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For suitable choice of a one-parameter family of functions $\{ g_w:\mathbb{R} \rightarrow \mathbb{C} \}_{w\in \mathbb{R}}$, the following two statements are equivalent (modulo sets of measure $0$):

$\begin{align} I) \quad \tilde{f}(w) = \int_\mathbb{R} dx\ f(x) \overline{g_w (x)} \quad \forall w\in \mathbb{R}\\ II) \quad f(x) = \int_\mathbb{R} dw\ \tilde f(w) g_x (w) \quad \forall x\in \mathbb{R} \end{align}$

Here, $f\in L^2(\mathbb{R})$.

I am looking for examples of such functions $\{ g_w \}$. Here are two examples:

$\begin{align*} (1)\quad &g_w(x) = (2\pi)^{-1/2} e^{i w x} \quad \text{(Fourier transform)}\\ (2)\quad &g_w(x) = \frac{1}{\pi} \int_0^\infty dt\ \cos [\frac{1}{3}t^3 + (w+x)t] \quad \text{(Airy functions)} \end{align*}$

Edit: Let me add one more qualifier. I want $\{ g_w \}$ to also have the following property:

Property: there is some differential operator $L$ s.t. $g_w$ is an eigenfunction of $L\ \forall w\in \mathbb{R}$ -- and no two $g_w$ have the same eigenvalue.

In (1), $L = \frac{d}{dx}$. In (2), $L= \frac{d^2}{dx^2} - x$.

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Have you read about wavelets? There is a reasonable article on Wikipedia about them. –  Geoff Robinson Dec 27 '11 at 0:31
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