On a remarkable class of entire functions

Integrability along a Line for a Class of Entire Functions

provided that {X„} is an increasing sequence of real numbers such that |X„ — n\ 0. This is the L2-analogue of the theorem, proved in different ways by Duffin and Schaeffer [6] and by B. Levin [8], that f(x) is bounded if {/(X„)} is bounded ; the cases of the two theorems when \n = n are due respectively to Plancherel and Pólya [il] and Cartwright [3]. The object of this pap...

The Asymptotic Developments of a Class of Entire Functions

The discrete analogue of a class of entire functions

Discrete analytic functions are coinplex valued functions defined at points of the z-plane with integer coordinates. The real and imaginary parts of these functions are required to satisfy difference equations analogous to the Cauchy-Riemann equations. The pseudo power z() is a discrete analytic function which is asymptotic to the ordinary power z for large z. The central topic of this paper is...

Entire Approximations for a Class of Truncated and Odd Functions

We solve the problem of finding optimal entire approximations of prescribed exponential type (unrestricted, majorant and minorant) for a class of truncated and odd functions with a shifted exponential subordination, minimizing the L1(R)-error. The class considered here includes new examples such as the truncated logarithm and truncated shifted power functions. This paper is the counterpart of t...

An extremal problem for a class of entire functions

Let f be an entire function of the exponential type, such that the indicator diagram is in [−iσ, iσ ], σ > 0. Then the upper density of f is bounded by cσ , where c≈ 1.508879 is the unique solution of the equation log (√ c2 + 1 + c) =√1 + c−2. This bound is optimal. To cite this article: A. Eremenko, P. Yuditskii, C. R. Acad. Sci. Paris, Ser. I 346 (2008). © 2008 Académie des sciences. Publishe...