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The construction of approximate solutions of [[Ill-posed problems|ill-posed problems]] that are stable with respect to small perturbations of the initial data (see also [[Regularization method|Regularization method]]).
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The construction of approximate solutions of [[Ill-posed problems|ill-posed problems]] that are stable with respect to small perturbations of the initial data (see also [[Regularization method|Regularization method]]).
  
 
====Comments====
 
====Comments====
 
The concept of "regularization" in mathematics is a quite general one, which extends far beyond regularization methods as are used to deal with ill-posed problems. It encompasses at least the following two intermingling ideas.
 
The concept of "regularization" in mathematics is a quite general one, which extends far beyond regularization methods as are used to deal with ill-posed problems. It encompasses at least the following two intermingling ideas.
  
1) The systematic replacement of a mathematical object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080920/r0809201.png" /> by a more regular one <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080920/r0809202.png" />, usually in such a way that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080920/r0809203.png" />.
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1) The systematic replacement of a mathematical object $  A $
 +
by a more regular one $  A ^ { \mathop{\rm reg} } $,  
 +
usually in such a way that $  ( A ^ { \mathop{\rm reg} } ) ^ { \mathop{\rm reg} } = A ^ { \mathop{\rm reg} } $.
  
2) The definition of a value of a function or other concept for objects where that value or concept is a priori undefined (or infinite, undetermined, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080920/r0809204.png" />). This is often done by placing the object in a suitable family (a [[Deformation|deformation]]) in such a way that the function value or concept is defined for all objects in the family near the original one, and then taking a suitable limit. Another technique consists in the removal of "systematic infinities" . The details of various regularization methods that are used depend very much on the particular context. Instead of the word "regularization" , one also finds such methods and techniques labelled by words and phrases like "normalization" , "renormalization" , "desingularization" , "resolution of singularities" , <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080920/r0809205.png" />.
+
2) The definition of a value of a function or other concept for objects where that value or concept is a priori undefined (or infinite, undetermined, $  \dots $).  
 +
This is often done by placing the object in a suitable family (a [[Deformation|deformation]]) in such a way that the function value or concept is defined for all objects in the family near the original one, and then taking a suitable limit. Another technique consists in the removal of "systematic infinities" . The details of various regularization methods that are used depend very much on the particular context. Instead of the word "regularization" , one also finds such methods and techniques labelled by words and phrases like "normalization" , "renormalization" , "desingularization" , "resolution of singularities" , $  \dots $.
  
 
Examples of regularizations in the sense of 1) or 2) above (or both) are: regularized sequences (cf. [[Regularization of sequences|Regularization of sequences]]), regularized operators and regularized solutions (cf. [[Ill-posed problems|Ill-posed problems]]; [[Regularization method|Regularization method]]; [[Integral equations, numerical methods|Integral equations, numerical methods]]; [[Fredholm equation, numerical methods|Fredholm equation, numerical methods]]), penalty function and other regularization techniques in optimization theory (cf. [[Mathematical programming|Mathematical programming]]; [[Penalty functions, method of|Penalty functions, method of]]), various renormalization schemes (cf. [[Renormalization|Renormalization]]), the normalization and desingularization of schemes and varieties (cf. [[Normal scheme|Normal scheme]]; [[Resolution of singularities|Resolution of singularities]]), the regularization of distributions (cf. [[Generalized function|Generalized function]]), the regularized trace of a Sturm–Liouville operator (cf. [[Sturm–Liouville problem|Sturm–Liouville problem]]), and the regularized characteristic determinant of a [[Hilbert–Schmidt operator|Hilbert–Schmidt operator]].
 
Examples of regularizations in the sense of 1) or 2) above (or both) are: regularized sequences (cf. [[Regularization of sequences|Regularization of sequences]]), regularized operators and regularized solutions (cf. [[Ill-posed problems|Ill-posed problems]]; [[Regularization method|Regularization method]]; [[Integral equations, numerical methods|Integral equations, numerical methods]]; [[Fredholm equation, numerical methods|Fredholm equation, numerical methods]]), penalty function and other regularization techniques in optimization theory (cf. [[Mathematical programming|Mathematical programming]]; [[Penalty functions, method of|Penalty functions, method of]]), various renormalization schemes (cf. [[Renormalization|Renormalization]]), the normalization and desingularization of schemes and varieties (cf. [[Normal scheme|Normal scheme]]; [[Resolution of singularities|Resolution of singularities]]), the regularization of distributions (cf. [[Generalized function|Generalized function]]), the regularized trace of a Sturm–Liouville operator (cf. [[Sturm–Liouville problem|Sturm–Liouville problem]]), and the regularized characteristic determinant of a [[Hilbert–Schmidt operator|Hilbert–Schmidt operator]].
  
