Optimal quadrature formulas for computing of Fourier integrals in a Hilbert space

TL;DR

This paper develops optimal quadrature formulas for efficiently computing Fourier integrals within a specific Hilbert space, providing explicit coefficients and numerical validation.

Contribution

It introduces new optimal quadrature formulas in the sense of Sard for Fourier integrals in a Hilbert space, with explicit coefficient formulas.

Findings

Explicit formulas for quadrature coefficients are derived.

Numerical results demonstrate the effectiveness of the formulas.

The approach improves accuracy for Fourier integral computation.

Abstract

In the present paper the optimal quadrature formulas in the sense of Sard are constructed for numerical integration of the integral $\int_a^b e^{2\pi i\omega x}\varphi(x)d x$ with $\omega\in \mathbb{R}$ in the Hilbert space $W_2^{(2,1)}[a,b]$ of complex-valued functions. Furthermore, the explicit expressions for coefficients of the constructed optimal quadrature formulas are obtained. At the end of the paper some numerical results are presented.