Dimitar Kolev Dimitrov - Academia.edu (original) (raw)

Papers by Dimitar Kolev Dimitrov

arXiv (Cornell University), Apr 11, 2023

We study two variations of the classical one-delta problem for entire functions of exponential ty... more We study two variations of the classical one-delta problem for entire functions of exponential type, known also as the Carathéodory-Fejér-Turán problem. The first variation imposes the additional requirement that the function is radially decreasing while the second one is a generalization which involves derivatives of the entire function. Various interesting inequalities, inspired by results due to Duffin and Schaeffer, Landau, and Hardy and Littlewood, are also established.

Journal of Computational and Applied Mathematics, Mar 1, 2017

An algorithmic approach for generating generalised Zernike polynomials by differential operators ... more An algorithmic approach for generating generalised Zernike polynomials by differential operators and connection matrices is proposed. This is done by introducing a new order of generalised Zernike polynomials such that it collects all the polynomials of the same total degree in a column vector. The connection matrices between these column vectors composed by the generalised Zernike polynomials and a family of polynomials generated by a Rodrigues formula are given explicitly. This yields a Rodrigues type formula for the generalised Zernike polynomials themselves with properly defined differential operators. Another consequence of our approach is the fact that the generalised Zernike polynomials obey a rather simple partial differential equation. We recall also how to define Hermite-Zernike polynomials.

$Research paper thumbnail of An extremal problem related to generalizations of the Nyman-Beurling and B\'aez-Duarte criteria$

arXiv (Cornell University), Aug 28, 2016

We establish generalizations of the Nyman-Beurling and Báez-Duarte criteria concerning lack of ze... more We establish generalizations of the Nyman-Beurling and Báez-Duarte criteria concerning lack of zeros of Dirichlet L-functions in the semi-plane ℜ(s) > 1/p for p ∈ (1, 2]. We pose and solve a natural extremal problem for Dirichlet polynomials which take values one at the zeros of the corresponding L-function on the vertical line ℜ(s) = 1/p.

Transactions of the American Mathematical Society, 2019

Associated with a given suitable function, or a measure, on R \mathbb {R} , we introduce a correl... more Associated with a given suitable function, or a measure, on R \mathbb {R} , we introduce a correlation function so that the Wronskian of the Fourier transform of the function is the Fourier transform of the corresponding correlation function, and the same holds for the Laplace transform. We obtain two types of results. First, we show that Wronskians of the Fourier transform of a nonnegative function on R \mathbb {R} are positive definite functions and that the Wronskians of the Laplace transform of a nonnegative function on R + \mathbb {R}_+ are completely monotone functions. Then we establish necessary and sufficient conditions in order that a real entire function, defined as a Fourier transform of a positive kernel K K , belongs to the Laguerre–Pólya class, which answers an old question of Pólya. The characterization is given in terms of a density property of the correlation kernel related to K K , via classical results of Laguerre and Jensen and employing Wiener’s L 1 L^1 Tauberi...

Proceedings of the American Mathematical Society, 2015

We obtain the explicit form of the expansion of the Jacobi polynomial P (α,β) n (1 − 2x/β) in ter... more We obtain the explicit form of the expansion of the Jacobi polynomial P (α,β) n (1 − 2x/β) in terms of the negative powers of β. It is known that the constant term in the expansion coincides with the Laguerre polynomial L (α) n (x). Therefore, the result in the present paper provides the higher terms of the asymptotic expansion as β → ∞. The corresponding asymptotic relation between the zeros of Jacobi and Laguerre polynomials is also derived.

Zero sets of bivariate polynomials Bivariate Gaussian distribution Bivariate orthogonal polynomia... more Zero sets of bivariate polynomials Bivariate Gaussian distribution Bivariate orthogonal polynomials Hermite polynomials Algebraic plane curves We establish various properties for the zero sets of three families of bivariate Hermite polynomials. Special emphasis is given to those bivariate orthogonal polynomials introduced by Hermite by means of a Rodrigues type formula related to a general positive definite quadratic form. For this family we prove that the zero set of the polynomial of total degree n + m consists of exactly n + m disjoint branches and possesses n + m asymptotes. A natural extension of the notion of interlacing is introduced and it is proved that the zero sets of the family under discussion obey this property. The results show that the properties of the zero sets, considered as affine algebraic curves in R 2 , are completely different for the three families analyzed.

Numerical Algorithms, 2014

ABSTRACT Sharp bounds for the zeros of symmetric Kravchuk polynomials Kn(x;M) are obtained. The r... more ABSTRACT Sharp bounds for the zeros of symmetric Kravchuk polynomials Kn(x;M) are obtained. The results provide a precise quantitative meaning of the fact that Kravchuk polynomials converge uniformly to Hermite polynomials, as M tends to infinity. They show also how close the corresponding zeros of two polynomials from these sequences of classical orthogonal polynomials are.

