On Estimating the Approximation of Locally Summable Functions by Gegenbauer Singular Integrals

Abstract

Using the generalized shift operator (GSO) generated by the Gegenbauer differential operator we introduce the notion of a Lebesgue–Gegenbauer (L-G)-point of a summable function 𝑓 on the interval [1,∞) and prove that almost all points of this interval are (L-G)-points of 𝑓. Furthermore, we give an exact (by order) estimation of the approximation of locally summable functions by singular integrals generated by GSO (Gegenbauer singular integrals).

Some aspects of harmonic analysis related to Gegenbauer expansions on the half-line

“…It is well known that the maximal function M µ is weak (1, 1) and is bounded on L p (X, dµ) for 1 < p < ∞ (see [19]). The measure of maximal function M µ f (ch x) introduced at the beginning of Section 1…”