Rows and diagonals of the Walsh array for entire functions with smooth Maclaurin series coefficients

Abstract

Let$f(z): = \sum\nolimits_{j = 0}^\infty {a_j z^J }$ be entire, witha _j≠0,j large enough,$\lim _{J \to \infty } a_{j + 1} /a_J = 0$, and, for someq∈C,$q_j : = a_{j - 1} a_{j + 1} /a_j^2 \to q$ asj→∞. LetE _mn(f; r) denote the error in best rational approximation off in the uniform norm on |z‖≤r, by rational functions of type (m, n). We study the behavior ofE _mn(f; r) asm and/orn→∞. For example, whenq above is not a root of unity, or whenq is a root of unity, butq _m has a certain asymptotic expansion asm→∞, then we show that, for each fixed positive integern,

,m→∞. In particular, this applies to the Mittag-Leffler functions$f(z): = \sum\nolimits_{j = 0}^\infty {z^j /\Gamma (1 + j/\lambda )}$ and to$f(z): = \sum\nolimits_{j = 0}^\infty {z^j /(j!)^{I/\lambda } }$, λ>0. When |q‖<1, we also handle the diagonal case, showing, for example, that

,n→∞. Under mild additional conditions, we show that we can replace 1+0(1)ⁿ by 1+0(1). In all cases we show that the poles of the best approximants approach ∞ asm→∞.