Difference between revisions of "Locality principle"
From Encyclopedia of Mathematics
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| − | A collective concept that combines a number of assertions related mainly to elliptic (in some cases to hypo-elliptic) equations (operators) and that follows from the pointwise character of the singularity of a fundamental solution for this class of equations. For example, an elliptic operator | + | {{TEX|done}} |
| + | A collective concept that combines a number of assertions related mainly to elliptic (in some cases to hypo-elliptic) equations (operators) and that follows from the pointwise character of the singularity of a fundamental solution for this class of equations. For example, an elliptic operator $L(D,x)$ with variable coefficients, written in the form | ||
| − | + | $$L(D,x)\equiv\sum_{|\alpha|\leq m}a_\alpha(x)D^\alpha,\quad x\in\mathbf R^n,$$ | |
| − | can be represented, in an appropriate sense, in a neighbourhood of a point | + | can be represented, in an appropriate sense, in a neighbourhood of a point $x_0$ as a sum |
| − | + | $$L(D,x)=\sum_{|\alpha|\leq m}a_\alpha(x_0)D^\alpha+L'(x),$$ | |
| − | where the first term is an operator with constant coefficients, and | + | where the first term is an operator with constant coefficients, and $L'(x)$ is "sufficiently small" in the given neighbourhood. |
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> N. Dunford, J.T. Schwartz, "Linear operators. Spectral theory" , '''2''' , Interscience (1963)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> N. Dunford, J.T. Schwartz, "Linear operators. Spectral theory" , '''2''' , Interscience (1963)</TD></TR></table> | ||
Latest revision as of 17:23, 26 September 2014
A collective concept that combines a number of assertions related mainly to elliptic (in some cases to hypo-elliptic) equations (operators) and that follows from the pointwise character of the singularity of a fundamental solution for this class of equations. For example, an elliptic operator $L(D,x)$ with variable coefficients, written in the form
$$L(D,x)\equiv\sum_{|\alpha|\leq m}a_\alpha(x)D^\alpha,\quad x\in\mathbf R^n,$$
can be represented, in an appropriate sense, in a neighbourhood of a point $x_0$ as a sum
$$L(D,x)=\sum_{|\alpha|\leq m}a_\alpha(x_0)D^\alpha+L'(x),$$
where the first term is an operator with constant coefficients, and $L'(x)$ is "sufficiently small" in the given neighbourhood.
References
| [1] | N. Dunford, J.T. Schwartz, "Linear operators. Spectral theory" , 2 , Interscience (1963) |
How to Cite This Entry:
Locality principle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Locality_principle&oldid=16547
Locality principle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Locality_principle&oldid=16547
This article was adapted from an original article by A.A. Dezin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article