Multiwavelets and two-scale similarity transform

Abstract

An important object in wavelet theory is the scaling function φ(t), satisfying a dilation equation φ(t)=Σ C _k φ(t−k). Properties of a scaling function are closely related with the properties of the symbol or mask P(ω)=Σ C _k e ^−iwk. The approximation order provided by φ(t) is the number of zeros of P(ω) at ω=π, or in other words the number of factors (1 + e ^iw) in P(ω). In the case of multiwavelets P(ω) becomes a matrix trigonometric polynomial. The factors (1+e ^−iw) are replaced by a matrix factorization of P(ω), which defines the approximation order of the multiscaling function. This matrix factorization is based on the two-scale similarity transform (TST) which connects matrix polynomials P($\bar \omega$) and Q(ω) by the relation Q(ω)=M(2ω)P(ω)M ⁻¹(ω). In this paper we study properties of the TST and show how it is connected with the theory of multiwavelets. This approach leads us to new results on regularity, symmetry and orthogonality of multi-scaling functions and opens an easy way to their construction.