Spectral representation of multi-valued functions

Multi-valued functions can be compute by multi-valued neural network. In order to use linear algebra method for analysis of multi-valued function, this paper gives the spectral representation of multi-valued functions. By define a checksum function of n variables, this paper gives an group of orthogonal bases. Any multi-valued function in GF( p)^pn can be uniquely represented as a linear combination of the bases. The coefficients are the correlations of the multi-valued function and the bases. All the coefficients combine the generalized spectrum of the multi-valued function. For a multi-valued function that can be realized by a multi-valued neuron, it is computable by depth-2 polynomial-size neural networks.