The Experts below are selected from a list of 17070 Experts worldwide ranked by ideXlab platform
Karlheinz Gröchenig - One of the best experts on this subject based on the ideXlab platform.
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Gabor (Super)Frames with Hermite Functions
2013Co-Authors: Karlheinz Gröchenig, Yurii LyubarskiiAbstract:Abstract. We investigate vector-valued Gabor Frames (sometimes called Gabor superFrames) based on Hermite functions Hn. Let h = (H0, H1,..., Hn) be the vector of the first n + 1 Hermite functions. We give a complete characterization of all lattices Λ ⊆ R 2 such that the Gabor system {e 2πiλ2t h(t − λ1) : λ = (λ1, λ2) ∈ Λ} is a Frame for L 2 (R, C n+1). As a corollary we obtain sufficient conditions for a single Hermite function to generate a Gabor Frame and a new estimate for the lower Frame Bound. The main tools are growth estimates for the Weierstrass σ-function, a new type of interpolation problem for entire functions on the Bargmann-Fock space, and structural results about vector-valued Gabor Frames. 1
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Gabor (super)Frames with Hermite functions
Mathematische Annalen, 2009Co-Authors: Karlheinz Gröchenig, Yurii LyubarskiiAbstract:We investigate vector-valued Gabor Frames (sometimes called Gabor superFrames) based on Hermite functions H _ n . Let h = ( H _0, H _1, . . . , H _ n ) be the vector of the first n + 1 Hermite functions. We give a complete characterization of all lattices $${\Lambda \subseteq \mathbb{R} ^2}$$ such that the Gabor system $${\{ {\rm e}^{2\pi i \lambda _{2} t}{\bf h} (t-\lambda _1): \lambda = (\lambda _1, \lambda _2) \in \Lambda \}}$$ is a Frame for $${L^2 (\mathbb{R} , \mathbb{C} ^{n+1})}$$ . As a corollary we obtain sufficient conditions for a single Hermite function to generate a Gabor Frame and a new estimate for the lower Frame Bound. The main tools are growth estimates for the Weierstrass σ -function, a new type of interpolation problem for entire functions on the Bargmann–Fock space, and structural results about vector-valued Gabor Frames.
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note on b splines wavelet scaling functions and gabor Frames
IEEE Transactions on Information Theory, 2003Co-Authors: Karlheinz Gröchenig, A J E M Janssen, Norbert Kaiblinger, Gotz E PfanderAbstract:Let g be a continuous, compactly supported function on such that the integer translates of g constitute a partition of unity. We show that the Gabor system (g,a,b), with window g and time-shift and frequency-shift parameters a,b>0 has no lower Frame Bound larger than 0 if b=2,3,... and a>0. In particular, (g,a,b) is not a Gabor Frame if g is a continuous, compactly supported wavelet scaling function and if b=2,3,... and a>0. We give an example for our result for the case that g=B/sub 1/, the triangle function supported by [-1,1], by showing pictures of the canonical dual corresponding to (g,a,b) where ab=1/4 and b crosses the lines N=2,3,.
Li Chai - One of the best experts on this subject based on the ideXlab platform.
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Frame Bound computation of two dimensional filter bank Frames
International Conference on Control and Automation, 2013Co-Authors: Yu Pan, Li Chai, Yuxia ShengAbstract:The upper (lower) Bound of a Frame is an important index in the analysis and design of filter bank Frames. There is lack of effectively numerical methods to compute the Frame Bounds for two-dimensional (2-D) filter banks (FBs). This paper investigates the computation problem of Frame Bounds and provides a frequency-independent solution. Firstly, the state space realization of 2-D FIR discrete FBs is given in the form of Roesser model. Then an LMI based optimization method is presented by using the generalized Kalman-Yakubovich-Popov (KYP) lemma. Finally, various examples are given on wavelet and Laplacian pyramid Frames to demonstrate the effectiveness of the proposed method for 2-D Frames.
