Find a function $f$ in $S(\mathbb R)$ the schwarz class such that $\|f\|_{2} = 1$, and $\int x^k f(x) =0$ for any $k$

Question

Find a function $f$ in $S(\mathbb R)$ the schwarz class such that $\|f\|_{2} = 1$, and $\int x^k f(x) =0$ for any $k$. I somehow think this is related to Fourier transform. Is it?

Thoughts: Consider the Fourier transform of $\hat f(\eta) = \int f(x) e^{-2\pi i x\eta}dx$, then $\partial_k \hat f (0)= (-2\pi i)^k \int x^k f(x) = 0$. Then I have a characteriztion of $\hat f$'s derivatives at zero. Then I am aiming for a function with this property. The first thing that comes to my mind is $e^{\frac{1}{-x^2}}$, but it is not in the Schwarz Class I believe.

Answer 1

If $f\in \mathcal S,$ then so is $\hat f,$ and $f$ is the inverse Fourier transform of $\hat f.$ It follows that

$$D^kf(0) = i^k\int_{\mathbb R} x^k\hat f(x)\,dx,\,\, k=0,1,2,\dots,$$

modulo constants arising from the definition/normalization of the Fourier transform.

So take $f(x) = \exp (-1/[x(1-x)]),$ $0<x<1,$ $f=0$ elsewhere. Then $f\in \mathcal S,$ and all derivatives of $f$ at $0$ vanish. Since $\mathcal S\subset L^2,$ $\hat f/\|\hat f\|_2$ will have the desired properties.