Strong Approximation by Riesz Means on Compact Symmetric Spaces

The rates of strong approximation by Riesz means on compact symmetric spaces are established.

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 Cesaro 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 someδ〉0 and a1（＋∞）=···=ad（＋∞）=＋∞.Then,for each integrable function f∈L1（Id）,we have the a.e.relation for the Cesaro means limn→∞σαa（n）f=f and for the Riesz means limn→∞σα,γa（n）f=f for any 0〈αj≤1≤γj（j=1,...,d）.A straightforward consequence of our result is the so-called cone restricted a.e.convergence of the multidimensional Cesaro and Riesz means of integrable functions,which was proved earlier by Weisz.

Almost Everywhere Convergence of Riesz Means on Noncompact Symmetric Space SL(3,H)/SP(3)

Let G/K be the noncompact Riemannian symmetric space SL(3, H)/Sp(3). We shall prove in this paper that for f∈L^P(SL(3, H)/Sp(3)), 1≤p≤2. the Riesz means of order z of f with respect to the eigenfunctions expansion of Laplace operator almost everywhere converge to f for Rez】б(n, p). The critical index δ(n,p) is the same as in the classical Stein’s result for Euclidean space. and as in the noncompact symmetric spaces of rank one and of complex type.

L^2 NORM INEQUALITY WITH POWER WEIGHTS FOR THE MAXIMAL RIESZ SPHERICAL MEANS

We have pointed out in [1] that so far the L^2 norm inequalities with power weights for the Riesz means σ_R~δ(g)(x) of multiple Fourier integrals have been obtained only by Hirschman and J. L. Rubio de Francia respectively:

On the Rankin-Selberg problem:higher power moments of the Riesz mean error term

Let Δ1(x;φ) be the error term of the first Riesz mean of the Rankin-Selberg problem. We study the higher power moments of Δ1(x; φ) and derive an asymptotic formula for the 3-rd, 4-th and 5-th power moments by using Ivi?’s large value arguments and other techniques.

[ℓ_(p)]_(e.r) Euler-Riesz Difference Sequence Spaces

Bas¸ar and Braha[1],introduced the sequence spaces˘ℓ_(∞),c˘and c˘0 of EulerCesaro bounded,convergent and null difference sequences and studied their some´properties.Then,in[2],we introduced the sequence spaces[ℓ_(∞)]_(e.r),[c]_(e.r)and[c_(0)]_(e.r)of Euler-Riesz bounded,convergent and null difference sequences by using the composition of the Euler mean E1 and Riesz mean Rq with backward difference operator∆.The main purpose of this study is to introduce the sequence space[ℓ_(p)]_(e.r)of Euler-Riesz p−absolutely convergent series,where 1≤p<∞,difference sequences by using the composition of the Euler mean E1 and Riesz mean Rq with backward difference operator∆.Furthermore,the inclusionℓ_(p)⊂[ℓ_(p)]_(e.r)hold,the basis of the sequence space[ℓ_(p)]_(e.r)is constucted andα−,β−andγ−duals of the space are determined.Finally,the classes of matrix transformations from the[ℓ_(p)]_(e.r)Euler-Riesz difference sequence space to the spacesℓ_(∞),c and c0 are characterized.We devote the final section of the paper to examine some geometric properties of the space[ℓ_(p)]_(e.r).

A Marcinkiewicz criterion for L^(p)-multipliers related to Schrodinger operators with constant magnetic fields

In this paper,we follow Dappa’s work to establish the Marcinkiewicz criterion for the spectral multipliers related to the Schrdinger operator with a constant magnetic field.We prove that if m and m′are locally absolutely continuous on(0,∞)and ‖m‖∞+sup j∈Z2j 2i+1 r|m′′(r)|dr<∞,then the multiplier defined by m(t)is bounded on Lpfor 2n/(n+3)<p<2n/(n-3)with n 3.Our approach is based on the estimates for the generalized Littlewood-Paley functions of the spectral representation of the Schrdinger operator with a constant magnetic field.