On the Cholesky factorization of the Gram matrix of locally supported functions |
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Authors: | T N T Goodman C A Micchelli G Rodriguez S Seatzu |
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Institution: | (1) Department of Mathematical Sciences, University of Dundee, DD1 4HN Dundee, Scotland, U.K.;(2) IBM Research Division, T.J. Watson Research Center, P.O. Box 218, 10598 Yorktown Heights, NY, USA;(3) Department of Mathematics, University of Cagliari, Viale Merello 92, 09123 Cagliari, Italy |
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Abstract: | Cholesky factorization of bi-infinite and semi-infinite matrices is studied and in particular the following is proved. If a bi-infinite matrixA has a Cholesky factorization whose lower triangular factorL and its lower triangular inverse decay exponentially away from the diagonal, then the semi-infinite truncation ofA has a lower triangular Cholesky factor whose elements approach those ofL exponentially. This result is then applied to studying the asymptotic behavior of splines obtained by orthogonalizing a large finite set of B-splines, in particular identifying the limiting profile when the knots are equally spaced.The first and second authors were partially supported by Nato Grant #920209, the second author also by the Alexander von Humboldt Foundation, and the last two authors by the Italian Ministry of University and Scientific and Technological Research. |
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Keywords: | Cholesky factorization Gram matrix orthogonal splines B-splines |
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