We apply a construction of generalized twisted convolution to investigate almost everywhere summability of expansions with respect to the orthonormal system of functions ${\ell }_n^a\left(x\right)={(n!/\Gamma (n+a+1))}^{1/2}{e}^{-x/2}{L}_n^a\left(x\right)$, n = 0,1,2,..., in $L^2({ℝ}_{+},x^{a}dx)$, a ≥ 0. We prove that the Cesàro means of order δ > a + 2/3 of any function $f\in L^p\left(x^{a}dx\right)$, 1 ≤ p ≤ ∞, converge to f a.e. The main tool we use is a Hardy-Littlewood type maximal operator associated with a generalized Euclidean convolution.

Using methods from [9] we prove the almost everywhere convergence of the Cesàro means of Laguerre series associated with the system of Laguerre functions $L_n^a\left(x\right)={(n!/\Gamma (n+a+1))}^{1/2}{e}^{-x/2}{x}^{a/2}{L}_n^a\left(x\right)$, n = 0,1,2,..., a ≥ 0. The novel ingredient we add to our previous technique is the $A_p$ weights theory. We also take the opportunity to comment and slightly improve on our results from [9].

We extend to locally compact abelian groups, Fejer's theorem on pointwise convergence of the Fourier transform. We prove that lim φU * f(y) = f (y) almost everywhere for any function f in the space (LP, l∞)(G) (hence in LP(G)), 2 ≤ p ≤ ∞, where {φU} is Simon's generalization to locally compact abelian groups of the summability Fejer Kernel. Using this result, we extend to locally compact abelian groups a theorem of F. Holland on the Fourier transform of unbounded measures of type q.

Let K be a closed Lie subgroup of the unitary group U(n) acting by automorphisms on the (2n+1)-dimensional Heisenberg group $H_n$. We say that $(K,H_n)$ is a Gelfand pair when the set $L_K^1\left(H_n\right)$ of integrable K-invariant functions on $H_n$ is an abelian convolution algebra. In this case, the Gelfand space (or spectrum) for $L_K^1\left(H_n\right)$ can be identified with the set $\Delta (K,H_n)$ of bounded K-spherical functions on $H_n$. In this paper, we study the natural topology on $\Delta (K,H_n)$ given by uniform convergence on compact subsets in $H_n$. We show that $\Delta (K,H_n)$ is a complete...

Nuclear groups form a class of abelian topological groups which contains LCA groups and nuclear locally convex spaces, and is closed with respect to certain natural operations. In nuclear locally convex spaces, weakly summable families are strongly summable, and strongly summable are absolutely summable. It is shown that these theorems can be generalized in a natural way to nuclear groups.