Orthogonal polynomials with ratio asymptotics
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A general construction is given for measures for which the corresponding orthogonal polynomials have ratio asymptotics. Let v be a positive Borel measure on the unit circle T and form the orthonormal polynomials pn(V, z) = y,(v)zn + , n = 0, 1 ..., with respect to v: J, (v, Z)Pm (v, z) dv (z) = 0nm . We say that these polynomials have ratio asymptotic behavior if (1) lim Pn+I(V, Z)/Pn(V, Z) = Z nf-*o0 uniformly on compact subsets of the exterior of the unit circle, and in this case we say shortly that v is in the class M. If v is given on the interval [-1, 1 ], then the corresponding class M(O, 1) is defined by the relation (2) lim Pn+I(v, Z)/Pn(V z) = Z+ z2 uniformly on compact subsets of C\[1, 1]. We mention, that the right-hand sides of (1) and (2) are the corresponding conformal mappings of the outer domains of the supports onto the exterior of the unit disk, and if a ratio asymptotic exists in the outer domains, then it must be of the form (1) or (2). There is a natural one-to-one correspondence between the measures in M(O, 1) and the even measures of M that is established by the usual projection of the unit circle onto [-1, 1] [7]. The importance of the classes M and M(O, 1) is explained by the connection between orthogonal polynomials, Pad6 approximation and continued fractions for Markov functions, and many investigations concerning the convergence properties of these quantities can be carried out (and have been carried out) for the case when the measure is in the class M or M(O, 1). In fact, several recent investigations by A. A. Gonchar and E. A. Rahmanov indicate Received by the editors August 7, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 42C05; Secondary 39A1 1, 41A99.
[1] W. Assche,et al. Sieved orthogonal polynomials and discrete measures with jumps dense in an interval , 1989 .
[2] Paul Nevai,et al. Distribution of zeros of orthogonal polynomials , 1979 .
[3] B. Simon,et al. From power pure point to continuous spectrum in disordered systems , 1985 .
[4] D. Lubinsky. Singularly continuous measures in Nevai’s class $M$ , 1991 .