A16

Secondary: 46E30

Milestones

Received: 19 August 1981

Published: 1 March 1983