## 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 $$\psi$$.

### MSC:

 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
Full Text: