Re: f and f' square integrable

[Stephen Montgomery-Smith]

Fedor wrote:
> Hi all,
>
>   suppose that f:R^n -->R is a smooth function such that f^2 and ( f '
> )^2 are integrable over R^n. Is it true that f(x) tends towards 0 when
> |x| tends towards infinity ? It is easy for n=1 but is it true for the
> general case ?
>
> Regards,
> fedor

I think not.

First consider a function like f(x)=log(1/|x|)^a for 0<a<1/2.  Check 
that f and f' are square integrable in R^2.

Next consider sum a_n f(x-(2n,0)) where a_n is square summable.