Convergence of generalized singular integrals to the unit, univariate case

In a recent paper, the second author (see 2) studied the degree of uniform approximation to the unit in terms of uniform moduli of smoothness, by the Jackson-type generalizations 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.