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 L₂(Ω ) with fewest number of terms, each of them is a combination of one-variable function. A system of nonlinear integral equations has been derived as an eigenvalue problem of gradient operator in the said space. It 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 fewest terms and exponential rapidity of convergence. It is also proved as a corollary, the greatest eigenvalue of the integral operators has multiplicity 1 if the dimension of the underlying space Rⁿ, n = 2, 4 and 6.