Using $(e^{ik\cdot x})$ as an orthonormal basis for $L^2(\mathbb{R}^d;\mathbb{C})$ to define trace.

Question

I know that $e^{ik\cdot x}$ are not elements of $L^2$. But I believe this is often used in quantum mechanics, and wondered if there is some justification for it.

For example, I have seen the trace of an operator $A$ on $L^2(\mathbb R^d;\mathbb C)$ defined by $$\text{Tr}(A)=\iint A\left(e^{ik\cdot x}\right)e^{-ik\cdot x}\;{\rm d}k\,{\rm d}x.$$

Can we view this orthonormal basis as a limit, or in some weak sense?