On multivariate discrete least squares

Abstract

For a positive integer n∈N we introduce the index set Nn:={1,2,…,n}. Let X:={xi:i∈Nn} be a distinct set of vectors in Rd, Y:={yi:i∈Nn} a prescribed data set of real numbers in R and F:={fj:j∈Nm},m<n, a given set of real valued continuous functions defined on some neighborhood O of Rd containing X. The discrete least squares problem determines a (generally unique) function f=∑j∈Nmcj⋆fj∈spanF which minimizes the square of the ℓ2−norm ∑i∈Nn(∑j∈Nmcjfj(xi)−yi)2 over all vectors (cj:j∈Nm)∈Rm. 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={xi:i∈Nn} and Y={yi:i∈Nn}.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 Rd. 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).