An extension of the ‘ 1/9 ’-problem

Abstract

We consider the rational approximation of a perturbed exponential function (1) f ( z ) = u 0 ( z ) + u 1 ( z ) e z on the negative real axis R − with u 0 and u 1 ≢ 0 being rational functions. Building upon results from the solution of the ‘ 1/9’-problem by Gonchar and Rakhmanov, it is shown that rational best approximants to f share the fast asymptotic convergence of rational approximants to the exponential function, despite the fact that the functions (1) may differ substantially from the exponential. Our main result is that for k ∈ Z fixed, we have (2) lim n → ∞ ‖ f − r n , n + k ∗ ‖ R − 1 / n = 1 9.28902549 … = H with H being the same Halphen constant as that already known from the solution of the ‘ 1/9’-problem and r n m ∗ = r n m ∗ ( f , R − ; ⋅ ) being a rational best approximant with denominator and numerator degrees at most n and m in the uniform norm on R − to the function f of type (1). Further, it is shown that the convergence on R − , which is characterized by (2), extends to compact subsets of the complex plane C and also to Hankel contours in C ∖ R − , and this extension holds with the same asymptotic rate of convergence H . Many of the results for rational best approximants extend also to rational close-to-best approximants, which by our definition are rational approximants with a convergence behavior on R − that is comparable in an n th root sense to that of the best approximants r n , n ∗ . The asymptotic distribution of the poles of approximants is studied and determined. In the present paper we present only results, complete proofs are not given, however, some of the main ideas of the proofs are sketched.