For a positive integer n∈Nn∈N we introduce the index set Nn:={1,2,…,n}Nn:={1,2,…,n}. Let X:={xi:i∈Nn}X:={xi:i∈Nn} be a distinct   set of vectors in RdRd, Y:={yi:i∈Nn}Y:={yi:i∈Nn} a prescribed   data set of real numbers in RR and F:={fj:j∈Nm},m<nF:={fj:j∈Nm},m<n, a given set of real valued continuous   functions defined on some neighborhood OO of RdRd containing XX. The discrete least squares problem determines a (generally unique) function f=∑j∈Nmcj⋆fj∈spanF which minimizes the square of the l2−l2−norm #R##N##R##N#∑i∈Nn(∑j∈Nmcjfj(xi)−yi)2#R##N##R##N#over all vectors (cj:j∈Nm)∈Rm(cj:j∈Nm)∈Rm. The value of ff at some s∈Os∈O may be viewed as the optimally predicted value (in the l2−l2−sense) of all   functions in spanFspanF from the given data X={xi:i∈Nn}X={xi:i∈Nn} and Y={yi:i∈Nn}Y={yi:i∈Nn}.#R##N##R##N#We ask “What happens if the components of XX and ss are nearly the same”. For example, when all these vectors are near the origin in RdRd. 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 FF.#R##N##R##N#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).
['Yeon Ju Lee', 'Charles A. Micchelli', 'Jungho Yoon']
On multivariate discrete least squares