Unexpected Uncertainty Principle for Disc Banach Spaces
K. Mahesh Krishna
TL;DR
The paper introduces an unexpected uncertainty principle for disc Banach spaces using unbounded continuous $p$-Schauder frames with $0<p<1$. It derives a lower bound on the product $\|\theta_f x\|_0\|\theta_g x\|_0$ in terms of cross-frame evaluations, specifically $\frac{1}{\left(\sup_{n,m}|f_n(\omega_m)|\right)^p\left(\sup_{n,m}|g_m(\tau_n)|\right)^p}$, via Garling’s inequality and frame expansions. This result is presented as counterintuitive compared with known bounded and unbounded uncertainty principles in Banach spaces. The findings illuminate a novel interaction between unbounded p-Schauder frames in the $0<p<1$ regime and suggest potential refinements connected to Tao’s uncertainty principle in prime dimensions.
Abstract
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.