Approximation with Rational Interpolants in A(-infinity)(D) for Dini Domains

摘要

Let D denote a Dini domain in C and let (C) over bar = C boolean OR {infinity}. For each n = 1,2,3..., take points A(n) = {a(ni)}(n)(i=0) in D and points B-n ={b(ni)}(n)(i=1)in (C) over bar D. Let alpha(n) and beta(n) be the normalized point counting measures of A(n) and B-n. Suppose that alpha(n)-> w*alpha, beta(n)-> w*beta and denote by alpha' and beta' their swept measures onto partial derivative D. Denote by U mu the logarithmic potential of the measure mu. We show that if alpha' = beta' and if {(n+1)(U-alpha n - U-alpha),{n(U-beta'n - U-beta')}uniformly have at most logarithmic growth at partial derivative D, then for every f is an element of A(-infinity)(D) and for the rational interpolants r(n),integral of degree n with poles at B-n interpolating to f at A(n) , we have r(n), f -> f in A(-infinity)(D) .