Convergence of generalized singular integrals to the unit, univariate case

Abstract

In a recent paper, the second author (see (2)) studied the degree of uniform approx- imation to the unit in terms of uniform moduli of smoothness, by the Jackson-type general- izations of Picard and of Gauss-Weierstrass singular integrals. In this paper we consider the L p -approximation, (1 p < +∞) by the above singular integrals in terms of the L p -moduli of smoothness, and both uniform and L p -approximation (in terms of the corresponding moduli of smoothness) by Jackson-type generalizations of the Poisson-Cauchy singular integrals.