This article is cited in 4 papers

On the Properties of Generalized Frames

A. A. Zakharova

M. V. Lomonosov Moscow State University

Abstract: In this paper, we introduce the notion of generalized frames and study their properties. Discrete and integral frames are special cases of generalized frames. We give criteria for generalized frames to be integral (discrete). We prove that any bounded operator $A$ with a bounded inverse acting from a separable space $H$ to $L_2(\Omega)$ (where $\Omega$ is a space with countably additive measure) can be regarded as an operator assigning to each element $x\in H$ its coefficients in some generalized frame.

Keywords: frame, tight frame, integral frame, bounded operator, separable Hilbert space, Lebesgue space, countably additive measure.

UDC: 517.518+517.982

Received: 30.05.2006
Revised: 21.03.2007

DOI: 10.4213/mzm4417

 English version: Mathematical Notes, 2008, 83:2, 190–200

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