Approximating the derivatives of \(L_ 2\)-Functionals by singular integrals (Q1917152)

Let \(G\) be an open set in \(\mathbb{R}^k\). Under natural conditions on the kernels \(\chi_n\) the author estimates the order of approximation by the singular integrals \[ f^s_n(x): = \int_G \chi_n(x,u) f(u)du, \quad x\in G,\;n\in \mathbb{N};\;f\in L_2(G). \] The subjects are considered from two points of view: as linear functionals on \(C^\infty_0\) and as distributions in the sense of L. Schwartz.