Abstract
Given a function F holomorphic on a neighborhood of some compact subset of the complex plane, we prove that if the zeros of the denominators of generalized Padé approximants (orthogonal Padé approximants and Padé–Faber approximants) for some row sequence remain uniformly bounded, then either F is a polynomial or F has a singularity in the complex plane. This result extends the known one for classical Padé approximants. Its proof relies, on the one hand, on difference equations where their coefficients relate to the coefficients of denominators of these generalized Padé approximants and, on the other hand, on a curious property of complex numbers.
Highlights
1 Introduction Currently, Padé approximation theory emphasizes inverse-type problems where we want to describe the analytic properties of the approximated function from the knowledge of the asymptotic behavior of the poles of the approximating functions
The object of the present paper is to investigate the relation between the boundedness of poles of row sequences of orthogonal Padé approximants and Padé–Faber approximants and the analyticity of the approximated function
In order to state a known result related to our study, we need to remind the reader of the definition of classical Padé approximants
Summary
Padé approximation theory emphasizes inverse-type problems where we want to describe the analytic properties of the approximated function from the knowledge of the asymptotic behavior of the poles of the approximating functions. The rational function Rn,m := P/Q is called the(n, m) classical Padé approximant of F It is well-known that, for any (n, m) ∈ N0 × N0, Rn,m always exists and is unique. The first two definitions are generalized Padé approximants constructed from the sequence of orthogonal polynomials (pn)n≥0. The two definitions are the ones of generalized Padé approximants constructed from the sequence of Faber polynomials (Φn)n≥0. For any integers n ≥ 0 and m ≥ 1, polynomials Qμn,m, Qμn,m, QEn,m, and Q En,m always exist but they may not be unique. Suetin [15, 16] was the first to give necessary and sufficient conditions for the convergence with geometric rate of the denominators of standard orthogonal Padé and standard Padé–Faber approximants on row sequences.
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