Weighted spaces of vector-valued continuous functions

Summary We study completeness of weighted spaces of vector-valued functions. Also, the dual of a weighted space of vector-valued continuous functions on a locally compact space is characterized as a space of vector valued Radon measures with values in the topological dual of the range space. Research supported by a grant from the National Science Foundation, GP 22713. Entrata in Redazione il 25 ottobre 1970.     

Riesz spaces of real continuous functions

An Archimedean Riesz space E is isomorphic to C(X) for some completely regular Hausdorff space X if and only if there exists a weak order unit e > 0 for which E is e-uniformly complete, e-semisimple, e-separating and 2-universally e-complete.     

Integral-type operators on continuous function spaces on the real line

In this paper we introduce some new sequences of positive linear operators, acting on a sufficiently large space of continuous functions on the real line, which generalize Gauss–Weierstrass operators.We study their approximation properties and prove an asymptotic formula that relates such operators to a second order elliptic differential operator of the form Lu?αu^′′+βu^′+γu.Shape-preserving and regularity properties are also investigated.     

Weighted approximation of functions on the real line by Bernstein polynomials

The authors give error estimates, a Voronovskaya-type relation, strong converse results and saturation for the weighted approximation of functions on the real line with Freud weights by Bernstein-type operators.     

Weighted composition operators on nonlocally convex weighted spaces of continuous functions

LetV be a system of weights on a completely regular Hausdorff spaceX and letB(E) be the topological vector space of all continuous linear operators on a general topological vector spaceE. LetCV ₀(X, E) andCV _b (X, E) be the weighted spaces of vector-valued continuous functions (vanishing at infinity or bounded, respectively) which are not necessarily locally convex. In the present paper, we characterize in this general setting the weighted composition operatorsW _π,? onCV ₀(X, E) (orCV _b (X, E)) induced by the operator-valued mappings π:X→B(E) (or the vector-valued mappings π:X→E, whereE is a topological algebra) and the self-map ? ofX. Also, we characterize the mappings π:X→B(E) (or π:x→E) and ?:X→X which induce the compact weighted composition operators on these weighted spaces of continuous functions.     

Classes of functions defined on the real line and their approximation by entire functions. I

We introduce classes of functions defined on the real line which in the periodic case become the classes defined by multipliers (·) and translations with respect to the argument equal to . We study the simplest properties of these classes needed for the study of their approximation characteristics which will be discussed in the second part of the paper.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 1, pp. 102–112, January, 1990.     

Steklov operators and semigroups in weighted spaces of continuous real functions

We consider Steklov operators in weighted spaces of continuous functions on the whole real line and on a bounded interval. We study the connections of these operators with some second order degenerate parabolic problems establishing a general Voronovskaja type formula.     

Solvability of systems of volterra integral equations of the first kind with piecewise continuous kernels

We construct an asymptotic approximation for solutions of systems of Volterra integral equations of the first kind with piecewise continuous kernels. We use the asymptotics as an initial approximation in the proposed method of successive approximations to the desired solutions. We prove the existence of a continuous solution depending on free parameters and establish sufficient conditions for the existence of a unique continuous solution. We illustrate the proved existence theorems with examples.     

Shift-invariant spaces on the real line

We investigate the structure of shift-invariant spaces generated by a finite number of compactly supported functions in . Based on a study of linear independence of the shifts of the generators, we characterize such shift-invariant spaces in terms of the semi-convolutions of the generators with sequences on . Moreover, we show that such a shift-invariant space provides -approximation order if and only if it contains all polynomials of degree less than .

     

Degenerate kernel schemes by wavelets for nonlinear integral equations on the real line

A degenerate kernel scheme is developed for nonlinear integral equations on the real line by approximation of kernels by wavelets. The rate of convergence of the approximate solutions is established in terms of the decay rate of the kernels andthe numbers of dilations and shifts used in approximation of the kernels. For linear equations, the Haar wavelet approximation is used and a numerical example is included.     

Nonautonomous locally Lipschitz continuous functional differential equations in spaces of continuous functions

A local existence and uniqueness result for the functional differential equation in a Banach space X (FDE) $$\begin{gathered} x\prime (t) \in f(t)x(t) + g(t)x_t ,0 \leqslant t \leqslant T, \hfill \\ x_0 = \phi \in C( - R,0;X) \hfill \\ \end{gathered} $$ is obtained, for the case where the operatorsf(t) satisfy only a local dissipativity condition and the operatorsg(t) are only locally Lipschitz continuous. The conditions include equations defined on cones.     

Solvability of convolution equations in the Beurling spaces of ultradifferentiable functions of mean type on an interval

We establish a solvability criterion for convolution equations in the Beurling classes of ultradifferentiable functions of mean type on an interval. Under study is also the question of degeneration of convolution equations into infinite-order equations with constant coefficients.     

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.