On estimating the approximation of locally summable functions by Gegenbauer singular integrals

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, infinity) 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).