To extend the Fourier analytic technique presented in Theorem 1.1 for constructing orthonormal expansions to domains other than the real line, one approach could be to consider generalizing the technique to multidimensional spaces. By adapting the concept of translation-invariant kernels to higher-dimensional spaces, such as Euclidean spaces or manifolds, the orthonormal basis functions can be constructed to capture the spatial relationships and symmetries present in those domains.
Additionally, the technique can be extended to non-Euclidean spaces, such as graphs or networks, by defining appropriate notions of translation invariance and adapting the Fourier analytic framework to these structures. This extension would enable the construction of orthonormal expansions for kernels defined on complex networks or irregular domains, opening up new possibilities for analyzing and modeling data in network science and graph theory.
Furthermore, considering domains with boundary conditions or irregular geometries would require modifications to the Fourier analytic technique to account for the specific characteristics of those domains. By incorporating boundary conditions or constraints into the construction of orthonormal bases, the technique can be adapted to handle a wider range of spatial configurations and enhance its applicability to diverse domains.
In summary, extending the Fourier analytic technique to construct orthonormal expansions for translation-invariant kernels on domains other than the real line involves adapting the methodology to higher-dimensional spaces, non-Euclidean structures, and domains with boundary conditions, thereby broadening its scope and utility in various fields of study.