On the approximation of measurable functions by continuous functions

We present a characterization of the completed Borel measure spaces for which every measurable function, with values in a separable Frechet space, is the almost everywhere limit of a sequence of continuous functions. From this characterization one can easily obtain results that have appeared recently in the literature, in a more general form. We also examine what happens when the range is a subset of an arbitrary Banach space, and show that this case often reduces to the separable case.     

On pointwise bounded approximation

Let G be an open subset in the extended complex plane and let A（G） denote the algebra of all functions analytic on G and continuous on G. We call a domain multi-nicely connected if there is a circular domain W and a conformal map ~ from W onto G such that the boundary value function of φ is univalent almost everywhere with respect to the arclength on aW. Suppose that every component of G is finitely connected and none of the components of G have single point boundary components. We show that for every bounded analytic function on G to be the pointwise limit of a bounded sequence of functions in A（G）, it is necessary and sufficient that each component of G is multi-nicely connected and the harmonic measures of G are mutually singular. This generalizes the corresponding result of Davie for the case when the components of G are simply connected.     

On strong approximation of continuous functions

An embedding theorem concerning strong approximation of continuous functions by their Fourier series is proved.     

Injective choice functions for countable families

This paper proves a necessary and sufficient condition for an arbitrary countable family of sets to have an injective choice function. Some applications are indicated.     

Uniform and tangential harmonic approximation of continuous functions on arbitrary sets

Necessary and sufficient conditions which must be imposed on a set E are derived, such that functions continuous on E G can be approximated by functions harmonic in a region G (Rⁿ.Translated from Matematicheskie Zametki, Vol. 9, No. 2, pp. 131–142, February, 1971.In conclusion I wish to thank my scientific director S. N. Mergelyan and also A. A. Gonchar for their valuable advice.     

On the pointwise estimations of approximation of functions and their derivatives by Hermite interpolation

The object of this paper is to establish the pointwise estimations of approximation of functions in C¹ and their derivatives by Hermite interpolation polynomials. The given orders have been proved to be exact in general.     

Uniform approximation of continuous functions by rational functions

Summary Viene data la condizione necessaria e sufficiente perchè le funzioni razionali di una variabile, aventi poli di ordine prefissato in assegnati punti del piano complesso, costituiscano un sistema completo iu C^o (0, 1). A Bruno Finzi nel suo70 ^mo compleanno. This research has been sponsored in part by the Aerospace Research Laboratoires through the European Office of Aerospace Research, OAR, United States Air Force, under Grant EOOAR-69-0066. Entrata in Redazione il 22 aprile 1970.     

On the degree of approximation of continuous functions

We show that the classical monotonicity conditions can be moderated in four theorems of P. Chandra.     

On statistical approximation in spaces of continuous functions

Our goal is to present approximation theorems for sequences of positive linear operators defined on C(X), where X is a compact metric space. Instead of the uniform convergence we use the statistical convergence. Examples and special cases are also provided.      

A polynomial approximation for arbitrary functions

We describe an expansion of Legendre polynomials, analogous to the Taylor expansion, for approximating arbitrary functions. We show that the polynomial coefficients in the Legendre expansion, and thus, the whole series, converge to zero much more rapidly compared to those in the Taylor expansion of the same order. Furthermore, using numerical analysis with a sixth-order polynomial expansion, we demonstrate that the Legendre polynomial approximation yields an error at least an order of magnitude smaller than that of the analogous Taylor series approximation. This strongly suggests that Legendre expansions, instead of Taylor expansions, should be used when global accuracy is important.     

On the approximation of continuous functions of two variables

It is shown that if P _m ^α,β (x) (α, β > ?1, m = 0, 1, 2, …) are the classical Jaboci polynomials, then the system of polynomials of two variables {Ψ _mn ^α,β (x, y)} _m,n=0 ^r = {P _m ^α,β (x)P _n ^α,β (y)} _{m, n=0} ^r (r = m + n ≤ N ? 1) is an orthogonal system on the set Ω _N×N = ?ub;(x _i , y _i ) _i,j=0 ^N , where x _i and y _i are the zeros of the Jacobi polynomial P _n ^α,β (x). Given an arbitrary continuous function f(x, y) on the square [?1, 1]², we construct the discrete partial Fourier-Jacobi sums of the rectangular type S _{m, n, N} ^α,β (f; x, y) by the orthogonal system introduced above. We prove that the order of the Lebesgue constants ∥S _{m, n, N} ^α,β ∥ of the discrete sums S _{m, n, N} ^α,β (f; x, y) for ?1/2 < α, β < 1/2, m + n ≤ N ? 1 is O((mn) ^{q + 1/2}), where q = max?ub;α,β?ub;. As a consequence of this result, several approximate properties of the discrete sums S _{m, n, N} ^α,β (f; x, y) are considered.     

Rate of pointwise approximation for locally bounded functions by Szász operators

In this paper the asymptotic behavior of Szász operators for locally bounded functions f is studied at points x where f(x+) and f(x−) exist. An asymptotic estimate of this type approximation is obtained by using some techniques and results of probability theory. The estimate essentially is the best possible for continuous points of f.     

On approximation of continuous functions by the Fejer sum of Fourier-Jacobi series

In the present paper, we study the degree of approximation of a continuous function f(x) by the Fejer sum F _n ^(α,β) (f,x) of its Fourier-Jacobi series, and also prove a simple inverse theorem.     

On the approximation of the continuous functions by some Hermite-Fejer interpolation polynomials

The object of this note is to improve Some wellknown results, which are related with the approximation problems of the continuous functions by Hermite-Fejér interpolation which based on the zeros of Chebyshev polynomials of the first or second kind.