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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2404.00910 |
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| _version_ | 1866917692025339904 |
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| author | Krishna, K. Mahesh |
| author_facet | Krishna, K. Mahesh |
| contents | Let $(\{f_n\}_{n=1}^\infty, \{τ_n\}_{n=1}^\infty)$ and $(\{g_n\}_{n=1}^\infty, \{ω_n\}_{n=1}^\infty)$ be unbounded continuous p-Schauder frames ($0<p<1$) for a disc Banach space $\mathcal{X}$. Then for every $x \in ( \mathcal{D}(θ_f) \cap\mathcal{D}(θ_g))\setminus\{0\}$, we show that \begin{align}\label{UB} (1) \quad \quad \quad \quad \|θ_f x\|_0\|θ_g x\|_0 \geq \frac{1}{\left(\displaystyle\sup_{n,m \in \mathbb{N} }|f_n(ω_m)|\right)^p\left(\displaystyle\sup_{n, m \in \mathbb{N}}|g_m(τ_n)|\right)^p}, \end{align} where \begin{align*} & θ_f: \mathcal{D}(θ_f) \ni x \mapsto θ_fx := \{f_n(x)\}_{n=1}^\infty\in \ell^p(\mathbb{N}), \quad θ_g: \mathcal{D}(θ_g) \ni x \mapsto θ_gx := \{g_n(x)\}_{n=1}^\infty\in \ell^p(\mathbb{N}). \end{align*} Inequality (1) is unexpectedly different from both bounded uncertainty principle arXiv:2308.00312v1 and unbounded uncertainty principle arXiv:2312.00366v1 for Banach spaces. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_00910 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Unexpected Uncertainty Principle for Disc Banach Spaces Krishna, K. Mahesh Functional Analysis Information Theory Mathematical Physics 42C15 Let $(\{f_n\}_{n=1}^\infty, \{τ_n\}_{n=1}^\infty)$ and $(\{g_n\}_{n=1}^\infty, \{ω_n\}_{n=1}^\infty)$ be unbounded continuous p-Schauder frames ($0<p<1$) for a disc Banach space $\mathcal{X}$. Then for every $x \in ( \mathcal{D}(θ_f) \cap\mathcal{D}(θ_g))\setminus\{0\}$, we show that \begin{align}\label{UB} (1) \quad \quad \quad \quad \|θ_f x\|_0\|θ_g x\|_0 \geq \frac{1}{\left(\displaystyle\sup_{n,m \in \mathbb{N} }|f_n(ω_m)|\right)^p\left(\displaystyle\sup_{n, m \in \mathbb{N}}|g_m(τ_n)|\right)^p}, \end{align} where \begin{align*} & θ_f: \mathcal{D}(θ_f) \ni x \mapsto θ_fx := \{f_n(x)\}_{n=1}^\infty\in \ell^p(\mathbb{N}), \quad θ_g: \mathcal{D}(θ_g) \ni x \mapsto θ_gx := \{g_n(x)\}_{n=1}^\infty\in \ell^p(\mathbb{N}). \end{align*} Inequality (1) is unexpectedly different from both bounded uncertainty principle arXiv:2308.00312v1 and unbounded uncertainty principle arXiv:2312.00366v1 for Banach spaces. |
| title | Unexpected Uncertainty Principle for Disc Banach Spaces |
| topic | Functional Analysis Information Theory Mathematical Physics 42C15 |
| url | https://arxiv.org/abs/2404.00910 |