## Annals of Functional Analysis

- Ann. Funct. Anal.
- Volume 8, Number 2 (2017), 177-189.

### Generalized shift-invariant systems and approximately dual frames

Ana Benavente, Ole Christensen, and María I. Zakowicz

#### Abstract

Dual pairs of frames yield a procedure for obtaining perfect reconstruction of elements in the underlying Hilbert space in terms of superpositions of the frame elements. However, practical constraints often force us to apply sequences that do not exactly form dual frames. In this article, we consider the important case of generalized shift-invariant systems and provide various ways of estimating the deviation from perfect reconstruction that occur when the systems do not form dual frames. The deviation from being dual frames will be measured either in terms of a perturbation condition or in terms of the deviation from equality in the duality conditions.

#### Article information

**Source**

Ann. Funct. Anal., Volume 8, Number 2 (2017), 177-189.

**Dates**

Received: 25 May 2016

Accepted: 12 August 2016

First available in Project Euclid: 14 January 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.afa/1484363069

**Digital Object Identifier**

doi:10.1215/20088752-3784315

**Mathematical Reviews number (MathSciNet)**

MR3597156

**Zentralblatt MATH identifier**

1362.42060

**Subjects**

Primary: 42C15: General harmonic expansions, frames

Secondary: 46E40: Spaces of vector- and operator-valued functions

**Keywords**

approximately dual frames frames generalized shift-invariant systems

#### Citation

Benavente, Ana; Christensen, Ole; Zakowicz, María I. Generalized shift-invariant systems and approximately dual frames. Ann. Funct. Anal. 8 (2017), no. 2, 177--189. doi:10.1215/20088752-3784315. https://projecteuclid.org/euclid.afa/1484363069