In what sense does the finite difference operator converge to the continuum differential operator in the hydrodynamic limit?

Question

For simplicity, let us set $V_n:=\{-1+1/2^n, \dotsc, 0 ,\dotsc, 1-1/2^n, 1\}$ with the periodic boundary conditions for each $n \in \mathbb{N}$ and think of the following vector space over this lattice:

\begin{equation} M(V_n):=\{f:V_n \to \mathbb{C} \mid f \text{ is the restriction of some Schwartz function on } \mathbb{R} \text{ to } V_n \}.\end{equation}

Then, clearly, $M(V_n)$ is a well-defined finite dimensional vector space over $\mathbb{C}$. Moreover, we can think of the direct limit of the collection $\{ M(V_n)\}_n$, which may be identified as the restriction of Schwartz space to the domain $(-1,1]$.

So, here are my questions. They are divided into two parts.

1. $M(V_n)$ is set-theoretically identical to the "set of all mappings from $V_n$ to $\mathbb{C}$". However, the "set of all mappings from $V_n$ to $\mathbb{C}$" cannot form a direct limit of the above kind since they can approximate singular functions such as the Heaviside function. What makes the crucial difference?

2. A finite symmetric difference operator $\partial_n$ on $M(V_n)$ defined by \begin{equation} \partial_n f(x):=2^{n+1}[f(x+1/2^n)-f(x-1/2^n)] \end{equation} is a linear operator. As $n \to \infty$, in what sense does this $\partial_n$ converge to the ordinary continuum differential operator?

These two questions seem quite subtle to me…. Could anyone please clarify?

Answer 1

"In what sense does this $\partial_n$ converge to the ordinary continuum differential operator?"

It converges in the sense that there exist injective linear maps $\iota_n:M(V_n)\to C^\infty_\text{periodic}([-1,1])$ that take the $2^{n+1}$ eigenspaces of $\partial_n$ to the first $2^{n+1}$ eigenspaces of $d/dx$, such that the $j$-th eigenvector of $\partial_n$ converges pointwise on $[-1,1]\cap \mathbb Z[\tfrac12]$ to the $j$-th eigenvector of $d/dx$, as $n\to \infty$.

Here, the sentence "first $2^{n+1}$ eigenspaces of $d/dx$" requires some clarification:

The eigenspaces of $d/dx$ are ordered by increasing absolute value of the eigenvalue, with the (arbitrary) convention that $+a$ comes before $-a$ whenever $a>0$.