TW 278

Adhemar Bultheel and Pablo González-Vera
Wavelets by orthogonal rational kernels


Suppose {φk}k=0..n is an orthonornal basis for the function space Ln of polynomials or rational functions of degree n with prescribed poles. Suppose n=2s and set Vs=Ln. Then kn(z,w)=Σk=0..2n φk(z)\overline{φk(w)}, is a reproducing kernel for Vs. 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 {ξnk}k=0..n, the functions {fn,k(z)=kn(z,xnk)}k=0..n will be an orthogonal basis for Vs. The orthogonal complement Ws=Vs+1-Vs is spanned by the functions {ψn,k(z)=ln(z,ynk)}k=0..n-1 for an appropriate choice of the parameters {ynk}k=0..n-1 where ln=kn+1-kn is the reproducing kernel for Ws. These observations form the basic ingredients for a wavelet type of analysis for orthogonal rational functions on the unit circle or the real line with respect to an arbitrary probability measure.

report.pdf / mailto: A. Bultheel