On a Theorem of Beurling and Livingston

Abstract

In their paper (1), Beurling and Livingston established a generalization of the Riesz-Fischer theorem for Fourier series in L^p using a theorem on duality mappings of a Banach space B into its conjugate space B*. It is our purpose in the present paper to give another proof of this theorem by deriving it from a more general result concerning monotone mappings related to recent results on non-linear functional equations in Banach spaces obtained by the writer (2, 3, 4, 5) and G. J. Minty (6).