On the Failure of Step-Response Tests to Certify Admissibility of Spectral Averaging Operators

TL;DR

This paper demonstrates that step-response tests are unreliable for certifying the admissibility of spectral averaging operators, revealing fundamental limitations and providing explicit counterexamples.

Contribution

It provides a sharp characterization of when periodic convolution maps [0,1]^N into itself and analyzes standard Fourier averaging methods, exposing their blind spots.

Findings

Step-response diagnostics can be misleading for periodic convolution operators.

A convolution preserves boundedness iff its kernel is nonnegative and rows are probability vectors.

Explicit counterexamples show spectral truncation can violate boundedness despite zero overshoot.

Abstract

We study translation-invariant smoothing operators on finite cyclic groups Z/NZ, expressed as periodic convolutions and analyzed via the discrete Fourier transform on Z/NZ. Step responses are widely used as a practical diagnostic for whether a smoothing operator preserves bounded ranges, for example whether it maps signals in [0,1]^N back into [0,1]^N. We show step-based diagnostics can be fundamentally misleading for periodic convolution operators. First, we give a sharp characterization: a constant-preserving periodic convolution maps [0,1]^N into [0,1]^N if and only if its convolution kernel is componentwise nonnegative, equivalently each row of the associated matrix is a probability vector. The proof is constructive and yields worst-case witnesses in {0,1}^N together with certified lower bounds on the magnitude of boundedness violation. We then analyze three standard Fourier-domain averaging constructions, Fejer (Ces\`aro) averaging, sharp spectral truncation, and a signed spectral control, and demonstrate a Nyquist-adjacent blind spot: at cutoff K = Nyquist - 1, the canonical step can exhibit essentially zero overshoot while the operator remains strongly non-admissible, with explicit binary inputs yielding outputs below 0 and above 1. Reproducible artifacts and a self-contained demo are provided.