Stability of trigonometric approximation in $L^p$ and applications to prediction theory
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866917935350546432 |
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| author | Klotz, Lutz Frank, Michael |
| author_facet | Klotz, Lutz Frank, Michael |
| contents | Let $Γ$ be an LCA group and $(μ_n)$ be a sequence of bounded regular Borel measures on $Γ$ tending to a measure $μ_0$. Let $G$ be the dual group of $Γ$, $S$ be a non-empty subset of $G \setminus \{ 0 \}$, and $[{\mathcal T}(S)]_{μ_n,p}$ the subspace of $L^p(μ_n)$, $p \in (0,\infty)$, spanned by the characters of $Γ$ which are generated by the elements of $S$. The limit behaviour of the sequence of metric projections of the function $1$ onto $[{\mathcal T}(S)]_{μ_n,p}$ as well as of the sequence of the corresponding approximation errors are studied. The results are applied to obtain stability theorems for prediction of weakly stationary or harmonizable symmetric $p$-stable stochastic processes. Along with the general problem the particular cases of linear interpolation or extrapolation as well as of a finite or periodic observation set are studied in detail and compared to each other. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2104_13132 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Stability of trigonometric approximation in $L^p$ and applications to prediction theory Klotz, Lutz Frank, Michael Statistics Theory Probability 60G25 (Primary) 62M10, 91B84 (Secondary) Let $Γ$ be an LCA group and $(μ_n)$ be a sequence of bounded regular Borel measures on $Γ$ tending to a measure $μ_0$. Let $G$ be the dual group of $Γ$, $S$ be a non-empty subset of $G \setminus \{ 0 \}$, and $[{\mathcal T}(S)]_{μ_n,p}$ the subspace of $L^p(μ_n)$, $p \in (0,\infty)$, spanned by the characters of $Γ$ which are generated by the elements of $S$. The limit behaviour of the sequence of metric projections of the function $1$ onto $[{\mathcal T}(S)]_{μ_n,p}$ as well as of the sequence of the corresponding approximation errors are studied. The results are applied to obtain stability theorems for prediction of weakly stationary or harmonizable symmetric $p$-stable stochastic processes. Along with the general problem the particular cases of linear interpolation or extrapolation as well as of a finite or periodic observation set are studied in detail and compared to each other. |
| title | Stability of trigonometric approximation in $L^p$ and applications to prediction theory |
| topic | Statistics Theory Probability 60G25 (Primary) 62M10, 91B84 (Secondary) |
| url | https://arxiv.org/abs/2104.13132 |