Asymptotic behaviour of orthogonal polynomials on the unit circle with asymptotically periodic reflection coefficients

[摘要] Let {a(n))(n is an element of)N-9 with a(n) is an element of C, a(n+N)=a(n) and /a(n)/<1 for all n is an element of N-0, be the periodic sequence of reflection coefficients and let {P-n}n be the associated sequence of orthogonal polynomials generated by P-n+1=zP(n)-<(a)over bar (n) P-n*>. Furthermore let {b(n)}(n is an element of N0) be an asymptotically periodic sequence of reflection coefficients which arises by a perturbation of the sequence {a}n is an element of n and thus satisfies the conditions lim(v-->x)b(j+vN)=a(j) for j=0,..., N-1, and /b(n)/ <1 for all n is an element of N0. Let {<(P (n))(n is an element of N0) generated by <(P)over tilde (n+1)>= the disturbed orthogonal polynomials. Using the ''periodic'' polynomials {P-n}(n is an element of N0) as a comparison system we derive so-called comparative asymptotics for the disturbed polynomials on and off the support of the disturbed orthogonality measure, which consists essentially of several arcs of the unit circle. As a by-product of these results we obtain asymptotically a description of the location of the zeros of {<(P)over tilde (n)}n is an element of N-0. Finally, a representation for the absolutely continuous part of the disturbed orthogonality measure is derived, and it is shown that there are at most finitely many point measures if the b(n)'s converge geometrically fast to the a(n)'s. (C) 1997 Academic Press.

[发布日期] 1997-03-01