Affine AP-frames and Stationary Random Processes
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866913133116784640 |
|---|---|
| author | Centeno, Hernán Diego Medina, Juan Miguel |
| author_facet | Centeno, Hernán Diego Medina, Juan Miguel |
| contents | It is known that, in general, an affine or Gabor AP-frame is an $L^2(\mathbb{R})$-frame and conversely. In part as a consequence of the Ergodic Theorem, we prove a necessary and sufficient condition for an affine (wavelet) system $\mathcal{A}=\{a^{j/2} ψ_{j,k}(t):=a^{-j/2} ψ(a^{-j} t -k) :j\in\mathbb{Z}, k\in\mathbb{K}:=b\mathbb{Z}\}$ to be an affine AP-Frame in terms of Gaussian stationary random processes expanding in this way what we have done recently for Gabor systems. Likewise, we study a connection between the decay of the associated stationary sequences $\{\langle{X,ψ_{j,k}}\rangle : k\in\mathbb{K}\}$ for each $j\in\mathbb{Z}$, and a smoothness condition on a Gaussian stationary random process $X=(X(t))_{t\in\mathbb{R}}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_15090 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Affine AP-frames and Stationary Random Processes Centeno, Hernán Diego Medina, Juan Miguel Probability Functional Analysis (Primary) 42C15, 42C40, 60G10, (Secondary) 46E35 It is known that, in general, an affine or Gabor AP-frame is an $L^2(\mathbb{R})$-frame and conversely. In part as a consequence of the Ergodic Theorem, we prove a necessary and sufficient condition for an affine (wavelet) system $\mathcal{A}=\{a^{j/2} ψ_{j,k}(t):=a^{-j/2} ψ(a^{-j} t -k) :j\in\mathbb{Z}, k\in\mathbb{K}:=b\mathbb{Z}\}$ to be an affine AP-Frame in terms of Gaussian stationary random processes expanding in this way what we have done recently for Gabor systems. Likewise, we study a connection between the decay of the associated stationary sequences $\{\langle{X,ψ_{j,k}}\rangle : k\in\mathbb{K}\}$ for each $j\in\mathbb{Z}$, and a smoothness condition on a Gaussian stationary random process $X=(X(t))_{t\in\mathbb{R}}$. |
| title | Affine AP-frames and Stationary Random Processes |
| topic | Probability Functional Analysis (Primary) 42C15, 42C40, 60G10, (Secondary) 46E35 |
| url | https://arxiv.org/abs/2507.15090 |