Some properties of periodic B-spline collocation matrices

SynopsisIn three recent papers by Cavaretta et al., progress has been made in understanding the structure of bi-infinite totally positive matrices which have a block Toeplitz structure. The motivation for these papers came from certain problems of infinite spline interpolation where total positivity played an important role.In this paper, we re-examine a class of infinite spline interpolation problems. We derive new results concerning the associated infinite matrices (periodic B-spline collocation matrices) which go beyond consequences of the general theory. Among other things, we identify the dimension of the null space of these matrices as the width of the largest band of strictly positive elements.

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Introduction

Totally Nonnegative Matrices ◽

10.23943/princeton/9780691121574.003.0001 ◽

2011 ◽

Author(s):

Shaun M. Fallat ◽

Charles R. Johnson

Keyword(s):

Total Positivity ◽

Nonnegative Matrices ◽

Extant Literature ◽

Totally Positive ◽

Positive Matrices ◽

Class Of Matrices ◽

Totally Positive Matrices

This introductory chapter is an overview of totally positive (or nonnegative) matrices (TP or TN matrices). Positivity has roots in every aspect of pure, applied, and numerical mathematics. The subdiscipline, total positivity, also is entrenched in nearly all facets of mathematics. At first it may appear that the notion of total positivity is artificial; however, this class of matrices arises in a variety of important applications. Historically, the theory of totally positive matrices originated from the pioneering work of Gantmacher and Krein in 1960. The chapter explores the extant literature on total positivity since then, before proceeding to the definitions and notations to be used in the rest of this volume. It also provides a brief overview of the succeeding chapters.

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Sign non‐reversal property for totally non‐negative and totally positive matrices, and testing total positivity of their interval hull

Bulletin of the London Mathematical Society ◽

10.1112/blms.12475 ◽

2021 ◽

Author(s):

Projesh Nath Choudhury ◽

M. Rajesh Kannan ◽

Apoorva Khare

Keyword(s):

Total Positivity ◽

Totally Positive ◽

Positive Matrices ◽

Interval Hull ◽

Totally Positive Matrices

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Some Conjectures for Immanants

Canadian Journal of Mathematics ◽

10.4153/cjm-1992-066-1 ◽

1992 ◽

Vol 44 (5) ◽

pp. 1079-1099 ◽

Cited By ~ 14

Author(s):

J. R. Stembridge

Keyword(s):

Total Positivity ◽

Supporting Evidence ◽

Combinatorial Structures ◽

The Third ◽

Totally Positive ◽

Positive Matrices ◽

Nonnegative Coefficients ◽

Totally Positive Matrices ◽

Real Matrices

AbstractWe present a series of conjectures for immanants, together with the supporting evidence we possess for them. The conjectures are loosely organized into three families. The first concerns inequalities involving the immanants of totally positive matrices (Le.,real matrices with nonnegative minors). This includes, for example, the conjecture that immanants of totally positive matrices are nonnegative. The second family involves the immanants of Jacobi-Trudi matrices. These conjectures were suggested by a previous conjecture of Goulden and Jackson (recently proved by C. Greene) that the immanants of Jacobi-Trudi matrices are polynomials with nonnegative coefficients. The third family involves geometric and combinatorial structures associated with total positivity and paths in acyclic digraphs.

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