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improving Frame Bound ratio for Frames generated by oversampled filter banks
International Conference on Acoustics Speech and Signal Processing, 2008Co-Authors: Li Chai, Jingxin Zhang, Cishen Zhang, E MoscaAbstract:This paper presents a simple method to improve the Frame-Bounds-ratio of perfect reconstruction (PR) oversampled filter banks (FBs) by adjusting the gain of each subband filter. For a given analysis PRFB, a finite convex optimization algorithm is presented to redesign the subband gains such that the Frame-Bounds-ratio of the FB is minimized. The algorithm also provides an effective way to compute the Frame Bounds. Examples show the effectiveness of the presented method.
E Mosca - One of the best experts on this subject based on the ideXlab platform.
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improving Frame Bound ratio for Frames generated by oversampled filter banks
International Conference on Acoustics Speech and Signal Processing, 2008Co-Authors: Li Chai, Jingxin Zhang, Cishen Zhang, E MoscaAbstract:This paper presents a simple method to improve the Frame-Bounds-ratio of perfect reconstruction (PR) oversampled filter banks (FBs) by adjusting the gain of each subband filter. For a given analysis PRFB, a finite convex optimization algorithm is presented to redesign the subband gains such that the Frame-Bounds-ratio of the FB is minimized. The algorithm also provides an effective way to compute the Frame Bounds. Examples show the effectiveness of the presented method.
Yurii Lyubarskii - One of the best experts on this subject based on the ideXlab platform.
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Gabor (Super)Frames with Hermite Functions
2013Co-Authors: Karlheinz Gröchenig, Yurii LyubarskiiAbstract:Abstract. We investigate vector-valued Gabor Frames (sometimes called Gabor superFrames) based on Hermite functions Hn. Let h = (H0, H1,..., Hn) be the vector of the first n + 1 Hermite functions. We give a complete characterization of all lattices Λ ⊆ R 2 such that the Gabor system {e 2πiλ2t h(t − λ1) : λ = (λ1, λ2) ∈ Λ} is a Frame for L 2 (R, C n+1). As a corollary we obtain sufficient conditions for a single Hermite function to generate a Gabor Frame and a new estimate for the lower Frame Bound. The main tools are growth estimates for the Weierstrass σ-function, a new type of interpolation problem for entire functions on the Bargmann-Fock space, and structural results about vector-valued Gabor Frames. 1
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Gabor (super)Frames with Hermite functions
Mathematische Annalen, 2009Co-Authors: Karlheinz Gröchenig, Yurii LyubarskiiAbstract:We investigate vector-valued Gabor Frames (sometimes called Gabor superFrames) based on Hermite functions H _ n . Let h = ( H _0, H _1, . . . , H _ n ) be the vector of the first n + 1 Hermite functions. We give a complete characterization of all lattices $${\Lambda \subseteq \mathbb{R} ^2}$$ such that the Gabor system $${\{ {\rm e}^{2\pi i \lambda _{2} t}{\bf h} (t-\lambda _1): \lambda = (\lambda _1, \lambda _2) \in \Lambda \}}$$ is a Frame for $${L^2 (\mathbb{R} , \mathbb{C} ^{n+1})}$$ . As a corollary we obtain sufficient conditions for a single Hermite function to generate a Gabor Frame and a new estimate for the lower Frame Bound. The main tools are growth estimates for the Weierstrass σ -function, a new type of interpolation problem for entire functions on the Bargmann–Fock space, and structural results about vector-valued Gabor Frames.
Markus Faulhuber - One of the best experts on this subject based on the ideXlab platform.
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Minimal Frame Operator Norms Via Minimal Theta Functions
Journal of Fourier Analysis and Applications, 2018Co-Authors: Markus FaulhuberAbstract:We investigate sharp Frame Bounds of Gabor Frames with chirped Gaussians and rectangular lattices or, equivalently, the case of the standard Gaussian and general lattices. We prove that for even redundancy and standard Gaussian window the hexagonal lattice minimizes the upper Frame Bound using a result by Montgomery on minimal theta functions.
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Minimal Operator Norms via Minimal Theta Functions
The journal of fourier analysis and applications, 2017Co-Authors: Markus FaulhuberAbstract:We investigate sharp Frame Bounds of Gabor Frames with chirped Gaussians and rectangular lattices or, equivalently, the case of the standard Gaussian and general lattices. We prove that for even redundancy and standard Gaussian window the hexagonal lattice minimizes the upper Frame Bound using a result by Montgomery on minimal theta functions.
