A theorem on bounded analytic functions

Michael Cannon Mooney

Vol. 43 (1972), No. 2, 457–463

DOI: 10.2140/pjm.1972.43.457

Abstract

The purpose of this paper is to prove the following

Theorem: Let ϕ₁,ϕ₂,⋯ be an infinite sequence of functions in L¹([0,2π]) such that L(f) = lim_n→∞∫ ₀^2πf(e^i𝜃)ϕ_n(𝜃)d𝜃 exists for every f ∈ H^∞. Then there is a ϕ ∈ L¹([0,2π]) such that L(f) = ∫ ₀^2πf(e^i𝜃)ϕ(𝜃)d𝜃 for all f ∈ H^∞.

Mathematical Subject Classification 2000

Primary: 46J15

Secondary: 30A98

Milestones

Received: 15 April 1971

Revised: 3 February 1972

Published: 1 November 1972