On the Existence of Nonlinear Padé–Chebyshev Approximations for Analytic Functions

We present examples of two functions that are analytic on the interval $[-1,1]$ and satisfy the condition that, for any $n=2,3,\dots$, the first of them does not have nonlinear Padé–Chebyshev approximations of type $(n,2)$ and the second function does not have nonlinear Padé–Chebyshev approximations of type $(n,n)$ (i.e., does not have diagonal approximations). Because of the existence criterion for nonlinear Padé–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 Padé 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.