Orthonormal scaling functions generating fractional Hilbert transforms of an orthonormal wavelet (Harmonic Analysis and Nonlinear Partial Differential Equations)

Abstract

The Hilbert transform is an important transform not only in Mathematics, but also in some applications. Since a wavelet function has zero average, the Hilbert transform of it is a good function in many cases. It is well-known that many wavelet functions, especially important ones, can be generated from scaling functions in the framework of multiresolution analysis (MRA. Hence, it is an important problem what is the scaling function from which the Hilbert transform of the wavelet function is generated. We consider two families of unitary operators. One is a family of extensions of the Hilbert transform called fractional Hilbert transforms. The other is a new family of operators which are a kind of modified translation operators. A fractional Hilbert transform of a given orthonormal wavelet (resp. scaling) function is also an orthonormal wavelet (resp. scaling) function, although a fractional Hilbert transform of a scaling function has bad localization in many cases. We show that a modified translation of a scaling function is also a scaling function, and it generates a fractional Hilbert transform of the corresponding wavelet function. Further, we show a good localization property of the modified translation operators. The modified translation operators act on the Meyer scaling functions as the ordinary translation operators. We give a class of scaling functions, on which the modified translation operators act as the ordinary translation operators.