APPROXIMATION BY WALSH-KACZMARZ-FEJR MEANS ON THE HARDY SPACE

The main aim of this paper is to find necessary and sufficient conditions for the convergence of Walsh-Kaczmarz-Fejr means in the terms of the modulus of continuity on the Hardy spaces Hp, when 0 p ≤ 1/2.     

COMPOSITION TYPE OPERATORS FROM HARDY SPACES TO μ-BLOCH SPACES ON THE UNIT BALL

Let φ be a holomorphic self-map of Bn and ψ ∈ H（Hn）. A composition type operator is defined by Tψ,φ（f） = ψf o φ for f ∈ H（Bn）, which is a generalization of the multiplication operator and the composition operator. In this article, the necessary and sufficient conditions are given for the composition type operator Tψ,φ to be bounded or compact from Hardy space HP（Bn） to μ-Bloch space Bμ（Bn）. The conditions are some supremums concerned with ψ,φ, their derivatives and Bergman metric of Bn. At the same time, two corollaries are obtained.     

COMMUTATORS OF MULTIPLIERS ON HARDY SPACES

Let T be the multiplier operator associated to a multiplier m, and [b, T] be the commutator generated by T and a BMO function b. In this paper, the authors have proved that [b,T] is bounded from the Hardy space H^1（R^n） into the weak L^1 （R^n） space and from certain atomic Hardy space Hb^1 （R^n） into the Lebesgue space L^1 （R^n）, when the multiplier m satisfies the conditions of Hoermander type.     

A NOTE ON THE WALSH-FEJR MEANS

The main aim of this paper is to prove that the maximal operator σ # is not bounded from the martingale Hardy space Hp (G) to the martingale Hardy space Hp (G) for 0 < p ≤ 1.     

TWO PROBLEMS ABOUT COMPOSITION OPERATORS ON HARDY SPACE

In this paper, two problems about composition operator on Hardy space are considered. Firstly, a new estimation of the norm of a class of composition operators is given. Secondly, the cyclic behavior of the adjoint operator of a composition operator is discussed.     

ON THE APPROXIMATION OF TCHEBYCHEFF-FOURIER SERIES IN THE C SPACE AND L_p~ω SPACE

The degree of approximation to a function f(x)∈C[-1,1] by (U, λ) means and f(x) ∈ Lpw by (Jr) means are discussed, some results in the literatures [1],[2],[3] have been improved.     

SOME CONVERSE RESULTS ON ONESIDED APPROXIMATION ： JUSTIFICATIONS

The present paper deals with best onesided approximation rate in Lp spaces ^~En（f）Lp of f∈C2π.Although it is clear that the estimate ^~E(f)Lp≤C‖f‖Lp cannot be correct for all f∈L^p2π in case p＜∞, the question whether ~En(f)Lp≤Cω(f,n^-l)Lp or ^~En(f)Lp≤CEn(f)Lp holds for f∈ C2π remains totally untouched.Therefore it forms a basic problem to justify onesided approximation. The present paper will provide an answer to settle down the basis.     

ON THE DEGREE OF APPROXIMATION BY WAVELET EXPANSIONS

In this paper we estimate the degree of approximation of wavelet expansions. Our result shows that the degree has the exponential decay for function f(x) ∈ L2 continuous in a finite interval (a,b) which is much superior to those of approximation by polynomial operators and by expansions of classical orthogonal series.     

ON THE APPROXIMAMATION OF DIFFERENTIABLE FUNCTIONS BY EULER MEANS

Let f∈C_(2π)~r.Denote by _n(f,x)the n-th Euler mean of f(x).This paper gives theasympto ic representations of the deviation _n(f,x)-f(x)and the quantity | _a(f,x)-f(x)|.Additionally,some applications of these asymptotic representations are obtained.     

ON THE EXACT DEGREE OF APPROXIMATION BY HERMITE-FEJER INTERPOLATION BASED ON THE LAGUERRE ABSCISSAS

For the nonpositive Hermite-Fejér interpolation based on the Laguerre abscissas, a pointwise two-sided estimate of the degree of approximation in the aleatoric interval [0, A] is first established.     

ON THE EXACT DEGREE OF APPROXIMATION BY HERMITE-FEJIER INTERPOLATION BASED ON THE LAGUERRE ABSCISSAS

For the nonpositive Hermite-Fejér interpolation based on the Laguerre abscissas, a pointwise two-sided estimate of the degree of approximation in the aleatoric interval [0, A] is first established.     

THE CODIMENSION FORMULA ON QUASI-INVARIANT SUBSPACES OF THE FOCK SPACE

Let M be an approximately finite codimensional quasi-invariant subspace of the Fock space. This paper gives a formula to calculate the codimension of such spaces and uses this formula to study the structure of quasi-invariant subspaces of the Fock space. Especially, as one of applications, it is showed that the analogue of Beurling's theorem is not true for the Fock space L_a~2 in the case of n > 2.     

