论文标题
非架构和padic功能性韦尔奇边界
Non-Archimedean and p-adic Functional Welch Bounds
论文作者
论文摘要
我们证明了非Archimedean（分别为P-ADIC）BANACH空间版本的非Archimedean（p-Adic）Welch边界最近由Krishna M. Krishna获得。更确切地说，我们证明了以下结果。 1。令$ \ mathbb {k} $为一个非架构（完整）值的值，满足$ \ weft | \ sum_ {j = 1}^{n}^{n}λ_j^2 \ right | = \ max_ {1个1 \ leq j \ leq n $，对于所有$ n \ in \ mathbb {n}。$ let $ \ m artercal {x} $ as a $ d $ -dimensional nonarchimedean banach banach Space aby $ \ mathbb {K k} $。如果$ \ \ {j = 1}^n $是$ \ Mathcal {x} $和$ \ {f_j \} _ {j = 1}^n $是$ \ {f_j \}^n $是$ \ \ \ {对于所有$ 1 \ leq j \ leq n $和运算符$ s_ {f，τ}：\ text {sym}^m（\ Mathcal {x}）\ ni x \ ni x \ mapsto \ sum_ {j = 1} \ text {sym}^m（\ Mathcal {x}）$，是对角aliz的，然后\ begin {align} \ text {（non-Archimedean functional wellch bounds）} \ quad \ quad \ quad \ ax_ {1 \ max {1 \ leq j，k \ leq n，k \ leq n，k \ leq n，j \ neq k} \ neq K} \ n \ neq k} \ n \ n \ n \ n \ n | n | f_j（τ_k）f_k（τ_j）|^{m} \} \ geq \ frac {| n |^2} {\ left | {d+m-1 }。 \ end {align} 2。对于prime $ p $，令$ \ mathbb {q} _p $为p-adic编号字段。令$ \ mathcal {x} $为$ d $ -dimensional p-adic banach Space，$ \ mathbb {q} _p $。如果$ \ \ {j = 1}^n $是$ \ Mathcal {x} $和$ \ {f_j \} _ {j = 1}^n $是$ \ {f_j \}^n $是$ \ \ \ {对于所有$ 1 \ leq j \ leq n $，并且存在$ b \ in \ mathbb {q} _p $，这样，$ \ sum_ {j = 1}^{n}^{n} f_j^{\ otimes m}（x） \ text {sym}^m（\ Mathcal {x}），$然后\ begin {align} \ text {（p-adic functional welch bounds）} \ quad \ quad \ quad \ max_ {1 \ leq j，k \ leq j \ leq n，j \ leq n，j \ neq k} \ neq k} \ { | f_j（τ_k）f_k（τ_j）|^{m} \} \ geq \ frac {| n |^2} {\ left | {d+m-1 }。 \ end {align}我们制定了非架构的功能和p-adic功能Zauner猜想。
We prove the non-Archimedean (resp. p-adic) Banach space version of non-Archimedean (resp. p-adic) Welch bounds recently obtained by M. Krishna. More precisely, we prove following results. 1. Let $\mathbb{K}$ be a non-Archimedean (complete) valued field satisfying $\left|\sum_{j=1}^{n}λ_j^2\right|=\max_{1\leq j \leq n}|λ_j|^2$ for all $ λ_j \in \mathbb{K}, 1\leq j \leq n$, for all $n \in \mathbb{N}.$ Let $\mathcal{X}$ be a $d$-dimensional non-Archimedean Banach space over $\mathbb{K}$. If $\{τ_j\}_{j=1}^n$ is any collection in $\mathcal{X}$ and $\{f_j\}_{j=1}^n$ is any collection in $\mathcal{X}^*$ (dual of $\mathcal{X}$) satisfying $f_j(τ_j) =1$ for all $1\leq j \leq n$ and the operator $S_{f, τ} : \text{Sym}^m(\mathcal{X})\ni x \mapsto \sum_{j=1}^nf_j^{\otimes m}(x)τ_j^{\otimes m} \in \text{Sym}^m(\mathcal{X})$, is diagonalizable, then \begin{align} \text{(Non-Archimedean Functional Welch Bounds)} \quad \max_{1\leq j,k \leq n, j \neq k}\{|n|, |f_j(τ_k)f_k(τ_j)|^{m} \}\geq \frac{|n|^2}{\left|{d+m-1 \choose m}\right| }. \end{align} 2. For a prime $p$, let $\mathbb{Q}_p$ be the p-adic number field. Let $\mathcal{X}$ be a $d$-dimensional p-adic Banach space over $\mathbb{Q}_p$. If $\{τ_j\}_{j=1}^n$ is any collection in $\mathcal{X}$ and $\{f_j\}_{j=1}^n$ is any collection in $\mathcal{X}^*$ (dual of $\mathcal{X}$) satisfying $f_j(τ_j) =1$ for all $1\leq j \leq n$ and there exists $b \in \mathbb{Q}_p$ such that $ \sum_{j=1}^{n}f_j^{\otimes m}(x) τ_j^{\otimes m} =bx$ for all $ x \in \text{Sym}^m(\mathcal{X}),$ then \begin{align} \text{(p-adic Functional Welch Bounds)} \quad \max_{1\leq j,k \leq n, j \neq k}\{|n|, |f_j(τ_k)f_k(τ_j)|^{m} \}\geq \frac{|n|^2}{\left|{d+m-1 \choose m}\right| }. \end{align} We formulate non-Archimedean functional and p-adic functional Zauner conjectures.
