摘要
It is intended to find the best representation of high-dimensional functions or multivariate data in L2(W) 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 anorthonormal system, and any function of L2(W) can beexpanded with fewest terms and exponential rapidity of convergence. It is also proved as a corollary, thegreatest eigenvalue of the integral operators hasmultiplicity 1 if the dimension of the underlying space nn = 2, 4 and 6.
It is intended to find the best representation of high-dimensional functions or multivariate data in L2(W) 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 anorthonormal system, and any function of L2(W) can beexpanded with fewest terms and exponential rapidity of convergence. It is also proved as a corollary, thegreatest eigenvalue of the integral operators hasmultiplicity 1 if the dimension of the underlying space nn = 2, 4 and 6.
