The convergence of rational functions of best approximation to the exponential function

Abstract

The object of the paper is to establish convergence throughout the entire complex plane of sequences of rational functions of prescribed types which satisfy a certain degree of approximation to the function a e y z a{e^{yz}} on the disk | z | ≦ ρ |z| \leqq \rho . It is assumed that the approximating rational functions have a bounded number of free poles. Estimates are given for the degree of best approximation to the exponential function by rational functions of prescribed types. The results obtained in the paper imply that the successive rows of the Walsh array for a e y z a{e^{yz}} on | z | ≦ ρ |z| \leqq \rho converge uniformly to a e y z a{e^{yz}} on each bounded subset of the plane.