I've been musing about the following question for a while now. Given an integral operator $G$ defined by $$ (Gf)(x) = \int_0^1 G(x,u) f(u)\,du $$ Is it possible to decompose this into two separate "one-sided" integral operators $L$ and $R$ such that $$ (Gf)(x) = \int_x^1 L(x,u) \int_0^x R(u,v) f(v)\,dv\,du $$ If so, under what assumptions on $G$? And are there expressions for the left and right operators $L$ and $R$ in terms of $G$?
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imho, if you can rewrite the equation in a suitable basis, you obtain an infinite matrix and may try the QR decomposition. Try wavelet basis or a parent of this family of functions like curvelets, ridglets... regarding properties of the operators $G$. If $G$ is a pseudo or Fourier integral operator, these bases are known to give almost diagonal representations.