Integral Equations
Fredholm's theorem for integral equations is expressed as follows. Let be an integral kernel, and consider the homogeneous equations
and its complex adjoint
Here, denotes the complex conjugate of the complex number, and similarly for . Then, Fredholm's theorem is that, for any fixed value of, these equations have either the trivial solution or have the same number of linearly independent solutions, .
A sufficient condition for this theorem to hold is for to be square integrable on the rectangle (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 to integral operators on multi-dimensional spaces, including, for example, Riemannian manifolds.
Read more about this topic: Fredholm's Theorem
Famous quotes containing the word integral:
“Self-centeredness is a natural outgrowth of one of the toddler’s major concerns: What is me and what is mine...? This is why most toddlers are incapable of sharing ... to a toddler, what’s his is what he can get his hands on.... When something is taken away from him, he feels as though a piece of him—an integral piece—is being torn from him.”
—Lawrence Balter (20th century)