- Fredholm's theorem
In
mathematics , Fredholm's theorems are a set of celebrated results ofIvar Fredholm in theFredholm theory ofintegral equations . There are several closely related theorems, which may be stated in terms of integral equations, in terms oflinear algebra , or in terms of theFredholm operator onBanach space s.The
Fredholm alternative is one of the Fredholm theorems.Linear algebra
Fredholm's theorem in linear algebra is as follows: if "M" is a matrix, then the
orthogonal complement of therow space of "M" is thenull space of "M"::operatorname{row } M)^ot = ker M
Similarly, the orthogonal complement of the column space of "M" is the null space of the adjoint:
:operatorname{col } M)^ot = ker overline{M}
Integral equations
Fredholm's theorem for integral equations is expressed as follows. Let K(x,y) be an
integral kernel , and consider thehomogeneous equation s:int_a^b K(x,y) phi(y) ,dy = lambda phi(x)
and its complex adjoint
:int_a^b psi(x) overline{K(x,y)} , dx = overline {lambda}psi(y)
Here, overline{lambda} denotes the
complex conjugate of thecomplex number lambda, and similarly for overline{K(x,y)}. Then, Fredholm's theorem is that, for any fixed value of lambda, these equations have either the trivial solution psi(x)=phi(x)=0 or have the same number oflinearly independent solutions phi_1(x),cdots,phi_n(x), psi_1(y),cdots,psi_n(y).A sufficient condition for this theorem to hold is for K(x,y) to be
square integrable on the rectangle a,b] imes [a,b] (where "a" and/or "b" may be minus or plus infinity).Here, the integral is expressed as a one-dimensional integral on the real number line. In
Fredholm theory , this result generalizes tointegral operator s on multi-dimensional spaces, including, for example,Riemannian manifold s.Existence of solutions
One of the Fredholm theorem's closely related to the
Fredholm alternative , concerns the existence of solutions to the inhomogeneousFredholm equation :lambda phi(x)-int_a^b K(x,y) phi(y) ,dy=f(x)
Solutions to this equation exist if and only if the function f(x) is
orthogonal to the complete set of solutions psi_n(x)} of the corresponding homogeneous adjoint equation::int_a^b overline{psi_n(x)} f(x) ,dx=0
where overline{psi_n(x)} is the complex conjugate of psi_n(x) and the former is one of the complete set of solutions to
:lambdaoverline{psi(y)} -int_a^b overline{psi(x)} K(x,y) ,dx=0
A sufficient condition for this theorem to hold is for K(x,y) to be
square integrable on the rectangle a,b] imes [a,b] .References
* E.I. Fredholm, "Sur une classe d'equations fonctionnelles", "Acta Math." , 27 (1903) pp. 365–390.
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