## $\ell^2(\mathbb{Z}) \simeq \ell^2(\{ \sqrt{n} : n \in \mathbb{N}\})$

Here is a result of Viazovska from 2017:

There exists a collection of even Schwartz functions $$a_n: \mathbb{R} \to \mathbb{R}$$ such that for any even function $$f: \mathbb{R} \to \mathbb{R}$$ and any $$x \in \mathbb{R}$$ we have: $$f(x) = \sum_{n = 0}^\infty a_n(x) \, f(\sqrt{n}) + \sum_{n=0}^\infty \hat{a}_n(x)\, \hat{f}(\sqrt{n})$$ where the right hand side converges absolutely.

Then she defines a space $$\mathfrak{s}$$ of rapidly decaying sequences of numbers. There is an isomorphism between the space of even Schwartz functions and the kernel of an operator $$L$$:

• $$\Psi: f \mapsto (f(\sqrt{n}))_{n > 0} \oplus (\hat{f}(\sqrt{n}))_{n > 0}$$
• $$L: (x,y) \mapsto \sum_{n \in \mathbb{Z}} x_{n^2} – \sum_{n \in \mathbb{Z}} y_{n^2}$$

In Fourier Analysis class, we learn that $$L^2(\mathbb{R}/\mathbb{Z}) \simeq \ell^2(\mathbb{Z})$$ is that correct? I think we also have that all Hilbert spaces are Unitary equivalent. Here’s the statement from Stein’s textbook on Real Analysis:

Cor 2.5 Any two infinite dimensional Hilbert spaces are unitarily equivalent.

The example is that $$\ell^2(\mathbb{N})\simeq \ell^2(\mathbb{Z})$$ for all countable sets, the proof is to re-order the basis.

Then we sholud also have $$\ell^2(\mathbb{Z})\simeq$$ for the square root $$M = \{ \sqrt{n} : n > 0, \; n \in \mathbb{Z}\}$$ or even $$M = \{ \sqrt{m} + \sqrt{n} : m,n > 0 ,\; m,n \in \mathbb{Z} \}$$.

The above theorem discusses the behavior of elements of $$f \in L^2(\mathbb{R})$$ on the subset of the square-roots of integers: $$\{ \sqrt{n} : n \in \mathbb{N}\} \subset \mathbb{R}$$ and discusses the behaviors of $$f$$ and $$\hat{f}$$ combined.

What is it called when we handle both $$f \oplus \hat{f}$$ combined? In physics the same wavefunction $$f$$ could be evaluated at a point $$\langle f|x\rangle$$ or a momentum $$ and that $$(x,p) \simeq \mathbb{R}^2$$ could be combined in to a single plane. Something like symplectic geometry.

The Hamiltonian of the Harmonic oscillator could be written $$H = x^2 + p^2$$ and is symmetric under a rotations of the $$(x,p)$$ plane. This “physics” approach is likely to come up short compared to Viazovska’s careful analysis. Could this theorem deduced using microlocal analysis?

## Limit of sequence of vectors in $\ell^2$ with coefficients approaching $0$

Let $$\{v_m\}_{m \in \mathbb{N}} \subset \ell^2$$ be a sequence in $$\ell^2$$ over the complex plane $$\mathbb{C}$$ such that: $$\{v_m\}_{m \in \mathbb{N}}$$ is linearly independend and $$v_m \to v$$

Let $$V= \operatorname{span} \{v_m\}_{m \in \mathbb{N}}$$

Let $$\{u_p\}_{p \in \mathbb{N}} \subset V$$ be a sequence in $$V$$ such that $$u_p \to u$$ so we have $$\forall p \in \mathbb{N}: u_p = \sum_{m=1}^\infty \left( a_{p,m} \cdot v_m \right)$$ with $$a_{m,p} \in \mathbb{C}$$ and for each fixed $$p \in \mathbb{N}$$ there are only finitely many m with $$a_{p,m} \neq 0$$

Further, we have for each fixed $$m \in \mathbb{N}$$ $$\lim_{p \to \infty} a_{p,m} =0$$ My question is if it is true that: $$\lim_{p \to \infty} u_p = a \cdot v$$ with $$a \in \mathbb{C}$$

Thanks.