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For all $x \in \mathbb{R}$, $$b(x) =\int_{-\infty}^\infty b(s)a(x,s)ds.$$ If it helps, we can assume that $a, b$ are continuous, nonnegative, and $\int_{-\infty}^\infty$ of $a$ or $b$ are both bounded.

Two questions: (1) Is $a$ unique? and (2) to what extent can we solve for $a$?

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    This looks like a special case of Fredholm Integral Equations of the Second Kind, with f(x) = 0, $\lambda$ = 1, and where $\phi$ is known instead of the Kernel. That might be useful.2012-07-14
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    A corresponding matrix problem: Given an eigenvector $b$ for an $n \times n$ matrix $A$, solve for the matrix. Answer: no, $A$ is not unique.2012-07-14
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    This sounds helpful, but I'm not sure what you mean: in what sense is that a corresopnding matrix problem? Thanks for your help.2012-07-14
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    Ah, because $\int_{-\infty}^\infty$ is an inner product. That's a very interesting way to look at this problem.2012-07-14

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