Title: Favard theorem for reproducing kernels
Authors: Bultheel, Adhemar ×
González-Vera, Pablo
Hendriksen, Erik
Njåstad, Olav
Issue Date: May-1992
Publisher: Department of Computer Science, K.U.Leuven
Series Title: TW Reports vol:TW170
Abstract: Consider for n=0,1,... the nested spaces L_n of rational functions of degree n at most
with given poles 1/α_i^* |α_i| < 1, i=1,...,n.
Let L = ∪_{i=0}^∞ L_n. Given a finite positive measure μ on the unit circle, we associate with it an inner product on L by ⟨f,g⟩ = ∫ f g^* dμ . Suppose k_n(z,w) is the reproducing kernel for L_n, i.e., ⟨f(z),k)n(z,w)⟩ = f(w) , for all f ∈ L_n, |w| < 1, then it is known that they satisfy a coupled recurrence relation.

In this report we shall prove a Favard type theorem which says that
if you have a sequence of kernel functions k_n(z,w)
which are generated by such a recurrence,
then there will be a measure μ supported on the unit circle
so that k_n is the reproducing kernel for L_n.
The measure is unique under certain extra conditions on the points α_i.
Publication status: published
KU Leuven publication type: IR
Appears in Collections:NUMA, Numerical Analysis and Applied Mathematics Section
× corresponding author

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