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I'm currently studying tight frames and came across the following two systems of ONB in $\mathbb{C}^N$. The question is how to prove that both systems are $A$-tight frames and what the frame bounds are. As I am somewhat new to the topic, I would like to know whether my argumentation is right.

The systems are :

  1. $\{e_1, e_1, e_1, e_2, e_2, e_2, ..., e_N, e_N, e_N\}$
  2. $\left \{\frac{e_1}{1}, \frac{e_2}{\sqrt{2}}, \frac{e_2}{\sqrt{2}}, \frac{e_3}{\sqrt{3}}, \frac{e_3}{\sqrt{3}}, \frac{e_3}{\sqrt{3}}, ..., \frac{e_N}{\sqrt{N}}, \frac{e_N}{\sqrt{N}}, ..., \frac{e_N}{\sqrt{N}}\right \}$

We are in the Hilbert space $\mathbb{C}^N$ and $\{e_n\}_{n = 1}^N$ is the ONB.

My approach : Proof by calculation.

1st System

An $A$-tight frame is defined here as $A \cdot \left\| f \right\|_{\mathbb{C}^N}^2 \leq \sum_{n = 1}^N |\langle f, e_n \rangle|^2 \leq B \cdot \left\| f \right\|_{\mathbb{C}^N}^2$ with $A = B$.

Accordingly, I would define the operator $T : \mathbb{C}^N \rightarrow \mathbb{C}^N$ which describes the mapping of the ONB $e = \{e_n\}_{n = 1}^N$ to the 1st system. Then, I could write

$\left\| T \cdot e \right\|^2 = |e_1|^2 + |e_1|^2 + |e_1|^2 + |e_2|^2 + |e_2|^2 + |e_2|^2 + ... = 3 \cdot |e_1|^2 + 3 \cdot |e_2|^2 + ... = 3 \cdot \sum_{i = 1}^N |e_i|^2 = 3$

This holds even with equality, therefore also on both sides and hence it is a tight frame with $A = 3$.

2nd System

I can use the same approach with the corresponding operator $T^{'}$ and write

$\left\| T \cdot e \right\|^2 = \sum_{i = 1}^N \frac{i}{\sqrt{i}} |e_i|^2 \leq \sum_{i = 1}^N \frac{i}{\sqrt{i}} \cdot \sum_{i = 1}^N |e_i|^2 = \sum_{i = 1}^N \sqrt{i} \cdot \sum_{i = 1}^N |e_i|^2$

So I would have $A = \sum_{i = 1}^N \sqrt{i}$ which can be upper bounded as shown in this post Sum of Square roots formula to $A = \frac{2}{3} \cdot (N + 1/2)^{3/2}$. However, I have seen that this system forms a Parsevalframe, i.e. $A = 1$. So I must be wrong here!

So my concrete questions would be :

  • Is this enough to show that the systems form $tight$ frames?
  • Are the frame bounds correct (especially need help for system 2)?

I appreciate any comments, I am mainly trying to understand what I have done here. :-) Unfortunately I do not have a solution to these questions in my book!

banachBoy
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  • You should clarify a few things, otherwise it's impossible to answer your question: What is $T$? I guess it should be the analysis or the synthesis operator, in which case it does not(!) map $\mathbb{C}^N$ into itself. What do you mean by $|e_1|$, the Euclidean norm of the vector $e_1$? If yes, why do you use two different notations for the norm, namely $|\cdot|$ and $|\cdot|$? – Jake28 Jul 11 '22 at 13:25

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