Wavelets by orthogonal rational kernels
Adhemar Bultheel, Pablo González-Vera · Contemporary mathematics - American Mathematical Society · 1999
Let L_n be the space of polynomials or rational functions of degree n at most with poles in a prescribed set of numbers inside the unit disk. Consider a complex inner product with respect to a finite positive measure on the unit circle. Suppose {φ_k:k=0,...,n} is the corresponding orthonornal basis for L_n. Set n=2^s and V_s=L_n. Then k_n(z,w)=∑_{k=0,...,n} φ_k(z)[φ_k(w)]*, is a reproducing kernel for V_s. For fixed w, such reproducing kernels are known to be functions localized in the neighborhood of z=w. Moreover, by an appropriate choice of the parameters {ξ_{nk}:k=0,...,n}, the functions {φ_{n,k}(z)=k_n(z,ξ_{n,k}):k=0,...,n} will be an orthogonal basis for V_s. The orthogonal complement W_s=V_{s+1} - V_s is spanned by the functions {ψ_{n,k}(z)=l_n(z,η_{n,k}:k=0,...,n-1} for an appropriate choice of the parameters {η_{n,k}:k=0,...,n-1} where l_n=k_{2n}-k_n is the reproducing kernel for W_s. These observations form the basic ingredients for the construction of rational wavelets on the unit circle with respect to an arbitrary positive measure.