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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

In order to approximate functions defined on (0, +∞), the authors consider suitable Lagrange polynomials and show their convergence
in weighted *L*
^{p}-spaces.

## 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

The paper is related to the lower and upper estimates of the norm for Mercer kernel matrices. We first give a presentation
of the Lagrange interpolating operators from the view of reproducing kernel space. Then, we modify the Lagrange interpolating
operators to make them bounded in the space of continuous function and be of the de la Vallée Poussin type. The order of approximation
by the reproducing kernel spaces for the continuous functions is thus obtained, from which the lower and upper bounds of the
Rayleigh entropy and the *l*
^{2}-norm for some general Mercer kernel matrices are provided. As an example, we give the *l*
^{2}-norm estimate for the Mercer kernel matrix presented by the Jacobi algebraic polynomials. The discussions indicate that the
*l*
^{2}-norm of the Mercer kernel matrices may be estimated with discrete orthogonal transforms.

## Abstract

We consider biorthogonal systems of functions associated to derivatives of orthogonal polynomials in the case of general weights. For Freud polynomials, it is proved that the derivatives of any orders of them constitute Hilbertian bases in the space of weighted square integrable functions.

## Abstract

The aim of this paper is to give weighted function spaces in which the sequence of Cesàro means of the Jacobi-Fourier series are uniformly convergent. Error estimate for the approximation will also be considered.

## Abstract

*p*

_{n}

^{(α,β)}and Jacobi weights

*w*

^{(a,b)}depending on

*α,β, a, b*> −1, where the subsets

*U*

_{k}(

*x*) ⊂ [−1, 1] located around

*x*and are given by

*U*

_{k}(

*x*). This approach uses estimates of Jacobi polynomials modified Jacobi weights initiated by Totik and Lubinsky in [1]. Various bounds for integrals involving Jacobi weights will be derived. The results of the present paper form the basis of the proof of the uniform boundedness of (

*C*, 1) means of Jacobi expansions in weighted sup norms in [3].

## Abstract

We give a weighted Hermite-Fejr-type interpolatory method on the real line, which is a positive operator on “good” matrices. We give an example on “good” interpolatory matrix by weighted Fekete points. To prove the convergence theorem we need the generalization of “Rodrigues’ property”.

## Abstract

Lubinsky and Totik’s decomposition [11] of the Cesàro operators *σ*
_{n}
^{(α,β)} of Jacobi expansions is modified to prove uniform boundedness in weighted sup norms, i.e., ‖*w*
^{(a,b)}
*σ*
_{n}
^{(α,β)}‖_{∞} ≦ *C*‖*w*
^{(a,b)}
*f*‖_{∞}, whenever *α,β* ≧ −1/2 and *a, b* are within the square around (*α*/2 + 1/4, *α*/2 + 1/4) having a side length of 1. This approach uses only classical results from the theory of orthogonal polynomials and
various estimates for the Jacobi weights. The present paper is concerned with the main theorems and ideas, while a second
paper [7] provides some necessary estimations.

## 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.