By Hans G. Feichtinger, Thomas Strohmer
The utilized and Numerical Harmonic research (ANHA) ebook sequence goals to supply the engineering, mathematical, and medical groups with major advancements in harmonic research, starting from summary har monic research to uncomplicated functions. 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. therefore, from our standpoint, the interleaving of thought and purposes and their inventive symbi otic evolution is axiomatic. Harmonic research is a wellspring of rules and applicability that has flour ished, built, and deepened through the years inside of many disciplines and by way of inventive cross-fertilization with varied components. The problematic and basic courting among harmonic research and fields akin to sig nal processing, partial differential equations (PDEs), and snapshot processing is mirrored in our state-of-the-art ANHA sequence. Our imaginative and prescient of contemporary harmonic research contains mathematical components akin to wavelet conception, Banach algebras, classical Fourier research, time frequency research, and fractal geometry, in addition to the varied themes that impinge on them.
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Additional info for Advances in Gabor Analysis
Collection traitement du signal, Hermes, 1993.  G. B. Folland, Harmonic analysis in phase space, Princeton Univ. Press, Princeton, NJ, 1989.  G. B. Folland and A. Sitaram, The uncertainty principle: a mathematical survey, J. Fourier Anal. Appl. 3 (1997), no. 3, 207-238.  E. D. thesis, Univ. of Connecticut, 2000.  K. Grochenig, An uncertainty principle related to the Poisson summation formula, Studia Math. 121 (1996), no. 1,87-104.  K. Grochenig, Foundations of time-frequency analysis, Birkhiiuser, Boston, 2001.
Then if2 ~ p < 00, if 1 ~ p ~ 2. 4) Lieb's inequalities are quite deep, their proof requires the sharp constants in the theorems of Young and Hausdorff-Young. It is not surprising that with Lieb's inequalities we can derive a better lower estimate for the size of the essential support of a time-frequency representation. 3. (, Thm. i IVgf(x, wWdxdw 2: (1 - c) Ilfll~lIgll~, for all p> 2. 2d then IIfl12 = IIgl12 = lUI ~ (I-c)~ (~)"-2 1. If In particular, Proof: We first apply Holder's inequality with exponents q = ~ and q' = ~, and then we use Lieb's inequality for p > 2 in the second step: IIfll~llglI~ ~ (1 - c) < < ffu IVgf(x, wWdxdw (fLu IVgf(x,wW·~dXdW) 2/p (fL2d xu (x,w)ql dXdW) ~ 2d (~) p (lIf11211g112f IUI~ .
In signal analysis one usually considers time-frequency representations in Cohen's class . These are sesquilinear forms C(f,g)(x,w) on S(I~d) x S(JRd) with the covariance property C(Mc,Tzf, M(,Tzf) (x, w) = C(f, f)(x-z, w-(). Under a weak continuity assumption it can be shown rigorously that all such time-frequency representations are of the form C(f, f) = W f * ell for some tempered distribution ell E S' [18, Thm. 1]. According to Metatheorem C the uncertainty principles discussed so far can be adapted to yield uncertainty principles for time-frequency representations of the form C(f, f) = W f * ell.