On a p-stable multiple integral. II

Summary We construct a multiple strictly p-stable stochastic integral as a Dunford type integral with respect to a L ^q -valued vector measure, 1qp2.     

On the convergence to the multiple Wiener-Itô integral

We study the convergence to the multiple Wiener-Itô integral from processes with absolutely continuous paths. More precisely, consider a family of processes, with paths in the Cameron-Martin space, that converges weakly to a standard Brownian motion in C₀([0,T]). Using these processes, we construct a family that converges weakly, in the sense of the finite dimensional distributions, to the multiple Wiener-Itô integral process of a function f∈L²(n[0,T]). We prove also the weak convergence in the space C₀([0,T]) to the second-order integral for two important families of processes that converge to a standard Brownian motion.     

On the Feynman integral in the space of continuous functions on a compact set. I

The Feynman integral in the space of continuous functions defined on a compact set is defined as the analytic continuation of the generalized Wiener integral in the sense of J. Kuelbs. We prove a theorem on the computation of Feynman integrals of functions that depent on linear functionals. Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 37, 1994, pp. 32–36.     

Solution of a certain multiple integral equation

An explicit solution is derived formally for a certain multiple integral equation involving a multidimensional fractional integral of Riemann-Liouville type. The main inversion theorem proved here provides a generalization of a result due to W. L. Wainwright [in “Fractional Calculus and Its Applications” (B. Ross, Ed.), pp. 298–305, Springer-Verlag, Berlin/Heidelberg/New York, 1975]. A simple illustration of the theorem, involving the classical Laguerre function, is also presented.     

On Henstock integral of fuzzy-number-valued functions (I)

In this paper, we define and discuss the (FH) integral for fuzzy-number-valued functions. Using a concrete structure into which we embed the fuzzy number space E¹, several necessary and sufficient conditions of integrability for fuzzy-number-valued functions are given by means of abstract function theory.     

Change of variables in a multiple Lebesgue integral

Necessary and sufficient conditions are found for a change of variables in a multiple Lebesgue integral for a class of quasibijective (one-to-one everywhere except on a set whose image has measure zero) mappings in the Euclidean space Rⁿ.Translated from Matematicheskie Zametki, Vol. 14, No. 1, pp. 39–48, July, 1973.     

On a new unified integral

In the present paper we derive a unified new integral whose integrand contains products of FoxH-function and a general class of polynomials having general arguments. A large number of integrals involving various simpler functions follow as special cases of this integral.     

An incompressibility condition for a certain class of integral functionals. I

One considers the integral functional (y), depending on the mapping of the domain R^m into R^m, on the set of mappings y, subjected to the incompressibility condition: det y=1. One computes its first and second variations. The obtained results are compared with the formulas arising from the formal application of the method of the undetermined Lagrange multipliers. One gives an application to problems of elasticity theory.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 115, pp. 203–214, 1982.     

Algebraic and multiple integral identities

In this paper we develop some identities involving symmetric products in an abstract algebra which was formerly introduced by Rimark Ree to investigate the shuffle product and relations with skew symmetric (Lie) products. His motivation was partially the characterization of homogeneous Lie polynomials in noncommuting variables, while our motivation is derived from problems in systems theory. The main link in these applications is the need for identities involving multiple integrals of functions of many variables. The relation between these identities and some of the abstract identities developed here is also worked out and some of the applications to systems theory reviewed.     

On spline approximation for a class of integral equations. I: Galerkin and collocation methods with piecewise polynomials

We consider the numerical solution of a class of one-dimensional non-compact integral equations by Galerkin and collocation methods and their iterated variants, using piecewise polynomials as basis functions. In particular, we obtain new results for the stability of the approximation methods, without any restriction on the norm of the integral operators. Furthermore, we extend results of Chandler and Graham^4,6 concerning error estimates and superconvergence to a more general class of operators.