I am working in a Hilbert space which is the closure of $C_{c}^{\infty}(\mathbb{R})$ with respect to the inner product given by \begin{equation} \tau(f,g)=\Re{\int_{0}^{\infty}p \hat{f}(-p)\hat{g}(p)}dp, \end{equation} where $\hat{f}$ denotes the Fourier transform of $f$. Call $K$ this Hilbert space.
I have a function $g(x)=\theta(t-x)h(x)$ where $\theta$ the Heaviside step function and $h\in C_{c}^{\infty}(\mathbb{R})$ such that $h(t)=0$, then $g$ is picewise smooth function with compact support, but ¿How can I prove that $g\in K$? I tried to approximate $g$ by convolution with standerd mollifiers, but it didn't work to me.
Moreover I am interesting in know if any picewise smooth function with compact support is in $K$.