Complex Gaussian Quadrature Rules for Hankel Transforms of Integer Order

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.

Quotes

None.

Key Insights Distilled From

Complex generalized Gauss-Radau quadrature rules for Hankel transforms of integer order

by Haiyong Wang... at arxiv.org 03-29-2024

https://arxiv.org/pdf/2403.19328.pdf

Deeper Inquiries

How can the proposed complex generalized Gauss-Radau quadrature rules be extended to handle Hankel transforms of fractional order

The proposed complex generalized Gauss-Radau quadrature rules can be extended to handle Hankel transforms of fractional order by considering the weight function ˜wµ,ν(x) = x^µK_ν(x), where µ ∈ N0 when ν ∈ (-1, 0) and µ ∈ N0 and µ > ν - 1 when ν > 0. The Gaussian quadrature rule can be defined using this weight function, and a quadrature rule for (H_νf)(ω) can be constructed based on this. However, it is important to note that this extension will not result in a generalized Gauss-Radau quadrature rule anymore, as the weight function and the conditions for µ and ν are different for fractional orders.

What are the potential limitations or drawbacks of the complex quadrature rules compared to other numerical methods for Hankel transforms, such as the digital linear filter or integration-summation-extrapolation methods

One potential limitation of the complex quadrature rules compared to other numerical methods for Hankel transforms, such as the digital linear filter or integration-summation-extrapolation methods, is the complexity of implementation. The construction of complex generalized Gauss-Radau quadrature rules involves specific conditions and considerations, which may require more computational effort and expertise to implement correctly. Additionally, the asymptotic error estimates for the complex quadrature rules focus on the behavior as the frequency ω approaches infinity, and may not provide as accurate results for finite or moderate values of ω. This could be a drawback when dealing with practical applications where the frequency range is limited.

The content discusses the asymptotic behavior of the quadrature rules as the frequency ω approaches infinity. How do the rules perform for finite, moderate values of ω, and are there any strategies to improve their efficiency in that regime

For finite, moderate values of ω, the complex generalized Gauss-Radau quadrature rules may still provide accurate results, but their efficiency could be impacted by the number of nodes and weights used in the quadrature rule. To improve efficiency in this regime, strategies such as adaptive quadrature methods could be employed. These methods adjust the number of nodes and weights based on the behavior of the integrand, focusing computational resources where they are most needed. By dynamically changing the quadrature rule based on the frequency ω and the characteristics of the function being integrated, the efficiency of the complex quadrature rules can be enhanced for finite and moderate values of ω.