This paper provides a product integration rule for highly oscillating integrands of the type $\int_{-a}^a e^{-\ii \omega (x-y)} f(x) dx, \quad a&gt;0, \quad \ii=\sqrt{-1}, \quad y \in [-a,a], \quad \omega \in \RR^+,$ 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.

### A product integration rule on equispaced nodes for highly oscillating integrals

#### Abstract

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.
##### Scheda breve Scheda completa Scheda completa (DC)
2023
File in questo prodotto:
File
AML_paper_v2.pdf

accesso aperto

Descrizione: Pre-print Manuscript
Tipologia: Documento in Pre-print
Licenza: Non definito
Dimensione 313.96 kB
Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11563/160466