Showing papers in "Integral Transforms and Special Functions in 1999"
TL;DR: In this article, a unified and systematic approach to sampling analysis for functions which are basically square-integrable is presented, which is valid under minimal and natural hypotheses, particularly to that connected with Kramer's lemma, and an application is a clean approach to exponential sampling theory for signals which possess certain integral representations, but also for Mellin-bandlimited signals as well as for those which are only approximately Mellinbandlimited.
Abstract: In several papers the authors introduced a self-contained approach (independent of Fourier or Laplace transform theory) to classical Mellin transform theory as well as to a new finite Mellin transform in case the functions in question are absolutely (Lebesgue) integrable. In this paper the matter is considered for functions which are basically square-integrable. The unified and systematic approach presented, which is valid under minimal and natural hypotheses, is applied to sampling analysis, particularly to that connected with Kramer's lemma. An application is a clean approach to exponential sampling theory (of optical circles) for signals which possess certain integral representations, but also for Mellin-bandlimited signals as well as for those which are only approximately Mellin-bandlimited.
61 citations
TL;DR: The notion of the adjoint Ore ring was introduced in this paper, and a definition of an adjoint polynomial, operator and equation was given for integrating solutions of Ore equations.
Abstract: We introduce the notion of the adjoint Ore ring and give a definition of an adjoint polynomial, operator and equation. We apply this for integrating solutions of Ore equations. ∗
43 citations
TL;DR: In this paper, the q-polynomials on the non-uniform exponential lattice x(s)= c 1 qs +c 3 were studied and the main data, i.e., the coefficients of the three-term recurrence relation, structure relation, the square of the norm, etc, in the exponential lattices, respectively.
Abstract: The main goal of this paper is to continue the sutudy of the q-polynomials on non-uniform lattices by using the approach introduced by Nikiforov and Uvarov in 1983. We consider the q-polynomials on the non-uniform exponential lattice x(s)= c 1 qs +c 3 and study some of their properties (differentiation formulas, structure relations, represntation in terms of hypergeometric and basic hypergeometric functions, etc). Special emphasis is given to q-analogues of the Charlier orthogonal polynomials. For these polynomials (Charlier) we compute the main data, i.e., the coefficients of the three-term recurrence relation, structure relation, the square of the norm, etc, in the exponential lattices , respectively.
25 citations
TL;DR: In this paper, for any given a > 0b> 0, determining conditions on c and β such that the convolution zF(a,b,c,z)∗f(z) is starlike if f ∊ R(β) is also treated.
Abstract: For β 0 for z∊ Δ. This paper deals with the following problem: for any given a> 0b> 0, determine conditions on c and β such that the convolution zF(a,b;c;z)∗f(z) is starlike if f ∊ R(β). The same question for the function zΠ(a;c;z) ∗f(z) is also treated. Here F(a,b;c;z) and Π(a;c;z) are the Gaussian and confluent hypergeometric functions, respectively. ∗ .
25 citations
TL;DR: In this article, the authors present a survey on the inversion of potential-type integral transforms based on the method of approximating inverse operators, which is applied to the problem of inverting potential type integral transforms with densities in Lp -spaces and to the characterization of the function spaces of fractional smoothness, connected with the degenerated differential operation.
Abstract: In this survey we present some results on the inversion of potential-type integral transforms based on the method of approximating inverse operators. This method is applied to the problem of inverting potential-type integral transforms with densities in Lp -spaces and to the characterization of the function spaces of fractional smoothness, connected with the degenerated differential operation. A previous survey [21] was devoted to the method of hypersingular integrals in application to inversion problems. ∗
20 citations
TL;DR: The Laplace transform representation of the incomplete beta function Bx (a,b) obtain a simple asymptotic expansion, in ascending powers of 1/a, whose truncation error can be easily estimated as mentioned in this paper.
