The approach presented by the authors could yield new insights and efficient computational methods for other classes of translation-invariant kernels, particularly those with specific properties or structures that can be leveraged in the orthonormal expansion framework. Some classes of translation-invariant kernels that could benefit from this approach include:
Periodic Kernels: For kernels that exhibit periodicity or cyclical behavior, the orthonormal expansions derived using Fourier analysis can provide a systematic way to represent the periodic components and extract relevant information. This can be valuable in applications such as time series analysis, where periodic patterns need to be identified and analyzed efficiently.
Anisotropic Kernels: Kernels that exhibit anisotropy, meaning their behavior varies along different directions, can be effectively represented using orthonormal bases that capture the directional components of the kernel. By extending the approach to handle anisotropic kernels, new insights can be gained into the directional characteristics of the kernels and their impact on data analysis and modeling.
Sparse Kernels: For kernels that have sparse or localized structures, the orthonormal expansions can provide a sparse representation that focuses on the essential components of the kernel while reducing computational complexity. By applying the approach to sparse kernels, efficient computational methods can be developed for tasks such as kernel approximation, interpolation, and regression.
By exploring these and other classes of translation-invariant kernels, the authors' approach can lead to novel insights, improved computational techniques, and enhanced understanding of kernel-based methods in various domains of science and engineering.