Rational wavelets on the real line

Suppose {phi(k)}(k=0)(n) is an orthonormal basis for the function space L-n of polynomials or rational functions of degree n with prescribed poles. Suppose n = 2(5) and set V-s = L-n. Then k(n)(z, w) = Sigma(k=0)(n)phi k(z)phi(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 {xi nk}(k=0)(n), the functions {alpha(n,k)(z) = k(n)(z, xi(nk))}(k=0)(n) will be an orthogonal basis for V-s. The orthogonal complement W-s = Vs+1 - V-s is spanned by the functions {psi(n,k)(z) = l(n)(z,eta(nk))}(k=0)(n-1) for an appropriate choice of the parameters {eta(nk)}(k=0)(n-1) where l(n) = k(n+1) - k(n) is the reproducing kernel for W-s. These observations form the basic ingredients for a wavelet type of analysis for orthogonal rational functions on the real line with respect to an arbitrary probability measure.