Parametrized and trigonometric derived uniform approximation by various smooth singular integral operators

Abstract. In this work we continue with the study of smooth Gauss–Weierstrass,
Poisson–Cauchy and Trigonometric singular integral operators that started in [3], see there chapters 10–14. This time the foundation of our research is a trigonometric Taylor's formula. We prove the parametrized univariate uniform convergence of our operators to the unit operator with rates via Jackson type parametrized inequalities involving the first modulus of continuity. Of interest here is a residual appearing term. Note that our operators are not in general positive.