## 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

Publisher Copyright:© 2016

## Keywords

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