Let $\epsilon > 0$. I would like to know if there exists $c < \infty$ such that for all $d \in \mathbb{N}$ the following holds. If $x \in \mathbb{R}^d$ let $N_x$ be the standard Gaussian centered at $x$. Let $\mathcal{V}$ be the subspace of $L^2(\mathbb{R}^d)$ spanned by the vectors $\{N_x: x \in \mathbb{Z}^d,||x||_2 \geq c\}$. Then the $L^2$ norm of the orthogonal projection of $N_0$ onto $\mathcal{V}$ is less than $\epsilon$.

I am also interested in the version of this question where the functions $N_x$ are the normalized indicators of balls of radius $\sqrt{d}$.

An idea that might be useful is as follows. In order to obtain the desired bound, it suffices to bound the norm of the projection onto the closed span $\mathcal{W}$ of $\{N_x :x \in \mathbb{R}^d,||x||_2 \geq c \}$. Since both $N_0$ and $\mathcal{W}$ are rotation-invariant, the projection of $N_0$ onto $\mathcal{W}$ must be rotation-invariant. Intuitively, this suggests that the projection should be a positive scalar multiple of

$f = \int_{\mathrm{O}(d)} u \cdot N_{\overline{x}} \, \mathrm{d} \sigma(u)$

where $\sigma$ is the Haar measure on the orthogonal group $\mathrm{O}(d)$ and $\overline{x} \in \mathbb{R}^d$ is any fixed point with $||\overline{x}||_2 = c$. Now,

$\langle N_0,f \rangle = \int_{\mathbb{R}^d} N_0(x) f(x) \, \mathrm{d}x = \int_{\mathbb{R}^d} N_0(x) \left( \int_{\mathrm{O}(d)} u \cdot N_{\overline{x}} \, \mathrm{d} \sigma(u) \right)(x) \, \mathrm{d}x \\ \int_{\mathrm{O}(d)} \int_{\mathbb{R}^d} N_0(x)( u \cdot N_{\overline{x}})(x)\, \mathrm{d}x \, \mathrm{d} \sigma(u) \\ = \int_{\mathrm{O}(d)} \int_{\mathbb{R}^d} (u \cdot N_0)(x) N_{\overline{x}}(x) \, \mathrm{d}x \,\mathrm{d}\sigma(u) \\ = \int_{\mathbb{R}^d} N_0(x) N_{\overline{x}}(x) \, \mathrm{d}x \leq \delta(c)$

where $\delta:[0,\infty) \to [0,\infty)$ is a function independent of $d$ with $\lim_{c \to \infty} \delta(c) = 0$.

Let $p$ be the projection of $N_0$ on $\mathcal{W}$ and let $\alpha > 0$ be such that $p = \alpha f$. We have $||p||^2 = \langle N_0,p \rangle = \langle N_0,\alpha f \rangle \leq \alpha \delta(c)$. Since $||p||_2 \leq 1$, we have $\alpha \leq 1/||f||_2$. Thus one would need to show that $||f||_2 > \beta$ for some absolute constant $\beta > 0$, as well as confirming the hypothesis about $f$.