Using a basis for $L^2(\mathbb{R}^d)$ to get a basis for $L^2(\{x\}\times\mathbb{R}^{d-k})$ for all $x$

Question

I am in the situation where it would be very convenient if I could take a basis $\{f_i\}_{i = 1}^\infty$ for $L^2(\mathbb{R}^n)$ and manipulate it in some way to get a basis for the square-integrable functions on a collection of slices, namely $L^2(\{x\}\times\mathbb{R}^{n-k})$ for each $x \in \mathbb{R}^{k}$. I know of course that I could do it on for each $x$, using Gram-Schmidt, but I need to relate the bases for each $x$.

How would I best do that?