Some notes on an expansion theorem of Paley and Wiener

R. J. Duffin, J. J. Eachus
1942 Bulletin of the American Mathematical Society  
have formulated a criterion for a set of functions {g n } to be "near" a given orthonormal set {/ w }. The interest of this criterion is that it guarantees the set {g n } to have expansion properties similar to an orthonormal set. 2 In particular, they show that the set \g n ) approximately satisfies Parseval's formula. In the first part of this paper we show that, conversely, if a set {g n } approximately satisfies Parseval's formula then there exists at least one orthonormal set which it is
more » ... ear." 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. 3 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
doi:10.1090/s0002-9904-1942-07797-4 fatcat:cf4b2u4jebhjbkgfdiv3gn3hwu