Structure of Invertible (Bi)infinite Totally Positive Matrices
TLDR
In this paper, it was shown that a non-invertible totally positive matrix A has one and only one "main diagonal", which is the property that all finite sections of A principal with respect to this diagonal are invertible and their inverses converge boundedly and entrywise to A I.About:
This article is published in Linear Algebra and its Applications.The article was published on 1982-10-01 and is currently open access. It has received 15 citations till now. The article focuses on the topics: Anti-diagonal matrix & Band matrix.read more
Citations
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Journal ArticleDOI
Totally positive matrices
TL;DR: A concise survey on totally positive matrices and related topics can be found in this paper, where the authors present a unified method for computing the total positivity of a matrices.
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On the existence of interpolating projections onto spline spaces
TL;DR: In this paper, sufficient conditions for the existence of a bounded interpolating projection onto subspaces of C[0, 1] are found, and for spaces of piecewise polynomial functions the projection can be bounded by the B-spline basis condition number.
Journal ArticleDOI
Cholesky factorization of positive definite bi-infinite matrices
TL;DR: A Cholesky factorization for positive definite bi-infinite matrices is presented and applications to block Toeplitz matrices, signal processing, and spline interpolation are derived.
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LU -factorization of order bounded operators on Banach sequence spaces
Kevin T. Andrews,Joseph D. Ward +1 more
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factorization of operators on
TL;DR: In this paper, it was shown that uniform invertibility of the compressions of an operator is not sufficient to insure an LU-factorization of the operator, thus answering a question of de Boor, Jia, and Pinkus.
References
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Inverses of Band Matrices and Local Convergence of Spline Projections
TL;DR: In this paper, it was shown that the size of the entries in the inverse of a band matrix can be bounded in terms of the norm of the matrix, the norm norm of its inverse and the bandwidth.
Book ChapterDOI
Odd-degree spline interpolation at a biinfinite knot sequence
TL;DR: It is shown that for an arbitrary strictly increasing knot sequence t = (t sub 1) infinity to minus infinity and for every i, there exists exactly one fundamental spline L sub i, of order 2r whose r-th derivative is square integrable.
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A factorization theorem for banded matrices
TL;DR: A factorization theorem for strictly m-banded totally positive matrices is proved, which shows that such a matrix is a product of m one- banded matrices with positive entries.
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The Inverse of a Totally Positive Biinfinite Band Matrix.
TL;DR: In this paper, it was shown that a bounded bi-infinite banded totally positive matrix A is boundedly invertible iff there is one and only one bounded sequence mapped by A to the sequence ((-)')■, i.e., the inverse is the bounded pointwise limit of inverses of finite sections of A principal with respect to a particular diagonal.
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On the Solvability of Certain Systems of Linear Difference Equations
TL;DR: For a certain class of block Toeplitz matrices, the smallest sector containing the zeros of the determinant for the corresponding symbol was identified in this article, where the smallest sectors are the smallest regions containing the smallest zeros for each symbol.