On multivariate discrete least squares

For a positive integer n∈N we introduce the index set N_n:={1,2,…,n}. Let X:={x_i:i∈N_n} be a distinct set of vectors in R^d, Y:={y_i:i∈N_n} a prescribed data set of real numbers in R and F:={f_j:j∈N_m},m<n, a given set of real valued continuous functions defined on some neighborhood O of R^d containing X. The discrete least squares problem determines a (generally unique) function f=∑_j∈Nmc_j^⋆f_j∈spanF which minimizes the square of the ℓ²−norm ∑i∈N_n(∑j∈N_mc_jf_j(x_i)−y_i)² over all vectors (c_j:j∈N_m)∈R^m. The value of f at some s∈O may be viewed as the optimally predicted value (in the ℓ²−sense) of all functions in spanF from the given data X={x_i:i∈N_n} and Y={y_i:i∈N_n}. We ask “What happens if the components of X and s are nearly the same”. For example, when all these vectors are near the origin in R^d. From a practical point of view this problem comes up in image analysis when we wish to obtain a new pixel value from nearby available pixel values as was done in [2], for a specified set of functions F. This problem was satisfactorily solved in the univariate case in Section 6 of Lee and Micchelli (2013). Here, we treat the significantly more difficult multivariate case using an approach recently provided in Yeon Ju Lee, Charles A. Micchelli and Jungho Yoon (2015).