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

Justin Grieshop

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 (Cesàro) 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.

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

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 (Cesàro) 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.
Paper Structure (46 sections, 4 theorems, 58 equations, 3 figures)

This paper contains 46 sections, 4 theorems, 58 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Contributions
  3. Scope and standing assumptions
  4. Setting, Notation, and the Admissibility Requirement
  5. A Characterization of Admissibility for Periodic Convolution
  6. Spectral Averaging Operators and a Step-Response Blind Spot
  7. A near-Nyquist blind spot for step-response testing
  8. Why the step test can look "perfect" anyway
  9. Quantitative separation between step overshoot and guaranteed failure
  10. Mechanism: kernel nonnegativity and a step-test blind spot
  11. What a step test actually measures
  12. What admissibility requires (and why negativity is fatal)
  13. The blind spot: cancellation of high-frequency negative lobes under step averaging
  14. Connection to classical Fourier summation: Dirichlet versus Fejér
  15. Certified violations across the cutoff sweep
  16. ...and 31 more sections

Key Result

Theorem 3.1

Let $h \in \mathbb{R}^N$ define a periodic convolution operator $T_h$. Assume the row-sum condition $\sum_{m} h[m] = 1$. Then the following are equivalent:

Figures (3)

  • Figure 1: Hero summary. Observed step-test violation versus spectral cutoff $K$, compared against the certified worst-case lower bound obtained from the constructive witness. The plot highlights a near-Nyquist "blind spot" where step overshoot vanishes while admissibility fails by a fixed amount.
  • Figure 2: Kernel triptych for three spectral averaging constructions at representative cutoffs $K$. Fejér averaging yields a nonnegative kernel (admissible). Sharp and signed spectral cutoffs produce oscillatory kernels with negative coefficients (non-admissible), even when their step responses appear visually well-behaved.
  • Figure 3: Observed step-test violation versus certified worst-case violation across a dense cutoff sweep. The near-Nyquist region shows a pronounced gap: step overshoot can be near zero while the guaranteed violation remains bounded away from zero.

Theorems & Definitions (10)

  • Definition 2.1: Unit mass / constant preservation
  • Definition 2.2: Admissibility on the unit interval
  • Theorem 3.1: Admissibility $\Leftrightarrow$ kernel nonnegativity for periodic convolution
  • proof
  • Corollary 3.2: Certified worst-case violation from negative mass
  • Proposition 4.1: Closed-form kernels at $K = \mathrm{Nyquist}-1$
  • proof
  • Corollary 4.2: Guaranteed violation magnitude at $K = \mathrm{Nyquist}-1$
  • proof
  • proof