Real-variables characterization of generalized Stieltjes functions

We obtain a characterization of generalized Stieltjes functions of any order λ>0$\lambda >0$ in terms of inequalities for their derivatives on (0,∞)$(0,\infty )$. When λ=1$\lambda =1$, this provides a new and simple proof of a characterization of Stieltjes functions first obtained by Widder in 1938.     

Approximation of bandlimited functions by finite exponential sums

For a compact set K in ℝ ⁿ , let B ² _K be the set of all functions f ∈ L ²(ℝ²) bandlimited to K, i.e., such that the Fourier transform f̂ of f is supported by K. We investigate the question of approximation of f ∈ B ² _K by finite exponential sums

Some new nonlinear inequalities and applications to boundary value problems

In this paper, we establish some new nonlinear integral inequalities of the Gronwall–Bellman–Ou-Iang-type in two variables. These on the one hand generalizes and on the other hand furnish a handy tool for the study of qualitative as well as quantitative properties of solutions of differential equations. We illustrate this by applying our new results to certain boundary value problem.     

Series like Taylor's series

A theorem and some examples are given concerning the convergence, in a space of generalized functions, of power series whose terms contain successive derivatives of a given function. One example is the Euler-Maclaurin sum formula, where there are some novelties.     

Paley–Wiener and Boas theorems for singular Sturm–Liouville integral transforms

This paper deals with a class of integral transforms arising from a singular Sturm–Liouville problem y″−q(x)y=−λy, x(a,b), in the limit-point case at one end or both ends of the interval (a,b). The paper completely solves the problem of characterization of the image of a function that has compact support (Paley–Wiener theorem) and also of a function that vanishes on some interval (Boas problem) under this class of transforms. The characterizations are obtained with no restriction on q(x) other than being locally integrable.     

Kolmogorov's problem on the class of multiply monotone functions

In this paper we give necessary and sufficient conditions for the system of positive numbers _Mk1,_Mk2,…,_Mkd$M_{k_1},M_{k_2},\dots ,M_{k_d}$, 0≤_k1<…<_kd≤r$0\le k_1<\dots <k_d\le r$, to guarantee the existence of an r  -monotone function defined on the negative half-line _R−${\mathbb{R}}_{-}$ and such that _{‖x(ki)‖∞}=_Mki${\Vert x^{(k_i)}\Vert }_{\infty }=M_{k_i}$, i=1,2,…,d$i=1,2,\dots ,d$. We also discuss some applications of the obtained results and connections with other problems.     

Complete monotonicity of some functions involving polygamma functions

In the present paper, we establish necessary and sufficient conditions for the functions x^α|ψ⁽ⁱ⁾(x+β)| and α|ψ⁽ⁱ⁾(x+β)|−x|ψ⁽ⁱ⁺¹⁾(x+β)| respectively to be monotonic and completely monotonic on (0,∞), where i∈N, α>0 and β≥0 are scalars, and ψ⁽ⁱ⁾(x) are polygamma functions.     

Two-sided inequalities for the Struve and Lommel functions

Abstract

Mathematical inequalities and other results involving such widely- and extensively-studied special functions of mathematical physics and applied mathematics as (for example) the Bessel, Struve and Lommel functions as well as the associated hypergeometric functions are potentially useful in many seemingly diverse areas of applications, especially in situations in which these functions are involved in solutions of mathematical, physical and engineering problems which can be modeled by ordinary and partial di?erential equations. With this objective in view, our present investigation is motivated by some open problems involving inequalities for a number of particular forms of the hypergeometric function ₁F₂(a; b, c; z). Here, in this paper, we apply a novel approach to such problems and obtain presumably new two-sided inequalities for the Struve function, the associated Struve function and the modified Struve function by first investigating inequalities for the general hypergeometric function ₁F₂(a; b, c; z). We also briefly discuss the analogous new inequalities for the Lommel function under some conditions and constraints. Finally, as special cases of our main results, we deduce several inequalities for the modified Lommel function and the normalized Lommel function.     

Computing Eigenvalues of Lommel-Type Equations by the Sampling Method

Using the semiclassical approximation for where and as , we shall show that theeigenvalues are the zeros of a certain function in a Paley–Wienerspace, which allows us to use the Whittaker–Shannon–Kotelnikovsampling theorem to approximate the eigenvalues. We show how thedistribution of the eigenvalues depends on the asymptotics of thecoefficients and as . After abrief discussion on the truncation error, numerical examples are provided.     

