On an extremal problem in Hp and prediction of p-stable processes 0

This paper provides new and interesting features on an extremal problem in the space of Hardy functions, $H^p, \; 0<p<1$, and then proceeds to give exact recipe formulas for extrapolating one or two steps ahead of current observations from a discrete-parameter stable harmonizable stochastic process of index $p, \; 0<p<1$. The existence of a best approximation $\tilde\phi _N$ of $z^{-N}\phi $ in $H^p$ for $\phi \in H^p$ together with an expression for $\phi (z)-z^N{\tilde{\phi}_N(z)}$ is given. It is observed that best approximation is not unique in this case, in contrast to the case $1\leq p$. For $N=1,2$, the authors provide explicit formulas for the best approximation and consequently for the best linear extrapolators. The paper lacks a working example.