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The Asymptotics of a Continuous Analogue of Orthogonal Polynomials

✍ Scribed by H. Widom

Publisher
Elsevier Science
Year
1994
Tongue
English
Weight
358 KB
Volume
77
Category
Article
ISSN
0021-9045

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✦ Synopsis

SzegΓΆ polynomials are associated with weight functions on the unit circle. M. G. Krein introduced a continuous analogue of these, a family of entire functions of exponential type associated with a weight function on the real line. An investigation of the asymptotics of the resolvent kernel of (\sin (x-y) / \pi(x-y)) on ([0, s]) leads to questions of the asymptotics of the Krein functions associated with the characteristic function of the complement of the interval ([-1,1]). Such asymptotics are determined here, and this leads to answers to certain questions involving the abovementioned kernel, questions arising in the theory of random matrices. (C) 1994 Academic Press, Inc.

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We consider asymptotics of orthogonal polynomials with respect to weights w(x)dx = e -Q(x) dx on the real line, where Q(x) = βˆ‘ 2m k=0 q k x k , q 2m > 0, denotes a polynomial of even order with positive leading coefficient. The orthogonal polynomial problem is formulated as a Riemann-Hilbert problem