A method of approximation in $H^p$, $0<p\leqslant 1$

A summability method for series, which is called in this paper the generalized Abel–Poisson method, is introduced. For functions in $H^p$, $0<p\leqslant 1$, it is shown that the rate of approximation of the boundary function by the generalized Abel–Poisson means is equivalent to the modulus of smoothness of fractional order. All estimates are carried out in the $L_{2\pi}^p$.