A product integration rule on equispaced nodes for highly oscillating integrals

This paper provides a product integration rule for highly oscillating integrands of the type \[ \int_{-a}^a e^{-\ii \omega (x-y)} f(x) dx, \quad a>0, \quad \ii=\sqrt{-1}, \quad y \in [-a,a], \quad \omega \in \RR^+, \] based on the approximation of $f$ by means of the Generalized Bernstein polynomials $\bar B_{m,\ell}f$. The rule involves the samples of $f$ at $m+1$ equally spaced points of $[-a,a],$ and differently from the classical Bernstein polynomials, the suitable modulation of the parameter $\ell\in \NN$ allows to increase the accuracy of the product rule, as the smoothness of $f$ increases. Stability and error estimates are proven for $f$ belonging to the space of continuous functions and their Sobolev-type subspaces. Finally, some numerical tests which confirm such theoretical estimates are shown.