Frames, operator representations, and open problems

Ole Christensen, Marzieh Hasannasab

Research output: Chapter in Book/Report/Conference proceedingBook chapterResearchpeer-review

13 Downloads (Pure)

Abstract

A frame in a Hilbert space H is a countable collection of elements in H that allows each f ϵ H to be expanded as an (infinite) linear combination of the frame elements. Frames generalize the wellknown orthonormal bases, but provide much more exibility and can often be constructed with properties that are not possible for orthonormal bases. We will present the basic facts in frame theory with focus on their operator theoretical characterizations and discuss open problems concerning representations of frames in terms of iterations of a fixed operator. These problems come up in the context of dynamical sampling, a topic that has recently attracted considerably interest within harmonic analysis. The goal of the paper is twofold, namely, that experts in operator theory will explore the potential of frames, and that frame theory will benefit from insight provided by the operator theory community.
Original languageEnglish
Title of host publicationOperator Theory: Advances and Applications
Volume268
PublisherSpringer
Publication date2018
Pages155-165
ISBN (Print)9783319759968
DOIs
Publication statusPublished - 2018
SeriesOperator Theory
Volume268
ISSN0255-0156

Keywords

  • Frames
  • Dual frames
  • Dynamical sampling
  • Operator theory

Cite this

Christensen, O., & Hasannasab, M. (2018). Frames, operator representations, and open problems. In Operator Theory: Advances and Applications (Vol. 268, pp. 155-165). Springer. Operator Theory, Vol.. 268 https://doi.org/10.1007/978-3-319-75996-8_8
Christensen, Ole ; Hasannasab, Marzieh. / Frames, operator representations, and open problems. Operator Theory: Advances and Applications. Vol. 268 Springer, 2018. pp. 155-165 (Operator Theory, Vol. 268).
@inbook{8c83ff950d594612bc83ae1b8d261c5f,
title = "Frames, operator representations, and open problems",
abstract = "A frame in a Hilbert space H is a countable collection of elements in H that allows each f {\"I}µ H to be expanded as an (infinite) linear combination of the frame elements. Frames generalize the wellknown orthonormal bases, but provide much more exibility and can often be constructed with properties that are not possible for orthonormal bases. We will present the basic facts in frame theory with focus on their operator theoretical characterizations and discuss open problems concerning representations of frames in terms of iterations of a fixed operator. These problems come up in the context of dynamical sampling, a topic that has recently attracted considerably interest within harmonic analysis. The goal of the paper is twofold, namely, that experts in operator theory will explore the potential of frames, and that frame theory will benefit from insight provided by the operator theory community.",
keywords = "Frames, Dual frames, Dynamical sampling, Operator theory",
author = "Ole Christensen and Marzieh Hasannasab",
year = "2018",
doi = "10.1007/978-3-319-75996-8_8",
language = "English",
isbn = "9783319759968",
volume = "268",
pages = "155--165",
booktitle = "Operator Theory: Advances and Applications",
publisher = "Springer",

}

Christensen, O & Hasannasab, M 2018, Frames, operator representations, and open problems. in Operator Theory: Advances and Applications. vol. 268, Springer, Operator Theory, vol. 268, pp. 155-165. https://doi.org/10.1007/978-3-319-75996-8_8

Frames, operator representations, and open problems. / Christensen, Ole; Hasannasab, Marzieh.

Operator Theory: Advances and Applications. Vol. 268 Springer, 2018. p. 155-165 (Operator Theory, Vol. 268).

Research output: Chapter in Book/Report/Conference proceedingBook chapterResearchpeer-review

TY - CHAP

T1 - Frames, operator representations, and open problems

AU - Christensen, Ole

AU - Hasannasab, Marzieh

PY - 2018

Y1 - 2018

N2 - A frame in a Hilbert space H is a countable collection of elements in H that allows each f ϵ H to be expanded as an (infinite) linear combination of the frame elements. Frames generalize the wellknown orthonormal bases, but provide much more exibility and can often be constructed with properties that are not possible for orthonormal bases. We will present the basic facts in frame theory with focus on their operator theoretical characterizations and discuss open problems concerning representations of frames in terms of iterations of a fixed operator. These problems come up in the context of dynamical sampling, a topic that has recently attracted considerably interest within harmonic analysis. The goal of the paper is twofold, namely, that experts in operator theory will explore the potential of frames, and that frame theory will benefit from insight provided by the operator theory community.

AB - A frame in a Hilbert space H is a countable collection of elements in H that allows each f ϵ H to be expanded as an (infinite) linear combination of the frame elements. Frames generalize the wellknown orthonormal bases, but provide much more exibility and can often be constructed with properties that are not possible for orthonormal bases. We will present the basic facts in frame theory with focus on their operator theoretical characterizations and discuss open problems concerning representations of frames in terms of iterations of a fixed operator. These problems come up in the context of dynamical sampling, a topic that has recently attracted considerably interest within harmonic analysis. The goal of the paper is twofold, namely, that experts in operator theory will explore the potential of frames, and that frame theory will benefit from insight provided by the operator theory community.

KW - Frames

KW - Dual frames

KW - Dynamical sampling

KW - Operator theory

U2 - 10.1007/978-3-319-75996-8_8

DO - 10.1007/978-3-319-75996-8_8

M3 - Book chapter

SN - 9783319759968

VL - 268

SP - 155

EP - 165

BT - Operator Theory: Advances and Applications

PB - Springer

ER -

Christensen O, Hasannasab M. Frames, operator representations, and open problems. In Operator Theory: Advances and Applications. Vol. 268. Springer. 2018. p. 155-165. (Operator Theory, Vol. 268). https://doi.org/10.1007/978-3-319-75996-8_8