On the Existence of Nonlinear Pade-Chebyshev Approximations for Analytic Functions

摘要

We present examples of two functions that are analytic on the interval [-1,1] and satisfy the condition that, for any n = 2,3,..., the first of them does not have nonlinear Pade-Chebyshev approximations of type (n, 2) and the second function does not have nonlinear Pade-Chebyshev approximations of type (n, n) (i.e., does not have diagonal approximations). Because of the existence criterion for nonlinear Pade-Faber approximations, which is obtained in the present paper, both of these examples follow from the respective well-known V. I. Buslaev counterexamples to the Baker-Graves-Morris conjecture and to the Baker- Gammel-Wills conjecture about the Pade approximations of a power series. In particular, the first of these functions is a rational function of type (2,3), and the second function is also defined by an explicit analytic expression.