On Riesz means in the multi-dimensional divisor problem

An estimate of the remainder term in an asymptotic formula for Riesz means of the multidimensional divisor function is obtained with an arbitrary value of the parameter α > 0.     

Asymptotics of the spectral function of an elliptic differential operator of second order

In the work the Riesz means of the spectral function of a boundary-value problem for an elliptic differential operator of second order are studied for the case of a geodesically concave boundary. Asymptotic formulas for the Riesz means with an estimate of the remainder term that is uniform over the entire domain are obtained in terms of the formal asymptotics of Green's function of the problem considered. An explicit expression is obtained for the spectral function in a neighborhood of the diagonal from which there follows, in particular, a precise estimate of the remainder term in the Weyl formula.The present work contains a detailed treatment of the results published in [1],Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 89, pp. 152–203, 1979.The author thanks V. M. Babich for posing the problem and for major resistance in the work and V. B. Filippov for useful discussions.     

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。     

Riesz Means on Compact Riemannian Symmetric Spaces

We study approximation properties of the Riesz means on compact symmetric spaces of rank one. To do so we establish equivalences between the Riesz means and Peetre K-moduli and estimate the weak type and the uniform approximation of the Riesz means at the critical index. The relations between the Riesz means and the best approximation as well as the Cesàro means are also considered.     

On the order of approximation by Riesz means in multiplicative systems in the classes E X [ɛ]

We establish the order of approximation by Riesz means of the Fourier series in a multiplicative system of a class of functions with given majorant of the sequence of best approximations. In some cases, approximations by Riesz means and best approximations are considered in a specific space, but, in other cases, approximations by Riesz means are considered in spaces with a stronger norm.     

Seminorms on ordered vector spaces that extend to Riesz seminorms on larger Riesz spaces

As a generalization of the notion of Riesz seminorm, a class of seminorms on directed partially ordered vector spaces is introduced, such that (1) every seminorm in the class can be extended to a Riesz seminorm on every larger Riesz space that is majorized and (2) a seminorm on an order dense linear subspace of a Riesz space is in the class if and only if it can be extended to a Riesz seminorm on the Riesz space. The latter property yields that if a directed partially ordered vector space has an appropriate Riesz completion, then a seminorm on the space is in the class if and only if it is the restriction of a Riesz seminorm on the Riesz completion. An explicit formula for the extension is given. The class of seminorms is described by means of a notion of solid unit ball in partially ordered vector spaces. Some more properties concerning restriction and extension are studied, including extension to L- and M-seminorms.     

The divergence of spectral decompositions connected with homogeneous elliptic operators

In this work an asymptotic formula is obtained for the means of the Riesz spectral function of a homogeneous elliptic operator of arbitrary order m≥2 with constant coefficients. Conditions are obtained under which localization (and convergence) of the means of the Riesz spectral decompositions of functions from the Hölder class is absent.     

AN f-ALGEBRA APPROACH TO THE RIESZ TENSOR PRODUCT OF ARCHIMEDEAN RIESZ SPACES

Abstract

We construct the Riesz tensor product of Archimedean Riesz spaces and derive its properties using functional calculus and f-algebras. We improve results on the approximation of elements in the Riesz tensor product by means of elements in the vector space tensor product in such a way that the order density property is a consequence of the improved approximation result.     

带非线性源项的Riesz回火分数阶扩散方程的预估校正方法

本文针对带非线性源项的Riesz回火分数阶扩散方程,利用预估校正方法离散时间偏导数,并用修正的二阶Lubich回火差分算子逼近Riesz空间回火的分数阶偏导数,构造出一类新的数值格式.给出了数值格式在一定条件下的稳定性与收敛性分析,且该格式的时间与空间收敛阶均为二阶.数值试验表明数值方法是有效的.     

Equiconvergence Theorems for Sturm Liouville Expansions and Sets of Divergence for Bochner Riesz Means in Sobolev Spaces

We state some equiconvergence results between Bochner Riesz means of expansions in eigenfunctions of suitable Sturm Liouville operators. Then we determine the Hausdorff dimension of the divergence set of Bochner Riesz means of radial functions in Sobolev classes on Euclidean and non Euclidean spaces.     

