Bandlimited signals of several variables

The author considers that any multi-dimensional signal is a distribution of several variables. The definition of a distribution in n-variables is defined. Next, the Fourier transform of a distribution is introduced. The author then proceeds to the consideration of the frequency band of a signal and the concept of a bandlimited signal. The frequency spectrum is a linear functional on the function space Z and its support must be the frequency band of a signal. As the element of Z is a complex function of n-variables and hence the support will be a point set. But the space Z consists of entire (analytic) functions and therefore it is generally impossible to define the support of a frequency spectrum