Trigonometric background multivariate smooth trigonometric singular integrals approximations

In this article we apply the uniform and \(L_{p}\) $L_p$ , \(1\le p<\infty \) $1\le p<\infty$ approximation properties of general smooth multivariate singular integral operators over \({\mathbb {R}}^{N}\) ${\mathbb{R}}^N$ , \(N\ge 1\) $N\ge 1$ . 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 \(L_{p}\) $L_p$ moduli of continuity.