I need to solve an integral equation of the form(adsbygoogle = window.adsbygoogle || []).push({});

$$\forall \omega \in [0,1], ~ \int_{\mathbb{R}} K(\omega,y)f(y)dy = \omega$$

where

- f is known and positive with $$\int_{\mathbb{R}} f(y)dy = 1$$

- K: [0,1] x R -> [0,1] is the unknown kernel

I am looking for a solution other than K(omega,y) = omega. I do not know if such a solution exists, so I am looking either for a solution, or for a proof that K(omega,y) = \omega is the only solution.

----------------------

Let me tell you more about my attempts :

- looking for K(omega,y) = a(omega)b(y) fails because it leads to a(omega) = \omega and b(y) = 1

- looking for K(omega,y) = a(omega-y) and trying to solve the convolution equation $$\forall \omega \in [0,1], ~ (a \star f)(\omega) = \omega$$ fails (I proved it with Fourier transforms, although I'm not 100% confident about my proof)

- setting K(omega,y) = omega + H(omega,y) and trying to solve $$\int_{\mathbb{R}} H(\omega,y) f(y) = 0$$ with either H(omega,y) = a(omega)b(y) or H(omega,y) = a(y-omega) fails as well

There may not be a solution but I hope you can help me find one or prove that it's impossible ! Thank you

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# Integral equation with bounded unknown kernel

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