On Extremal Functions in Inequalities for Entire Functions

Let B (sigma) , sigma > 0 , be the class of entire functions of exponential type <= sigma bounded on the real line. For a number tau is an element of R and a sequence {c(k)}(k is an element of Z) of complex numbers satisfying the condition Sigma(k is an element of Z) |c(k)|< + infinity , the operator H on B-sigma defined by H ( f )( x ) = Sigma(k is an element of Z) ckf ( x - tau + k pi sigma) is considered. Obviously, |H(f)(x)| ( f )( x ) | <= kappa ||f|| infinity , x is an element of R , f is an element of B sigma, sigma , kappa =Sigma (k is an element of Z) |c(k)| The main purpose of the paper is to describe all extremal functions for this inequality. Theorem 1 proved in the paper asserts that if (1) c s c s+1 < 0 for some s is an element of Z and (2) there exists an epsilon is an element of C with |epsilon| = 1 such that epsilon ck(-1)k k ( - 1) k >= 0 for all k is an element of Z , then the set of all extremal functions for the above inequality coincides with the set of functions of the form f ( t ) = mu e i sigma t + nu e-i sigma t, - i sigma t , mu, nu is an element of C . The proof of Theorem 1 essentially uses Theorem 2, which says that if f is an element of B sigma sigma and there exists a point xi is an element of R for which |f(xi)| ( xi ) | = ||f||infinity and f ( xi + pi/sigma) ) = -f ( xi ) , then f ( t ) = mu e i sigma t + nu e-i sigma t, - i sigma t , mu, nu is an element of C . Theorem 3 gives general examples of operators satisfying both conditions of Theorem 1. In particular, such is the fractional derivative operator H ( f )( x ) = f (r, beta ) ( x ) for r >= 1 and beta is an element of R .