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 ∫−aae−iω(x−y)f(x)dx,a>0,i=−1,y∈[−a,a],ω∈R+,based on the approximation of f by means of the Generalized Bernstein polynomials B̄m,ℓf. 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 ℓ∈N 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.