Integral transforms - convergence

[Pere Callahan]

Hi,

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

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

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

Thanks

Pere

 

[fresh_42]

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.