Tight wavelet frames on the space of M-positive vectors

Abstract

Wavelets on the sets of [Formula: see text]-positive vectors in the Euclidean space are studied. These sets are multidimensional analogs of the half-line in the Walsh analysis. Following the ideas of the Walsh analysis, the space of [Formula: see text]-positive vectors is equipped with a coordinate-wise addition. Harmonic analysis on this space is also similar to the Walsh harmonic analysis, and the Fourier transform is such that there exists a class of so-called test functions (with a compact support of the function itself and of its Fourier transform). Tight wavelet frames consisting of the test functions are studied. A complete description of the masks generating such frames is given, and an algorithmic method for constructing them is developed. These frames may be very useful for applications to signal processing because some examples of such systems on the half-line were already investigated in this aspect, and it appeared that they have an advantage over classical wavelet systems when used for processing fractal signals and images.