Stationary Process Invertibility and the Unilateral Shift Operator
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915911609352192 |
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| author | Ganesh, Anand Bose, Babhrubahan Rajagopalan, Anand |
| author_facet | Ganesh, Anand Bose, Babhrubahan Rajagopalan, Anand |
| contents | The bilateral shift operator $B$ has been the mainstay of stationary process modeling whereas we argue that the unilateral shift operator $T$ may be better suited to analyze invertibility. While doing so, we partially unify the notion of stationary process invertibility (associated with a sufficent but not necessary $\ell^1$ condition) with the algebraic invertibility of the transfer function $f(T)$. We establish a rigorous operator theoretic foundation for these arguments proving that for $f \in \mathbb{W}_+$, the Wiener algebra, $f(T)$ is well defined, that $\| f(T) \| = \| f \|_{\infty}$ and that $f(T) = T_f$, the Toeplitz operator. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_02336 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Stationary Process Invertibility and the Unilateral Shift Operator Ganesh, Anand Bose, Babhrubahan Rajagopalan, Anand Functional Analysis Statistics Theory 60G10 G.3 The bilateral shift operator $B$ has been the mainstay of stationary process modeling whereas we argue that the unilateral shift operator $T$ may be better suited to analyze invertibility. While doing so, we partially unify the notion of stationary process invertibility (associated with a sufficent but not necessary $\ell^1$ condition) with the algebraic invertibility of the transfer function $f(T)$. We establish a rigorous operator theoretic foundation for these arguments proving that for $f \in \mathbb{W}_+$, the Wiener algebra, $f(T)$ is well defined, that $\| f(T) \| = \| f \|_{\infty}$ and that $f(T) = T_f$, the Toeplitz operator. |
| title | Stationary Process Invertibility and the Unilateral Shift Operator |
| topic | Functional Analysis Statistics Theory 60G10 G.3 |
| url | https://arxiv.org/abs/2604.02336 |