Inverse problems, Sobolev–Chebyshev polynomials and asymptotics

Abstract

UDC 517.9 Let ( u , v ) be a pair of quasidefinite and symmetric linear functionals with { P n } n ≥ 0 and { Q n } n ≥ 0 as respective sequences of monic orthogonal polynomial (SMOP). We define a sequence of monic polynomials { R n } n ≥ 0 as follows: P n + 2 ' ( x ) n + 2 + b n P n ' ( x ) n - Q n + 1 ( x ) = d n R n - 1 ( x ) , n ≥ 1. We give necessary and sufficient conditions for { R n } n ≥ 0 to be orthogonal with respect to a quasidefinite linear functional w . In addition, we consider the case where { P n } n ≥ 0 and { Q n } n ≥ 0 are monic Chebyshev polynomials of the first and second kinds, respectively, and study the relative outer asymptotics of Sobolev polynomials orthogonal with respect to the Sobolev inner product 〈 p , q 〉 S = ∫ -1 1 p q ( 1 - x 2 ) - 1 / 2 ⅆ x + λ 1 ∫ -1 1 p ' q ' ( 1 - x 2 ) 1 / 2 ⅆ x + λ 2 ∫ -1 1 p ' ' q ' ' ⅆ μ ( x ) , where μ is a positive Borel measure associated with w and λ 1 , λ 2 > 0 , λ 2 is a linear polynomial of λ 1 .