Orthonormal wavelet basis with arbitrary real dilation factor

摘要

Daubechies posed the following problem in Ten Lectures on Wavelets (SIAM, Philadelphia, PA, 1992): "It is an open question whether there exist orthonormal wavelet bases (not necessarily associated with a multiresolution analysis), with good time-frequency localization, and with irrational a" (that is, for an arbitrary irrational dilation factor a > 1, with appropriate wavelet function psi and constant b > 0, whether can {a(j/2)psi(a(j)t-bk) : j,k is an element of Z} construct an orthonormal wavelet basis with good time-frequency localization?). Our answer is "Yes". In this paper, we introduce a new type of orthonormal wavelet basis having an arbitrary real dilation factor greater than 1. This orthonormal wavelet basis requires an infinite number of wavelet shapes when its dilation factor is irrational.