Optimal quadrature evaluation of integrals of rapidly oscillating functions in Lipschitz interpolation class in case of strong oscillation
The authors consider the following integrals \[ I_{1}(\omega)=\int_{0}^{1}f(x)\sin\omega x dx,\,\,\,I_{2}(\omega)=\int_{0}^{1}f(x)\cos\omega x dx \tag{1} \] where \(f\in F\) (\(F\) is a given class of functions), \(\omega\) an arbitrary real number, \(|\omega|\geq 2\pi\), and the information about \(f\) is available at no more than \(N\) points. They find optimal estimates and derive accuracy-optimal quadrature formulas to evaluate the Fourier transform of a compact function on the assumption that \(f\in C_{L,N}\), where \(C_{L,N}\) is an interpolation class of functions that satisfy the Lipschitz property (\(|f(x_{1})-f(x_{2})| \leq L|x_{1}-x_{2}|\), \(x_{1},x_{2} \in [0,1] \)) and are defined by \(2N\) fixed values. The case of strong oscillation of the integrand is considered. The optimality is substantiated based on the boundary function method, namely, constructing the Chebyshev center and Chebyshev radius in the uncertainty domain of the problem solution.
scientific article; zbMATH DE number 5790971