Frame scalings: A condition number approach

Casazza, Peter G.

doi:10.1016/j.laa.2017.02.020

Cited by 8 publications

(5 citation statements)

References 24 publications

(38 reference statements)

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“…i=1 in an n-dimensional Hilbert space H n is a frame if there are constants 0 < A ≤ B < ∞ satisfying: 2 , for all φ ∈ H n .…”

Section: A Family Of Vectors {φ I } Mmentioning

confidence: 99%

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Controlled scaling of Hilbert space frames for R^2

Casazza,

2020

Preprint

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A Hilbert space frame on R n is scalable if we can scale the vectors to make them a tight frame. There are known classifications of scalable frames. There are two basic questions here which have never been answered in any R n :(1) Given a frame in R n , how do we scale the vectors to minimize the condition number of the frame? I.e. How do we scale the frame to make it as tight as possible? (2) If we are only allowed to use scaling numbers from the interval [1 − ǫ, 1 + ǫ], how do we scale the frame to minimize the condition number? We will answer these two questions in R 2 to begin the process towards a solution in R n .

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“…i=1 in an n-dimensional Hilbert space H n is a frame if there are constants 0 < A ≤ B < ∞ satisfying: 2 , for all φ ∈ H n .…”

Section: A Family Of Vectors {φ I } Mmentioning

confidence: 99%

“…A frame is scalable if we can change the lengths of the frame vectors to form a tight frame. Since then, much work has been done on this problem [1,2,5,6,7,8,9,10]. The reason we like this is because the condition number heavily determines the complexity of reconstruction.…”

Section: A Family Of Vectors {φ I } Mmentioning

confidence: 99%

Controlled scaling of Hilbert space frames for R^2

Casazza,

2020

Preprint

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“…We refer the reader to, e.g., weighted and controlled frames [37], which are under very mild conditions equivalent to classical Hilbert frames. Nonetheless, they have applications for example in the implementation of wavelets on the sphere [53, 54], and nowadays become important for the scaling of frames [55, 56]. As a trivial example, look at , where is an orthonormal basis for .…”

Section: Stevenson Frames Revisitedmentioning

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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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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“…Definition 1.1. A family of vectors {x i } m i=1 in an n-dimensional Hilbert space H n is a frame if there are constants 0 < A ≤ B < ∞ satisfying: 2 , for all x ∈ H n .…”

Section: Introductionmentioning

confidence: 99%

“…There is a lot of literature on the subject of scalable frames [1,2,3,6,8,9,10,11,13,14]. Unfortunately the frames for which this standard scaling process is possible are very few.…”

Section: Introductionmentioning

confidence: 99%

Piecewise scalable frames

Casazza¹,

Carli²,

Tran³

2022

Preprint

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In this paper we define "piecewise scalable frames". This new scaling process allows us to alter many frames to Parseval frames which is impossible by the previous standard scaling. We give necessary and sufficient conditions for a frame to be piecewise scalable. We show that piecewise scalability is preserved under unitary transformations. Unlike standard scaling, we show that all frames in R 2 and R 3 are piecewise scalable. We also show that if the frame vectors are close to each other, then they might not be piecewise scalable. Several properties of scaling constants are also presented.

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Frame scalings: A condition number approach

Cited by 8 publications

References 24 publications

Controlled scaling of Hilbert space frames for R^2

Controlled scaling of Hilbert space frames for R^2

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

Piecewise scalable frames

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