Optimization of approximate integrals of rapidly oscillating functions in the Hilbert space

Abstract

In this work, we construct an optimal quadrature formula in the sense of Sard based on a functional approach for numerical calculation of integrals of rapidly oscillating functions. To solve this problem, we will use Sobolev’s method.

To do this, we first solve the boundary value problem for an extremal function. To solve the boundary value problem, we use direct and inverse Fourier transforms and find the fundamental solution of the given differential operator. Using the extremal function, we find the norm of the error functional. For the given nodes, we find the minimum value of the error functional norm along the coefficients.

This quadrature formula is exact for the hyperbolic functions $sinh\left(x\right),cosh\left(x\right)$ and a constant term. In this work, we consider the case $\omega h\notin Z$ and $\omega \in R$ in the Hilbert space ${\text{K}}_2^{\left(3\right)}(0,1)$.

We apply the constructed quadrature formula for reconstruction of a Computed Tomography image.