Order of uniform approximation to analytic functions by rational trigonometric and weighted rational functions

Upper estimates on the order of uniform rational trigonometric approximation are established for continuous 2{dollar}pi{dollar}-periodic functions that are analytic except at a finite number of singularities in each period. The estimates are similar to those given by J. Szabados (1969) for the rational case. Rational trigonometric functions of degree n are constructed which provide an order {dollar}O(esp{lcub}-csqrt{lcub}n{rcub}{rcub}){dollar} if the function being approximated satisfies certain analyticity conditions and has a modulus of continuity which is {dollar}O(hsplambda){dollar} as h {dollar}to{dollar} 0{dollar}sp+{dollar} (c and {dollar}lambda{dollar} are positive constants.).; In order to obtain the rational trigonometric results, approximation on ({dollar}-{dollar}1,1) by functions of the form {dollar}sqrt{lcub}1-xsp2{rcub}R(x){dollar} with rational R is first considered. More generally, for fixed {dollar}phi{dollar} in C (a, 1) ({dollar}-1{dollar} {dollar}le{dollar} a {dollar}<{dollar} 1) rational functions R are constructed to make {dollar}phi{dollar}R uniformly close to a function f in C (a, 1). f and the weight {dollar}phi{dollar} both have analytic continuations onto the open unit disc and {dollar}phi{dollar} is nonzero there. An estimate given by A. A. Goncar (1967) for the order of rational approximation of an analytic function with endpoint singularities is extended to the weighted rational case. Estimates for the approximation order are made for all degrees of the rational function in addition to finding asymptotic results.