Almost Everywhere Convergence of Orthogonal Expansions of Several Variables

Almost Everywhere Convergence of Orthogonal Expansions of Several Variables

For weighted L space on the unit sphere of R, in which the weight functions are invariant under finite reflection groups, a maximal function is introduced and used to prove the almost everywhere convergence of orthogonal expansions in h-harmonics. The result applies to various methods of summability, including the de la Vallée Poussin means and the Cesàro means. Similar results are also establi...

Summability of orthogonal expansions of several variables

Cesàro Means of Orthogonal Expansions in Several Variables

Cesàro (C, δ) means are studied for orthogonal expansions with respect to the weight function ∏d i=1 |xi |2κi on the unit sphere, and for the corresponding weight functions on the unit ball and the Jacobi weight on the simplex. A sharp pointwise estimate is established for the (C, δ) kernel with δ >−1 and for the kernel of the projection operator, which allows us to derive the exact order for t...

Almost Everywhere Convergence of Series in L

We answer positively a question of J. Rosenblatt (1988), proving the existence of a sequence (ci) with ∑∞ i=1 |ci| = ∞, such that for every dynamical system (X,Σ, m, T ) and f ∈ L1(X), ∑∞i=1 cif(T ix) converges almost everywhere. A similar result is obtained in the real variable context.

About the Almost Everywhere Convergence of the Spectral Expansions of Functions

Abstract. In this paper we study the almost everywhere convergence of the expansions related to the self-adjoint extension of the LaplaceBeltrami operator on the unit sphere. The sufficient conditions for summability is obtained. The more general properties and representation by the eigenfunctions of the Laplace-Beltrami operator of the Liouville space L 1 is used. For the orders of Riesz means...