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    This study introduces reproducing kernel Hilbert spaces (RKHS) using Fourier optics and coherence theory. It details RKHS construction, properties, and applications in wave optics, offering a foundation for advanced functional analysis.

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    Area of Science:

    • Mathematical Physics
    • Functional Analysis
    • Optics

    Background:

    • Reproducing kernel Hilbert spaces (RKHS) are fundamental in various scientific fields.
    • Understanding RKHS requires advanced mathematical concepts.

    Purpose of the Study:

    • To introduce RKHS using tools from Fourier optics and coherence theory.
    • To illustrate the construction, properties, and applications of RKHS.

    Main Methods:

    • Defining RKHS and exploring inner product variations across function spaces.
    • Presenting the construction rules and fundamental properties of RKHS.
    • Utilizing eigenfunctions and eigenvalues to derive the integral representation of the reproducing kernel.

    Main Results:

    • Demonstrated how inner products vary based on function spaces.
    • Illustrated the construction and properties of RKHS.
    • Derived integral representations of the reproducing kernel, enabling pseudomodal expansions and generalized sampling.

    Conclusions:

    • RKHS provide a powerful mathematical framework with significant applications in wave optics.
    • The study offers a bridge between functional analysis and optical applications.
    • An appendix provides further insights into advanced functional analysis techniques for RKHS.