ON THE ORDER OF APPROXIMATION FOR THE RATIONAL INTERPOLATION TO ｜x｜

The order of approximation for Newman-type rational interpolation to |x| is studied in this paper. For general set of nodes, the extremum of approximation error and the order of the best uniform approximation are estimated. The result illustrates the general quality of approximation in a different way. For the special case where the interpolation nodes are $x_i = \left( {\frac{i}{n}} \right)^r (i = 1,2, \cdots ,n;r > 0)$x_i = \left( {\frac{i}{n}} \right)^r (i = 1,2, \cdots ,n;r > 0) , it is proved that the exact order of approximation is O( \frac1n ),O( \frac1nlogn ) and O( \frac1n^r )O\left( {\frac{1}{n}} \right),O\left( {\frac{1}{{n\log n}}} \right) and O\left( {\frac{1}{{n^r }}} \right) , respectively, corresponding to 01.     

ON THE CONVERGENCE OF APPROXIMATION TO CAUCHY SINGULAR INTEGRALS BY WAVELETS

本文讨论了用周期小波对Ｈｏ．．ｌｄｅｒ空间进行的特征刻划，给出了Ｈｏ．．ｌｄｅｒ空间中的若干小波逼近性质，证明了用周期小波逼近Ｃａｕｃｈｙ型奇异积分时的一些收敛定理．     

THE GLOBAL APPROXIMATION BY LEFT-BERNSTEIN-DURRMEYER QUASI-INTERPOLANTS IN Lp[0, 1]

In this paper,we will use the 2r-th Ditzian-Totik modulus of smoothness wp^2r(f,t)p to discuss the direct and inverse theorem of approximation by Left-Bernstein-Durrmeyer quasi-interpolants Mn^[2r-1]f for functions of the space Lp[0,1](1≤p≤ ∞)。     

ON THE BEST APPROXIMATION OF FUNCTIONS IN THE Ep(1

SHBN XIECHANG 《数学年刊B辑(英文版)》1980,1(1):51-62

ON THE SHEAF OF GENERALIZED FUNDAMENTAL GROUP AND COVERING SPACE

ＯＮ ＴＨＥ ＳＨＥＡＦ ＯＦ ＧＥＮＥＲＡＬＩＺＥＤ ＦＵＮＤＡＭＥＮＴＡＬ ＧＲＯＵＰ ＡＮＤ ＣＯＶＥＲＩＮＧ ＳＰＡＣＥ￥ＧＡＮＤＡＮＹＡＮＡｂｓｔｒａｃｔ：Ｔｈｅａｕｔｈｏｒｃｏｎｓｔｒｕｃｔｓｔｈｅｓｈｅａｆｏｆｇｅｎｅｒａｌｉｚｅｄｆｕ...     

ADJACENCY PRESERVING MAPS ON THE SPACE OF SELF-ADJOINT OPERATORS

§1. Introduction and Statement of ResultsThe study of the geometry of matrices was initiated by Hua [11–18] in the middle oftwenty century. In this geometry, the points of the space are a certain kind of matricesof a given size (rectangular matrices, symmetric matrices, Hermitian matrices, etc.). Witheach such space of matrices, we associate a group of motions. The main problem is tocharacterize this group by a geometric invariant called adjacency (see [23]). Recently, someauthors have gen…     

BOUNDEDNESS OF STEIN＇S SQUARE FUNCTIONS ASSOCIATED TO OPERATORS ON HARDY SPACES

Let(X, d, μ) be a metric measure space endowed with a metric d and a nonnegative Borel doubling measure μ. Let L be a second order non-negative self-adjoint operator on L2(X). Assume that the semigroup e-tLgenerated by L satisfies the Davies-Gaffney estimates. Also, assume that L satisfies Plancherel type estimate. Under these conditions,we show that Stein's square function Gδ(L) arising from Bochner-Riesz means associated to L is bounded from the Hardy spaces Hp L(X) to Lp(X) for all 0 p ≤ 1.     

ON CONSERVATIVE APPROXIMATION BY LINEAR POLYNOMIAL OPERATORS AN EXTENSION OF THE BERNSTEIN''''S OPERATOR

In this work we slwly linear polynomial operators preserving some consecutive i-convexities and leaving in-verant the polynomtals up to a certain degree. First we study the existence of an incompatibility between the conservation of cenain i-cotivexities and the invariance of a space of polynomials. Interpolation properties are obtained and a theorem by Berens and DcVore about the Bernstein's operator ts extended. Finally, from these results a genera'ized Bernstein's operator is obtained.