On a Problem of Leont'ev and Representing Systems of Exponentials

We study whether an entire function of exponential type has totally regular growth if its derivative increases sufficiently fast on the zero set of the function itself. In particular, for a function with a trigonometrically convex (or positive) lower indicator, we obtain a solution of a well-known problem of Leont'ev. As an application, we refine some already known results concerning the characterization of exponents of the representing systems of exponentials.     

Entire functions with a given sequence of zeros and of regular behavior on the real axis. I

We demonstrate that if a sequence of positive numbers increasing to infinity has S-density then it can be prolonged to some sequence that is the zero set of the Weierstrass product of regular behavior on the real axis.     

Strongly regular growth of entire functions of order zero

For entire functions of order zero we introduce a new concept of regularity of growth, which is shown to possess properties similar to those which characterize the concept of totally regular growth of entire functions of finite order in the sense of Levin-Pflüger. Translated fromMatematicheskie Zameiki, Vol. 63, No. 2, pp. 196–208, February, 1998. This research was partially supported by the International Science Foundation under grant No. UCR000.     

Generalization of weierstrass canonical integrals

In this paper we prove that a subharmonic function in ℝ ^m of finite λ-type can be represented (within some subharmonic function) as the sum of a generalized Weierstrass canonical integral and a function of finite λ-type which tends to zero uniformly on compacts of ℝ ^m . The known Brelot-Hadamard representation of subharmonic functions in ℝ ^m of finite order can be obtained as a corollary from this result. Moreover, some properties of R-remainders of λ-admissible mass distributions are investigated.     

On the distribution of products of members of a sequence with positive density

Periodica Mathematica Hungarica -     

On some criteria for completely regular growth of entire functions of exponential type

This paper sharpens the author’s previous results concerning the completely regular growth of an entire function of exponential type all of whose zeros are simple, forming a sequence Λ = {λ_k} _k=1 ^∞ . For a function with real zeros, we write the growth regularity conditions (on the real axis and on the entire plane) in terms of lower bounds only for the absolute value of the derivative at the points λ_k. We also obtain an analog of Krein’s theorem concerning the functions whose inverse can be expanded in the corresponding series of simple fractions.     

Existence of an Angular Density of Roots of Entire Functions of Order Zero

We obtain sufficient conditions for the set of roots of an entire function of order zero of strongly regular growth to possess an angular density.     

Free products with amalgamation and HNN-extensions of uniformly exponential growth

We prove that, with rare exceptions, groups that are free products with amalgamation or HNN-extensions are of uniformly exponential growth. Translated fromMatematicheskie Zametki, Vol. 67, No. 6, pp. 811–815, June, 2000.     

On the Growth of Entire Functions of Exponential Type near a Straight Line

Let be a sequence of complex numbers such that as . For close to the imaginary axis, upper bounds of the indicator of a nonzero entire function of exponential type with minimal growth vanishing on is obtained. These bounds give sufficient conditions for the system of exponents to be incomplete in an unbounded domain in .     

On the zeros of polynomials

In this paper we extend a classical result due to Cauchy and its improvement due to Datt and Govil to a class of lacunary type polynomials.     

Some lower bounds for the canonical volume of 3-folds of general type

Let V be a smooth projective 3-fold of general type. We show that the canonical volume K3 of V has a lower bound1/10 provided q(V) 0 and K3≥3/40 provided P4(V)≥5. We also give some clean and better form for Noether inequalities for minimal 3-folds of general type.     

On Oscillation of Functions with Bounded Spectral Band

In this paper, we consider extremal oscillatory properties of functions with bounded spectrum, i.e., with bounded support (in the sense of distributions) of the Fourier transform. For such functions f, we give criteria of extendability of }f} from the real axis to a function F on the complex plane with derivatives F ^(m) having no real zeros and without enlarging the width of spectrum. In particular, we give examples of functions $f$ from the real Paley–Wiener space such that every function f ^(m), m=0, 1,..., has a finite number of real zeros.     

On Generators in Ideals of Entire Functions of Finite Order and Type on the Plane

We study ideals in the space of entire functions of finite order and type on the plane. We obtain necessary and sufficient conditions for the ideal of functions vanishing at the common zeros of a system of N functions to be generated by the latter.     

关于f~((k))-af~n的零点

设 f (z) 为平面内超越亚纯函数 ,a≠ 0 为常数 ,证明了当 n≥ k+3 时 ,f( k) -afn有无穷多个零点 .     

On computing distributions of products of non-negative independent random variables

We introduce a new functional representation of probability density functions (PDFs) of non-negative random variables via a product of a monomial factor and linear combinations of decaying exponentials with complex exponents. This approximate representation of PDFs is obtained for any finite, user-selected accuracy. Using a fast algorithm involving Hankel matrices, we develop a general numerical method for computing the PDF of the sums, products, or quotients of any number of non-negative independent random variables yielding the result in the same type of functional representation. We present several examples to demonstrate the accuracy of the approach.