On multivariate discrete least squares

Abstract

For a positive integer n is an element of N we introduce the index set N-n := {1, 2,..., n}. Let X := {x(i) : i is an element of N-n} be a distinct set of vectors in R-d, Y := {y(i) : i is an element of N-n} a prescribed data set of real numbers in R and F := {f(j) : j is an element of 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 = Sigma(j is an element of Nm) c(j)(star) f(j) is an element of spanF which minimizes the square of the l(2)-norm Sigma(i is an element of Nn) (Sigma(j is an element of Nm) c(j)f(j)(x(i)) - y(i))(2) over all vectors (c(j) : j is an element of N-m) is an element of R-m. The value of f at some s is an element of O may be viewed as the optimally predicted value (in the l(2)-sense) of all functions in spanF from the given data X = {x(i) : i is an element of N-n} and Y = {y(i) : i is an element of 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). (C) 2016 Published by Elsevier Inc.