Spectral Regimes of Truncated sinc-Power and Pulse-Train Kernels: Edge Exponents, Per-Edge Plunges, and the Landau-Widom Limit

Abstract

A signal band-limited to one rectangular band and observed through a finite window is the classical concentration problem of Slepian, Landau, Pollak, and Widom. The eigenvalues of the operator cluster near one and near zero, separated by a transition called the plunge. The plunge width grows like the logarithm of the window length, with a universal coefficient of one over pi squared set by the band-edge jump. This paper follows the clusters and the plunge as the band is deformed. An elementwise power of the kernel smooths the band edge, and the plunge gives way to a power-law edge with exponent one less than the power. Disjoint copies of the band widen the plunge in proportion to the number of band edges, as the multiple-interval theorem of Landau and Widom implies, so the coefficient is a per-edge constant. Shrinking each band to a line gives the classical finite-rank operator. Overlapping bands give a multilevel staircase, and a self-similar band gives a cluster hierarchy governed by the Cantor dimension. At the smooth edge, rescaled eigenvalues do not collapse onto one curve. The edge descriptor passes from a constant to an exponent to an integer to a fractal dimension. For the single band, classical results place all eigenfilter zeros on the unit circle, with more of them inside the band as the order rises, so the dominant eigenvectors act as lowpass filters.