Namespaces
Variants
Actions

Difference between revisions of "Oscillating kernel"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
 
(One intermediate revision by one other user not shown)
Line 1: Line 1:
A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070480/o0704801.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070480/o0704802.png" />, such that for any points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070480/o0704803.png" />, which (when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070480/o0704804.png" />) include at least one interior point, the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070480/o0704805.png" /> is an [[Oscillating matrix|oscillating matrix]].
+
<!--
 +
o0704801.png
 +
$#A+1 = 5 n = 0
 +
$#C+1 = 5 : ~/encyclopedia/old_files/data/O070/O.0700480 Oscillating kernel
 +
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}}
 +
{{TEX|done}}
  
 
+
A function  $  K( x, s) $,
====Comments====
+
$  a \leq  x, s \leq  b $,
 
+
such that for any points  $  x _ {1} \dots x _ {n} \in [ a, b] $,
 +
which (when  $  n= 2 $)
 +
include at least one interior point, the matrix  $  \| K( x _ {i} , x _ {k} ) \| _ {1}  ^ {n} $
 +
is an [[oscillating matrix]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F.R. Gantmakher,   M.G. Krein,   "Oscillation matrices and kernels and small vibrations of mechanical systems" , Dept. Commerce USA. Joint Publ. Service  (1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Karlin,   "Total positivity" , Stanford Univ. Press  (1960)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  F.R. Gantmakher, M.G. Krein, "Oscillation matrices and kernels and small vibrations of mechanical systems" , Dept. Commerce USA. Joint Publ. Service  (1961)  (Translated from Russian)</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Karlin, "Total positivity" , Stanford Univ. Press  (1960)</TD></TR>
 +
</table>

Latest revision as of 17:54, 27 July 2024


A function $ K( x, s) $, $ a \leq x, s \leq b $, such that for any points $ x _ {1} \dots x _ {n} \in [ a, b] $, which (when $ n= 2 $) include at least one interior point, the matrix $ \| K( x _ {i} , x _ {k} ) \| _ {1} ^ {n} $ is an oscillating matrix.

References

[a1] F.R. Gantmakher, M.G. Krein, "Oscillation matrices and kernels and small vibrations of mechanical systems" , Dept. Commerce USA. Joint Publ. Service (1961) (Translated from Russian)
[a2] S. Karlin, "Total positivity" , Stanford Univ. Press (1960)
How to Cite This Entry:
Oscillating kernel. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Oscillating_kernel&oldid=13036
This article was adapted from an original article by V.I. Lomonosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article