Some Notes on an Expansion Theorem of Paley and Wiener

Paley and Wiener have formulated a criterion for a set of functions {gn} to be "near" a given orthonormal set {/w}. The interest of this criterion is that it guarantees the set {gn} to have expansion properties similar to an orthonormal set. In particular, they show that the set \gn) approximately satisfies Parseval's formula. In the first part of this paper we show that, conversely, if a set {gn} approximately satisfies Parseval's formula then there exists at least one orthonormal set which it is "near." In the second part of the paper we consider sets which are on the borderline of being near a given orthonormal set. The last part of this paper gives a simple formula for constructing sets near a given orthonormal set. As an application of this formula we obtain new properties of the so called non-harmonic Fourier series. We shall handle these problems abstractly, using the notation of Hubert space. Subscript variables are assumed to range over all positive integers and ]>j shall mean a sum over all positive integers. By a finite sequence shall be meant a sequence with only a finite number of nonzero members. For application to the space L2 the norm of a function f(x) is defined in the usual way as ||/|| = (fa \f(x) \ dx). A complete set which satisfies the Paley-Wiener criterion shall be termed strongly complete. The principal novelty in the proof is the association of a linear transformation G with each set of elements {gn}Thus if {ypn} is an orthonormal setwe defineG^Ard'n =T^^ngnfor every finite sequence of constants {an}. The norm of G is the limit superior of \\GX\\ for elements x such that \\x\\ = 1. With this definition of norm the aggregate of bounded linear transformations clearly forms a normed linear