On the order of approximation of functions of \(H^ p({\mathbb{R}})\) \((0<p\leq 1)\) by certain means of Fourier integrals (Q1100386)

The author considers the order of approximation in \(L^ p({\mathbb{R}})\) to functions in \(H^ p({\mathbb{R}})\), \(0<p\leq 1\), by means of the form \(M_{\epsilon,\phi}(f,x)=\int^{\infty}_{0}\phi (\epsilon t)\hat f(t)e^{2\pi ixt}dt,\) where \(\epsilon <0\), \(x\in {\mathbb{R}}\), and \(\phi\) is a continuous function decreasing exponentially. Examples include the Riesz means, Gauss-Weierstrass means and the Abel-Poisson means. The results obtained are analogous to the ones obtained earlier by Storozhenko for the spaces \(H^ p(D)\).