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By Hans G. Feichtinger, Thomas Strohmer

The utilized and Numerical Harmonic research (ANHA) publication sequence goals to supply the engineering, mathematical, and medical groups with major advancements in harmonic research, starting from summary har­ monic research to easy purposes. The name of the sequence displays the im­ portance of purposes and numerical implementation, yet richness and relevance of purposes and implementation count essentially at the constitution and intensity of theoretical underpinnings. hence, from our perspective, the interleaving of concept and functions and their artistic symbi­ otic evolution is axiomatic. Harmonic research is a wellspring of rules and applicability that has flour­ ished, constructed, and deepened over the years inside of many disciplines and through artistic cross-fertilization with different components. The problematic and basic dating among harmonic research and fields akin to sig­ nal processing, partial differential equations (PDEs), and photograph processing is mirrored in our state-of-the-art ANHA sequence. Our imaginative and prescient of recent harmonic research comprises mathematical parts reminiscent of wavelet concept, Banach algebras, classical Fourier research, time­ frequency research, and fractal geometry, in addition to the various issues that impinge on them.

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4 such a 9 cannot be strictly convex on [0,00). The example just given is more or less typical for all cases where J(t, v) = for some t =j:. 0,1 and v =j:. o,~, 1 with 9 as above. 24) can only vanish for v = 0, ~, 1 and for v of the form pi M with integer p, M and gcd(p, M) = 1. In the latter case we also need that d k = d k +1 for k = 1,2, ... unless k is a multiple of M. By convexity of b this implies that b is linear on each of the intervals [mM + 1- t, (m + l)M + t], m = 0,1, ... (in order that this latter condition can hold with an integrable, non-trivial b we also need that t :S ~).

6, 723-726. [28] J. Lakey, Personal communication. [29] E. H. Lieb, Integral bounds for radar ambiguity functions and Wigner distributions, J. Math. Phys. 31 (1990), no. 3, 594-599. [30] E. Wilczock, Zur Funktionalanalysis der Wavelet- und Gabortransformation, Thesis, TV Miinchen, 1998. H. Feichtinger et al. M. 2) n,m where 1111 and ( , ) denote the standard norm and inner product of £2 (JR). We refer to [1]-[3] for generalities about frames for a Hilbert space and Gabor frames in particular, and how they give rise to methods of efficient signal representation.

27] A. J. E. M. Janssen, Proof of a conjecture on the supports of Wigner distributions, J. Fourier Anal. Appl. 4 (1998), no. 6, 723-726. [28] J. Lakey, Personal communication. [29] E. H. Lieb, Integral bounds for radar ambiguity functions and Wigner distributions, J. Math. Phys. 31 (1990), no. 3, 594-599. [30] E. Wilczock, Zur Funktionalanalysis der Wavelet- und Gabortransformation, Thesis, TV Miinchen, 1998. H. Feichtinger et al. M. 2) n,m where 1111 and ( , ) denote the standard norm and inner product of £2 (JR).

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