Core Concepts
Complex Gaussian quadrature rules can be constructed for Hankel transforms of integer order by incorporating value and derivative information at the left endpoint. These rules achieve optimal asymptotic error decay and their existence is guaranteed.
Abstract
The content discusses the construction of complex generalized Gauss-Radau quadrature rules for evaluating Hankel transforms of integer order.
Key highlights:
- Hankel transforms appear in many physical problems but are difficult to evaluate numerically due to the oscillatory and slowly decaying behavior of the integrand.
- Complex Gaussian quadrature rules can achieve optimal asymptotic error decay for Hankel transforms, but their existence is only guaranteed for orders in the range [0, 1/2].
- By incorporating value and derivative information at the left endpoint, the authors show that complex generalized Gauss-Radau quadrature rules can be constructed for Hankel transforms of integer order, with guaranteed existence.
- The authors also investigate the existence and properties of orthogonal polynomials related to these quadrature rules, proving that they exist for even degrees when the order and weight exponent difference are both integers, and their zeros lie on the imaginary axis.
- Numerical experiments confirm the optimal asymptotic error decay of the proposed quadrature rules.
Stats
The content does not provide any specific numerical data or metrics to support the key claims. The focus is on the theoretical construction and analysis of the quadrature rules.