Inverse problems, Sobolev–Chebyshev polynomials and asymptotics - Molano, Luis Alejandro Molano (Ukrains’kyi Matematychnyi Zhurnal, 2023)
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_nP_n’(x)n-Q_n+1(x)=d_nR_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 productequation*p,q_S=_-1^1pq(1-x^2)^-1/2dx+λ_1_-1^1p’q’(1-x^2)^1/2dx+λ_2_-1^1p’’q’’dμ(x),equation* 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.