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Predictability of sequences and subsequences with spectrum degeneracy at periodically located points

Nikolai Dokuchaev

TL;DR

The paper tackles the problem of predicting sequences whose spectra exhibit degeneracy at periodically located points on the unit circle, and it demonstrates that finite-horizon, linear predictors based on explicit convolution kernels can achieve uniform predictability for these degenerate-spectrum classes.A core contribution is showing that m-periodic subsequences are predictable when m divides M, and that predictability extends to compound processes formed from such components, even when traditional spectrum gaps are absent.It further introduces braided spectrum degeneracy and proves that a class of sequences with this degeneracy is uniformly recoverable from subsequences, with the striking result that a class ${\cal P}_m$ is everywhere dense in $\ell_2$, enabling approximation of any square-summable sequence from periodic subsequences.Together, these results provide both theoretical guarantees and practical kernel-based methods for prediction and recovery in signals with periodic spectrum structure, along with discussions of numerical implementation and robustness to noise.

Abstract

The paper established sufficient conditions of predictability with degeneracy for the spectrum at $M$-periodically located isolated points on the unit circle. It is also shown that $m$-periodic subsequences of these sequences are also predictable if $m$ is a divisor of $M$. The predictability can be achieved for finite horizon with linear predictors defined by convolutions with certain kernels. As an example of applications, it is shown that there exists a class of sequences that is everywhere dense in the class of all square-summable sequences and such that its members can be recovered from their periodic subsequences. This recoverability is associated with certain spectrum degeneracy of a new kind.

Predictability of sequences and subsequences with spectrum degeneracy at periodically located points

TL;DR

The paper tackles the problem of predicting sequences whose spectra exhibit degeneracy at periodically located points on the unit circle, and it demonstrates that finite-horizon, linear predictors based on explicit convolution kernels can achieve uniform predictability for these degenerate-spectrum classes.A core contribution is showing that m-periodic subsequences are predictable when m divides M, and that predictability extends to compound processes formed from such components, even when traditional spectrum gaps are absent.It further introduces braided spectrum degeneracy and proves that a class of sequences with this degeneracy is uniformly recoverable from subsequences, with the striking result that a class ${\cal P}_m$ is everywhere dense in $\ell_2$, enabling approximation of any square-summable sequence from periodic subsequences.Together, these results provide both theoretical guarantees and practical kernel-based methods for prediction and recovery in signals with periodic spectrum structure, along with discussions of numerical implementation and robustness to noise.

Abstract

The paper established sufficient conditions of predictability with degeneracy for the spectrum at -periodically located isolated points on the unit circle. It is also shown that -periodic subsequences of these sequences are also predictable if is a divisor of . The predictability can be achieved for finite horizon with linear predictors defined by convolutions with certain kernels. As an example of applications, it is shown that there exists a class of sequences that is everywhere dense in the class of all square-summable sequences and such that its members can be recovered from their periodic subsequences. This recoverability is associated with certain spectrum degeneracy of a new kind.

Paper Structure

This paper contains 9 sections, 16 theorems, 81 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The main results
  3. Robustness of prediction
  4. Predictability of subsequences
  5. On detecting predictable sequences
  6. Predictability of compound processes
  7. On recovery of sequences from their subsequences
  8. Proofs
  9. On numerical implementation

Key Result

Theorem 2.2

For any $q>0$, $r>0$, $\nu\in {\mathbb{Z}}_1^+$, $\beta\in(-\pi,\pi]$, $s\in{\mathbb{Z}}$, and $\rho>0$, the class of sequences ${\cal X}_{m\nu,\beta}(q,c,r)$ is uniformly predictable on finite horizon in the sense of Definition defP with $\psi(t)=e^{i\theta t}$ and $\theta=(\beta-\pi)/(m\nu)$.

Figures (2)

  • Figure 1: The distance $|\widehat{H}_4\left(e^{i\omega}\right)-e^{2 i\omega}|$ for $\gamma=3$ on $[-\pi,\pi]$.
  • Figure 4: Process $g$: a truncated path a Gaussian process.

Theorems & Definitions (25)

  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3
  • Theorem 2.4
  • Definition 2.5
  • Lemma 2.6
  • Corollary 2.7
  • Corollary 2.8
  • Corollary 2.9
  • Lemma 2.10
  • ...and 15 more