Abstract. We obtain growth comparison results of logarithmic differences, difference quotients and logarithmic derivatives for finite order meromorphic functions. Our results are both generalizations and extensions of previous results. We construct examples showing that the results obtained are best possible in certain sense. Our findings show that there are marked differences between the growt...

Let f be an entire function of positive order and finite type. The subject of this note is the convergence acceleration of polynomial approximants of f by incorporating information about the growth of f(z) for z → ∞. We consider “near polynomial approximation” on a compact plane set K, which should be thought of as a circle or a real interval. Our aim is to find sequences (fn)n of functions whi...

We prove here some polynomial approximation theorems, somewhat related to the Szasz-Mfintz theorem, but where the domain of approximation is the integers, by dualizing a gap theorem of C. l ~ Y I for periodic entire functions. In another Paper [7], we shall prove, by similar means, a completeness theorem ibr some special sets of entire functions. I t is well known (see, for example [l]) tha t i...

it is immaterial which value of z is used in (2). If (1) holds in a region of the s-plane, for example in an angle, ƒ(z) is said to be of exponential type c in that region. Functions of exponential type have been extensively studied, both for their own sake and for their applications. I shall discuss here a selection of their properties, chosen to illustrate how the restriction (1) on the growt...

Since this series is absolutely convergent everywhere in the plane, lanl must approach zero as n approaches infinity. Consequently, there exists for each a, an index n(a) for which lanl is a maximal coefficient. B. Lepson [3]1 raised the question of characterizing entire functions for whidi n (a) is bounded in a. 2 In the sequel we shall consider certain interesting variations of Lepson's probl...

Let G(k) = ∫ 1 0 g(x)e kxdx, g ∈ L1(0, 1). The main result of this paper is the following theorem. THEOREM 1. There exists g 6≡ 0, g ∈ C∞ 0 (0, 1), such that G(kj) = 0, kj < kj+1, limj→∞ kj =∞, limk→∞ |G(k)| does not exist, lim supk→+∞ |G(k)| = ∞. This g oscillates infinitely often in any interval [1− δ, 1], however small δ > 0 is. MSC: 30D15, 42A38, 42A63

in this paper we introduce entire sequence spaces defined by a sequence of modulus functions . we study some topological properties of these spaces and prove some inclusion relations.

Since this series is absolutely convergent everywhere in the plane, the terms \an\ must approach 0. Consequently, there exists for each a, an index n0 = n(a) ior which \an\ is a maximal coefficient. B. Lepson [2] raised the problem of characterizing entire functions for which n(a) is bounded. The latter are called functions of bounded index. In what follows, we shall give a partial solution to ...