Approximation by Kantorovich-Bernstein polynomials inL p(0<p<1)-metric

The purpose of the present paper is to evaluate the error of the approximation of the function f∈L₁[0,1] by Kantorovich-Bernstein polynomials in L_p-metric (0<p<1).     

Approximation of compressions of periodic functions in the spaceL p,p<1

We investigate the behavior of the best approximation of compressions of functions by trigonometric polynomials in the spaceL _p, p<1.Translated from Ukrainskii Matematicheskii Zhurnal, Vol.47, No. 12, pp. 1708–1711, December, 1995.This work was financially supported by the Foundation for Fundamental Research of the Ukrainian State Committee on Science and Technology.     

Hardy Approximation to L p Functions on Subsets of the Circle with 1< =p< = infinity

We consider approximation of L ^p functions by Hardy functions on subsets of the circle for . After some preliminaries on the possibility of such an approximation which are connected to recovery problems of the Carleman type, we prove existence and uniqueness of the solution to a generalized extremal problem involving norm constraints on the complementary subset. December 6, 1995. Date revised: August 26, 1996.     

A characterization of L p(ℝ) by local trigonometric bases with 1<p<∞

We show that the local trigonometric bases introduced by Malvar, Coifman and Meyer constitute bases, but not unconditional bases, for L^p(ℝ) with 1<p<∞, p≠2. In addition, we characterize the functions in L^p(ℝ) for 1<p<∞ in terms of their local trigonometric basis coefficients. Dedicated to Dr. Charles A. Micchelli for his 60th birthday Mathematics subject classification (2000) 42C15. Supported by Prof. Y. Xu under his grant in program of “One Hundred Distinguished Chinese Scientists” of the Chinese Academy of Sciences, the National Natural Science Foundation of China (No. 10371122), and the second author is supported by Tianyuan Fund for Mathematics (No. A0324648).     

Представление функций классовL p [0, 1], 0<p<1, ортогональными рядами

u . uuu u     

球面Hardy空间上Riesz平均的逼近

引进了球面Hrady空间上Riesz平均算子及Peetre K模。讨论了Riesz平均算子在Hardy空间上的逼近性质。证明了Riesz平均算子与Peetre K模的强渐近等价关系。所得结果表明Peetre K模完全刻划了Riesz平均的逼近。     

Korovkin-Type Approximation Theory in Riesz Spaces

In this paper, we prove the following Riesz spaces’ version of the Korovkin theorem. Let E and F be two Archimedean Riesz spaces with F uniformly complete, let W be a nonempty subset of \(E^{+}\), and let \((T_{n})\) be a given sequence of (r-u)-continuous elements of \(\mathcal {L(}E,F)\), such that \(\left| T_{n}-T_{m}\right| x=\left| (T_{n}-T_{m})x\right| \mathcal {\ }\)for all \(x\in E^{+},\) \(m,n\ge n_{0}\) (for a given \(n_{0}\in \mathbb {N} )\). If the sequence \((T_{n}x)_{n}\) \((r-u)\)-converges for every \(x\in W\), then \((T_{n})\) \((r-u)\)-converges also pointwise on the ideal \(E_{W}\), generated by W, to a linear operator \(S_{0}:E_{W}\rightarrow F\). We also prove a similar Korovkin-type theorem for nets of operators. Some applications for f-algebras and orthomorphisms are presented.     

Gaussian estimates and L p -boundedness of Riesz means

We show that suitable upper estimates of the heat kernel are sufficient to imply the L ^p boundedness of several families of operators associated with the Schr?dinger group in various situations. This generalizes results by Sj?strand and others in the Euclidean case, and by Alexopoulos in the case of Lie groups and Riemannian manifolds. RID="*" ID="*"Research partially supported by the European Commission (European TMR Network "Harmonic Analysis" 1998-2001, Contract ERBFMRX-CT97-0159).     

集值Subpramart的另一类Riesz逼近

假定(X,‖.‖)为可分的Banach空间,X*为其对偶空间,X*可分。设(Ω,Φ,P)为完备的概率空间,{Φn,n≥1}为Φ的上升子σ-域族,且Φ=∨Φn本文首先讨论了集值Subpramart若干性质,在此基础上,给出了集值Subpramart的一类Riesz鞅逼近。     

Approximation by Weakly Compact Operators in L1

For every bounded linear operator T in L₁[0, 1] there is an element of best approximation in the ideal of weakly compact operators. We also give some sufficient conditions for ‖T+S‖ = ‖T‖ + ‖S‖, where S and T are L₁-operators.     

Polynomial approximation inL p (0<p<1)

We prove that forf∈L _p , 0<p<1, andk a positive integer, there exists an algebraic polynomialP _n of degree ≤n such that $$\left\| {f - P_n } \right\|_p \leqslant C\omega _k^\varphi \left( {f,\frac{1}{n}} \right)_p $$ whereω _k ^? (f,t)p is the Ditzian-Totik modulus of smoothness off inL _p , andC is a constant depending only onk andp. Moreover, iff is nondecreasing andk≤2, then the polynomialP _n can also be taken to be nondecreasing.     

紧对称空间上Riesz平均的强逼近

本文在紧对称空间中给出了全测度集上Ｒｉｅｓｚ平均强逼近的阶     

On the Approximation by Modified Interpolation Polynomials in Spaces L p

We consider certain modified interpolation polynomials for functions from the space L _p[0, 2], 1 p . An estimate for the rate of approximation of an original function f by these polynomials in terms of its modulus of continuity is obtained. We establish that these polynomials converge almost everywhere to f.     

On Approximation by Reciprocals of Spherical Harmonics in L p Norm

Let S^1-1,q≥2,be the surface of the unit sphere in the Euclidean space R^1,f（x）∈L^p（S^q-1）,f（x）≥0,f absohutely unegual to 0,1≤p≤＋∞,Then,it is proved in the present paper that there is a spherical harmonics PN（x） of order≤N and a constant C〉0 such that where ω（f,δ）L^p=sup 0〈t≤δ‖St（f）-f‖L^p is a kind of moduli of continuity and ^‖f-1/PN‖L^p≤Cω（f,N^-1）L^p,St（f,μ）=1/｜S^q-2｜Sin^2λt ∫-μμ’=t f（μ＇）dμ＇ is a translation operator.     

Subspaces ofL p (G) spanned by characters: 0<p<1

LetG be an infinite compact abelian group,μ a Borel measure onG with spectrumE, and 0<p<1. We show that ifμ is not absolutely continuous with respect to Haar measure, thenL _E ^P (G), the closure inL ^p (G) of theE-trigonometric polynomials, does not have enough continuous linear functionals to separate points. Ifμ is actually singular, thenL _E ^p (G) does not have any nontrivial continuous linear functionals at all. Our methods recover the classical F. and M. Riesz theorem, and a related several variable result of Bochner; they reveal the existence of small sets of characters that spanL ^P (T), where T is the unit circle; and they show that theH ^p spaces of the “big disc algebra” have one-dimensional dual.