Singularly continuous measures in Nevai’s class 𝑀

Abstract

Abstract.Let dv be a nonnegative Borel measure on \-n, n], with 0 < f*n du < co and with support of Lebesgue measure zero. We show that there exist {Vj}%¡ C (0, oo) and {tj}jZ\ C (-it, n) such that if 00 dß(6) :=Y/1jdu(d + tj), 6 e[-n , ti], j=i (with the usual periodic extension dv(d ± 2n) = dv(6)), then the leading coefficients {Kn(dß)}^L0 of the orthonormal polynomials for dß satisfy \im>K"(dp)/Kn+x(dp)= 1.As a consequence, we obtain pure singularly continuous measures da on [-1,1] lying in Nevai's class M .

A Singular Riesz Product in the Nevai Class and Inner Functions with the Schur Parameters in ∩p>2lp

“…The first example of a singular measure _ with lim n a n =0 was constructed by D. Lubinsky [6]. In the present paper we show that the Nevai class contains also singular Riesz products which are very close to measures satisfying the Szego conditioǹ…”

Schur's Algorithm, Orthogonal Polynomials, and Convergence of Wall's Continued Fractions in L2(T)

“…By Rakhmanov's theorem Nevai's class contains Erdo s' class. There are examples of pure jump measures [5,31,33], pure singular continuous measures [32], including some singular Riesz products [28], in Nevai's class. Totik [59] constructed further important examples of measures in Nevai's class.…”

Singular Measures on the Unit Circle and Their Reflection Coefficients

“…The converse statement is not at all true: there are plenty of singular measures with lim n Ä a n =0 (cf. [17], [18], [21]). However, the following partial converse is valid.…”