I have a question regarding the possibility of constructing a Discrete Wavelet Transform based on a scaling function having Gaussian decay (and no more decay than that). More specifically, I am thinking in using a scaling function which might be a finite linear combination of Hermite functions.
I recall that the Hermite functions $\left\lbrace h_n\right\rbrace_{n=1}^{+\infty}$ defined by $$h_n(x)=(-1)^n (2^n n! \sqrt{n})^{-\frac{1}{2}}e^{\frac{x^2}{2}} \tfrac{d^n}{dx^n}e^{-x^2}$$ form an orthonormal basis in $L^2(\mathbb{R})$.
According to the book Ten lectures on Wavelets by I. Daubechies (see p. 145), a strategy to build examples of new orthonormal wavelet bases may have as a starting point a choice of a suitable scaling function $\phi$ which satisfies the following properties:
- $\phi$ and its Fourier transform $\hat{\phi}$ have a reasonable decay,
- there exists a sequence $c\in l^2(\mathbb{Z})$ such that $$\phi(x)= \sum_n c_n \phi(2x-n), $$
- there are numbers $0<\alpha$, $\beta<+\infty$ such that $$\alpha \leq \sum_{l\in \mathbb{Z}} |\hat{\phi}(\xi+2\pi l)|^2\leq \beta,$$
- $\int_{\mathbb{R}}\phi(x)\,dx \neq 0$.
My question is: is it known if a scaling function satisfying these properties and such that it is a finite linear combination of Hermite functions exist?
The fourth condition imply that we could only take even Hermite functions $h_{2n}$ since $h_{2n+1}$ are odd functions; but it is not clear to me whether we can find finite linear combinations of even Hermite functions such that the second and third conditions hold.
I find that this is a rather natural question since Hermite functions have gaussian decay and are eigenfunctions of the Fourier transform, thus, they have good space-frequency localization properties. However, I was not able to find an answer to my question in the literature.
I am aware about the Mexican Hat Wavelet (MHW), which is nothing else than the second derivative of the Gaussian function, but everything I found in the literature regarding the MHW in an Euclidean setting is related to Continuous Wavelet Transforms, not DWT.
