## A note on finite dual frame pairs.(English)Zbl 1365.42023

Let $$\mathbb{K}^d$$ denote either one of the finite dimensional spaces $$\mathbb{R}^d$$ or $$\mathbb{C}^d$$. A sequence $$\{e_n\}_{n=1}^N\subset \mathbb{K}^d$$ is said to be a frame for $$\mathbb{K}^d$$, with frame bounds $$A,B>0$$, provided that $A\|x\|^2\leq \sum_{n=1}^N|\langle x,e_n\rangle|^2\leq B\|x\|^2,\qquad x\in \mathbb{K}^d.$ The frame $$\{e_n\}_{n=1}^N$$ is said to be tight if one can take $$A=B$$ in the above inequality.
For any given frame $$\{e_n\}_{n=1}^N$$, it is always possible to find at least one sequence $$\{f_n\}_{n=1}^N\subset \mathbb{K}^d$$ such that $x=\sum_{n=1}^N\langle x,e_n\rangle f_n=\sum_{n=1}^N\langle x,f_n\rangle e_n,\qquad x\in \mathbb{K}^d.$ This $$\{f_n\}_{n=1}^N$$ is also a frame and the sequences $$\{e_n\}_{n=1}^N$$ and $$\{f_n\}_{n=1}^N$$ are called dual frames. A tight frame $$\{e_n\}_{n=1}^N$$ has $$\{\frac{1}{A}e_n\}_{n=1}^N$$ for (its canonical) dual frame. The following important result characterizes the possible norms of elements of a tight frame:
Given a sequence $$\{a_n\}_{n=1}^N\subset [0,\infty)$$ with $$N\geq d$$, (1) there exists a tight frame $$\{e_n\}_{n=1}^N$$ for $$\mathbb{K}^d$$ such that $$a_n=\langle e_n,e_n\rangle$$, $$n=1,\ldots,N$$, if and only if (2) $$\max_{1\leq n\leq N}a_n\leq \frac{1}{d}\sum_{n=1}^Na_n$$.
Inequality (2) is known as the fundamental inequality of tight frames.
The paper extends this result to a pair of dual frames $$\{e_n\}_{n=1}^N$$ and $$\{f_n\}_{n=1}^N$$ by characterizing the possible values of $$\langle e_n,f_n\rangle$$. This is Theorem 3.1 of the paper, which states that given $$\{\alpha_n\}_{n=1}^N\subset\mathbb{K}$$ with $$N>d$$, the following statements are equivalent:
(1)
There exist dual frames $$\{e_n\}_{n=1}^N$$ and $$\{f_n\}_{n=1}^N$$ for $$\mathbb{K}^d$$ such that $$\alpha_n=\langle e_n,f_n\rangle$$, $$1\leq n\leq N$$.
(2)
There exists a tight frame $$\{g_n\}_{n=1}^N$$ and a corresponding dual frame $$\{h_n\}_{n=1}^N$$ for $$\mathbb{K}^d$$ such that $$\alpha_n=\langle g_n,h_n\rangle$$, $$1\leq n\leq N$$.
(3)
$$d=\sum_{n=1}^N\alpha_n$$.
The assumption $$N>d$$ is due to the fact that for $$\{e_n\}_{n=1}^N$$ to be a frame for $$\mathbb{K}^d$$ it is necessary that $$N\geq d$$, and if $$N=d$$, any frame $$\{e_n\}_{n=1}^N$$ is a basis and has a unique dual frame, namely, $$\{\|e_n\|^{-2}e_n\}_{n=1}^N$$, in which case $$\alpha_n=1$$ for all $$n$$.
The paper then studies to a good extent the frames $$\{e_n\}_{n=1}^N$$ for $$\mathbb{K}^d$$ having (what the authors name) the full range property, that is, those frames $$\{e_n\}_{n=1}^N$$ such that for every $$\{\alpha_n\}_{n=1}^N\subset\mathbb{K}$$ with $$\sum_{n=1}^N\alpha_n=d$$, one can find a dual frame $$\{f_n\}_{n=1}^N$$ satisfying that $$\alpha_n=\langle e_n,f_n\rangle$$ for $$1\leq n\leq N$$.

### MSC:

 42C15 General harmonic expansions, frames
Full Text:

### References:

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