Generalized frame operator, lower semiframes, and sequences of translates

Abstract

AbstractGiven an arbitrary sequence of elements of a Hilbert space , the operator is defined as the operator associated to the sesquilinear form , for . This operator is in general different from the classical frame operator but possesses some remarkable properties. For instance, is always self‐adjoint with regard to a particular space, unconditionally defined, and, when ξ is a lower semiframe, gives a simple expression of a dual of ξ. The operator and lower semiframes are studied in the context of sequences of integer translates of a function of . In particular, an explicit expression of is given in this context, and a characterization of sequences of integer translates, which are lower semiframes, is proved.