1
$\begingroup$

I am trying to solve the equation:

$\phi(x)=\int_{-\infty}^\infty K(x, t)\phi(t)dt$

for $K$ given $\phi$. This closely resembles a Fredholm Integral of the Second Kind, which has the form:

$\phi(x)=f(x)+\lambda \int_{-\infty}^\infty K(x, t)\phi(t)dt$

with $\phi$ unknown and $\lambda$, $K$, $f$ known. Is Fredholm Theory helpful, even though I'm solving for a different term in the expression than usual?

Improve this question
$\endgroup$
2
  • 2
    $\begingroup$ The finite-dimensional analog of this problem would be to find a matrix $K$ given only that one particular vector is an eigenvector with eigenvalue 1. That problem is obviously very underdetermined (as soon as the dimension exceeds 1). In the infinite-dimensional case of integral kernels, the problem is even "more" underdetermined. That's why Robert Israel's answer gives lots of $K$'s that work. $\endgroup$
    – Andreas Blass
    Jul 15, 2012 at 18:10
  • $\begingroup$ Seconding @Andreas Blass' comment, and noting how simple Bob Israel's solution is, I might go so far as to speculate that there's some misunderstanding about the context of the question... and perhaps upon closer examination the question should be somewhat different than as posed. $\endgroup$
    – paul garrett
    Jul 29, 2012 at 18:27

1 Answer 1

Reset to default
2
$\begingroup$

One family of solutions is $K(x,t) = \phi(x) h(t)$ where $h(t)$ is any function such that $\int_{-\infty}^\infty h(t) \phi(t)\ dt = 1$.

Somewhat more generally, if $\phi(x) = \sum_{n=1}^N \phi_n(x)$ take $K(x,t) = \sum_{n=1}^N \phi_n(x) h_n(t)$ where $\int_{-\infty}^\infty h_n(t) \phi(t)\ dt = 1$ for each $n$.

You could also add any linear combination of functions $u(x) v(t)$ where $\int_{-\infty}^\infty v(t) \phi(t)\ dt = 0$.

Improve this answer
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.