Difference between revisions of "Fredholm theorems"
(Importing text file) |
Ulf Rehmann (talk | contribs) m (tex encoded by computer) |
||
Line 1: | Line 1: | ||
+ | <!-- | ||
+ | f0414701.png | ||
+ | $#A+1 = 18 n = 0 | ||
+ | $#C+1 = 18 : ~/encyclopedia/old_files/data/F041/F.0401470 Fredholm theorems | ||
+ | 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}} | ||
+ | |||
''for integral equations'' | ''for integral equations'' | ||
Line 4: | Line 16: | ||
The homogeneous equation | The homogeneous equation | ||
− | + | $$ \tag{1 } | |
+ | \phi ( x) - \lambda \int\limits _ { a } ^ { b } K ( x, s) \phi ( s) ds = 0 | ||
+ | $$ | ||
and its transposed equation | and its transposed equation | ||
− | + | $$ \tag{2 } | |
+ | \psi ( x) - \lambda \int\limits _ { a } ^ { b } K ( s, x) \psi ( s) ds = 0 | ||
+ | $$ | ||
− | have, for a fixed value of the parameter | + | have, for a fixed value of the parameter $ \lambda $, |
+ | either only the trivial solution, or have the same finite number of linearly independent solutions: $ \phi _ {1} \dots \phi _ {n} $; | ||
+ | $ \psi _ {1} \dots \psi _ {n} $. | ||
===Theorem 2.=== | ===Theorem 2.=== | ||
For a solution of the inhomogeneous equation | For a solution of the inhomogeneous equation | ||
− | + | $$ \tag{3 } | |
+ | \phi ( x) - \lambda \int\limits _ { a } ^ { b } K ( x, s) \phi ( s) ds = f ( x) | ||
+ | $$ | ||
to exist it is necessary and sufficient that its right-hand side be orthogonal to a complete system of linearly independent solutions of the corresponding homogeneous transposed equation (2): | to exist it is necessary and sufficient that its right-hand side be orthogonal to a complete system of linearly independent solutions of the corresponding homogeneous transposed equation (2): | ||
− | + | $$ \tag{4 } | |
+ | \int\limits _ { a } ^ { b } f ( x) \psi _ {j} ( x) dx = 0,\ j = 1 \dots n. | ||
+ | $$ | ||
===Theorem 3.=== | ===Theorem 3.=== | ||
− | (the Fredholm alternative). Either the inhomogeneous equation (3) has a solution, whatever its right-hand side | + | (the Fredholm alternative). Either the inhomogeneous equation (3) has a solution, whatever its right-hand side $ f $, |
+ | or the corresponding homogeneous equation (1) has non-trivial solutions. | ||
===Theorem 4.=== | ===Theorem 4.=== | ||
The set of characteristic numbers of equation (1) is at most countable, with a single possible limit point at infinity. | The set of characteristic numbers of equation (1) is at most countable, with a single possible limit point at infinity. | ||
− | For the Fredholm theorems to hold in the function space | + | For the Fredholm theorems to hold in the function space $ L _ {2} [ a, b] $ |
+ | it is sufficient that the kernel $ K $ | ||
+ | of equation (3) be square-integrable on the set $ [ a, b] \times [ a, b] $( | ||
+ | $ a $ | ||
+ | and $ b $ | ||
+ | may be infinite). When this condition is violated, (3) may turn out to be a [[Non-Fredholm integral equation|non-Fredholm integral equation]]. When the parameter $ \lambda $ | ||
+ | and the functions involved in (3) take complex values, then instead of the transposed equation (2) one often considers the adjoint equation to (1): | ||
− | + | $$ | |
+ | \psi ( x) - \overline \lambda \; \int\limits _ { a } ^ { b } \overline{ {K ( s, x) }}\; \psi ( s) ds = 0. | ||
+ | $$ | ||
In this case condition (4) is replaced by | In this case condition (4) is replaced by | ||
− | + | $$ | |
+ | \int\limits _ { a } ^ { b } f ( x) \overline{ {\psi _ {j} ( x) }}\; dx = 0,\ \ | ||
+ | j = 1 \dots n. | ||
+ | $$ | ||
These theorems were proved by E.I. Fredholm [[#References|[1]]]. | These theorems were proved by E.I. Fredholm [[#References|[1]]]. | ||
Line 39: | Line 73: | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> E.I. Fredholm, "Sur une classe d'equations fonctionnelles" ''Acta Math.'' , '''27''' (1903) pp. 365–390</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> E.I. Fredholm, "Sur une classe d'equations fonctionnelles" ''Acta Math.'' , '''27''' (1903) pp. 365–390</TD></TR></table> | ||
− | |||
− | |||
====Comments==== | ====Comments==== | ||
− | Instead of the phrases "transposed equation" and "adjoint equation" one sometimes uses "adjoint equation of a Fredholm integral equationadjoint equation" and "conjugate equation of a Fredholm integral equationconjugate equation" (cf. [[#References|[a4]]]); in the latter terminology | + | Instead of the phrases "transposed equation" and "adjoint equation" one sometimes uses "adjoint equation of a Fredholm integral equationadjoint equation" and "conjugate equation of a Fredholm integral equationconjugate equation" (cf. [[#References|[a4]]]); in the latter terminology $ \overline \lambda \; $ |
