On approximation of functions in the spaces \(L_p ( \mathbb R^{2})\) and \(L_p (\mathbb R_{+}^{2})\) by generalized singular integrals with positive kernels (Q613007)

The authors discuss the following question: How the behavior of \[ \Biggl\|\int_{F^2}\Delta^r_{u,v}(f;\cdot) \varphi_n(u)\psi_m(v)\,du\,dv\Biggr\|_{p,\mathbb R^2} \] as \(n,m\to\infty\), where \(F\subset\mathbb R\), \[ \Delta^r_{u,v}(f; x,y)= \sum^r_{i,j=1} (-1)^{i+j} C^i_r C^j_r f(x+ iu,y+ iv)- f(x,y), \] \(\varphi_n\) and \(\psi_m\) are positive kernels, is connected with the structural properties of \(f\) that are characterized by the moduli of continuity?