Eigenvalues and Eigenfunctions

Abstract

The results obtained in Chapters 2–5 can be used in the computation of eigenvalues of filters, which are given by translation-invariant linear operators. To recall, let A :$L^2(\mathbb{Z}_{N})\rightarrow L^2(\mathbb{Z}_{N})$ be a filters, i.e., a translation-invariant linear operator. Then the matrix (A)_S of A with respect to the standard basis S for $L^2(\mathbb{Z}_{N})$ is circulant. The filter A is in fact a convolution operator $C_{b}$ with impulse response b, where b is simply the first column of the matrix A. The filter A is also a Fourier multiplier $T_\sigma$ with symbol σ and σ = $\hat{b}$ The matrix (A)_F of the filter A with respect to the Fourier basis F for L²($\mathbb{Z}_N$) is diagonal, and is given by

$$(A)_F = \left(\begin{array}{ccccc} \lambda_0 & 0 & 0 & \ldots & 0\\ 0 & \lambda_1 & 0 & \ldots & 0\\ 0 & 0 & \lambda_2 & \ldots & 0\\ \vdots & \vdots & \vdots & \ldots & \vdots\\ 0 & 0 & 0 & \ldots & \lambda_{N-1}\\ \end{array}\right),$$

where λ_m is the eigenvalue of A corresponding to the eigenfunction F_m, m = 0, 1, …, N – 1.