I am reading the definition of wavelet frame given from Ole Christensen's book "An Introduction to Frames and Riesz Bases." The definition reads: "A frame for $L^2(\mathbb{R})$ of the form $\left\{a^{j / 2} \psi\left(a^j x-k b\right)\right\}_{j, k \in \mathbb{Z}}$ is called a wavelet frame," where $a>1$, $b>0$, and $\psi \in L^2(\mathbb{R})$.
In my exploration, I stumbled upon the following set: $\left\{\frac{1}{\sqrt{t}} \psi\left(\frac{x-u}{t}\right) \right\}_{t\in \mathbb{N}_{\neq 0}, u\in \mathbb Z}$, where $\psi$ is a constant function equal to 1 with support $[-1/2, 1/2]$.
The original definition requires $a>1$, and if we try to transform $\frac{1}{\sqrt{t}}$ into the term $a^{j/2}$, given that $t\geq 1$, we find ourselves with $a\leq 1$. What implications does this observation have on the nature of my set? Can it still be considered a wavelet frame for $L^2(\mathbb{R})$?
This leads me to ponder: does the inability to meet the $a>1$ condition rule out the possibility of my set being a wavelet frame? Could it, at the very least, be classified as a frame, even if not strictly fitting the traditional wavelet frame definition?
Thanks!