Multivariate rational approximation

Abstract

We estimate the error in approximating a function f f by rational functions of degree n n in the norm of L q ( Ω ) , Ω := [ 0 , 1 ] d {L_q}(\Omega ),\,\Omega : = {[0,\,1]^d} . Among other things, we prove that if f f is in the Sobolev space W p k ( Ω ) W_p^k(\Omega ) and if k / d − 1 / p + 1 / q > 0 k/d - 1/p + 1/q > 0 , then f f can be approximated by rational functions of degree n n to an order O ( n − k / d ) O({n^{ - k/d}}) .