Spectral decomposition

Gaussian kernel matrix can be factorized into $(\Phi \text{X}{)}^{\text{H}}\Phi \text{X}={\text{X}}^{\text{H}}{\Phi }^{\text{H}}\Phi \text{X}={\text{X}}^{\text{H}}\text{X}$, where $\Phi$ is Gaussian kernel basis matrix and $\text{X}$ is coefficients matrix of reproducing kernel Hilbert space $K(\cdot ,x)\in H_K$ https://www.jkangpathology.com/post/reproducing-kernel-hilbert-space/.

A matrix is a system. A system takes input and gives output. A matrix is a linear system. Differentiation and Integration are linear systems. Fourier transformation matches input basis and operator (differentiation) basis. Z transformation matches input digital signal and infinite impulse response filter.

A time-domain sequence can be transformed into a frequency domain by discrete Fourier transformation. The dimension of the discrete Fourier matrix is determined by the length of the time-domain sequence.