相伴于非散度型椭圆算子的Riesz变换的L~p有界性

本文利用小波方法在一般阶的非散度椭圆算子的系数BMO模非常小的情形下,证明了广义 Riesz变换的 L~p(2≤p<+∞)有界性。     

Operators in pre-Riesz spaces: moduli and homomorphisms

We focus on two topics that are related to moduli of elements in partially ordered vector spaces. First, we relate operators that preserve moduli to generalized notions of lattice homomorphisms, such as Riesz homomorphisms, Riesz* homomorphisms, and positive disjointness preserving operators. We also consider complete Riesz homomorphisms, which generalize order continuous lattice homomorphisms. Second, we characterize elements with a modulus by means of disjoint elements and apply this result to obtain moduli of functionals and operators in various settings. On spaces of continuous functions, we identify those differences of Riesz* homomorphisms that have a modulus. Many of our results for pre-Riesz spaces of continuous functions lead to results on order unit spaces, where the functional representation is used.

     

On the boundedness of the maximal operators of double Walsh-logarithmic means of Marcinkiewicz type

The main aim of this paper is to investigate the (H _p , L _p )-type inequality for the maximal operators of Riesz and Nörlund logarithmic means of the quadratical partial sums of Walsh-Fourier series. Moreover, we show that the behavior of Nörlund logarithmic means is worse than the behavior of Riesz logarithmic means in our special sense.     

On a multidimensional form of F. Riesz's ``rising sun" lemma

A multidimensional version of the Riesz rising sun lemma is proved by means of a generalized dyadic process.

     

Spectral properties of the operator of Riesz potential type

For the operator of the Riesz potential type we estimate the second term in the asymptotic behavior of the spectrum and find its ``regularized' trace.

     

Nörlund and Riesz mean of sequences of fuzzy real numbers

In this article we study some properties of the Nörlund and Riesz mean of sequences of fuzzy real numbers. We establish necessary and sufficient conditions for the Nörlund and Riesz means to transform convergent sequences of fuzzy numbers into convergent sequences of fuzzy numbers with limit preserving.     

On multiple Fourier integrals of piecewise smooth functions with discontinuities of the second kind

It is well known that the Riesz means of eigenfunction expansions of piecewise smooth functions of order s>(n−3)/2 converge uniformly on compacts where these functions are smooth. In 2000 L. Brandolini and L. Colzani considered eigenfunction expansions of piecewise smooth functions with discontinuities of the second kind across smooth surfaces. They showed that the Riesz means of these functions of order s>(n−3)/2 may diverge even at certain points where these functions are smooth. Here it is argued that this effect depends on the measure of the singularity area, i.e. we consider functions with singularities across more limited areas and prove that the Riesz means of their eigenfunction expansions of order s>(n−3)/2 converge uniformly on compacts where these functions are continuous.     

Growth properties for generalized Riesz potentials of functions satisfying Orlicz conditions

Riesz decomposition theorem says that superharmonic functions on the punctured unit ball are represented as the sum of generalized (Newtonian) potentials and harmonic functions. In this paper we study growth properties near the origin of spherical means for generalized Riesz potentials of functions satisfying Orlicz conditions in the punctured unit ball.     

Almost Everywhere Convergence of Sequences of Cesàro and Riesz Means of Integrable Functions with Respect to the Multidimensional Walsh System

The aim of this paper is to prove the a.e.convergence of sequences of the Cesa`ro and Riesz means of the Walsh–Fourier series of d variable integrable functions.That is,let a=(a1,...,ad):N→Nd(d∈P)such that aj(n+1)≥δsupk≤n aj(k)(j=1,...,d,n∈N)for someδ0 and a1(+∞)=···=ad(+∞)=+∞.Then,for each integrable function f∈L1(Id),we have the a.e.relation for the Ces`aro means limn→∞σαa(n)f=f and for the Riesz means limn→∞σα,γa(n)f=f for any 0αj≤1≤γj(j=1,...,d).A straightforward consequence of our result is the so-called cone restricted a.e.convergence of the multidimensional Ces`aro and Riesz means of integrable functions,which was proved earlier by Weisz.     

Tauberian conditions connecting Cesaro and Riesz discrete means methods

Tauberian conditions for Cesaro methods and discrete Riesz means are discussed.