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### TW 278

Adhemar Bultheel and Pablo González-Vera

*Wavelets by orthogonal rational kernels*

**Abstract**

Suppose {φ_{k}}_{k=0..n}
is an orthonornal basis for the function space
**L**_{n} of polynomials or rational functions
of degree n with prescribed poles.
Suppose n=2^{s} and
set **V**_{s}=**L**_{n}.
Then
k_{n}(z,w)=Σ_{k=0..2n}
φ_{k}(z)\overline{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 {f_{n,k}(z)=k_{n}(z,x_{nk})}_{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,y_{nk})}_{k=0..n-1}
for an appropriate choice of the parameters {y_{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 unit circle or the real line with
respect to an arbitrary probability measure.