edit: More precise explanation
Let's consider a tight frame $w_1, w_2, w_3$ for all $v \in \Bbb{R}^4$. What happens if I perform the following operation: $\sum_{i=1}^{I} b_i \sum_{n=1}^{3} a_{n,i} w_{n}$ where all $b_i,a_{n,i} \in \Bbb{R}$
Is this still a tight frame? And if not in the general case, is there conditions under which it will be? For instance constraints on a and b, or a different sum operation like mentioned below. This is really a question about developing intuition, so even indirect comments would be highly appreciated!
Let $\{f_k^{(i)} \}_{k \in \mathbb{N}} \subset \mathcal{H}$, where $i = 1,..., N$ be tight frames for $\mathcal{H}$ with frame bounds $A_i$. By definition, it holds for all $i = 1, ..., N$ that $$ A_i \|f\|^2 = \sum_{k \in \mathbb{N}} |\langle f, f_k^{(i)} \rangle |^2. $$ Thus $$ \sum_{i = 1}^N A_i \|f \|^2 = \sum_{i = 1}^N \sum_{k \in \mathbb{N}} |\langle f, f_k^{(i)} \rangle |^2, $$ and hence $\cup_{i = 1}^N \{f_k^{(i)} \}_{k \in \mathbb{N}}$ is a tight frame for $\mathcal{H}$ with frame bound $\sum_{i = 1}^N A_i$.