  We denote by $\bbt$ the unit circle and by $\bbd$ the unit disc of $\bbc$.
Let $s$ be a non-negative real and $\omega$ a weight such that $\omega(n) =
(1+n)^{s} \quad (n \geq 0)$ and such that the sequence $\dsp
\Big(\frac{\omega(-n)}{(1+n)^{s}} \Big)_{n \geq 0}$ is non-decreasing. We
define the Banach algebra $$ A_{\omega}(\bbt) = \Big\{f \in \calc(\bbt) :
\big\| f \big\|_{\omega} = \sum_{n = -\infty}^{+\infty} | \hat{f}(n) |
\omega(n) < +\infty \Big\}, $$ If $I$ is a closed ideal of $A_{\omega}(\bbt)$,
we set $h^{0}(I) = \Big\{z \in \bbt : f(z) = 0 \quad (f \in I) \Big\}$. We
describe here all closed ideals $I$ of $A_{\omega}(\bbt)$ such that $h^{0}(I)$
is at most countable. A similar result is obtained for closed ideals of the
algebra $A_{s}^{+}(\bbt) = \Big\{f \in A_{\omega}(\bbt) : \hat{f}(n) = 0 \quad
(n<0) \Big\}$ without inner factor. Then, we use this description to establish
a link between operators with countable spectrum and interpolating sets for
$\textrm{{\LARGE $a$}}^{\infty}$, the space of infinitely differentiable
functions in the closed unit disc $\bar{\bbd}$ and holomorphic in $\bbd$.
