  We consider the problem of optimal linear estimation of the functional $A
\xi~=~\sum_{j = 0}^{\infty} a_j \xi_j$ that depends on the unknown values
$\xi_j,j=0,1,\dots, $ of a random sequence $\{\xi_j,j\in\mathbb Z\}$ from
observations of the sequence
  $\{\xi_k+\eta_k,k\in\mathbb Z\}$ at points $k = -1, -2, \dots$, where
$\{\xi_k,k\in\mathbb Z\}$ and $\{\eta_k,k\in\mathbb Z\}$ are mutually
independent harmonizable symmetric $\alpha$-stable random sequences which have
the spectral densities $f(\theta)>0$ and $g(\theta)>0$ satisfying the
minimality condition.
  The problem is investigated under the condition of spectral certainty as well
as under the condition of spectral uncertainty.
  Formulas for calculating the value of the error and the spectral
characteristic of the optimal linear estimate of the functional are derived
under the condition of spectral certainty where spectral densities of the
sequences are exactly known.
  In the case of spectral uncertainty where spectral densities of the sequences
are not exactly known, while a class of admissible spectral densities is given,
relations that determine the least favorable spectral densities and the minimax
spectral characteristic are derived.
