Approximation and Extension in Normed Spaces of Infinitely Differentiable Functions

We consider questions of rational and polynomial approximation,and related extension questions, for various normed spaces ofinfinitely differentiable functions on perfect compact subsetsof the complex plane C and the real line R. We obtain an approximationtheorem for compact planar sets which are, in a precise sense,locally radially self-absorbing. All smoothly-bounded compactsets are of this type. We give a variety of results and counterexampleson extension, mainly in the one-dimensional case. We also provepolynomial approximation theorems for totally-disconnected sets,linear sets, and some others.     

Approximation of Infinitely Differentiable Periodic Functions by Interpolation Trigonometric Polynomials in Integral Metric

We obtain asymptotic equalities for the upper bounds of approximations of periodic infinitely differentiable functions by interpolation trigonometric polynomials in the metric of L₁ on the classes of convolutions.     

Pointwise Estimates for the Coconvex Approximation of Differentiable Functions

We obtain pointwise estimates for the coconvex approximation of functions of the class W ^r , r > 3.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 1, pp. 47–59, January, 2005.     

On the Representation of Infinitely Differentiable Functions by Series of Exponentials

We consider the problem of representing elements of a weighted space of infinitely differentiable functions on the real line by series of exponentials.     

Adaptive Cross Approximation of Multivariate Functions

In this article we present and analyze a new scheme for the approximation of multivariate functions (d=3,4) by sums of products of univariate functions. The method is based on the Adaptive Cross Approximation (ACA) initially designed for the approximation of bivariate functions. To demonstrate the linear complexity of the schemes, we apply it to large-scale multidimensional arrays generated by the evaluation of functions.     

Approximation of Differentiable Periodic Functions by Their Biharmonic Poisson Integrals

We determine the exact values and asymptotic decompositions of upper bounds of approximations by biharmonic Poisson integrals on classes of periodic differentiable functions.     

Surjectivity of Linear Differential Operators in a Weighted Space of Infinitely Differentiable Functions

We present a sufficient condition for the surjectivity of a linear differential operator of infinite order with constant coefficients in a weighted space of infinitely differentiable functions on the real line.     

On the Exponential Convergence of Spline Approximation Methods for Wiener-Hopf Equations

This paper deals with the study of some new multidimensional operators of fractional calculus. These operators are considered in certain spaces of test and generalized functions. Some mapping properties and product rules are obtained.