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Apr 25, 2019 at 9:20 comment added Pietro Majer To put it in a slightly more general frame: your operator is a rank-one perturbation of the identity (hence Fredholm-$0$): $f\mapsto f-\lambda \langle\phi,f\rangle u$. Here with $u(x):=e^x$ and $\langle\phi,f\rangle:=\int_a^b e^{-y}f(y)dy$; the same conclusions hold in general.
Apr 24, 2019 at 17:06 comment added jonathan wolf @PietroMajer , thank you so much for your valuable comment! :)
Apr 23, 2019 at 10:08 comment added Pietro Majer in fact the last "another idea" is correct, so you already answered!
Apr 23, 2019 at 9:24 history asked jonathan wolf CC BY-SA 4.0