Still another example is the zeta-function regularization used to define certain (quotients of) infinite determinants in functional integration and quantum field theory. This goes as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080920/r0809206.png" /> be a suitable operator, e.g. a Laplace or Laplace–Beltrami operator. Define its generalized zeta-function
+
Still another example is the zeta-function regularization used to define certain (quotients of) infinite determinants in functional integration and quantum field theory. This goes as follows. Let $  A $
 +
be a suitable operator, e.g. a Laplace or Laplace–Beltrami operator. Define its generalized zeta-function
  
<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/r/r080/r080920/r0809207.png" /></td> </tr></table>
+
$$
 +
\zeta _ {A} ( s)  = \sum _ { n } \lambda _ {n}  ^ {-} s ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080920/r0809208.png" /> runs over the spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080920/r0809209.png" /> (counting multiplicities). At least formally, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080920/r08092010.png" />, which provides the opportunity to try to define the zeta-function regularized determinant by
+
where $  \lambda _ {n} $
 +
runs over the spectrum of $  A $(
 +
counting multiplicities). At least formally, $  \zeta  ^  \prime  ( s) \mid  _ {s=} 0 = - \sum _ {n}  \mathop{\rm log} ( \lambda _ {n} ) $,  
 +
which provides the opportunity to try to define the zeta-function regularized determinant by
  
<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/r/r080/r080920/r08092011.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm det} ( A)  =   \mathop{\rm exp} ( - \zeta  ^  \prime  ( s)) \mid  _ {s=} 0 .
 +
$$
  
 
For more details (and other related schemes) cf. [[#References|[a1]]], [[#References|[a2]]].
 
For more details (and other related schemes) cf. [[#References|[a1]]], [[#References|[a2]]].
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Two somewhat different uses of the word "regularizing" in mathematics are as follows.
 
Two somewhat different uses of the word "regularizing" in mathematics are as follows.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080920/r08092012.png" /> is a bounded linear operator between normed spaces, then a bounded linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080920/r08092013.png" /> is called a "regularizer of a bounded linear operatorregularizer of K" if there are compact operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080920/r08092014.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080920/r08092015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080920/r08092016.png" />. This concept is of importance in the context of singular integral operators, cf. e.g. [[#References|[a3]]]. I.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080920/r08092017.png" /> is an inverse of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080920/r08092018.png" /> modulo compact operators.
+
If $  K $
 +
is a bounded linear operator between normed spaces, then a bounded linear operator $  R $
 +
is called a "regularizer of a bounded linear operatorregularizer of K" if there are compact operators $  A, B $
 +
such that $  RK = I- A $,  
 +
$  KR = I- B $.  
 +
This concept is of importance in the context of singular integral operators, cf. e.g. [[#References|[a3]]]. I.e. $  R $
 +
is an inverse of $  K $
 +
modulo compact operators.
  
A similar idea, but with deviating terminology, occurs in the theory of pseudo-differential operators. In that context a (pseudo-differential, integral) operator is called regularizing if it takes (extends to an operator that takes) distributions to smooth functions. Given a pseudo-differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080920/r08092019.png" />, an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080920/r08092020.png" /> is called a right (left) parametrix of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080920/r08092021.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080920/r08092022.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080920/r08092023.png" />), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080920/r08092024.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080920/r08092025.png" />) is regularizing; cf. [[#References|[a4]]] for a variety of precise statements and results concerning parametrices.
+
A similar idea, but with deviating terminology, occurs in the theory of pseudo-differential operators. In that context a (pseudo-differential, integral) operator is called regularizing if it takes (extends to an operator that takes) distributions to smooth functions. Given a pseudo-differential operator $  P $,  
 +
an operator $  R $
 +
is called a right (left) parametrix of $  P $
 +
if $  PR = I+ K $(
 +
$  RP = I+ K  ^  \prime  $),  
 +
where $  K $(
 +
respectively, $  K  ^  \prime  $)  
 +
is regularizing; cf. [[#References|[a4]]] for a variety of precise statements and results concerning parametrices.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> S.W. Hawking, "Zeta function regularization of path integrals" ''Comm. Math. Phys.'' , '''55''' (1977) pp. 133–148 {{MR|0524257}} {{ZBL|0407.58024}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R.E. Gamboa Saravi, M.A. Muschietti, J.E. Solomin, "On the quotient of the regularized determinant of two elliptic operators" ''Comm. Math. Phys.'' , '''110''' (1987) pp. 641–654 {{MR|895221}} {{ZBL|0648.35086}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> R. Kress, "Linear integral equations" , Springer (1989) pp. Chapt. 5 {{MR|1007594}} {{ZBL|0671.45001}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> F. Trèves, "Pseudodifferential and Fourier integral operators" , '''1–2''' , Plenum (1980) {{MR|0597145}} {{MR|0597144}} {{ZBL|0453.47027}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> S.W. Hawking, "Zeta function regularization of path integrals" ''Comm. Math. Phys.'' , '''55''' (1977) pp. 133–148 {{MR|0524257}} {{ZBL|0407.58024}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R.E. Gamboa Saravi, M.A. Muschietti, J.E. Solomin, "On the quotient of the regularized determinant of two elliptic operators" ''Comm. Math. Phys.'' , '''110''' (1987) pp. 641–654 {{MR|895221}} {{ZBL|0648.35086}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> R. Kress, "Linear integral equations" , Springer (1989) pp. Chapt. 5 {{MR|1007594}} {{ZBL|0671.45001}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> F. Trèves, "Pseudodifferential and Fourier integral operators" , '''1–2''' , Plenum (1980) {{MR|0597145}} {{MR|0597144}} {{ZBL|0453.47027}} </TD></TR></table>

Revision as of 08:10, 6 June 2020


The construction of approximate solutions of ill-posed problems that are stable with respect to small perturbations of the initial data (see also Regularization method).

