On Baker’s Patchwork Conjecture for Diagonal Padé Approximants

Abstract

We prove that for entire functions f of finite order, there is a sequence of integers $$\mathcal {S}$$ such that as $$n\rightarrow \infty $$ through S, $$\begin{aligned} \min \left\{ \left| f-\left[ n/n\right] \right| \left( z\right) ,\left| f-\left[ n-1/n-1\right] \right| \left( z\right) \right\} ^{1/n}\rightarrow 0 \end{aligned}$$ uniformly for z in compact subsets of the plane. More generally this holds for sequences of Newton–Pade approximants and for functions whose errors of approximation by rational functions of type $$\left( n,n\right) $$ decay sufficiently fast. This establishes George Baker’s patchwork conjecture for large classes of entire functions.