Approximative K-atomic decompositions and frames in Banach spaces

Abstract

K-frames and atomic systems for an operator K in Hilbert spaces were introduced by Gavruta [17] and further studied by Xio, Zhu and Gavruta [35]. In this paper, we have introduced the notion of an approximative atomic system for an operator K in Banach spaces and obtained interesting results. A complete characterization of family of approximative local atoms of subspace of Banach space has been obtained. Also, a necessary and sufficient condition for the existence of an approximative atomic system for an operator K is given. Finally, explicit methods are given for the construction of an approximative atomic systems for an operator K from a given Bessel sequence and approximative X d -Bessel sequence.∞ k=1 f, f k f k , f ∈ H. The frame operator S is bounded, linear and invertible on H. Thus, a frame for H allows each vector in H to be written as a linear combination of the elements in the frame, but the linear independence between the elements is not required; i.e for each vector f ∈ H we have,

Atomic decomposition of finite signed measures on compacts of R^n

“…Atomic decomposition are also used to solve some eigenvalue problems, as done in [10], where some second-order ordinary differential equations are solved by using atomic decompositions in L p . Finally, let us also recall that generalization of atomic decompositions (and of frames of Banach spaces) have been provided, as in [19].…”