We discuss the Newton-Gregory interpolation process based on the geometric mesh 1, q, q 2 ,. . .,... more We discuss the Newton-Gregory interpolation process based on the geometric mesh 1, q, q 2 ,. . ., with a quotient q ∈ C, |q| < 1, for rational functions with a single pole ζ ∈ C. It is shown that the sequence of interpolating polynomials converges in the disc {z : |z| < |ζ|.

Applied Numerical Mathematics, 2011

In an attempt to answer a long standing open question of Al-Salam we generate various beautiful f... more In an attempt to answer a long standing open question of Al-Salam we generate various beautiful formulae for convolutions of orthogonal polynomials similar to U n (x) = n k=0 P k (x)P n−k (x), where U n (x) are the Chebyshev polynomials of the second kind and P k (x) are the Legendre polynomials. The results are derived both via the generating functions approach and a new convolution formulae for hypergeometric functions. We apply some addition formulae similar to the well-known expansion H n (x + y) = 2 −n/2 n k=0 n k H k (√ 2x)H n−k (√ 2 y) for the Hermite polynomials, due to Appell and Kampé de Fériet, to obtain new interesting inequalities about the zeros of the corresponding orthogonal polynomials.

SIAM Journal on Numerical Analysis, 2016

The present paper is a continuation of a recent article [SIAM J. Numer. Anal., 52 (2014), pp. 186... more The present paper is a continuation of a recent article [SIAM J. Numer. Anal., 52 (2014), pp. 1867--1886], where we proposed an algorithmic approach for approximate calculation of sums of the form sumj=1Nf(j)\sum_{j=1}^{N} f(j)sumj=1Nf(j). The method is based on a Gaussian type quadrature formula for sums, which allows the calculation of sums with a very large number of terms NNN to be reduced to sums with a much smaller number of summands nnn. In this paper we prove that the Weierstrass--Dochev--Durand--Kerner iterative numerical method, with explicitly given initial conditions, converges to the nodes of the quadrature formula. Several methods for computing the nodes of the discrete analogue of the Gaussian quadrature formula are compared. Since, for practical purposes, any approximation of a sum should use only the values of the summands f(j)f({j})f(j), we implement a simple but efficient procedure to additionally approximate the evaluations at the nodes by local natural splines. Explicit numerical examples are provided. Moreove...

SIAM Journal on Scientific Computing, 2020

We explore the computational issues concerning a new algorithm for the classical least-squares ap... more We explore the computational issues concerning a new algorithm for the classical least-squares approximation of NNN samples by an algebraic polynomial of degree at most nnn when the number NNN of t...

Pre-publicaciones del …, 2004

We establish sufficient conditions for a matrix to be almost totally positive, thus extending a r... more We establish sufficient conditions for a matrix to be almost totally positive, thus extending a result of Craven and Csordas who proved that the corresponding con-ditions guarantee that a matrix is strictly totally positive. Then we apply our main result in order to obtain new sufficient ...

Proceedings of the American Mathematical Society, 2021

We study the behaviour of the smallest possible constants dn and cn in Hardy's inequalities n k=1... more We study the behaviour of the smallest possible constants dn and cn in Hardy's inequalities n k=1 4 − c ln n < dn, cn < 4 − c ln 2 n , c > 0 are established.

Proceedings of the American Mathematical Society, 1998

The celebrated Turán inequalities P n 2 ( x ) − P n − 1 ( x ) P n + 1 ( x ) ≥ 0 P_{n}^{2}(x) - P_... more The celebrated Turán inequalities P n 2 ( x ) − P n − 1 ( x ) P n + 1 ( x ) ≥ 0 P_{n}^{2}(x) - P_{n-1}(x) P_{n+1}(x) \geq 0 , x ∈ [ − 1 , 1 ] x \in [-1,1] , n ≥ 1 n \geq 1 , where P n ( x ) P_{n}(x) denotes the Legendre polynomial of degree n n , are extended to inequalities for sums of products of four classical orthogonal polynomials. The proof is based on an extension of the inequalities γ n 2 − γ n − 1 γ n + 1 ≥ 0 \gamma _{n}^{2} - \gamma _{n-1} \gamma _{n+1} \geq 0 , n ≥ 1 n \geq 1 , which hold for the Maclaurin coefficients of the real entire function ψ \psi in the Laguerre-Pólya class, ψ ( x ) = ∑ n = 0 ∞ γ n x n / n ! \psi (x) = \sum _{n=0}^{\infty } \gamma _{n} x^{n}/n! .

