S-families, S-bases and the Fourier expansion theorem

In this paper we introduce the concept of ultra-linear combination for a certain class of families of tempered distributions. These new concept give a precise mathematical meaning to the tool of \linear superposition" used in physics and, in particular, in quantum mechanics as we can see in [Di], [Fu], [Pa] and [Sha]. More precisely the new concepts introduced in the paper are: S-family of tempered distributions, operator generated by an S-family, ul- tralinear combinations of an S-family, the S-ultralinear span of a family of class S, system of S-generators, S-ultralinear independence, S-bases, superposition of an S-family with respect to a family of distributions. Moreover, we state and prove a Fourier-type theorem and we give one its generalization.