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The paper analyzes the boundedness of the coefficients involved in Gaussian expansion series. These series arise from the reconstruction of bandlimited functions, applying the sampling theorem with Gaussians as reconstruction filters. The boundedness of the ideal coefficients is a previous requirement that should be imposed to the approximation function. This is due to the fact that the coefficient sequence should be absolutely summable. With this sort of requirements, the targeted function is guaranteed to exhibit finite energy so that it will be manageable from the viewpoint of the approximation theory. On the other hand, the bounds of the coefficients affect considerably to the approximation errors and consequently to the accuracy of the estimation. The major result of this work is formalized in a series of propositions where it is stated how the coefficients are upper bounded by a signal ''sinus cardinalis'' (sinc). Finally, an energy measure of the approximation error is determined as a mean square error. In this line, a number of results are presented in both the univariate and the multivariate case showing how these errors strongly depend on the coefficients in the Gaussian expansion.