CHEBYSHEV RATIONAL APPROXIMATION TO ENTIRE FUNCTIONS IN [ 0 , oc ]

Introduction : Quite recently Chebyshev rational approximation to certain entire functions on the whole positive axis has attracted the attention of many mathematicians. In this respect the papers ([3-7, 9]) are worth mentioning. All these papers have been devoted only to entire functions of finite order. On the other hand, methods developed and used in these papers are valid only to entire functions of finite order . In this paper we develop a method by which we can get results for functions of zero, finite as well as for infinite orders. We also obtain lower bounds for Ao,,,, the Chebyshev constants for 1/f on [0, co) . Besides this, we obtain much more precise information in the case of functions of zero order . In fact we give an example which shows clearly how much closely one can approximate entire functions of small growth . Notation. For any non-negative integer n, -rn denotes the collection of real polynomials of degree at most n . Then let 2o,n= inf 1,1IfIIP. Il[o,-) P En,,