Harmonic wavelet analysis. (English) Zbl 0793.42020

By observing that the Fourier transform \(\widehat\phi_ s\) of the sinc function \(\phi_ s(x)=\sin \pi x/\pi x\) is the characteristic function of \([-\pi,\pi)\), it is clear that the family \(\phi_ s(x- k)\), \(k\in\mathbb{Z}\), is orthonormal and that \(\phi_ s\) generates a multiresolution analysis (MRA) of \(L^ 2= L^ 2(-\infty,\infty)\). In addition, by shifting \(\widehat\phi_ s\) to the right by \(\pi\) and \(3\pi\) to give \(\widehat\phi(\omega)= \widehat\phi_ s(\omega- \pi)\) and \(\widehat\psi(\omega)= \widehat\phi_ s(\omega-3\pi)\), respectively, we have \(\widehat\phi({1\over 2}\omega)= \widehat\phi(\omega)+ \widehat\psi(\omega)\); i.e., the ideal lowpass filter \(\widehat\phi({1\over 2}\omega)=\chi_{[0,4\pi)}(\omega)\) is the sum of an ideal lowpass filter \(\widehat\phi=\chi_{[0,2\pi)}\) and an ideal bandpass filter \(\widehat\psi= \chi_{[2\pi,4\pi)}\). Consequently, \(\phi(x)- (e^{i2\pi x}-1)/i2\pi x\) also generates an MRA of \(L^ 2\) and \(\psi(x)= (e^{i4\pi x}- e^{i2\pi x})/i2\pi x\) generates the orthogonal complementary subspaces of this MRA. In addition, since the family \(\psi(x- k)\), \(k\in\mathbb{Z}\), is also orthonormal, we see that \(\psi\) is an orthonormal wavelet. This wavelet differs from the Shannon wavelet \(\psi_ s\) only in that the negative passband of \(\widehat\psi_ s\) is set to be zero. The author calls \(\psi(x)= \omega(x)\) a harmonic wavelet, and applies FFT to the implementation of the wavelet coefficients, i.e., the wavelet transform at the time-scale locations \((k2^{-j},2^{-j})\) relative to the analyzing wavelet \(\psi\).


42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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