Namespaces
Variants
Actions

Difference between revisions of "Total positivity"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (tex encoded by computer)
m (Undo revision 49003 by Ulf Rehmann (talk))
Tag: Undo
Line 1: Line 1:
<!--
+
A real-valued function (or kernel) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110030/t1100301.png" /> on two linearly ordered sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110030/t1100302.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110030/t1100303.png" /> is called totally positive if for each positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110030/t1100304.png" /> and for all
t1100301.png
 
$#A+1 = 32 n = 0
 
$#C+1 = 32 : ~/encyclopedia/old_files/data/T110/T.1100030 Total positivity
 
Automatically converted into TeX, above some diagnostics.
 
Please remove this comment and the {{TEX|auto}} line below,
 
if TeX found to be correct.
 
-->
 
  
{{TEX|auto}}
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110030/t1100305.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
{{TEX|done}}
 
 
 
A real-valued function (or kernel) $  K ( x,y ) $
 
on two linearly ordered sets  $  X $
 
and  $  Y $
 
is called totally positive if for each positive integer  $  m $
 
and for all
 
 
 
$$ \tag{a1 }
 
x _ {1} < \dots < x _ {m} ;  y _ {1} < \dots < y _ {m} ,  x _ {i} \in X,  y _ {j} \in Y,
 
$$
 
  
 
one has
 
one has
  
$$ \tag{a2 }
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110030/t1100306.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
K \left (
 
  
If both $  X $
+
If both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110030/t1100307.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110030/t1100308.png" /> are finite sets, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110030/t1100309.png" /> can be considered a finite [[Matrix|matrix]] and then one calls it a totally positive matrix. Consequently, a totally positive matrix is a matrix whose minors (cf. [[Minor|Minor]]) are all non-negative. When all the inequalities are strict in these definitions, the matrix or kernel is called strictly totally positive. Some authors replace the term  "total positivity"  by  "totally non-negative matrixtotally non-negative kerneltotal non-negativity"  and  "strict total positivity"  by  "total positivity" . When (a2) holds only for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110030/t11003010.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110030/t11003011.png" /> is a positive integer, the kernel is called totally positive of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110030/t11003013.png" />.
and $  Y $
 
are finite sets, then $  K $
 
can be considered a finite [[Matrix|matrix]] and then one calls it a totally positive matrix. Consequently, a totally positive matrix is a matrix whose minors (cf. [[Minor|Minor]]) are all non-negative. When all the inequalities are strict in these definitions, the matrix or kernel is called strictly totally positive. Some authors replace the term  "total positivity"  by  "totally non-negative matrixtotally non-negative kerneltotal non-negativity"  and  "strict total positivity"  by  "total positivity" . When (a2) holds only for $  m \leq  r $,  
 
where $  r $
 
is a positive integer, the kernel is called totally positive of order $  r $.
 
  
 
Sign-regularity is an extension of the concept of total positivity. For it, (a2) is replaced by the weaker requirement that
 
Sign-regularity is an extension of the concept of total positivity. For it, (a2) is replaced by the weaker requirement that
  
$$ \tag{a3 }
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110030/t11003014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>
\epsilon _ {m} K \left (
 
  
for a given sequence $  ( \epsilon _ {1} \dots \epsilon _ {r} ) $
+
for a given sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110030/t11003015.png" /> with each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110030/t11003016.png" />. In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110030/t11003017.png" /> is said to be sign-regular of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110030/t11003020.png" />, with signature sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110030/t11003021.png" />. Analogously one can define strict sign-regularity.
with each $  \epsilon _ {m} = \pm  1 $.  
 
In this case $  K $
 
is said to be sign-regular of order $  r $,  
 
with signature sequence $  ( \epsilon _ {1} \dots \epsilon _ {r} ) $.  
 
Analogously one can define strict sign-regularity.
 
  
If $  K $
+
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110030/t11003022.png" /> is a totally positive function of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110030/t11003023.png" /> of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110030/t11003024.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110030/t11003025.png" /> is a called a Pólya frequency function (respectively, a Pólya frequency sequence) of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110030/t11003026.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110030/t11003027.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110030/t11003028.png" />, the integers).
is a totally positive function of order $  r $
 
of the form $  K ( x,y ) = f ( x - y ) $,  
 
then $  f $
 
is a called a Pólya frequency function (respectively, a Pólya frequency sequence) of order $  r $
 
when $  X = Y = \mathbf R $(
 
respectively, $  X = Y = \mathbf Z $,  
 
the integers).
 
  
Simple examples of totally positive functions are $  K ( x,y ) = e ^ {xy } $
+
Simple examples of totally positive functions are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110030/t11003029.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110030/t11003030.png" /> and
with $  X = Y = \mathbf R $
 
and
 
  
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110030/t11003031.png" /></td> </tr></table>
K ( x,y ) = \left \{
 
\begin{array}{l}
 
{1 \  \textrm{ if  }  a \leq  x \leq  y \leq  b, } \\
 
{0 \  \textrm{ if  }  a \leq  x \leq  y \leq  b, }
 
\end{array}
 
\right .
 
$$
 
  
with $  X = Y = [ a,b ] $.  
+
with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110030/t11003032.png" />. A well-known example of a strictly totally positive matrix on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110030/t11003033.png" /> is the Vandermonde matrix (cf. also [[Vandermonde determinant|Vandermonde determinant]])
A well-known example of a strictly totally positive matrix on $  X = Y = \mathbf R $
 
is the Vandermonde matrix (cf. also [[Vandermonde determinant|Vandermonde determinant]])
 
  
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110030/t11003034.png" /></td> </tr></table>
V ( x _ {1} \dots x _ {n} ) = \left ( x _ {i}  ^ {j} \right ) _ {1 \leq  i \leq  n;0 \leq  j \leq  m }  ,
 
$$
 
  
defined for $  x _ {1} < \dots < x _ {n} $.
+
defined for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t110/t110030/t11003035.png" />.
  
