Best approximation by an angle and approximation by an angle of singular integrals of functions \(f\in L_ p(R_ n)\), \(2<p<\infty\) (Q1116023)

Let (\({\mathcal H}_ j)_{j=1}\) be a sequence of kernels such that \(\int^{\infty}_{-\infty}{\mathcal H}_ j(t)dt=\sqrt{2\pi},\int^{\infty}_{-\infty}| {\mathcal H}_ j(t)| dt\leq M,\) (M independent of j) and \(\lim_{j\to \infty}\int_{| t| \geq \delta >0}| {\mathcal H}_ j(t)| dt=0\). Denote by \[ I_{\ell_ j}f=\frac{1}{2\pi}\int^{\infty}_{-\infty}f(x_ 1,...,x_{j-1},x_ j-t_ j,x_{j+1},...,x_ n){\mathcal H}_{\ell_ j}(t_ j)dt_ j, \] \(\ell_ j\in {\mathbb{N}}\), and \(f\in L_ p({\mathbb{R}}^ m)\), \(2<p<\infty\), and by \(I_{\ell_ 1\ell_ 2...\ell_ m}f=I_{\ell_ 1}I_{\ell_ 2}...I_{\ell_ m}f,\) \(m\leq n\). An m- dimensional angle for f is defined by: \[ y_{\ell_ 1\ell_ 2...\ell_ m}f=I_{\ell_ 1}f+...+I_{\ell_ m}f-I_{\ell_ 1\ell_ 2}f-I_{\ell_ 1\ell_ 3}f-...-I_{\ell_{m-1}\ell_ m}f+...+(-1)^{m-1}I_{\ell_ 1\ell_ 2...\ell_ m}f. \] The author gives estimates for the norm \(\| y_{\ell_ 1...\ell_ m}-f\|_ p\) in the terms of the best approximation of f by elements in the space of entire functions of exponential type. The corresponding results for the case \(1<p\leq 2\) was established by the same author in: [Mat. Vesn. 35, 289-304 (1983; Zbl 0537.41012)].