On a new Segal algebra

Feichtinger, Hans G.

doi:10.1007/bf01320058

Cited by 282 publications

(255 citation statements)

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“…The most natural measures of concentration are given by norms of the modulation spaces, which are Banach spaces invented and extensively studied by Feichtinger, with some of the main references being [10,11,12,13,14]. For a detailed development of the theory of modulation spaces and their weighted counterparts, we refer to the original literature mentioned above and to [16,.…”

Section: Application To Gabormentioning

confidence: 99%

Redundancy for localized frames

2011

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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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Section: Application To Gabormentioning

confidence: 99%

Redundancy for localized frames

2011

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“…A sufficient condition is, R X ∈ S 0 (Ê + × Ê + ) and Φ ∈ S 0 (Ê + × Ê), which denotes Feichtingers algebra [11], [12]. In this case, we define S 0 on the locally compact abelian (LCA) groups Ê + × Ê + and Ê + × Ê.…”

Section: Introductionmentioning

confidence: 99%

“…Let Φ be real-valued and Φ ∈ S 0 (Ê + × Ê), let Ψ be defined by (10) and fulfill (12). Then, we have the equality…”

Section: Theoremmentioning

confidence: 99%

“…Now, if Φ ∈ S 0 (Ê + × Ê) and Φ is real-valued, then from the invariance properties of S 0 , and by the fact that the Mellin transform is the abstract Fourier transform of [12] when we work on the LCA group Ê + × Ê + , Then, Ψ ∈ S 0 (Ê + × Ê + ) and by (11) the kernel Ψ satisfies the Hermitian property. Thus, since S 0 ⊂ L 2 [11], Ψ is the kernel of compact self-adjoint operator on L 2 (Ê + ×Ê + ) [22].…”

Section: Multiple Window Siws Estimationmentioning

confidence: 99%

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Optimal Scale Invariant Wigner Spectrum Estimation of Gaussian Locally Self-Similar Processes Using Hermite Functions

Maleki

2017

J Theor Probab

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This paper investigates the mean square error optimal estimation of scale invariant Wigner spectrum for the class of Gaussian locally self-similar processes, by the multitaper method. In this method, the spectrum is estimated as a weighted sum of scale invariant windowed spectrograms. Moreover, it is shown that the optimal multitapers are approximated by the quasi Lamperti transformation of Hermite functions, which is computationally more efficient. Finally, the performance and accuracy of the estimation is studied via simulation. keywords:Locally self-similar processes scale invariant Wigner spectrum multitaper method Hermite functions time-frequency analysis. 60G18 60G99

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“…< q2 We follow the definition of positive definite measures given by Dupuis in [6], but using the Segal algebra S0(G) which is equivalent to the space of translation bounded quasimeasures [9]. The advantage is that for e S0(G)' its Fourier transform $ belongs to S0(F)' [I0] and for f e L we have, as proven in [8,2], that…”

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confidence: 99%