Computation of the Riesz-Herglotz transform and its application to quadrature formulas over the unit circle (Q1849715)

A method is proposed for the computation of the Riesz-Herglotz transform. Numerical experiments show the effectiveness of this method. The author studies its application to the computation of integrals over the unit circle in the complex plane of analytic functions. This approach leads to the integration by Taylor polynomials. On the other hand, with the goal of minimizing the quadrature error bound for analytic functions, in the set of quadrature formulas of Hermite interpolatory type, the author finds that this minimum is attained by the quadrature formula based on the integration of the Taylor polynomial. These two different approaches suggest the effectiveness of this formula. Numerical experiments comparing with other quadrature methods with the same domain of validity, or even greater such as Szegö formulas, traditionally considered as the counterpart of the Gauss formulas for integrals on the unit circle, confirm the superiority of the numerical estimations.