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On multivariate discrete least squares

Title
On multivariate discrete least squares
Authors
Lee Y.J.Micchelli C.A.Yoon J.
Ewha Authors
윤정호
SCOPUS Author ID
윤정호scopus
Issue Date
2016
Journal Title
Journal of Approximation Theory
ISSN
0021-9045JCR Link
Citation
vol. 211, pp. 78 - 84
Keywords
Collocation matrixMultivariate least squaresMultivariate Maclaurin expansionWronskian
Publisher
Academic Press Inc.
Indexed
SCI; SCIE; SCOPUS WOS scopus
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). © 2016
DOI
10.1016/j.jat.2016.07.005
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자연과학대학 > 수학전공 > Journal papers
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