On expansion with respect to Gabor frames generated by the Gaussian function (Q520622)

The paper focuses on expansion formulas for an arbitrary function \(f \in L_2(\mathbb{R})\) when one wishes to expand \(f\) in terms of a Gabor frame arising from an exponential window function. Earlier it has been shown by other authors that the system \[ g_{km}(x, \alpha_1, \alpha_2) = \exp \left(-\frac{(x - k\alpha_1)^2}{2}\right) e^{i m \alpha_2 x}, \quad k, m \in \mathbb{Z}, \tag{1} \] generates a Gabor frame with the window function \(g(x) = e^{-x^2/2}\) if for all positive numbers \(\alpha_1,\) \(\alpha_2\) the condition \(\alpha_1 \alpha_2 < 2\pi\) is satisfied. In this paper, explicit formulas are proposed for the case \( \alpha_1 \alpha_2 = \pi/n,\) \(n \in \mathbb{N},\) when the window function is the window function of the dual frame of (1). Such explicit formulas are claimed to improve existing algorithms for obtaining expansions with respect to such frames.