 
Generalizations of all these concepts can be found in [[#References|[a4]]]. This useful book has, for many years, been the only one entirely devoted to the theory of total positivity. Interesting historical facts concerning the development of this subject can be found scattered throughout its chapters.
 
Generalizations of all these concepts can be found in [[#References|[a4]]]. This useful book has, for many years, been the only one entirely devoted to the theory of total positivity. Interesting historical facts concerning the development of this subject can be found scattered throughout its chapters.

Revision as of 14:53, 7 June 2020

A real-valued function (or kernel) on two linearly ordered sets and is called totally positive if for each positive integer and for all

(a1)

one has

(a2)

If both and are finite sets, then can be considered a finite matrix and then one calls it a totally positive matrix. Consequently, a totally positive matrix is a matrix whose minors (cf. Minor) are all non-negative. When all the inequalities are strict in these definitions, the matrix or kernel is called strictly totally positive. Some authors replace the term "total positivity" by "totally non-negative matrixtotally non-negative kerneltotal non-negativity" and "strict total positivity" by "total positivity" . When (a2) holds only for , where is a positive integer, the kernel is called totally positive of order .

Sign-regularity is an extension of the concept of total positivity. For it, (a2) is replaced by the weaker requirement that

(a3)

for a given sequence with each . In this case is said to be sign-regular of order , with signature sequence . Analogously one can define strict sign-regularity.

If is a totally positive function of order of the form , then is a called a Pólya frequency function (respectively, a Pólya frequency sequence) of order when (respectively, , the integers).

Simple examples of totally positive functions are with and

with . A well-known example of a strictly totally positive matrix on is the Vandermonde matrix (cf. also Vandermonde determinant)

defined for .

Generalizations of all these concepts can be found in [a4]. This useful book has, for many years, been the only one entirely devoted to the theory of total positivity. Interesting historical facts concerning the development of this subject can be found scattered throughout its chapters.

The theory of totally positive matrices was developed in the 1930s by F.R. Gantmacher and M.G. Krein in connection with vibrations of mechanical systems. Green functions of certain differential operators provide important examples of totally positive matrices. Independently, I.J. Schoenberg developed the theory of total positivity in connection with the variation diminishing properties of matrices, giving rise to spline theory. Schoenberg also initiated the study of Pólya frequency functions and sequences. Much of this work was influenced by the work of O.D. Kellog, G. Pólya, M. Fekete, T.S. Motzkin to name only a few of the many mathematicians who worked in various aspects of this subject. Spectral properties of totally positive kernels were studied by Gantmacher and Krein. An interesting and detailed discussion of the history of this development with a self-contained presentation of the main results can be found in [a5].

Starting in the late 1950s, S. Karlin began to demonstrate in numerous publications on the subject the breath of application and depth of mathematical importance of total positivity. Under Karlin's influence, many authors have been motivated to investigate the application of total positivity to approximation theory, analysis, statistics, biology, and geometric modeling. See [a6] for a valuable reference source for information on total positivity.

Totally positive matrices form a subclass of the non-negative matrices. See [a1] for a very interesting survey of totally positive matrices up to 1987; it contains many of their properties available up to that time. In 1996 a new book, [a2], appeared, devoted entirely to the theory and applications of total positivity. It is an excellent source of information on the exciting recent developments in the subject with a special emphasis on its applications. For example, it has chapters devoted to the connections between spline functions and total positivity, others to Chebychev spaces, totally positive matrices, combinatorics, statistics, integral equations, geometric modeling, complex analysis and nonlinear analysis.

One of the most recent fields where total positivity has interesting applications is that of geometric modeling. In the design of curves it is useful to be able to predict and control the shape of the curve from the shape of the control polygon of the curve, determined by certain points. There are several articles in [a2] which address this important practical problem.

Totally positive functions arise quite naturally in the theory of diffusion stochastic processes (cf. also Diffusion process). For a birth-and-death process, the transition probability function is totally positive. These questions will be addressed in Volume II of [a4] which, unfortunately, has yet (1996) to appear. A series of papers written by Karlin and J.L. McGregor address this fascinating connection to total positivity, see, for example, [a3].

References

[a1] T. Ando, "Totally positive matrices" Linear Alg. & Its Appl. , 90 (1987) pp. 165–219
[a2] "Total positivity and its applications" M. Gasca (ed.) C.A. Micchelli (ed.) , Kluwer Acad. Publ. (1996)
[a3] S. Karlin, J.L. McGregor, "Classical difussion processes and total positivity" J. Math. Anal. Appl. , 1 (1960) pp. 163–183
[a4] S. Karlin, "Total positivity" , I , Stanford Univ. Press (1968)
[a5] A. Pinkus, "Spectral properties of totally positive kernels and matrices" M. Gasca (ed.) C.A. Micchelli (ed.) , Total Positivity and its Applications , Kluwer Acad. Publ. (1996) pp. 477–511
[a6] L.L. Schumaker, "Spline functions: Basic theory" , Wiley (1981)
How to Cite This Entry:
Total positivity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Total_positivity&oldid=49003
This article was adapted from an original article by M. GascaCh. Micchelli (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article