# How to solve this special case of Fredholm Integral Equation of the First Kind

General form of 'Fredholm Integral Equation of the First Kind'

$f(x) = \int_a^b{K(x,t)\phi(t)} dt$

Where $\phi(t)$ is the unkown

My special case is

$1 = \int_a^b{k(t)\phi(t)} dt$

A trivial solution is of course

$\phi(t) = \frac{1}{\int_a^b k(t)dt}$

I wonder whether there are some general nontrivial solutions.

• Something like $\phi(t)=\frac{1}{k(t_0)}\delta(t_0-t)$ if you allow distributions as solutions. – krvolok Aug 17 '15 at 14:42
• I certainly allow distributions (i even expect to find solutions of their kind) – tgoossens Aug 17 '15 at 14:50