  As is well known the kernel of the orthogonal projector onto the polynomials
of degree $n$ in $L^2(w_{\a,\b}, [-1, 1])$ with $w_{\a,\b}(t) =
(1-t)^\a(1+t)^\b$ can be written in terms of Jacobi polynomials. It is shown
that if the coefficients in this kernel are smoothed out by sampling a
$C^\infty$ function then the resulting function has almost exponential (faster
than any polynomial) rate of decay away from the main diagonal. This result is
used for the construction of tight polynomial frames for $L^2(w_{\a,\b})$ with
elements having almost exponential localization.
