Expansion of entire functions into Lagrange series

This paper is devoted to the problem of representing entire functions, in spaces described by the order and the type of these functions, by Lagrange series that converge in the natural topology in these spaces; this topology is stronger than the topology of compact convergence. Translated fromMatematicheskie Zametki, Vol. 61, No. 1, pp. 119–124, January, 1997. Translated by M. A. Shishkova     

Infinite order differential operators in spaces of entire functions

Differential operators ?(Δ_θ,ω), where ? is an exponential type entire function of a single complex variable and Δ_θ,ω=(θ+ωz)D+zD², D=∂/∂z, , θ?0, , acting in the spaces of exponential type entire function are studied. It is shown that, for ω?0, such operators preserve the set of Laguerre entire functions provided the function ? also belongs to this set. The latter consists of the polynomials possessing real nonpositive zeros only and of their uniform limits on compact subsets of the complex plane . The operator exp(aΔ_θ,ω), a?0 is studied in more details. In particular, it is shown that it preserves the set of Laguerre entire functions for all . An integral representation of exp(aΔ_θ,ω), a>0 is obtained. These results are used to obtain the solutions to certain Cauchy problems employing Δ_θ,ω.     

Refined-mesh interpolation method for functions with a boundary-layer component

Linear and quadratic spline interpolation methods for a one-variable function with a boundary-layer component are examined. It is shown that the interpolation method for such a function leads to considerable errors when applied on a uniform mesh. The error of linear and quadratic spline interpolations on meshes that are refined in the boundary layer is estimated. Numerical results are presented.     

On interpolation with products of positive definite functions

In this paper we consider the problem of scattered data interpolation for multivariate functions. In order to solve this problem, linear combinations of products of positive definite kernel functions are used. The theory of reproducing kernels is applied. In particular, it follows from this theory that the interpolating functions are solutions of some varational problems.     

Migration of zeros for successive derivatives of entire functions

It is shown that if is an entire function of order less than one, all of whose zeros are real, then the minimal root of is an increasing function of which accelerates as increases.

     

On exposed functions in Bernstein spaces

For σ > 0, the Bernstein space {ie427-01} consists of those L ¹(ℝ) functions whose Fourier transforms are supported by [−σ, σ]. Since {ie427-02} is separable and dual to some Banach space, the closed unit ball {ie427-03} of {ie427-04} has sufficiently large sets of both exposed and strongly exposed points: {ie427-05} coincides with the closed convex hull of its strongly exposed points. We investigate some properties of exposed points, construct several examples, and obtain as corollaries relations between the sets of exposed, strongly exposed, weak* exposed, and weak* strongly exposed points of {ie427-06}.     

Convolution operators and zeros of entire functions

Let be a real entire function of order less than with only real zeros. Then we classify certain distribution functions such that the convolution has only real zeros.

     

Geometric conditions for interpolation in weighted spaces of entire functions

We use L² estimates for the equation to find geometric conditions on discrete interpolating varieties for weighted spaces A_p(ℂ) of entire functions such that |f(z)|≤Ae^Bp(z) for some A, B>0. In particular, we give a characterization when p(z)=e^|z| and more generally, when In p(e^r) is convex andIn p(r) is concave. Acknowledgements and Notes. The author wishes to thank X. Massaneda for useful talks and remarks.     

A note about Volterra operators on weighted Banach spaces of entire functions

We characterize boundedness, compactness and weak compactness of Volterra operators acting between different weighted Banach spaces of entire functions with sup‐norms in terms of the symbol g; thus we complement recent work by Bassallote, Contreras, Hernández‐Mancera, Martín and Paul 3 for spaces of holomorphic functions on the disc and by Constantin and Peláez 16 for reflexive weighted Fock spaces.     

A note on exponent of convergence of zeros of entire functions

If f(z) is an entire function with ρ ₁ > 0 as its exponent of convergence of zeros and if 0 ≤ α < ρ ₁, then we prove the existence of entire functions each having α as its exponent of convergence of zeros.      

A relationship between the maximum modulus and Taylor coefficients of entire functions of several complex variables

We present results on the relationship between the growth of the maximum modulus and the decay of Taylor coefficients of entire functions of several variables. The results are obtained by two different methods, the first of which had been proposed earlier by Oskolkov for the one-dimensional case, and the second is based on the use of the Legendre-Jung-Fenchel conjugates of the weight functions. Attention is mainly paid to the characterization of the growth of entire functions with respect to the conjunction of variables; however, some results are obtained for the case in which there is different growth with respect to different variables. Translated fromMatematicheskie Zametki, Vol. 62, No. 2, pp. 238–258, August, 1997. Translated by N. K. Kulman     

Interpolation and basicity: Several variables entire functions of exponential type

The paper investigates the interpolation and basicity problem in Banach spaces of entire functions of exponential type in several variable. Some sufficient conditions for solvability of these problem and the explicit analytic form of the solutions are obtained.     

A generalization of Laguerre's theorems on zeros of entire functions

We prove some results generalizing the classical Laguerre theorems about the multiplicity and the number of zeros of the function

Application of the Borel transform to the study of the spectrum of integral equations whose kernels are entire functions of exponential type

Using the Borel transform, we study the spectrum of a class of non-compact integral operators whose kernels are of exponential type and square integrable on the real line. Our method also enables us to obtain an interesting characterization of a well-known integral equation involving the Bessel function

     

Stieltjes moments of entire functions of exponential type

The lacunary homogeneous moment problem

Convolution operators on spaces of entire functions

We show that nontrivial convolution operators on certain spaces of entire functions on E are frequently hypercyclic when E is a normed space and when E is the strong dual of a Fréchet nuclear space. We also obtain results of existence and approximation for convolution equations on certain spaces of entire functions on arbitrary locally convex spaces.     

Multi-point Taylor expansions of analytic functions

Taylor expansions of analytic functions are considered with respect to several points, allowing confluence of any of them. Cauchy-type formulas are given for coefficients and remainders in the expansions, and the regions of convergence are indicated. It is explained how these expansions can be used in deriving uniform asymptotic expansions of integrals. The method is also used for obtaining Laurent expansions in several points as well as Taylor-Laurent expansions.

     

On the Fatou components of two permutable transcendental entire functions

Let f and g be two distinct permutable transcendental entire functions. Suppose further that q(g)=aq(f)+b for some nonconstant polynomial q(z) and constants a(≠0), . In this article, we will investigate the dynamical properties of f and g and show that they have the same Fatou sets with the same components.