A Class of Nonharmonic Fourier Series

Redundancy is the qualitative property which makes Hilbert space frames so useful in practice. However, developing a meaningful quantitative notion of redundancy for infinite frames has proven elusive. Though quantitative candidates for redundancy exist, the main open problem is whether a frame with redundancy greater than one contains a subframe with redundancy arbitrarily close to one. We will answer this question in the affirmative for 1 -localized frames. We then specialize our results to Gabor multi-frames with generators in M 1 (R d ), and Gabor molecules with envelopes in W (C, l 1 ). As a main tool in this work, we we answer a longstanding question in finite dimensional frame theory by showing there is a universal function g(x, y) so that for every > 0, every Parseval frame {f i } M i=1 for an N -dimensional Hilbert space H N has a subset of fewer than (1 + )N elements which is a frame for H N with lower frame bound g( , M N ). The result of this work is the first meaningful quantitative notion of redundancy for a large class of infinite frames.

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“…where K a is as in (9), and the convolution sequence u t r s v ∈ l 1 (G). This means (F # , a, E # ) is l 1 localized.…”

Section: Proofmentioning

confidence: 99%

Redundancy for localized frames

2011

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“…Finally, Kadec in [8] proved that the optimal value of this constant (called the Paley-Wiener constant) is L K = 1 4 (earlier, Levinson in [10] proved that for δ = 1 4 one can perturb the orthonormal Fourier basis to a noncomplete set). The stability question of Fourier frames was considered by Duffin and Schaeffer in their seminal paper [6]. They used a type (1) inequality with µ = 0 and they obtained…”

Section: If F Is a Riesz Basis Then G Is Also A Riesz Basismentioning

confidence: 99%

“…The first result is an enhancement of the Paley-Wiener type constant for nonharmonic series given by Duffin and Schaefer in [6] and used recently in some applications (see [3]). In the case of an orthonormal basis our estimate reduces to Kadec' optimal 1/4 result.…”

Section: Introductionmentioning

confidence: 99%

Stability theorems for Fourier frames and wavelet Riesz bases

Bălan

1997

The Journal of Fourier Analysis and Applications

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In this paper we present two applications of a Stability Theorem of Hilbert frames to nonharmonic Fourier series and wavelet Riesz basis. The first result is an enhancement of the Paley-Wiener type constant for nonharmonic series given by Duffin and Schaefer in [6] and used recently in some applications (see [3]). In the case of an orthonormal basis our estimate reduces to Kadec' optimal 1/4 result. The second application proves that a phenomenon discovered by Daubechies and Tchamitchian [4] for the orthonormal Meyer wavelet basis (stability of the Riesz basis property under small changes of the translation parameter) actually holds for a large class of wavelet Riesz bases.

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“…Hilbert 空间(向量)框架首先是由 R. Duffin 和 A. Schaeffer 为研究非调和 Fourier 级数及其一些重要的应用而引入的 [3] ；近 20 年来，它在理论研究和应用方面都得到迅速发展，文献 [4]对其作了比较详细的介绍。…”

Section: *-Cunclassified