ON THE CESARO SUMMABILITY OF ULTRASPHERICAL SERIES

In this paper the Cesaro summability of the ultraspherical series has been investigated which extend and generalize the results of Wang [5, 6] of Fourier sen es.

Singular Integrals on Ultraspherical Series

One of the main uses of harmonic analysis on the sphere is to discover new theorems about series of ultraspherical (Gegenbauer) polynomials. In this paper, we will construct singular integral operators from scalar functions on the sphere to vector functions. These operators when restricted to zonal functions give Lp-bounded (1 < p < ∞ ) operators on ultraspherical series.We will use [7, Chapter 9] as our main reference. Let G denote a compact group, with identity e, and Ĝ its dual, the set of equivalence classes of continuous irreducible unitary representations of G.

On Riesz logarithmic summability of the ultraspherical series

Let f{θ, φ) be a function defined for the range 0 ≤ φ π, 0 ≤ φ ≤ 2π on a sphere S. The ultraspherical series associated with this function iswhereand the ultraspherical polynomials are defined by the following expansion.