Szegő-type Asymptotics for Ray Sequences of Frobenius-padé Approximants
چکیده
Let σ̂ be a Cauchy transform of a possibly complex-valued Borel measure σ and {pn} be a system of orthonormal polynomials with respect to a measure μ, supp(μ)∩ supp(σ) = ∅. An (m,n)-th Frobenius-Padé approximant to σ̂ is a rational function P/Q, deg(P) 6m, deg(Q) 6 n, such that the first m+n+ 1 Fourier coefficients of the linear form Qσ̂−P vanish when the form is developed into a series with respect to the polynomials pn. We investigate the convergence of the Frobenius-Padé approximants to σ̂ along ray sequences n n+m+1 → c > 0, n− 1 6m, when μ and σ are supported on intervals on the real line and their Radon-Nikodym derivatives with respect to the arcsine distribution of the respective interval are holomorphic functions.