Convergence of the truncation error in calculating singular convolutions (Q1360824)

For the approximation of singular convolutions of the type \[ z(t)=(2\pi)^{-1}\int_{-\infty}^\infty\exp(-i\omega t) \widehat{L}(\omega) \widehat{u}(\omega)d\omega, \] the author considers regularization methods based on replacing \(\widehat{L}\) by \(\widehat{H}_\alpha=\widehat{L} R_\alpha\), \(R_\alpha\) for \(\alpha\in \mathbb{R}\) being a suitable family of functions. In particular, he studies the error of the trapezoidal quadrature formula for such regularized singular convolutions. The main results are conditions on \(R_\alpha\), which depend on the growth of \(\widehat{L}\), for the convergence of the discretised to the regularised convolution, as well as error bounds.