Journal of Mathematical Analysis and Applications, 2021

Abstract For parameters c ∈ ( 0 , 1 ) and β > 0 , let l 2 ( c , β ) be the Hilbert space of re... more Abstract For parameters c ∈ ( 0 , 1 ) and β > 0 , let l 2 ( c , β ) be the Hilbert space of real functions defined on N (i.e., real sequences), for which ‖ f ‖ c , β 2 : = ∑ k = 0 ∞ ( β ) k k ! c k [ f ( k ) ] 2 ∞ . We study the best (i.e., the smallest possible) constant γ n ( c , β ) in the discrete Markov-Bernstein inequality ‖ Δ P ‖ c , β ≤ γ n ( c , β ) ‖ P ‖ c , β , P ∈ P n , where P n is the set of real algebraic polynomials of degree at most n and Δ f ( x ) : = f ( x + 1 ) − f ( x ) . We prove that (i) γ n ( c , 1 ) ≤ 1 + 1 c for every n ∈ N and lim n → ∞ ⁡ γ n ( c , 1 ) = 1 + 1 c ; (ii) For every fixed c ∈ ( 0 , 1 ) , γ n ( c , β ) is a monotonically decreasing function of β in ( 0 , ∞ ) ; (iii) For every fixed c ∈ ( 0 , 1 ) and β > 0 , the best Markov-Bernstein constants γ n ( c , β ) are bounded uniformly with respect to n. A similar Markov-Bernstein inequality is proved for sequences, and a relation between the best Markov-Bernstein constants γ n ( c , β ) and the smallest eigenvalues of certain explicitly given Jacobi matrices is established.

Journal d'Analyse Mathématique, 2019

We prove that the signs of the Maclaurin coefficients of a wide class of entire functions that be... more We prove that the signs of the Maclaurin coefficients of a wide class of entire functions that belong to the Laguerre-Pólya class posses a regular behaviour.

Potential Analysis, 2018

In this note we extend the classical relation between the equilibrium configurations of unit mova... more In this note we extend the classical relation between the equilibrium configurations of unit movable point charges in a plane electrostatic field created by these charges together with some fixed point charges and the polynomial solutions of a corresponding Lamé differential equation. Namely, we find similar relation between the equilibrium configurations of unit movable charges subject to a certain type of rational or polynomial constraint and polynomial solutions of a corresponding degenerate Lamé equation, see details below. In particular, the standard linear differential equations satisfied by the classical Hermite and Laguerre polynomials belong to this class. Besides these two classical cases, we present a number of other examples including some relativistic orthogonal polynomials and linear differential equations satisfied by those.

Proceedings of the American Mathematical Society, 2016

Geometric properties of the classical Lommel and Struve functions, both of the first kind, are st... more Geometric properties of the classical Lommel and Struve functions, both of the first kind, are studied. For each of them, three different normalizations are applied in such a way that the resulting functions are analytic in the unit disk of the complex plane. For each of the six functions we determine the radius of starlikeness precisely.

Journal of Mathematical Analysis and Applications, 2016

Geometric properties of the Jackson and Hahn-Exton q-Bessel functions are studied. For each of th... more Geometric properties of the Jackson and Hahn-Exton q-Bessel functions are studied. For each of them, three different normalizations are applied in such a way that the resulting functions are analytic in the unit disk of the complex plane. For each of the six functions we determine the radii of starlikeness and convexity precisely by using their Hadamard factorization. These are q-generalizations of some known results for Bessel functions of the first kind. The characterization of entire functions from the Laguerre-Pólya class via hyperbolic polynomials play an important role in this paper. Moreover, the interlacing property of the zeros of Jackson and Hahn-Exton q-Bessel functions and their derivatives is also useful in the proof of the main results. We also deduce a sufficient and necessary condition for the close-to-convexity of a normalized Jackson q-Bessel function and its derivatives. Some open problems are proposed at the end of the paper. 2010 Mathematics Subject Classification. 30C45, 30C15. Key words and phrases. Bessel functions; q-Bessel functions; univalent, starlike functions; convex functions; radius of starlikeness; radius of convexity; zeros of q-Bessel functions; Laguerre-Pólya class of entire functions; Laguerre inequality; interlacing property of zeros.

In this short paper I try to answer questions raised by my teacher Borislav Bojanov which concern... more In this short paper I try to answer questions raised by my teacher Borislav Bojanov which concern interlacing of zeros of real polynomials and consider two specific topics. The first one concerns one of his favorite results, a theorem due to Vladimir Markov which states that the derivatives of two polynomials with real interlacing zeros posses zeros which also interlace. The second is a problem about monotonicity of zeros of classical orthogonal polynomials and Sturm's comparison theorem for solutions of Sturm-Liouville differential equations.