By Elias M. Stein
ISBN-10: 0691032165
ISBN-13: 9780691032160
This booklet includes an exposition of a few of the most advancements of the final 20 years within the following parts of harmonic research: singular fundamental and pseudo-differential operators, the speculation of Hardy areas, L\sup\ estimates regarding oscillatory integrals and Fourier vital operators, kin of curvature to maximal inequalities, and connections with research at the Heisenberg crew.
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Extra resources for Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals
Sample text
By using such complex wavelets we can separate the phase and amplitude components within the signal. Actually, we can easily make a complex version of the Mexican hat wavelet by taking its Fourier transform, setting the negative frequency components in the Fourier domain to zero and then performing an inverse Fourier transform to get the complex wavelet. However, in this section, we focus on the most commonly used complex wavelet, the Morlet wavelet, which is defined as ðtÞ ¼ ÿ1=4 ðei2f0 t ÿ eÿð2f0 Þ 2 =2 Þ eÿt 2 =2 ð2:36Þ where f0 is the central frequency of the mother wavelet.
Location D: This is similar to location B. As the wavelet traverses the discontinuity further, the left-hand lobe of the signal produces a negative contribution to the integral, the right-hand portion of the wavelet produces an equal positive contribution, as with location B. This time, however, the central portion of the wavelet Copyright © 2002 IOP Publishing Ltd. 12. The wavelet decomposition of a signal discontinuity. (a) A signal with a step discontinuity. (b) Plots of Tða; bÞ against b at four arbitrary a scales for the discontinuity: a ¼ 1, 2, 6 and 16.
A) A composite signal (bottom right) constructed from the noise, curtailed sinusoid and exponential decay signals shown. (b) Wavelet transform plot of the composite signal (Mexican hat wavelet). (c) Modulus maxima plot corresponding to the transform plot in (b). 21) and the analysis of multifractal signals in chapter 7. 9 The signal energy: wavelet-based energy and power spectra The total energy contained in a signal, xðtÞ, is defined as its integrated squared magnitude ð1 jxðtÞj2 dt ¼ jjxðtÞjj2 ð2:19Þ E¼ ÿ1 Copyright © 2002 IOP Publishing Ltd.