Real inversion formulas with rates of convergence

Some classical real inversion formulas, such as those concerning Fourier, Laplace and Stieltjes transforms, are unified in a way which allows us to give rates of convergence. As illustration, the case of the Fourier transform is considered. This revised version was published online in June 2006 with corrections to the Cover Date.     

Asymptotic Expansions of Generalized Stieltjes Transforms of Algebraically Decaying Functions

Asymptotic expansions of Stieltjes and generalized Stieltjes transforms of functions having an asymptotic expansion in negative integer powers of their variable have been exhaustively investigated by R. Wong. In this article, we extend this analysis to Stieltjes and generalized Stieltjes transforms of functions having an asymptotic expansion in negative rational powers of their variable. Distributional approach is used to derive asymptotic expansions of the Stieltjes and generalized Stieltjes transforms of this kind of functions for large values of the parameter(s) of the transformation. Error bounds are obtained at any order of the approximation for a large family of integrands. The asymptotic approximation of an integral involved in the calculation of the mass renormalization of the quantum scalar field and of the third symmetric elliptic integral are given as illustrations.     

Asymptotic Expansions of Generalized Stieltjes Transforms of Algebraically Decaying Functions

Asymptotic expansions of Stieltjes and generalized Stieltjes transforms of functions having an asymptotic expansion in negative integer powers of their variable have been exhaustively investigated by R. Wong. In this article, we extend this analysis to Stieltjes and generalized Stieltjes transforms of functions having an asymptotic expansion in negative rational powers of their variable. Distributional approach is used to derive asymptotic expansions of the Stieltjes and generalized Stieltjes transforms of this kind of functions for large values of the parameter(s) of the transformation. Error bounds are obtained at any order of the approximation for a large family of integrands. The asymptotic approximation of an integral involved in the calculation of the mass renormalization of the quantum scalar field and of the third symmetric elliptic integral are given as illustrations.     

Stieltjes integral equations in partially ordered sets

Stieltjes integral equations are considered in partially ordered sets of real valued functions, and connections are made using integral inequalities to the existence and uniqueness problems of hereditary systems which are not Lipschitz.     

Nonlinear eigenvalue problem for a system of ordinary differential equations subject to a nonlocal condition

For a system of linear ordinary differential equations supplemented with a nonlocal condition specified by the Stieltjes integral, the problem of calculating the eigenvalues belonging to a given bounded domain in the complex plane is examined. It is assumed that the coefficient matrix of the system and the matrix function in the Stieltjes integral are analytic functions of the spectral parameter. A numerically stable method for solving this problem is proposed and justified. It is based on the use of an auxiliary boundary value problem and formulas of the argument principle type. The problem of calculating the corresponding eigenfunctions is also treated.     

Non-local PDEs with a state-dependent delay term presented by Stieltjes integral

Parabolic partial differential equations with state-dependent delays (SDDs) are investigated. The delay term presented by Stieltjes integral simultaneously includes discrete and distributed SDDs. The singular Lebesgue–Stieltjes measure is also admissible. The conditions for the corresponding initial value problem to be well-posed are presented. The existence of a compact global attractor is proved.     

Addition in a class of nonlinear Stieltjes integrators

A product integral formula is established for the generation of an evolution system by the sum of two Stieltjes integrators. The class of Stieltjes integrators considered is the class recently introduced and studied by J.V. Herod.     

A strong approximation for double subfractional integrals

A Riemann–Stieltjes integral strong approximation to double Stratonovich integrals with respect to odd and even fractional Brownian motions is considered. We prove the convergence in quadratic mean, uniformly on compact time intervals, of the ordinary double integral process obtained by linear interpolation of the odd and even fractional Brownian motions, to the double Stratonovich integral. The deterministic integrands are continuous or are given by bimeasures.     

Stieltjes integral inequalities of Gronwall type and applications

Summary We obtain estimates for solutions of integral inequalities of Gronwall type involving Stieltjes integrals and their inverse inequalities. From these we obtain some new results for integral inequalities for Riemann integrals and functional integral inequalities. Extensions are also given to Bihari type integral inequalities.Research supported by NSERC Canada.     

A Parseval-type Theorem Applied to Certain Integral Transforms

In this paper, a Parseval-type theorem for Laplace and Stieltjestransforms is proved. The theorem yields a number of corollariesabout Laplace, Stieltjes, generalized Stieltjes, potential,Laguerre, and Weyl transforms. The theorem gives some methodsfor solving certain types of integral equations and evaluatinginfinite integrals.     

