Abstract
Let {ϕ n (dμ)} be a system of orthonormal polynomials on the unit circle with respect to a measuredμ. Szegö's theory is concerned with the asymptotic behavior ofϕ n (dμ) when logμ'∈L 1. In what follows we will discuss the asymptotic behavior of the ratio φn(dμ 1)/φn(dμ 2) off the unit circle in casedμ 1 anddμ 2 are close in a sense (e.g.,dμ 2=g dμ 1 whereg≥0 is such thatQ(e it)g(t) andQ(e it)/g(t) are bounded for a suitable polynomialQ) and μ ′1 >0 almost everywhere or (a somewhat weaker requirement) lim n→∞Φ n (dμ 1,0)=0, for the monic polynomials Φ n . The consequences for orthogonal polynomials on the real line are also discussed.
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Communicated by Ronald A. DeVore.
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Máté, A., Nevai, P. & Totik, V. Extensions of Szegö's theory of orthogonal polynomials, II. Constr. Approx 3, 51–72 (1987). https://doi.org/10.1007/BF01890553
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DOI: https://doi.org/10.1007/BF01890553