# Search Results

## Abstract

The best rate of approximation of functions on the sphere by spherical polynomials is majorized by recently introduced moduli of smoothness. The treatment applies to a wide class of Banach spaces of functions.

Summary In his paper [1]P. Turán discovers the interesting behaviour of Hermite-Fejér interpolation (based on the Čebyšev roots) not describing the derivative values at “exceptional” nodes {η_{n}}_{n=1}
^{∞}. Answering to his question we construct such exceptional node-sequence for which the mentioned process is bounded for bounded functions whenever −1<*x*<1 but does not converge for a suitable continuous function at any point of the whole interval [−1, 1].

## Abstract

Let ℂ[−1,1] be the space of continuous functions on [−,1], and denote by Δ^{2} the set of convex functions *f* ∈ ℂ[−,1]. Also, let *E*
_{n}(*f*) and *E*
_{n}
^{(2)} (*f*) denote the degrees of best unconstrained and convex approximation of *f* ∈ Δ^{2} by algebraic polynomials of degree < *n*, respectively. Clearly, *En* (*f*) ≦ *E*
_{n}
^{(2)} (*f*), and Lorentz and Zeller proved that the inverse inequality *E*
_{n}
^{(2)} (*f*) ≦ *cE*
_{n}(*f*) is invalid even with the constant *c* = *c*(*f*) which depends on the function *f* ∈ Δ^{2}.
In this paper we prove, for every *α* > 0 and function *f* ∈ Δ^{2}, that

*c*(

*α*) is a constant depending only on

*α*. Validity of similar results for the class of piecewise convex functions having

*s*convexity changes inside (−1,1) is also investigated. It turns out that there are substantial differences between the cases

*s*≦ 1 and

*s*≧ 2.

## Abstract

We prove that the weighted error of approximation by the Szász-Mirakyan-type operator introduced in [1] is equivalent to the modulus of smoothness of the function. This result is analogous to previous results for Bernstein-type operators obtained by Ditzian-Ivanov and Szabados.

## Abstract

*f*∈

*C*[−1, 1]. Let the approximation rate of Lagrange interpolation polynomial of

*f*based on the nodes

_{n + 2}(

*f, x*). In this paper we study the estimate of Δ

_{n + 2}(

*f,x*), that keeps the interpolation property. As a result we prove that

*T*

_{n}(

*x*) = cos (

*n*arccos

*x*) is the Chebeyshev polynomial of first kind. Also, if

*f*∈

*C*

^{r}[−1, 1] with

*r*≧ 1, then

## Abstract

We construct a new kind of rational operator which can be used to approximate functions with endpoints singularities by algebric weights in [−1,1], and establish new direct and converse results involving higher modulus of smoothness and a very general class of step functions, which cannot be obtained by weighted polynomial approximation. Our results also improve related results of Della Vecchia [5].

## Abstract

We present direct and strong converse theorems for a general sequence of positive linear operators satisfying some functional equations. The results can be applied to some extensions of Baskakov and Szász–Mirakyan operators.

## Abstract

Let *f* be a real continuous 2*π*-periodic function changing its sign in the fixed distinct points *y*
_{i} ∈ *Y*:= {*y*
_{i}}_{i∈ℤ} such that for *x* ∈ [*y*
_{i}, *y*
_{i−1}], *f*(*x*) ≧ 0 if *i* is odd and *f*(*x*) ≦ 0 if *i* is even. Then for each *n* ≧ *N*(*Y*) we construct a trigonometric polynomial *P*
_{n} of order ≦ *n*, changing its sign at the same points *y*
_{i} ∈ *Y* as *f*, and

*N*(

*Y*) is a constant depending only on

*Y*,

*c*(

*s*) is a constant depending only on

*s*,

*ω*

_{3}(

*f, t*) is the third modulus of smoothness of

*f*and ∥ · ∥ is the max-norm.

## Abstract

Recently P. Mache and M. W. Müller introduced the Baskakov quasi-interpolants and obtained an approximation equivalence theorem. In this paper we consider simultaneous approximation equivalence theorem for Baskakov quasi-interpolants.

## Abstract

The uniform weighted approximation errors of Baskakov-type operators are characterized for weights of the form for *γ*
_{0},*γ*
_{∞}∊[−1,0]. Direct and strong converse theorems are proved in terms of the weighted *K*-functional.