Numerical methods for hypersingular integrals on the real line

In the present paper the authors propose two numerical methods to approximate Hadamard transforms of the type Hp( f wβ , t) = ∫ = R f (x) (x − t)p+1 wβ (x)d x , where p is a nonnegative integer and wβ (x) = e−|x | β , β > 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 {Hk( f wβ , 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.