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    We introduce a new discrete Hankel transform (DHT) based on Fourier-Bessel expansions. This novel transform is invertible and includes standard operational rules, offering an alternative to approximating the continuous Hankel transform.

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    Area of Science:

    • Mathematics
    • Signal Processing
    • Applied Physics

    Background:

    • Existing discrete Hankel transforms primarily approximate the continuous Hankel integral transform.
    • A need exists for a theoretically grounded discrete Hankel transform with inherent properties like orthogonality.

    Purpose of the Study:

    • To propose and evaluate a novel theory for a discrete Hankel transform (DHT).
    • To establish the orthogonality and invertibility of the proposed DHT.
    • To derive the standard transform rules (shift, modulation, multiplication, convolution).

    Main Methods:

    • Discretization scheme derived from Fourier-Bessel expansions.
    • Mathematical derivation of orthogonality and invertibility properties.
    • Development of standard transform rules for the DHT.

    Main Results:

    • A theoretically sound discrete Hankel transform (DHT) is established.
    • The proposed DHT exhibits orthogonality, ensuring its invertibility.
    • Standard shift, modulation, multiplication, and convolution rules are derived for the DHT.

    Conclusions:

    • The novel DHT provides a robust framework for discrete Hankel analysis.
    • This DHT can effectively approximate the continuous Hankel transform, analogous to the DFT's role for the Fourier transform.
    • The derived operational rules facilitate practical applications in signal and image processing.