Integral transforms - convergence

Hi,

I have a question which is most probably standard, but I don't have access to a textbook right now...

Given a function [itex]K(y,x):\mathbb{R}^2\to\mathbb{R}[/itex] and a funtion [itex]f_0:\mathbb{R}\to\mathbb{R}[/itex], under what circumstances does the sequence [itex](f_n)_{n\geq0}[/itex] with
[tex]
f_n(x) = \int_{\mathbb{R}}{dy\, f_{n-1}(y)K(y,x)}
[/tex]
converge?

I have a feeling [itex]|K(x,y)|<1[/itex] could be sufficient (maybe if one assumes that f_0 tends to zero fast enough...)

Thanks

Pere
 

fresh_42

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You will have to impose conditions on ##f_0##, yes, since otherwise we could set ##K=\frac{1}{2}## and chose a function ##f_0## which hasn't a finite integral.

In general I would look at ##L^1## spaces.
 

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