Let f be an integrable function on the unit sphere Σn−1 of Rn(n⩾3) and let σNδ be the Cesaro means of order σ of the Fourier-Laplace series of f. The special value λ:=n−2/2 of σ is known as the critical index. This paper proves that Open image in new window and Open image in new window where ω(f,t)p is the 1st-order modulus of continuity of f in Lp-metric which is defined in a way different than in the classical case of n=2.

The saturation rate and class of (0, m1, m2, ..., mq) trigonometric interpolation operators in\(\tilde C_{2\pi } \) spaces have been determined by Cavaretta and Selvaraj. In this paper, we consider the convergence and saturation problems of these operators in\(\tilde L_{2\pi }^p (1 \leqslant p< \infty )\) and obtain complete results.

Let Bσ,p,1 <-p<-∞, be the set of all functions from L8(R) which can be continued to entire functions of exponential type <-σ. The well known Shannon sampling theorem and its generalization [1] state that every f∈Bσ,p, 1<p<∞, can be represented as $$f(x) = \mathop \Sigma \limits_{j \in z} f(j\pi /\sigma )\tfrac{{sin\sigma (x - j\pi /\sigma )}}{{\sigma (x - j\pi /\sigma )}}, \sigma > 0$$ It is well known that the Shannon sampling theorem plays an important role in many applied fields [1,2,7]. In this paper we give an application of sampling theorem to approximation theory.

This paper deals with Kolmogorov criterion for best uniform coapproximation and strongly unique best uniform coapproximation. Some relations between best uniform approximation and best uniform coapproximation are obtained. Some equalities and best uniform coapproximation are connected.

Given a continuous function f defined on the unit cube ofR n and a convex function φt, φt(0)=0, φt(x)>0, for x>0, we prove that the set of best Lφt-approximations by monotone functions has exactly one element ft, which is also a continuous function. Moreover if the family of convex functions {φt}t>0 converges uniformly on compact sets to a function φ0, then the best approximation ft→f0 uniformly, as t→0, where f0 is the best approximation of f within the Orlicz space L0. The best approximations {ft} are obtained as well as minimizing integrals or the Luxemburg norm.

It is proved that the Chebyshev polynomial\(\overline T _n (x) = T_n (x\cos \tfrac{\pi }{{2n}})\), has the greatest uniform norm on [−1, 1] of its third derivative among the real polynomials of degree at most n, which are bounded by 1 in [−1, 1] and vanish in −1 and 1.

In this paper, the authors give the Marcinkiewicz—Zygmund inequality based on the zeros of the first kind Chebyshev polynomials in Orlicz norm. As application, the degree of approximation by two kinds of modified Lagrange interpolatory polynomials in Orlicz spaces is studied.

Some results from the theory of best (or best simultaneous) approximation in a normed linear space have been extended to a normed almost linear space [strong normed almost linear space].

In this paper the concept of positive definite bilinear matrix moment functional, acting on the space of all the matrix valued continuous functions defined on a bounded interval [a,b] is introduced. The best approximation matrix problem with respect to such a functional is solved in terms of matrix Fourier series. Basic properties of matrix Fourier series such as the Riemann—Lebesgue, matrix property and the bessel—parseval matrix inequality are proved. The concept of total set with respect to a positive definite matrix functional is introduced, and the totallity of an orthonormal sequence of matrix polynomials with respect to the functional is established.

The appriximation properties of generalized conic curves are studied in this paper. A generalized conic curve is defined as one of the following curves or their affine and translation equivalent curves: (i) conic curves, including parabolas, hyperbolas and ellipses; (ii) generalized monomial curves, including curves of the form x=yγ, γ∈R, γ≠0,1, in the x−y Cartesian coordinate system; (iii) exponential spiral curves of the form ρ(ϑ)=Aeγϑ, A>0, γ≠0, in the ρ-ϑ polar coordinate system.