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Difference between revisions of "Volterra kernel"

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A (matrix) function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096850/v0968501.png" /> of two real variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096850/v0968502.png" /> such that either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096850/v0968503.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096850/v0968504.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096850/v0968505.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096850/v0968506.png" />. If such a function is the kernel of a linear integral operator, acting on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096850/v0968507.png" />, and is itself square-integrable in the triangle in which it is non-zero, the operator generated by it is known as a Volterra integral operator (cf. [[Volterra operator|Volterra operator]]).
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A (matrix) function $K(s,t)$ of two real variables $s,t$ such that either $K(s,t)\equiv0$ if $a\leq s<t\leq b$ or $K(s,t)\equiv0$ if $a\leq t<s\leq b$. If such a function is the kernel of a linear integral operator, acting on the space $L_2(a,b)$, and is itself square-integrable in the triangle in which it is non-zero, the operator generated by it is known as a Volterra integral operator (cf. [[Volterra operator|Volterra operator]]).
  
  

Latest revision as of 19:03, 27 April 2014

A (matrix) function $K(s,t)$ of two real variables $s,t$ such that either $K(s,t)\equiv0$ if $a\leq s<t\leq b$ or $K(s,t)\equiv0$ if $a\leq t<s\leq b$. If such a function is the kernel of a linear integral operator, acting on the space $L_2(a,b)$, and is itself square-integrable in the triangle in which it is non-zero, the operator generated by it is known as a Volterra integral operator (cf. Volterra operator).


Comments

See also Volterra equation.

How to Cite This Entry:
Volterra kernel. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Volterra_kernel&oldid=31953
This article was adapted from an original article by A.B. Bakushinskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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