Parametrized trigonometric derived Lp degree of approximation by various smooth integral operators

In this work we continue with the study of smooth Gauss-Weierstrass, Poisson-Cauchy, and trigonometric singular integral operators that started in [Anastassiou, G.A., Intelligent Mathematics: Computational Analysis, Springer, Heidelberg, New York, Chapter 12, 2011], see there chapters 10-14. This time the foundation of our research is a trigonometric Taylor’s formula. We prove the parametrized univariate Lp convergence of our operators to the unit operator with rates via Jackson-type parametrized inequalities involving the first Lp modulus of continuity. Of interest here is a residual appearing term. Note that our operators are not in general positive.