## Abstract

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∈Nm}c_{j}^{⋆}f_{j}∈spanF which minimizes the square of the ℓ^{2}−norm ∑i∈N_{n}(∑j∈N_{m}c_{j}f_{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 ℓ^{2}−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).

Original language | English |
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Pages (from-to) | 78-84 |

Number of pages | 7 |

Journal | Journal of Approximation Theory |

Volume | 211 |

DOIs | |

State | Published - 1 Nov 2016 |

### Bibliographical note

Funding Information:Yeon Ju Lee was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning ( NRF-2015R1C1A1A02037556 ). Jungho Yoon was supported by the grants NRF-2015-R1A5A1009350 and NRF-2015-R1D1A1A09057553 through the National Research Foundation of Korea . Charles A. Micchelli was supported by US National Science Foundation grant DMS-1522339 .

Publisher Copyright:

© 2016

## Keywords

- Collocation matrix
- Multivariate least squares
- Multivariate Maclaurin expansion
- Wronskian