Integration with respect to Fractal Functions and Stochastic Calculus II

The link between fractional and stochastic calculus established in part I of this paper is investigated in more detail.We study a fractional integral operator extending the Lebesgue – Stieltjes integral and introduce a related concept of stochastic integral which is similar to the so – called forward integral in stochastic integration theory.The results are applied to ODE driven by fractal functions and to anticipative SDE whose noise processes possess absolutely continuous generalized covariation processes.A survey on this approach may be found in [21].     

第二拉普拉斯变换及其应用

本文提出了一种新的积分变换——第二拉普拉斯变换,大家所熟悉的单边拉普拉斯变换为本变换的一种特殊情况．关于第二拉普拉斯变换的应用,本文举例说明     

Certain singular integral operators revisited

This note presents an alternate approach to the analysis of the composition of some (classical) singular integral oprators that arise in a number of applications. It is based on the facts that certain naturally paired operators have the same range and are injective. In the course of this analysis the proofs of some classical identities are unified     

On the dominated convergence theorem for the Kurzweil–Stieltjes integral

This paper is concerned with a version of the Lebesgue dominated convergence theorem (DCT) which has been stated for the Kurzweil–Stieltjes integral of real functions. Our objective in this work is to analyze the extension of this result to include vector functions with values in Banach spaces. We establish that the mentioned convergence theorem for the Kurzweil–Stieltjes integral can be formulated in weaker versions for reflexive and separable Banach spaces, and spaces having the Schur property, nonetheless it is not verified in the general case.     

Stieltjes representation of the 3D Bruggeman effective medium and Padé approximation

The paper deals with Bruggeman effective medium approximation (EMA) which is often used to model effective complex permittivity of a two-phase composite. We derive the Stieltjes integral representation of the 3D Bruggeman effective medium and use constrained Padé approximation method introduced in [39] to numerically reconstruct the spectral density function in this representation from the effective complex permittivity known in a range of frequencies. The problem of reconstruction of the Stieltjes integral representation arises in inverse homogenization problem where information about the spectral function recovered from the effective properties of the composite, is used to characterize its geometric structure. We present two different proofs of the Stieltjes analytical representation for the effective complex permittivity in the 3D Bruggeman effective medium model: one proof is based on direct calculation, the other one is the derivation of the representation using Stieltjes inversion formula. We show that the continuous spectral density in the integral representation for the Bruggeman EMA model can be efficiently approximated by a rational function. A rational approximation of the spectral density is obtained from the solution of a constrained minimization problem followed by the partial fractions decomposition. We show results of numerical rational approximation of Bruggeman continuous spectral density and use these results for estimation of fractions of components in a composite from simulated effective permittivity of the medium. The volume fractions of the constituents in the composite calculated from the recovered spectral function show good agreement between theoretical and predicted values.     

一个积分等式的新证

通过傅立叶积分公式和筛选性质给出了关于积分等式∫ -∞ ^∞e^t^2/2dt=√2π的一种新证明方法．     

The Beesack–Darst–Pollard inequalities and approximations of the Riemann–Stieltjes integral

Utilising the Beesack version of the Darst–Pollard inequality, some error bounds for approximating the Riemann–Stieltjes integral are given. Some applications related to the trapezoid and mid-point quadrature rules are provided.     

A note on semipositone boundary value problems with nonlocal,nonlinear boundary conditions

We consider the existence of at least one positive solution to a semipositone boundary value problem with nonlocal, nonlinear boundary conditions, which can be quite general since the nonlinearity is realized as a Stieltjes integral. By assuming that the associated Stieltjes measure decomposes in a certain way, the classical Leray-Schauder degree is utilized to derive the existence result.     

Continued fractions and the polygamma functions

In a previous study we have shown that the polygamma functions (derivatives of the logarithm of the gamma function) relate to Stieltjes transforms in the square of the argument. These transforms in turn may be converted to Stieltjes continued fractions; in the background is a determined Stieltjes moment problem.In the present study we use the Hamburger form of the Stieltjes integral to produce a set of real monotonic increasing and monotonic decreasing approximants to each of the real and imaginary parts of a polygamma function when the argument is complex. The approximants involve rational fractions which appear to be new.Special attention is given to ln Γ(z) and the psi function.     

Adjoint classes of functions in the H<Subscript>1</Subscript> sense

Using the concept of the H₁-integral, we consider a similarly defined Stieltjes integral. We prove a Riemann-Lebesgue type theorem for this integral and give examples of adjoint classes of functions.