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Quasi-discrete Hankel transform.

L Yu, M Huang, M Chen

|December 18, 2007

Summary

A new quasi-discrete Hankel transform (QDHT) offers an efficient method for numerical calculations. This framework introduces a novel discrete Parseval's theorem and highlights the importance of the S factor for QDHT construction.

Area of Science:

Numerical Analysis
Applied Mathematics
Signal Processing

Background:

The Hankel transform is crucial in various scientific fields, including wave propagation and image reconstruction.
Existing numerical methods for the Hankel transform can be computationally intensive and lack efficiency.
There is a need for improved numerical frameworks to handle Hankel transform evaluations accurately and quickly.

Purpose of the Study:

To introduce a novel quasi-discrete Hankel transform (QDHT) as an efficient numerical evaluation framework.
To derive a discrete form of Parseval's theorem for the zero-order Hankel transform.
To analyze the QDHT matrix and identify critical components for its construction.

Main Methods:

Development of a quasi-discrete algorithm for the zero-order Hankel transform.

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Derivation and validation of a discrete version of Parseval's theorem.

Analysis of the transform matrix properties and the role of the S factor.

Main Results:

The proposed QDHT provides an efficient and accurate method for numerical Hankel transform evaluation.
A novel discrete Parseval's theorem for the zero-order Hankel transform has been successfully obtained.
The S factor, derived from a truncated radius, is identified as a critical parameter in the QDHT framework.

Conclusions:

The QDHT framework presents a significant advancement in the numerical computation of Hankel transforms.
The newly derived discrete Parseval's theorem offers valuable theoretical insights and practical applications.
The QDHT method, with careful consideration of the S factor, promises enhanced computational efficiency in relevant scientific domains.