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

Some results of Cameron-Storvick on the structure of the Banach algebra S of functionals on the space C^p[a, b] of p-dimensional vector-valued functions x(t), t [a, b], x(a)=0, are generalized to a Banach algebra containing functionals defined on the set C^p(Y), where Y is an infinite-dimensional compact set. In particular it is shown that analytic Feynman integrals of such functionals over the space C^p(Y) exist.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 3, 1997, pp. 45–49.     

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.     

Computation of certain continuous integrals in the space of continuous functions defined on a compact set

We give formulas for computing integrals of functionals that are functions of linear and quadratic functionals with respect to generalized Wiener measure in the sense of J. Kuelbs in the space of continuous functions defined on a compact set.Translated fromMatematicheskie Melody i Fiziko-Mekhanicheskie Polya, Issue 34, 1991, pp. 35–39.     

The conditional Chebyshev center of a compact set of continuous functions

We establish characteristic properties of a subspace L of finite codimension of the space C(T) that has a Chebyshev center and a Chebyshev net for every compact set from C(T). We show that these properties are the same as the conditions for the existence in L of an element of best approximation for every element from C(T).Translated from Matematicheskie Zametki, Vol. 14, No. 4, pp. 469–478, October, 1973.     

Iteration operators on a set of continuous functions of Baire space

Continuous functions on Baire space are considered. Iteration operators are defined on a set of continuous functions. The idea of a module of continuity of a function is introduced. The condition for the growth of module of continuity φ whose satisfaction guarantees that for any enumerable sequence of integration operators and any natural n there exists (n + 1) argument function with the module of continuity φ which cannot be obtained from n-argument functions with the module of continuity φ using any operator of this sequence is formulated. Examples of iteration operators are given.     

On Volterra partial integral equations in the space of continuous functions of three variables

It is well known that any Volterra integral equation of the second kind with compact operator is uniquely solvable. Partial integral operators are not compact, even in the general case of continuous kernels. Unique solvability conditions for Volterra partial integral equations of the second kind in the space of continuous functions of three variables are considered. Conditions for a Volterra partial integral equation to be equivalent to a three-dimensional Volterra integral equation with compact operator are obtained. Continuum analogues of matrix equations for some problems of scattering theory are reduced to the Volterra partial integral equations under examination. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 38, Suzdal Conference-2004, Part 3, 2006.     

Asymptotic properties of one compact set of smooth functions in the space of functions of boundedp-variation

Translated from Matematicheskie Zametki, Vol. 57, No. 2, pp. 214–227, February, 1995.     

On the space of subgroups of a compact group II

Let G be a compact group. Let S(G), C(G), N(G) be the spaces of closed subgroups, cosets of closed subgroups, normal closed subgroups (respectively) of G, with the Vietoris topology.Then: (1) S(G) and C(G) are never connected; (2) N(G) is always totally disconnected; (3) C(G) is totally disconnected if and only if G is totally disconnected; and (4) S(G) is totally disconnected if and only if G/Z(G) is totally disconnected.Further: for totally disconnected G (equivalently, profinite G) (5) S(G), C(G) and N(G) are κ-metrisable; (6) S(G), C(G) and N(G) are Dugundji compact if G has small weight; and (7) consequences for field extensions are derived.     

On set functions with values in a topological vector space

Summary Spaces of set functions defined on a σ-ring and taking values in a topological vector space are supplied with a certain weak topology. Their continuous duals are characterized; certain subsets (the set of simple measures) are singled out and studied and compactness conditions are given. Finally the properties of metrizability, normality, barreledness and of being semi-Montel are studied in connection with these spaces. Entrata in Redazione il 12 maggio 1971.