A generalized spectral concentration problem and the varying masks algorithm

TL;DR

This work generalizes the classical Slepian–Pollak–Landau spectral concentration problem to arbitrary space and Fourier masks, yielding a generalized concentration operator $\mathcal{K}$ that is Hilbert–Schmidt, self-adjoint, and compact. It provides a Strang-splitting representation for Gaussian-like masks, constructs commuting differential operators for quadratic domains, and demonstrates how quasi-modes can be built via BCH expansions, enabling analytical insight beyond spheres and Gaussians. To tackle numerical instability from eigenvalue clustering, the authors propose the varying masks algorithm, which evolves both space and Fourier masks to reliably approximate the leading eigenpairs in multiple dimensions and domain shapes. Numerical experiments in 1D and 2D show that the varying-masks approach delivers localized, symmetric eigenvectors with concentration ratios close to the true eigenvalues, outperforming standard eigendecomposition in challenging cases. The framework thus furnishes robust theory and practical algorithms for spatio–frequency concentration with general domains and masks, with potential broad applicability in signal processing and physics.

Abstract

In this paper we generalize the spectral concentration problem as formulated by Slepian, Pollak and Landau in the 1960s. We show that a generalized version with arbitrary space and Fourier masks is well-posed, and we prove some new results concerning general quadratic domains and gaussian filters. We also propose a more general splitting representation of the spectral concentration operator allowing to construct quasi-modes in some situations. We then study its discretization and we illustrate the fact that standard eigen-algorithms are not robust because of a clustering of eigenvalues. We propose a new alternative algorithm that can be implemented in any dimension and for any domain shape, and that gives very efficient results in practice.

Figures (15)

• Figure 1: 30 first eigenvalues of the matrix $\mathbf{K}(\varepsilon)$, with $N=150$ points of discretization, $\Omega = 0.1\cdot 2\pi$.
• Figure 2: First sixteen eigenvectors obtained for the initial discretized concentration problem ($\varepsilon=0$, solid blue), for the modified discretized concentration problem ($\varepsilon=2$, orange dashes), and the exact eigenvectors given by the DPSS (green dots).
• Figure 3: 16 first eigenvectors obtained with a standard eigendecomposition (solid blue curve) of $\mathbf{K}(0)$, compared to the exact eigenvectors (dashed orange curve). Here, $\widehat{m_{F}} = \mathbf{1}_{[-0.3\cdot 2\pi\mu(\varepsilon), 0.3\cdot 2\pi \mu(\varepsilon)]}$. $N_1 = 150, \eta = 10^{-10}$.
• Figure 4: 16 first eigenvectors obtained with the varying masks procedure (solid blue curve) of $\mathbf{K}(0)$, compared to the exact eigenvectors (dashed orange curve). Here, $\widehat{m_{F}} = \mathbf{1}_{[-0.3\cdot 2\pi\mu(\varepsilon), 0.3\cdot 2\pi\mu(\varepsilon)]}$. $N_1 = 150, \eta = 10^{-10}$.
• Figure 5: 16 first eigenvectors obtained with a standard eigendecomposition (solid blue curve) of $\mathbf{K}(0)$, compared to the exact eigenvectors (dashed orange curve). Here, $\widehat{m_F} = \mathbf{1}_{[-0.49 \cdot 2\pi\mu(\varepsilon), 0.49\cdot 2\pi\mu(\varepsilon)]}$. $N_1 = 150, \eta = 10^{-10}$.

Theorems & Definitions (32)

• Lemma 1
• proof
• Proposition 1
• proof
• Lemma 2: Symmetries
• proof
• Lemma 3: Translations with binary masks
• proof
• Lemma 4: Affine transformations with binary masks
• proof