On the convergence of the method for indefinite integration of oscillatory and singular functions
In 2007, the first author [Numer. Algorithms 46, No. 3, 219--251 (2007; Zbl 1131.65025)] proposed an efficient method for computing indefinite integrals of the form \[ I(y)=\int_{-1}^{y}G(x)K(x) dx\,\,\,(-1\leq y\leq 1), \tag{1} \] where \(G\) is a smooth function, while the function \(K\) is highly oscillatory and may possess singularity points. In the present paper the authors assume that \(K\) is absolutely integrable. It is considered the problem to approximate the integral (1) with a uniform error less then a given value of error tolerance. In this article are analyzed the properties of the difference operator related to the difference equation for the Chebyshev coefficients of a function that satisfies a given linear differential equation with polynomial coefficients. They are presented the results of numerical experiments verifying the reliability of the error estimations proved in this paper.
scientific article; zbMATH DE number 5707079
scientific article