An equivalent definition of $H^p$ spaces in the half-plane and some applications

Classes of functions that are holomorphic for $\operatorname{Im}z>0$ and satisfy
$$ \sup_{0<t<\pi}\int_0^\infty|f(re^{it})|^p\,dr<\infty,\qquad p\in(0,\infty), $$
are considered. It is proved that they coincide with the usual classes $H^p$ in the half-plane. This result is applied to an interpolation problem in $H^p$ in a strip and to the problem of basicity of exponential functions in the space $L^2$ on the line, with exponentially decreasing weight.
Bibliography: 8 titles.