Gram Matrices of Systems of Uniformly

Let AN N be the set of the Gram matrices of systems fejg N j formed by vectors ej of a Hilbert spaceH with norms kejkH j N Let BN K be the set of the Gram matrices of systems ffjg N j formed by functions fj L with kfjkL K j N It is shown that for any K the set BN K is narrower than AN as N More precisely it is proved that not every matrix A in AN can be represented as A B where B BN K and is a diagonal matrix In this paper we prove a theorem announced in First we recall some well known concepts and problems and introduce the notation used in what follows As usual RN N N is the N dimensional Euclidean space with the inner product h i and Euclidean norm j j Let B fx R jxj g S fx R jxj g For a set of vectors Z fzigpi RN denote by GZ the Gram matrix GZ fhzi zkigpi k Let ON be the rotation group of RN Clearly for ON and Z f zigpi we have G Z GZ Below we consider the Hilbert space L and the smaller space L Denote by the inner product