Abstract
In this article a new quadrature and cubature formulas are obtained. These formulas are formulas of Hermitian type, which exactly on some trigonometric polynomials. The question of the convergence of the quadrature process associ- ated with these formulas is investigated. Numerical analysis of these formulas is holds. The obtained formulas may have an advantage in integrating strongly oscillating functions.
Highlights
In 1814 Carl Friedrich Gauss published his famous quadrature formula of the highest algebraic degree of accuracy
The Gaussian formula opened the central direction in numerical analysis devoted to the construction of approximation methods of a given form that are exact for all polynomials of the highest possible degree
This article devoted to the quadrature and cubature formulas of Hermitian type, obtained by the author
Summary
In 1814 Carl Friedrich Gauss published his famous quadrature formula of the highest algebraic degree of accuracy. The Gaussian formula opened the central direction in numerical analysis devoted to the construction of approximation methods of a given form that are exact for all polynomials of the highest possible degree. By the Weierstrass theorem, an arbitrary continuous function can be approximated by polynomials with arbitrary accuracy. For this reason, every approximation method which is good for a broad class of polynomials was regarded as a good method. At that time Andrei Nikolaevich Kolmogorov posed the following problem in the 1940s: Construct a quadrature formula of a given type that has minimal error on a given class of functions. This article devoted to the quadrature and cubature formulas of Hermitian type, obtained by the author.
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