Localized polynomial frames on the interval with Jacobi weights

As is well known the kernel of the orthogonal projector onto the polynomials of degree $n$ in $L^2(w_{\text{a},\text{b}},[-1,1])$ 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 $C^{\mathrm{\infty }}$ 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 $L^2(w_{\text{a},\text{b}})$ with elements having almost exponential localization.