Comments

The concept of "regularization" in mathematics is a quite general one, which extends far beyond regularization methods as are used to deal with ill-posed problems. It encompasses at least the following two intermingling ideas.

1) The systematic replacement of a mathematical object $ A $ by a more regular one $ A ^ { \mathop{\rm reg} } $, usually in such a way that $ ( A ^ { \mathop{\rm reg} } ) ^ { \mathop{\rm reg} } = A ^ { \mathop{\rm reg} } $.

2) The definition of a value of a function or other concept for objects where that value or concept is a priori undefined (or infinite, undetermined, $ \dots $). This is often done by placing the object in a suitable family (a deformation) in such a way that the function value or concept is defined for all objects in the family near the original one, and then taking a suitable limit. Another technique consists in the removal of "systematic infinities" . The details of various regularization methods that are used depend very much on the particular context. Instead of the word "regularization" , one also finds such methods and techniques labelled by words and phrases like "normalization" , "renormalization" , "desingularization" , "resolution of singularities" , $ \dots $.

Examples of regularizations in the sense of 1) or 2) above (or both) are: regularized sequences (cf. Regularization of sequences), regularized operators and regularized solutions (cf. Ill-posed problems; Regularization method; Integral equations, numerical methods; Fredholm equation, numerical methods), penalty function and other regularization techniques in optimization theory (cf. Mathematical programming; Penalty functions, method of), various renormalization schemes (cf. Renormalization), the normalization and desingularization of schemes and varieties (cf. Normal scheme; Resolution of singularities), the regularization of distributions (cf. Generalized function), the regularized trace of a Sturm–Liouville operator (cf. Sturm–Liouville problem), and the regularized characteristic determinant of a Hilbert–Schmidt operator.

Still another example is the zeta-function regularization used to define certain (quotients of) infinite determinants in functional integration and quantum field theory. This goes as follows. Let $ A $ be a suitable operator, e.g. a Laplace or Laplace–Beltrami operator. Define its generalized zeta-function

$$ \zeta _ {A} ( s) = \sum _ { n } \lambda _ {n} ^ {-} s , $$

where $ \lambda _ {n} $ runs over the spectrum of $ A $( counting multiplicities). At least formally, $ \zeta ^ \prime ( s) \mid _ {s=} 0 = - \sum _ {n} \mathop{\rm log} ( \lambda _ {n} ) $, which provides the opportunity to try to define the zeta-function regularized determinant by

$$ \mathop{\rm det} ( A) = \mathop{\rm exp} ( - \zeta ^ \prime ( s)) \mid _ {s=} 0 . $$

For more details (and other related schemes) cf. [a1], [a2].

Two somewhat different uses of the word "regularizing" in mathematics are as follows.

If $ K $ is a bounded linear operator between normed spaces, then a bounded linear operator $ R $ is called a "regularizer of a bounded linear operatorregularizer of K" if there are compact operators $ A, B $ such that $ RK = I- A $, $ KR = I- B $. This concept is of importance in the context of singular integral operators, cf. e.g. [a3]. I.e. $ R $ is an inverse of $ K $ modulo compact operators.

A similar idea, but with deviating terminology, occurs in the theory of pseudo-differential operators. In that context a (pseudo-differential, integral) operator is called regularizing if it takes (extends to an operator that takes) distributions to smooth functions. Given a pseudo-differential operator $ P $, an operator $ R $ is called a right (left) parametrix of $ P $ if $ PR = I+ K $( $ RP = I+ K ^ \prime $), where $ K $( respectively, $ K ^ \prime $) is regularizing; cf. [a4] for a variety of precise statements and results concerning parametrices.

References

[a1] S.W. Hawking, "Zeta function regularization of path integrals" Comm. Math. Phys. , 55 (1977) pp. 133–148 MR0524257 Zbl 0407.58024
[a2] R.E. Gamboa Saravi, M.A. Muschietti, J.E. Solomin, "On the quotient of the regularized determinant of two elliptic operators" Comm. Math. Phys. , 110 (1987) pp. 641–654 MR895221 Zbl 0648.35086
[a3] R. Kress, "Linear integral equations" , Springer (1989) pp. Chapt. 5 MR1007594 Zbl 0671.45001
[a4] F. Trèves, "Pseudodifferential and Fourier integral operators" , 1–2 , Plenum (1980) MR0597145 MR0597144 Zbl 0453.47027
How to Cite This Entry:
Regularization. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Regularization&oldid=24548
This article was adapted from an original article by V.Ya. ArseninA.N. Tikhonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article