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Given an open subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130020/e1300201.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130020/e1300202.png" />, Yu.V. Egorov [[#References|[a1]]] defined the generalized function algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130020/e1300203.png" /> as the factor algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130020/e1300204.png" /> modulo the ideal of sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130020/e1300205.png" /> which vanish eventually on every compact subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130020/e1300206.png" />. The family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130020/e1300207.png" /> provides a [[Sheaf|sheaf]] of differential algebras on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130020/e1300208.png" />. Convolution with a sequence of mollifiers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130020/e1300209.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130020/e13002010.png" /> converges to the Dirac measure, gives an imbedding of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130020/e13002011.png" /> of compactly supported distributions into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130020/e13002012.png" /> which respects derivatives as well as supports. It can be extended as a sheaf morphism to an imbedding of the space of distributions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130020/e13002013.png" />.
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As a generalized function algebra, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130020/e13002014.png" /> can be employed to study non-linear partial differential equations. In particular, Egorov has used the algebra to construct generalized solutions to boundary value problems as well as evolution equations. In the latter case the spatial derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130020/e13002015.png" /> may be replaced by the difference operator
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Given an open subset $\Omega$ of ${\bf R} ^ { n }$, Yu.V. Egorov [[#References|[a1]]] defined the generalized function algebra $\mathcal{A} ( \Omega )$ as the factor algebra of $( \mathcal{C} ^ { \infty } ( \Omega ) ) ^ { \mathbf{N} }$ modulo the ideal of sequences $( u_j )_{ j \in \mathbf{N}}$ which vanish eventually on every compact subset of $\Omega$. The family $\{ \mathcal{A} ( \Omega ) : \Omega \text { open } \}$ provides a [[Sheaf|sheaf]] of differential algebras on ${\bf R} ^ { n }$. Convolution with a sequence of mollifiers $( \varphi _ { j } ) _ { j \in \mathbf{N} }$, where $\varphi_{j}$ converges to the Dirac measure, gives an imbedding of the space ${\cal E} ^ { \prime } ( \Omega )$ of compactly supported distributions into $\mathcal{A} ( \Omega )$ which respects derivatives as well as supports. It can be extended as a sheaf morphism to an imbedding of the space of distributions $\mathcal{D} ^ { \prime } ( \Omega )$.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130020/e13002017.png" /> denotes the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130020/e13002018.png" />th unit vector in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130020/e13002019.png" />. This way partial differential equations are approximated by ordinary difference-differential equations in the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e130/e130020/e13002020.png" />.
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As a generalized function algebra, $\mathcal{A} ( \Omega )$ can be employed to study non-linear partial differential equations. In particular, Egorov has used the algebra to construct generalized solutions to boundary value problems as well as evolution equations. In the latter case the spatial derivative $( \partial / \partial x _ { k } ) u ( x )$ may be replaced by the difference operator
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\begin{equation*} j \left( u \left( x + \frac { 1 } { j } e _ { k } \right) - u ( x ) \right), \end{equation*}
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where $e_k$ denotes the $k$th unit vector in ${\bf R} ^ { n }$. This way partial differential equations are approximated by ordinary difference-differential equations in the algebra $\mathcal{A} ( \Omega )$.
  
 
See also [[Generalized function algebras|Generalized function algebras]].
 
See also [[Generalized function algebras|Generalized function algebras]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  Yu.V. Egorov,  "A contribution to the theory of generalized functions"  ''Russian Math. Surveys'' , '''45''' :  5  (1990)  pp. 1–49</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top">  Yu.V. Egorov,  "A contribution to the theory of generalized functions"  ''Russian Math. Surveys'' , '''45''' :  5  (1990)  pp. 1–49</td></tr></table>

Latest revision as of 16:45, 1 July 2020

Given an open subset $\Omega$ of ${\bf R} ^ { n }$, Yu.V. Egorov [a1] defined the generalized function algebra $\mathcal{A} ( \Omega )$ as the factor algebra of $( \mathcal{C} ^ { \infty } ( \Omega ) ) ^ { \mathbf{N} }$ modulo the ideal of sequences $( u_j )_{ j \in \mathbf{N}}$ which vanish eventually on every compact subset of $\Omega$. The family $\{ \mathcal{A} ( \Omega ) : \Omega \text { open } \}$ provides a sheaf of differential algebras on ${\bf R} ^ { n }$. Convolution with a sequence of mollifiers $( \varphi _ { j } ) _ { j \in \mathbf{N} }$, where $\varphi_{j}$ converges to the Dirac measure, gives an imbedding of the space ${\cal E} ^ { \prime } ( \Omega )$ of compactly supported distributions into $\mathcal{A} ( \Omega )$ which respects derivatives as well as supports. It can be extended as a sheaf morphism to an imbedding of the space of distributions $\mathcal{D} ^ { \prime } ( \Omega )$.

As a generalized function algebra, $\mathcal{A} ( \Omega )$ can be employed to study non-linear partial differential equations. In particular, Egorov has used the algebra to construct generalized solutions to boundary value problems as well as evolution equations. In the latter case the spatial derivative $( \partial / \partial x _ { k } ) u ( x )$ may be replaced by the difference operator

\begin{equation*} j \left( u \left( x + \frac { 1 } { j } e _ { k } \right) - u ( x ) \right), \end{equation*}

where $e_k$ denotes the $k$th unit vector in ${\bf R} ^ { n }$. This way partial differential equations are approximated by ordinary difference-differential equations in the algebra $\mathcal{A} ( \Omega )$.

See also Generalized function algebras.

References

[a1] Yu.V. Egorov, "A contribution to the theory of generalized functions" Russian Math. Surveys , 45 : 5 (1990) pp. 1–49
How to Cite This Entry:
Egorov generalized function algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Egorov_generalized_function_algebra&oldid=49979
This article was adapted from an original article by Michael Oberguggenberger (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article