+ | is replaced by $ \lambda $. | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> I.C. Gohberg, S. Goldberg, "Basic operator theory" , Birkhäuser (1981)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> K. Jörgens, "Lineare Integraloperatoren" , Teubner (1970)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> V.I. Smirnov, "A course of higher mathematics" , '''4''' , Addison-Wesley (1964) (Translated from Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> P.P. Zabreiko (ed.) A.I. Koshelev (ed.) M.A. Krasnoselskii (ed.) S.G. Mikhlin (ed.) L.S. Rakovshchik (ed.) V.Ya. Stet'senko (ed.) T.O. Shaposhnikova (ed.) R.S. Anderssen (ed.) , ''Integral equations - a reference text'' , Noordhoff (1975) (Translated from Russian)</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> I.C. Gohberg, S. Goldberg, "Basic operator theory" , Birkhäuser (1981)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> K. Jörgens, "Lineare Integraloperatoren" , Teubner (1970)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> V.I. Smirnov, "A course of higher mathematics" , '''4''' , Addison-Wesley (1964) (Translated from Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> P.P. Zabreiko (ed.) A.I. Koshelev (ed.) M.A. Krasnoselskii (ed.) S.G. Mikhlin (ed.) L.S. Rakovshchik (ed.) V.Ya. Stet'senko (ed.) T.O. Shaposhnikova (ed.) R.S. Anderssen (ed.) , ''Integral equations - a reference text'' , Noordhoff (1975) (Translated from Russian)</TD></TR></table> |
Latest revision as of 19:40, 5 June 2020
for integral equations
Theorem 1.
The homogeneous equation
$$ \tag{1 } \phi ( x) - \lambda \int\limits _ { a } ^ { b } K ( x, s) \phi ( s) ds = 0 $$
and its transposed equation
$$ \tag{2 } \psi ( x) - \lambda \int\limits _ { a } ^ { b } K ( s, x) \psi ( s) ds = 0 $$
have, for a fixed value of the parameter $ \lambda $, either only the trivial solution, or have the same finite number of linearly independent solutions: $ \phi _ {1} \dots \phi _ {n} $; $ \psi _ {1} \dots \psi _ {n} $.
Theorem 2.
For a solution of the inhomogeneous equation
$$ \tag{3 } \phi ( x) - \lambda \int\limits _ { a } ^ { b } K ( x, s) \phi ( s) ds = f ( x) $$
to exist it is necessary and sufficient that its right-hand side be orthogonal to a complete system of linearly independent solutions of the corresponding homogeneous transposed equation (2):
$$ \tag{4 } \int\limits _ { a } ^ { b } f ( x) \psi _ {j} ( x) dx = 0,\ j = 1 \dots n. $$
Theorem 3.
(the Fredholm alternative). Either the inhomogeneous equation (3) has a solution, whatever its right-hand side $ f $, or the corresponding homogeneous equation (1) has non-trivial solutions.
Theorem 4.
The set of characteristic numbers of equation (1) is at most countable, with a single possible limit point at infinity.
For the Fredholm theorems to hold in the function space $ L _ {2} [ a, b] $ it is sufficient that the kernel $ K $ of equation (3) be square-integrable on the set $ [ a, b] \times [ a, b] $( $ a $ and $ b $ may be infinite). When this condition is violated, (3) may turn out to be a non-Fredholm integral equation. When the parameter $ \lambda $ and the functions involved in (3) take complex values, then instead of the transposed equation (2) one often considers the adjoint equation to (1):
$$ \psi ( x) - \overline \lambda \; \int\limits _ { a } ^ { b } \overline{ {K ( s, x) }}\; \psi ( s) ds = 0. $$
In this case condition (4) is replaced by
$$ \int\limits _ { a } ^ { b } f ( x) \overline{ {\psi _ {j} ( x) }}\; dx = 0,\ \ j = 1 \dots n. $$
These theorems were proved by E.I. Fredholm [1].
References
[1] | E.I. Fredholm, "Sur une classe d'equations fonctionnelles" Acta Math. , 27 (1903) pp. 365–390 |
Comments
Instead of the phrases "transposed equation" and "adjoint equation" one sometimes uses "adjoint equation of a Fredholm integral equationadjoint equation" and "conjugate equation of a Fredholm integral equationconjugate equation" (cf. [a4]); in the latter terminology $ \overline \lambda \; $ is replaced by $ \lambda $.
References
[a1] | I.C. Gohberg, S. Goldberg, "Basic operator theory" , Birkhäuser (1981) |
[a2] | K. Jörgens, "Lineare Integraloperatoren" , Teubner (1970) |
[a3] | V.I. Smirnov, "A course of higher mathematics" , 4 , Addison-Wesley (1964) (Translated from Russian) |
[a4] | P.P. Zabreiko (ed.) A.I. Koshelev (ed.) M.A. Krasnoselskii (ed.) S.G. Mikhlin (ed.) L.S. Rakovshchik (ed.) V.Ya. Stet'senko (ed.) T.O. Shaposhnikova (ed.) R.S. Anderssen (ed.) , Integral equations - a reference text , Noordhoff (1975) (Translated from Russian) |
Fredholm theorems. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fredholm_theorems&oldid=12814