论文标题
无条件框架对的定量界限
Quantitative bounds for unconditional pairs of frames
论文作者
论文摘要
我们制定了关于帧乘数的定量有限维猜想，并证明[SB2]中的猜想1等效。 然后，我们为某些类别的帧乘数提供了猜想的解决方案。特别是，我们证明存在一个通用常数$κ> 0 $，因此对于所有$ c，β> 0 $和$ n \ in \ mathbb {n} $ in \ mathbb {n} $中的$ n is the是正确的。令$（x_j）_ {j = 1}^n $和$（f_j）_ {j = 1}^n $为有限的尺寸hilbert空间中的序列，该空间满足$ \ | x_j \ | = \ | = \ | | f_j \ | $ \ big \ | \ sum_ {j = 1}^n \ varepsilon_j \ langle x，f_j \ rangle x_j \ big \ | \ | \ leq c \ | x \ | x \ |，\ qquad \ qquad \ qquad \ qquad \ qquad \ textrm {for所有$ x \ in \ ell_2^m $和$ | = 1 $ | | $$如果$（f_j）_ {j = 1}^n $的框架运算符具有eigenvalues $λ_1\ geq ... \geqλ_m$和$λ_1\ leqβm^{ - 1}}} C $。对于$（x_j）_ {j = 1}^n $，也是如此。
We formulate a quantitative finite-dimensional conjecture about frame multipliers and prove that it is equivalent to Conjecture 1 in [SB2]. We then present solutions to the conjecture for certain classes of frame multipliers. In particular, we prove that there is a universal constant $κ>0$ so that for all $C,β>0$ and $N\in\mathbb{N}$ the following is true. Let $(x_j)_{j=1}^N$ and $(f_j)_{j=1}^N$ be sequences in a finite dimensional Hilbert space which satisfy $\|x_j\|=\|f_j\|$ for all $1\leq j\leq N$ and $$\Big\|\sum_{j=1}^N \varepsilon_j\langle x,f_j\rangle x_j\Big\|\leq C\|x\|, \qquad\textrm{ for all $x\in \ell_2^M$ and $|\varepsilon_j|=1$}. $$ If the frame operator for $(f_j)_{j=1}^N$ has eigenvalues $λ_1\geq...\geqλ_M$ and $λ_1\leq βM^{-1}\sum_{j=1}^Mλ_j$ then $(f_j)_{j=1}^N$ has Bessel bound $κβ^2 C$. The same holds for $(x_j)_{j=1}^N$.
