The Degree of Convergence for Entire Functions

Consider a Doint set C in the complex plane whose complement K is connected and regular (i .e. K possesses a Green ' s function with pole at infini ty). Let denote the transfini te diameter of C . Recall that d m = l /U ' C®)! where <{>(z) irans >C onto tl ê exterior of the uni t circle. Equivalent ly , d^ = Lin | |T^(z) | Ij.] where | |g(z) | | c = max] <*(z) | ,zeC and T..(z) is t^e Tchebycheff polynomial (or Faber nolynooial) for C . Given f(z) defined on C , let P*(z) be the best polynomial approximation to f(z) on C i .e. , | |f(z)-P (z) | | c is minimized for polynomials p (z) of degree n . The purpose of this paper is to prove the THEOREM l, r e have if and only if f(z) is entire of order p > O and type O < T < ° ° The method of proof essent ially combines basic techniques from the theory of entire functions wi th machinery used to establish degree of convergence theoremsfor polynomial approximations to analyt ic functions. Thus it is shown that this degree of convergence is achieved (with a factor 1+e, arbitrary e>o) by a polynomial expansion of the form where p(z) is a polynomial of degree X and ^ ( z ) is of defree A-l . The level curves of |p(z)| define a lemniscate which approximates the boundary of C . A number of lemmas are established which relate the nature of the coefficient polynomials q v (z) to the order and type of f(z). OO f ( z ) I o k ( z ) P ( Z ) k-1 k= l THE DEGREE OF CONVERGENCE FOR ENTIRE FUNCTIONS