Abstract: The Laplace transform representation of the incomplete beta function Bx (a,b) obtain a very simple asymptotic expansion, in ascending powers of 1/a, whose truncation error can be easily estimated.
14 citations
TL;DR: In this article, a special class of Szego polynomials on the real axis with an indeterminate moment problem is presented, and a simple criterion of determinacy of the moment problem has been found.
Abstract: Szego polynomials orthogonal on the unit circle satisfy the recurrence relation with reflection parameters . We construct special classes of Szego polynomials . These polynomials appear to be orthogonal on the positive (or negative) real axis. We present explicit examples of such polynomials that are connected with the Askey–Wilson polynomials. In contrast to the case of the Szego polynomials on the unit circle, there exist Szego polynomials on the real axis with an indeterminate moment problem. A simple criterion of determinacy of the moment problem is found.
10 citations
TL;DR: In this paper, the authors give sharp forms of inclusion properties concerning the class P α(β) introduced and studied earlier by Kim et al. They also consider several applications of their results to a class of generalized hypergeometric functions.
Abstract: The object of the present paper is to give sharp forms of inclusion properties concerning the class P α(β) introduced and studied earlier by Kim et al. [4]. We also consider several applications of our results to a class of generalized hypergeometric functions.
8 citations
TL;DR: In this article, a new generalization of Bessel function is given, some properties, various integral representations and integral transform with this function are studied, and properties, integral representation and transform with Bessel functions are studied.
Abstract: Wide and various applications of the special functions in contemporary mathematics stimulate apperance of new types of special functions and generalization of known functions. The generalized hypergeometric functions especially — Bessel functions are the most important among the special functions. Some generalizations of Bessel functions are known [1–3], [5], [8–9], [13] etc. In this article new generalization of Bessel function is given, some properties, various integral representations and integral transform with this function are studied.
5 citations
TL;DR: The relativistic Jacobi polynomials (RJP) as discussed by the authors are an extension of the classical Jacobi orthogonal polynomial in the sense that they reduce to the latter in the non-relativistic limit.
Abstract: A new polynomials set, of generalized hypergeometric type, is defined. These polynomials, called relativistic Jacobi polynomials (RJP) and denoted by represent an extension of the classical Jacobi orthogonal polynomials in the sense that they reduce to the latter in the non-relativistic limit (N → ∞). Some basic properties of these polynomials, as well as for the RHP (see [6] and [7]) and the RLP (see [2] and [3]), are derived.
4 citations
TL;DR: In this article, the cosine integral and its associated functions are defined as locally summable functions on the real line and their derivatives are found as distributions, and some convolution products of these distributions and other functions are then found.
Abstract: The cosine integral Ci(λx) and its associated functions Ci+(λx) and C(λx) are defined as locally summable functions on the real line and their derivatives are found as distributions. Some convolution products of these distributions and other functions are then found.
TL;DR: In this paper, Tchebycheff polynomials of several variables are introduced for the case of three dimensions, and explicit formulas for three dimensions are given in the form of three-dimensional hyperbolic functions.
Abstract: Generalization of cosh and sinh hyperbolic functions known for a long time (see[1] and references therein) are used to introduce Tchebycheff polynomials of several variables.Explicit formulas aer given in the case of three dimensions.
TL;DR: In this article, a study of the function being closely connected with the modified Besssl function of the third kind K ν(x), and the corresponding integral transform Liouville reactional integration and differentiation is investigated.
Abstract: The paper is devoted to a study of the function being closely connected with the modified Besssl function of the third kind K ν(x), and the corresponding integral transform Liouville reactional integration and differentiation is investigated. Two differential equations of fractional order is solved. Mapping properties of the transform and its compositions with differentiation operatiors are proved on spaces of tested and generalized functions .
TL;DR: Liouville's method is applied for investigating the asymptotic behaviour of the Laguerre polynomials as discussed by the authors, and a weight estimate following from these results is obtained.
