A Guide to Localized Frames and Applications to Galerkin-Like Representations of Operators

Balázs, Péter; Gröchenig, Karlheinz

doi:10.1007/978-3-319-55550-8_4

Cited by 18 publications

(12 citation statements)

References 67 publications

(105 reference statements)

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“…For orthonormal sequences, it is well known that operators can be uniquely described by a matrix representation [36]. The same can be constructed with frames and their duals, see [6,9].…”

Section: Operator Representation In Frame Coordinatesmentioning

confidence: 99%

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Frames for the Solution of Operator Equations in Hilbert Spaces with Fixed Dual Pairing

Balázs

Harbrecht

2018

Numerical Functional Analysis and Optimization

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For the solution of operator equations, Stevenson introduced a definition of frames, where a Hilbert space and its dual are not identified. This means that the Riesz isomorphism is not used as an identification, which, for example, does not make sense for the Sobolev spaces and . In this article, we are going to revisit the concept of Stevenson frames and introduce it for Banach spaces. This is equivalent to -Banach frames. It is known that, if such a system exists, by defining a new inner product and using the Riesz isomorphism, the Banach space is isomorphic to a Hilbert space. In this article, we deal with the contrasting setting, where and are not identified, and equivalent norms are distinguished, and show that in this setting the investigation of -Banach frames make sense.

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“…For orthonormal sequences, it is well known that operators can be uniquely described by a matrix representation [36]. The same can be constructed with frames and their duals, see [6,9].…”

Section: Operator Representation In Frame Coordinatesmentioning

confidence: 99%

“…in [37,40,41,48,53]. Recently, such operator representations got also a more theoretical treatment [6,9,12].…”

Section: Introductionmentioning

confidence: 99%

Frames for the Solution of Operator Equations in Hilbert Spaces with Fixed Dual Pairing

Balázs

Harbrecht

2018

Numerical Functional Analysis and Optimization

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“…It is well known that each linear operator is represented by corresponding matrix, and the opposite is also true: each matrix generates corresponding linear operator. Some authors have constructed various types of matrix representation of operators using an orthonormal basis [20], frames and their canonical duals [18], Gabor frames [25], Bessel sequences [8] and localized frames [11]. The operators induced by matrices, with respect to Bessel sequences, are also introduced by Balazs [8].…”

Section: Generalized Bessel Multipliersmentioning

confidence: 99%

Generalized Bessel Multipliers in Hilbert Spaces

2018

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The notation of generalized Bessel multipliers is obtained by a bounded operator on ℓ 2 which is inserted between the analysis and synthesis operators. We show that various properties of generalized multipliers are closely related to their parameters, in particular it will be shown that the membership of generalized Bessel multiplier in the certain operator classes requires that its symbol belongs in the same classes, in special sense. Also, we give some examples to illustrate our results. As we shall see, generalized multipliers associated with Riesz bases are well-behaved, more precisely in this case multipliers can be easily composed and inverted. Special attention is devoted to the study of invertible generalized multipliers. Sufficient and/or necessary conditions for invertibility are determined. Finally, the behavior of these operators under perturbations is discussed.

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“…Not long ago we introduced a family of quasi-Banach spaces -which we called 'wave packet spaces' -that encompasses all these spaces, studied their properties, equipped them with Banach frames -which we called wave packet systems -and provided their atomic decompositions [2]. Efficiency of the approximation of functions [12] of various classes in terms of a combination of elements of an appropriately chosen wave packet system and their potential usefulness for discretisation of bounded linear operators [1], in general, and Fourier integral operators [5], in particular, by Galerkin method can be expected to depend crucially on how well these elements are localised. Herein we give a precise and thorough answer to this question.…”

Section: Introductionmentioning

confidence: 99%

On near orthogonality of the Banach frames of the wave packet spaces

Bytchenkoff

2020

Preprint

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In solving scientific, engineering or pure mathematical problems one is often faced with a need to approximate the function of a given class by the linear combination of a preferably small number of functions that are localised one way or another both in the time and frequency domain. Over the last seventy years or so a range of systems of thus localised functions have been developed to allow the decomposition and synthesis of functions of various classes. The most prominent examples of such systems are Gabor functions, wavelets, ridgelets, curvelets, shearlets and wave atoms. We recently introduced a family of quasi-Banach spaces -which we called wave packet spaces -that encompasses all those classes of functions whose elements have sparse expansions in one of the above-mentioned systems, supplied them with Banach frames and provided their atomic decompositions. Herein we prove that the Banach frames and sets of atoms of the wave packet spaces are well localised or, more specifically, that they are near orthogonal.

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A Guide to Localized Frames and Applications to Galerkin-Like Representations of Operators

Cited by 18 publications

References 67 publications

Frames for the Solution of Operator Equations in Hilbert Spaces with Fixed Dual Pairing

Frames for the Solution of Operator Equations in Hilbert Spaces with Fixed Dual Pairing

Generalized Bessel Multipliers in Hilbert Spaces

On near orthogonality of the Banach frames of the wave packet spaces

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