Henriksen?s contributions to residue class rings of analytic and entire functions

The author surveys, summarizes and generalizes results of Golasiński and Henriksen, and of others, concerning certain residue class rings.Let A(R) denote the ring of analytic functions over reals R and E(K) the ring of entire functions over R or complex numbers C. It is shown that if m is a maximal ideal of A(R), then A(R)/m is isomorphic either to the reals or a real-closed field that is η₁-set, while if m is a maximal ideal of E(K), then E(K)/m is isomorphic to one of these latter two fields or to complex numbers.     

Maximal order for quadratures usingn evaluations

The problem is to find approximationsI (f; h) to the integralI(f; h)= ₀ ^h f. Such an approximation has local orderp ifI(f; h)–I (f; h)=O(h ^p ) ash0. Let(n) denote the maximal local order possible for a method usingn evaluations of a function or its derivatives. We show that (n)=2n+1 if the information used is Hermitian. This is conjectured to be true in general. The conjecture is established for all methods using three or fewer evaluations.This research was supported in part by the National Science Foundation under Grant MCS75-222-55 and the Office of Naval Research under Contract N00014-76-C-0370, NR 044-422. Numerical results reported in this paper were obtained through the computing facilities of the University of Maryland.     

A generalisation of the functional approach to compactness

A topological space X is compact iff the projection π:X×Y→Y is closed for any space Y. Taking this as a definition and then asking that π maps α-closed subspaces of X×Y onto β-closed subspaces of Y, for different closures α and β, extends the notion of compactness to include also examples of “asymmetric compactness” pursued in the article.Categorical closure operators and a so-called “functional approach to general topology” are employed to define and prove fundamental properties of compact objects and proper maps in this generalised setting.     

A class of logarithmically completely monotonic functions and the best bounds in the second Kershaw's double inequality

In the article, the sufficient and necessary conditions such that a class of functions which involve the psi function ψ$\psi$ and the ratio Γ(x+t)/Γ(x+s)$\Gamma (x+t)/\Gamma (x+s)$ are logarithmically completely monotonic are established, the best bounds for the ratio Γ(x+t)/Γ(x+s)$\Gamma (x+t)/\Gamma (x+s)$ are given, and some comparisons with known results are carried out, where s and t   are two real numbers and x>-min{s,t}$x>-\min \{s,t\}$.     

Reproducing kernels for Hardy spaces on multiply connected domains

Associated with a boundedg-holed (g0) planar domainD are two types of reproducing kernel Hilbert spaces of meromorphic functions onD. We give explicit formulas for the reproducing kernel functions of these spaces. The formulas are in terms of theta functions defined on the Jacobian variety of the Schottky double of the regionD. As applications we settle a conjecture of Abrahamse concerning Nevalinna-Pick interpolation on an annulus and obtain explicit formulas for the curvature (in the sense of Cowen and Douglas) of rank 1 bundle shift operators.     

On the best constants in inequalities of the Markov and Wirtinger types for polynomials on the half-line

The main topic of the paper is best constants in Markov-type inequalities between the norms of higher derivatives of polynomials and the norms of the polynomials themselves. The norm is the L² norm with Laguerre weight. The leading term of the asymptotics of the constants is determined and tight bounds for the principal coefficient in this term, which is the operator norm of a Volterra operator, are given. For best constants in inequalities of the Wirtinger type, the limit is computed and an asymptotic formula for the error term is presented.     

On Hindman spaces and the Bolzano–Weierstrass property

We examine relationships between two classes of topological spaces defined with the aid of the Hindman ideal. We also do the same for another ideal—instead of sums, as in the Hindman ideal, we consider differences.     

Weak regularity and consecutive topologizations and regularizations of pretopologies

L. Foged proved that a weakly regular topology on a countable set is regular. In terms of convergence theory, this means that the topological reflection Tξ of a regular pretopology ξ on a countable set is regular. It is proved that this still holds if ξ is a regular σ-compact pretopology. On the other hand, it is proved that for each n<ω there is a (regular) pretopology ρ (on a set of cardinality c) such that k(RT)ρ>n(RT)ρ for each k<n and n(RT)ρ is a Hausdorff compact topology, where R is the reflector to regular pretopologies. It is also shown that there exists a regular pretopology of Hausdorff RT-order ?ω₀. Moreover, all these pretopologies have the property that all the points except one are topological and regular.     

CATEGORIES OF TOPOLOGICAL GROUPS

Abstract

This paper is a survey of recent (and some not so recent, results concerning categorical constructions on topological groups, with particular emphasis on free topological groups and coproducts (free products) of topological groups. An extensive bibliography is included.