Quadrature formulae for the positive real axis in the setting of Mellin analysis: Sharp error estimates in terms of the Mellin distance
Provides optimal error bounds for numerical integration on the positive real axis, relevant to researchers in approximation theory and numerical analysis.
The paper develops sharp error estimates for quadrature formulas on the positive real axis using Mellin analysis, showing that the remainder converges to zero at the best possible rate as a function of the Mellin distance to bandlimited functions. Numerical experiments confirm the theoretical results.
The general Poisson summation formula of Mellin analysis can be considered as a quadrature formula for the positive real axis with remainder. For Mellin bandlimited functions it becomes an exact quadrature formula. Our main aim is to study the speed of convergence to zero of the remainder for a function $f$ in terms of its distance from a space of Mellin bandlimited functions. The resulting estimates turn out to be of best possible order. Moreover, we characterize certain rates of convergence in terms of Mellin--Sobolev and Mellin--Hardy type spaces that contain $f$. Some numerical experiments illustrate and confirm these results.