Orthogonal Polynomials on Arcs of the Unit Circle II. Orthogonal Polynomials with Periodic Reflection Coefficients
- Peherstorfer, Franz
- Steinbauer, Robert
DOIAbstract
AbstractFirst we give necessary and sufficient conditions on a set of intervalsEl=∪lj=1[ϕ2j−1, 2j], 1<…<2l and 2l−ϕ1⩽2π, such that onElthere exists a real trigonometric polynomialτN(ϕ) with maximal number, i.e.,N+l, of extremal points onEl. The associated algebraic polynomialTN(z)=zN/2τN(z),z=eiϕ, is called the complex Chebyshev polynomial. Then it is shown that polynomials orthogonal onElhave periodic reflection coefficients if and only if they are orthogonal onElwith respect to a measure of the form[formula]certain point measures, whereAis a real trigonometric polynomial with no zeros onEland there exists a complex Chebyshev polynomial onEl. Let us point out in this connection that Geronimus has shown that orthogonal polynomials generated...
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