We consider the problem of optimal linear estimation of the functional $A ξ~=~\sum_{j = 0}^{\infty} a_j ξ_j$ that depends on the unknown values $ξ_j,j=0,1,\dots, $ of a random sequence $\{ξ_j,j\in\mathbb Z\}$ from observations of the sequence $\{ξ_k+η_k,k\in\mathbb Z\}$ at points $k = -1, -2, \dots$, where $\{ξ_k,k\in\mathbb Z\}$ and $\{η_k,k\in\mathbb Z\}$ are mutually independent harmonizable symmetric $α$-stable random sequences which have the spectral densities $f(θ)>0$ and $g(θ)>0$ 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.