  We present a class of weight functions $ w$ on the circle $ \mathbb{T}$,
called L\'evy-Khinchin-Schoenberg (LKS) weights, for which we are able to
completely characterize (in terms of a capacitary inequality) all Fourier
multipliers for the weighted space $ L^{2}(\mathbb{T},w)$. We show that the
multiplier algebra is nontrivial if and only if $ 1/w\in L^{1}(\mathbb{T})$,
and in this case multipliers satisfy the Spectral Localization Property (no
"hidden spectrum"). On the other hand, the Muckenhoupt $ (A_{2})$ condition
responsible for the basis property of exponentials $ (e^{ikx})$ is more or less
independent of the Spectral Localization Property and LKS requirements. Some
more complicated compositions of LKS weights are considered as well.
