Functions of Bounded mean Square, and Generalized Fourier-Stieltjes Transforms

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Abstract
A complex function on the real line is said to be bounded in mean square if it is locally in L2 (i.e. on each finite interval) and satisfies (1.1) The set of all such functions clearly forms a linear space over the complex numbers and is a Banach space B under the norm ‖·‖B defined by (1.1). This space, among others, has been discussed by Beurling in [1], where it was shown to be the dual, in the Banach space sense, of a certain Banach (convolution) algebra of functions. We have used Beurling's characterization of B and others of his results throughout this paper, and indeed the essence of one or two of the proofs has been derived from his theorems.

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