What is a waveform dictionary?

Question

If I google "waveform dictionary" or "waveform dictionaries" I find some papers that use these mathematical objects e.g. The Analysis of Foreign Exchange Data Using Waveform Dictionaries. For example, in the above-mentioned paper, we can read that

A waveform dictionary is a class of transforms that generalizes both windowed Fourier transforms and wavelets

Anyway, a formal mathematical definition seems to be missing. What is a waveform dictionary? Is it a frame (i.e. the generalization of Riesz basis)? I think it is not a basis because it should be overcomplete. References, papers, solutions are welcome.

Thanks!

Answer 1

By looking for the speaker giving the talk that I linked in my comment on your question, I was able to find this Arxiv paper that gives the following explicit definition of a waveform dictionary.

Let $\gamma = (t,\xi ,u) \in \Gamma = \Bbb R_{>0} \times \Bbb R^2$. A waveform dictionary $\mathcal G$ is a collection of $L^2(\Bbb R)$ functions of the form $$ G_\gamma(x) = \frac 1{\sqrt{t}} g\left(\frac{x-u}{t}\right)e^{2 \pi i \xi x}, $$ where $g \in L^2(\Bbb R)$ is called the window function that satisfies $\|g\|_{L^2(\Bbb R)} = 1, g(0) \neq 0,$ and the integral of $g$ is non-zero. The function $G_\gamma$ is known as a time-frequency atom.

In comparison to a Gabor system, we act on the window function not only with time-shift and frequency-shift operators, but also with some kind of "time dilation" operator $f(x) \mapsto \frac 1{\sqrt{t}}f(x/t)$.

The paper also cites the following reference, which seems promising:

Ole Christensen et al. An introduction to frames and Riesz bases, volume 7. Springer, 2003.