Trigonometric background multivariate smooth trigonometric singular integrals approximations

In this article we apply the uniform and L_p , 1≤ p<∞ approximation properties of general smooth multivariate singular integral operators over ℝ^N , N≥ 1 . It is a trigonometric based approach with detailed applications to the corresponding smooth multivariate trigonometric singular integral operators. The results are quantitative via Jackson type inequalities involving the first uniform and L_p moduli of continuity.