Optimal Representation of Multivariate Functions or Data in Visualizable Low-dimensional Spaces

It is intended to find the best representation of high-dimensional functions or multivariate data in the L₂(Ω) space with the fewest number of terms, each of them is a combination of one-variable function. A system of non-linear integral equations has been derived as an eigenvalue problem of gradient operator in the above-said space. It is proved that the complete set of eigenfunctions generated by the gradient operator constitutes an orthonormal system, and any function of L₂(Ω) can be expanded with the fewest terms and exponential rapidity of convergence. It is also proved as a Corollary, all eigenvalues of the integral operators has multiplicity equal to 1 if the dimension of the underlying space Rⁿ is n = 2, 4 and 6.