In his original optimal scheme of signal representation, Gabor implemented a Gaussian window and critical sampling density of the combined time-frequency (TF) space, and a complex-exponential, harmonic, kernel. This expansion was later investigated using different window functions, and/or sampling densities, and/or kernel functions. In the present work we investigate the application of two specific, non-standard, families of kernel functions.

The window function proposed by Gabor is the best in terms of the joint entropy, i.e. the compactness of the representation in the combined space. However, using this window function, or any other "well-behaved" localized function, with critical sampling of the combined Gabor space, entails a problem of stability, i.e., the set of Gabor elementary functions do not constitute a frame.

The purpose of this work is to examine advantages afforded by two specific families of non-exponential kernel functions. In particular, a "well localized" window function is used with these kernels in order to examine whether such a combination can overcome the problem of representation stability, inherent in the case of critical sampling.

We show that there is a solution to the problem of stability in the case of critical sampling by using unconventional kernel. In the case of the first kernel proposed in this study, we extract one of the unique properties of the solution, and generalize it with the goal to achieve stability. The set of Gabor elementary function does not constitute, in this case, a frame over the interval of its periodicity, but it does constitute a frame over subinterval of the periodicity. In the case of the second proposed kernel, we attempt to achieve stability by using variable resolution over the TF combined space, having low resolution for high frequencies and high resolution for low frequencies.

Using the first type of non-exponential kernel, we propose new ways of how to construct the best representation set for a given implementation. This is done by improving the frame properties and the TF localization of the representation set. It is shown that the localization in the frequency domain can be improved without effecting the time domain localization, but it can not be done in such a way that the frequency representation of the proposed elementary functions will be square integrable. Lastly, we also address the issue of controlling the redundancy in the L²(R) spanning set {g_mn(x)} by selection of the appropriate kernel function.