Concepts de base
The author explores the universality of almost periodicity in bounded discrete time series without using Fourier transform, presenting a recursive formula that converges to an almost periodic function.
Résumé
The content delves into constructing an almost periodic function for arbitrary bounded discrete time series without relying on Fourier transform. By discretizing the time series and employing permutation operators, the author establishes statistical features and conditional probabilities. The main theorem asserts the existence of a suitable function approximating the time series locally, leading to an almost periodic function as time progresses. Through detailed proofs and examples, the author demonstrates how this approach can be applied to dynamical systems and linear regression relations, ultimately showcasing the convergence to an almost periodic function.
Stats
For any n ∈ {1, 2, · · · , N}, there exists a δK ∈ [0, 1) such that for any n: P(ak(n) = ¯y(·) σn(ℓ) = ¯y(· − ℓ) for ℓ = 1, 2, · · · , L) ≥ 1 - δK.
Magnitude relation: For any K′ > 0 and t ∈ Z: aKk(nKt ) - aK+K'k(nK+K't ) < C/K.
There exists u : Z → [−1, 1] such that |u(t0 + t') - y(t0 + t')| < C/K ≥ (1 - δK)t for any K ∈ Z≥1 and t ∈ Z≥0.
Citations
"The nontrivial discovery is explicit construction of u (see (12)), and it tends to the discretized almost periodic function, without any use of Fourier transform."
"By applying the above properties, then we see that the determinant of the matrix X is nonzero."
"In some situations, this periodic chain may be different for different choices of t0."