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.

A product integration rule on equispaced nodes for highly oscillating integrals

Fermo L.
;
2023

Abstract

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.
Approximation by polynomials
Boolean iterated sums of Bernstein operators
Quadrature rules
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11584/347037
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