Abstract: Liouville's method is applied for investigating the asymptotic behaviour of the Laguerre polynomials. Asymptotic expressions are obtained and a reminder term is estimated. a weight estimate following from these results is obtained.
TL;DR: In this article, the classical convolution trasfrom of an ordinary is extended to a generalized function space consisting of complex functions analytic in strips around real axis, and a comparative of the transform defined here and those available in the literature is undertaken.
Abstract: The classical convolution trasfrom of an ordinary is extended to a generalized function space consisting of complex functions analytic in strips around real axis. An inversion theorem is also obtained. A comparative of the transform defined here and those available in the literature is undertaken.
TL;DR: In this article, an extension of the Leibniz rule given by Osler to derive a general formula, which is then used to derive the number of infinite sums for a specific case.
Abstract: In this paper we use an extension of the Leibniz rule given by Osler to derive a general formula, which is then used to derive a number of infinite sums Various specific cases and examples are presented Some results including those recently given by Al-Zamel and Kalla [1] follow as pargicular cases of our results
TL;DR: In this paper, a representation theorem for generalized Laplacians on certain types of bounded domains D of as applications was proved and a Levy-Khintchine formula for certain k-dimensional continuous polynomial hypergroups (D,∗) was given.
Abstract: In this work we prove a representation theorem for generalized Laplacians on certain types of bounded domains D of As applications we give a Levy-Khintchine formula for certain k-dimensional continuous polynomial hypergroups (D,∗), and we characterize convolution semi-groups on D.
TL;DR: Fourier transforms on finite intervals (finite Fourier transforms) are considered here, not as Fourier coefficients, but as functions of a continuous variables The tables of properties of finite Fourier exponential, Fourier sine, and Fourier consine transformations are composed by means of an extension of Graf's theorem for Bessel functions, a series of parabolic cylinder functions, and spectral relationships for Chebyshev and Legendre polynomials.
Abstract: Fourier transforms on finite intervals (finite Fourier transforms) are considered here, not as Fourier coefficients, but as functions of a continuous variables The tables of properties of finite Fourier exponential, Fourier sine, and Fourier consine transformations are composed The use of these properties is illustrated by means of an extension of Graf's theorem for Bessel functions, a series of the parabolic cylinder functions, and spectral relationships for Chebyshev and Legendre polynomials
TL;DR: In this article, it was shown that the main formula givev by Trembly and Gugere is in general a consequence of two much simpler differintegration formulae.
Abstract: It is show that the main formula givev by Trembly and Gugere is in general a consequence of two much simpler differintegration formulae.
TL;DR: In this article, the Schwartz distribution can be seen as a regularized model product in calssical distrubution theory, where the singular products of the distributions have coinciding point singularites.
Abstract: Schwartz distributions can be canonically imbedded in the algebra of Colombeau generalized functions and sometimes their singular products can be evaluated in terms of distributions again. Here we propose some results of that kind concerning the products of the distributions , that have coinciding point singularites. When restricted to dimension one, these results are also easily transformed into the setting of regularized model products in calssical distrubution theory.
TL;DR: In this article, necessary and sufficient conditions for a linear operator commuting with a certain type of generalized integration operator in order to be a continuous automorphism in the space of continuous functions on an arbitrary interval are found.
Abstract: Necessary and sufficient conditions for a linear operator commuting with a certain type of generalized integration operator in order to be a continuous automorphism in the space of continuous functions on an arbitrary interval are found. It is shown that these conditions are equivalent to the requirement that the images of non-zero constants are cyclic elements of the considered integration operator.
TL;DR: Theorems on asymptotic behaviour of a general intrgral transform of functions and distributions are proved in this article. And the concept of pseudo-asymptotic expansion (p.a.e) is introduced and a characterization is given.
Abstract: Theorems on asymptotic behaviour of a general intrgral transform of functions and distributions are proved. The concept of pseudo-asymptotic expansion (p.a.e) is introduced and a characterization is given. The p.a.e of the general inetgral transform is investigated.