Approximation on set of total measure of Bochner-Riesz means below the critical index

In this paper it has been considered convergence and approximation for functions inSobolev Spaces L_m~1(R~n)by Bochner-Riesz means below the critical index.A theorem that ofprecise approximation orders on set of total measure has been proved.     

The convergence of the Bochner-Riesz means at the critical index

In this paper, we study the pointwise convergence of the Bochner-Riesz means at the critical index on the space We weaken the hypothesis, is a Lebesgue point", which is required on some research results by instead considering the convergence of averages of the function over balls when the radials of the balls approach to 0.

     

Boundedness for commutators of Bochner-Riesz operators below a critical index

In this paper, the authors obtain a necessary and sufficient condition for the LP bound-edness of commutators generated by Bochner-Riesz operators below a critical index and Lipschitz functions in two dimensional case. The authors also discuss a similar problem for higher dimensions.     

Approximation for spherical functions by Bochner-Riesz means

This paper deals with approximation of functions on the unit sphere. The uniform and almost everywhere approximation properties of Bochner-Riesz means at and below the critical index for spherical harmonic expansions are studied. The results obtained here are analogous to those of multiple Fourier series. Supported by NSFC     

On the Bochner-Riesz means of critical order

Stein's well-known logarithmic asymptotics of the Lebesgue constants of the Bochner-Riesz means of critical order is extended to Lebesgue constants of more general linear means of multiple Fourier series. These means are generated by certain class of functions supported in convex domains with boundaries of non-vanishing Gaussian curvature.

     

Spherical jump of a function and the Bochner-Riesz means of conjugate multiple fourier series and fourier integrals

Mathematical Notes - We introduce the notion of spherical jump of a function of several variables at a given point with respect to a homogeneous harmonic polynomial. Here, if the function is...     

On almost everywhere summability of conjugate Bochner-Riesz means

In this paper, we discuss the conjugate Bochner-Riesz means and get 4 theorems.     

Approximation of spherical means of multiple Fourier integrals and Fourier series on set of full measure

In this paper we consider the approximation for functions in some subspaces of L² by spherical means of their Fourier integrals and Fourier series on set of full measure. Two main theorems are obtained. Supported by NNSFC.     

Convergence of generalized conjugate partial Fourier integrals

Generalized conjugate partial Fourier integrals are used to find jumps of functions. The rate of convergence is studied and sharp results are obtained.     

Approximation of fourier integral by Bochner-Riesz means under one-side condition

In this paper we study the approximation of Fourier integral by Bochner-Riesz means under one-side condition and improve the results in[1].     

On the localization and convergence of multiple Fourier integral by Bochner-Riesz means

In this paper we consider , in one case that f_{x 0} (t) is a ΛBMV function on [0, ∞], and in another case thatfεL ¹ _m-1(Rⁿ) and when |k|=m−1 and f(x)=0 when |x−x₀|<δ for some δ>0. Our theormes improve the results of Pan Wenjie ([1]).     

Approximation of conjugate differentiable functions by biharmonic Poisson integrals

We determine the exact values of upper bounds of approximations by biharmonic Poisson integrals on classes of conjugate differentiable functions in uniform and integral metrics.     

Approximation to continuous functions by Cesàro means of double Fourier series and conjugate series

We study the rate of uniform approximation to continuous functions ƒ(x, y), 2π-periodic in each variable, in Lipschitz classes Lip(α, β) and in Zygmund classes Z(α, β), 0 < α, β 1, by Cesàro means σ_mn^γδ(ƒ) of positive orders of the rectangular partial sums of double Fourier series. The rate of uniform approximation to the conjugate functions ^1,0, ^0,1 and ^1,1 by the corresponding Cesàro means is also discussed in detail. The difference between the classes Lip(α, β) and Z(α, β), similar to the one-dimensional case, appears again when max(α, β) = 1. (Compare Theorems 2 and 3 with Theorems 4 and 5.) One surprising result is the following: The uniform approximation rate by σ_mn^{γδ 1,0} to ^1,0 is worse in general than that by σ_mn^{γδ 1,1} to ^1,1 for ƒ ε Lip(1, 1). In fact, the appearance of an extra factor [log(n + 2)]² in the former case is unavoidable (see Theorem 6). All approximation rates we obtain, with one exception, are shown to be exact. Two conjectures are also included.     

Approximation by Fourier means and generalized moduli of smoothness

The quality of approximation by Fourier means generated by an arbitrary generator with compact support in the spaces L_p, 1 ≤ p ≤ +∞, of 2π-periodic pth integrable functions and in the space C of continuous 2π-periodic functions in terms of the generalized modulus of smoothness constructed froma 2π-periodic generator is studied. Natural sufficient conditions on the generator of the approximation method and values of smoothness ensuring the equivalence of the corresponding approximation error and modulus are obtained. As applications, Fourier means generated by classical kernels as well as the classical moduli of smoothness are considered.     

Weighted norm inequalities of Bochner-Riesz means

Let w be a Muckenhoupt weight and be the weighted Hardy spaces. We use the atomic decomposition of and their molecular characters to show that the Bochner-Riesz means are bounded on for 0<p?1 and δ>max{n/p−(n+1)/2,[n/p]rw⁻¹(rw−1)−(n+1)/2}, where rw is the critical index of w for the reverse Hölder condition. We also prove the boundedness of the maximal Bochner-Riesz means for 0<p?1 and δ>n/p−(n+1)/2.