Numerical methods for hypersingular integrals on the real line. (English) Zbl 1372.65067

Summary: In the present paper the authors propose two numerical methods to approximate Hadamard transforms of the type
\[ \mathbf{H}_p(fw_{\beta},t) =~~=\!\!\!\!\!\!\!\int_{\mathbb{R}} \frac{f(x)}{(x-t)^{p+1}} w_{\beta}(x)dx, \]
where \(p\) is a nonnegative integer and \(w_{\beta}(x) = e^{-|x|^{\beta}}\), \(\beta > 1\), is a Freud weight. One of the procedures employed here is based on a simple tool like the “truncated” Gaussian rule conveniently modified to remove numerical cancellation and overflow phenomena. The second approach is a process of simultaneous approximation of the functions \(\{\mathbf{H}_k (fw_{\beta}, t)\}^p_{k=0}\). This strategy can be useful in the numerical treatment of hypersingular integral equations. The methods are shown to be numerically stable and convergent and some error estimates in suitable Zygmund-type spaces are proved. Numerical tests confirming the theoretical estimates are given. Comparisons of our methods among them and with other ones available in literature are shown.


65D30 Numerical integration
41A55 Approximate quadratures
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