I am looking for non-negative functions f:R -> R, f(x) >= 0 which fulfill
\int_oo^oo f(x)dx = \int_oo^oo f^2(x)dx.
Does this integral equation have any solution besides f(x) = 0 and if yes
how would one solve such an equation (analytically)? One might need to
specify further properties of the functions but at the moment I am not
sure what I would want to allow or disallow.
Thanks in advance
Till
Let f(x)=1 for x in [a,b] (-oo < a <b < oo) and 0 elsewhere; this
works for infinitely many choices of a and b.
Until you add more conditions, the set of solutions is so large that
you're not going to be able to describe the solutions except by
saying that you're talking about functions with int f = int f^2.
Proof that the set of solutions is incredibly large: Suppose that
g is any non-negative function such that both integrals int g
and int g^2 are finite and strictly positive. Then there exists
a > 0 such that if f = ag then int f = int f^2.
>Thanks in advance
>
> Till
ok, it helps not to be blind ;) Thanks. What if I want f to be
p >= 1 times differentiable everywhere?
Best
Till
>Hi,
>
>ok, it helps not to be blind ;) Thanks. What if I want f to be
>p >= 1 times differentiable everywhere?
>
Try reading all the replies.