By Hans G. Feichtinger, Thomas Strohmer
The utilized and Numerical Harmonic research (ANHA) e-book 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 identify of the sequence displays the im portance of functions and numerical implementation, yet richness and relevance of functions and implementation rely essentially at the constitution and intensity of theoretical underpinnings. hence, from our standpoint, the interleaving of concept and purposes and their inventive symbi otic evolution is axiomatic. Harmonic research is a wellspring of principles and applicability that has flour ished, built, and deepened over the years inside of many disciplines and by way of inventive cross-fertilization with various parts. The tricky and basic dating among harmonic research and fields similar to sig nal processing, partial differential equations (PDEs), and picture processing is mirrored in our cutting-edge ANHA sequence. Our imaginative and prescient of recent harmonic research comprises mathematical components resembling wavelet conception, Banach algebras, classical Fourier research, time frequency research, and fractal geometry, in addition to the various themes that impinge on them.
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Extra info for Advances in Gabor Analysis
2) expresses indeed that Vgf is essentially supported on U. 2. Uncertainty Principles for Time-Frequency Representations 21 The weak uncertainty principle is a simple consequence of the CauchySchwarz inequality. Therefore, o and so lUI ~ 1 - c. A less trivial and stronger estimate on the size of the essential support can be deduced from Lieb's inequalities. These can also be seen as a version of the uncertainty principle. 2. (Lieb [29)) Assume that f,g E L 2 (Rd ). Then if2 ~ p < 00, if 1 ~ p ~ 2.
1. (Hardy) Let f E L2(JRd ), and assume that and for some a, b> O. Then three cases can occur. (i) If a b = 1, then f(x) (ii) If ab > 1, then f = ce- a7Tx2 . =O. (iii) If a b < 1, then any finite linear combination of Hermite functions satisfies these decay conditions. Once again this uncertainty principle can be rephrased in terms of the Rihaczek distribution: Assume that IRf(x,w)1 = O(e-7T(ax2+bw2)). If ab f(x) = C e- a7TX2 ,. if ab > 1, then f = O. = 1, then According to Metatheorem C we may conjecture a Hardy-like theorem for other time-frequency representations.
This completes the proof. 2 Let N = 2,3, .... 11) . 12) Proof: We have for t E [0,1) that le - tJ + 1 = leJ + 1 or leJ according as t :S e - leJ or t > e - l eJ . Case lcJ = 1. When 1 :S e:S 2 - N-l and t E [0,1), at least one of the numbers e - t - liN, l = 0,1, ... , N - 1, falls in the interval (e, 2](mod 1). 4) ;::: 1. When 2 - N- 1 < e < 2 and t E (1 N-l,e-l), all points e-t-lIN, l = 0, ... ,N -1, fall in the interval [1,e). M. Janssen vanishes at l/ = ~. 4) = O. Case lcJ > 1. 15) ensuring that the series stays away from 0 when l/ E [0,1).
Advances in Gabor Analysis by Hans G. Feichtinger, Thomas Strohmer