SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Proof 1 ("clustering-based"). Suppose that a function  $f_{*}$  is the uniform limit of functions  $f_{r} \in H_{N}^{(2)}$ . We identify the different frequency clusters associated with the sequence  $f_{r}$ . Let  $0 \leq \omega_{1,r} \leq \ldots \leq \omega_{N,r}$  be the  $N$  sorted frequencies appearing in the sine wave expansion of  $f_{r}$ ; we assume here without loss of generality that all the frequencies are nonnegative. For each  $n = 1, \ldots, N - 1$  one of the following three options holds:
1. either  $\omega_{n+1,r} - \omega_{n,r} \stackrel{r \to \infty}{\longrightarrow} \infty$ ;
2. or  $\sup_n (\omega_{n+1,r} - \omega_{n,r}) < \infty$ ;
3. or none of the above holds.
For each  $n$ , by taking a subsequence of  $f_{r}$ , we can exclude option 3. By taking subsequences  $N$  times, we can then assume without loss of generality that for each  $n$  either option 1 or option 2 holds. It follows that we can partition the set  $\{1,\dots ,N\}$  into contiguous subsets  $I_{1} = \{1,2,\dots ,i_{1}\} ,I_{2} = \{i_{1} + 1,\ldots ,i_{2}\} ,\ldots ,I_{s} = \{i_{s - 1} + 1,\ldots ,N\}$  such that for some constant  $\Omega$
$$
\left| \omega_ {n, r} - \omega_ {n ^ {\prime}, r} \right| \left\{\begin{array}{l l}\stackrel {{r \rightarrow \infty}} {{\longrightarrow}} \infty ,&\text {i f} \not \exists t: n, n ^ {\prime} \in I _ {t};\\\leq \Omega \forall r,&\text {i f} \exists t: n, n ^ {\prime} \in I _ {t}.\end{array}\right. \tag {18}
$$
The sets  $I_{t}$  thus represent the largest clusters of frequencies that remain at a bounded distance from each other in the limit  $r\to \infty$ .
Clearly, there is at most one cluster whose frequencies remain bounded in the limit  $r \to \infty$ , namely  $I_{t_0}$  with  $t_0 = 1$ . It will be convenient to assume that  $I_{t_0}$  has bounded frequencies but possibly is empty.
For any  $M = 1,2,\ldots$  and  $\Omega >0$  let us denote by  $H_{M,\Omega}^{(2)}$  the set of functions  $f:[0,1]\to \mathbb{C}$  of the form
$$
f (x) = \sum_ {n = 1} ^ {M} c _ {n} e ^ {i \omega_ {n} x}, \tag {19}
$$
where  $c_{n}\in \mathbb{C}$  are arbitrary amplitudes and  $\omega_{n}\in [-\Omega ,\Omega ]$
Lemma 1. For any  $f \in H_{M,\Omega}^{(2)}$  and  $n = 1,2,\ldots$
$$
\max  _ {x \in [ 0, 1 ]} | f ^ {(n)} (x) | \leq 2 ^ {M (M + 1)} (1 + \Omega) ^ {M \max  (n, M - 1)} \max  _ {x \in [ 0, 1 ]} | f (x) |. \tag {20}
$$
We remark that the constant in Eq. (20) is likely suboptimal, but it must certainly grow with  $M$ , since, as the proof of the sufficiency part of Theorem 4 shows, arbitrarily narrow frequency bands are sufficient to generate any degree- $M$  polynomial.
Proof. The function  $f$  satisfies the differential equation  $Df = 0$ , where
$$
D = \prod_ {n = 1} ^ {M} \left(\frac {d}{d x} - i \omega_ {n}\right) = \frac {d ^ {M}}{d x ^ {M}} - \sum_ {n = 0} ^ {M - 1} b _ {n} \frac {d ^ {n}}{d x ^ {n}} \tag {21}
$$
with some constants  $b_n$ . Consider the vector-valued function  $\mathbf{f}(x) = (f(x), f'(x), \ldots, f^{(M-1)}(x))^T$ . It satisfies the differential equation
$$
\frac {d}{d x} \mathbf {f} = A \mathbf {f} \tag {22}
$$
with the matrix  $A = (a_{nm})_{n,m = 0}^{M - 1}$ , where
$$
a _ {n m} = \left\{ \begin{array}{l l} 1, & n + 1 = m \text {a n d} n <   M - 1, \\ b _ {m}, & n = M - 1, \\ 0, & \text {o t h e r w i s e .} \end{array} \right. \tag {23}
$$
The solution of this equation is
$$
\mathbf {f} (x) = e ^ {A \left(x - x _ {0}\right)} \mathbf {f} \left(x _ {0}\right). \tag {24}
$$
If  $\| \cdot \|$  is any norm on the  $\mathbb{C}^M$ , then
$$
\left\| \mathbf {f} (x) - \mathbf {f} \left(x _ {0}\right) \right\| \leq \left(e ^ {\left\| A \right\| \left| x - x _ {0} \right|} - 1\right) \| \mathbf {f} \left(x _ {0}\right) \|, \tag {25}
$$
where  $\| A\|$  is the respective matrix norm. Let  $\| \cdot \|_{\infty}$  be the maximum  $(l^{\infty})$  norm, then
$$
\| A \| _ {\infty} = \max  _ {n} \sum_ {m} | a _ {n m} | = \max  \left(\sum_ {m} | b _ {m} |, 1\right) \leq \prod_ {m = 1} ^ {M} (1 + | \omega_ {m} |) \leq (1 + \Omega) ^ {M}. \tag {26}
$$
Now let  $x_0 = \arg \max_{x \in [0,1]} \| f(x) \|_\infty$  and let  $n_0 = \arg \max_{n=0,\dots,M-1} |f^{(n)}(x_0)|$  so that  $\| \mathbf{f}(x_0) \|_\infty = |f^{(n_0)}(x_0)|$ . If  $n_0 = 0$ , then
$$
\max  _ {x \in [ 0, 1 ]} | f ^ {\prime} (x) | \leq \max  _ {x \in [ 0, 1 ]} | f (x) |. \tag {27}
$$
Suppose that  $n_0 > 0$ . Observe that if  $|x - x_0| \leq \frac{\ln 2}{2} (1 + \Omega)^{-M}$ , then
$$
\begin{array}{l} \left| f ^ {\left(n _ {0}\right)} (x) - f ^ {n _ {0}} \left(x _ {0}\right) \right| \leq \left\| \mathbf {f} (x) - \mathbf {f} \left(x _ {0}\right) \right\| _ {\infty} (28) \\ \leq \left(e ^ {\| A \| _ {\infty} | x - x _ {0} |} - 1\right) \| \mathbf {f} \left(x _ {0}\right) \| _ {\infty} (29) \\ \leq \left(e ^ {\left(1 + \Omega\right) ^ {M} | x - x _ {0} |} - 1\right) \left| f ^ {\left(n _ {0}\right)} \left(x _ {0}\right) \right| (30) \\ \leq \frac {1}{2} \left| f ^ {\left(n _ {0}\right)} \left(x _ {0}\right) \right|. (31) \\ \end{array}
$$
Consider the complex unit direction  $z_0 = \frac{f^{(n_0)}(x_0)}{|f^{(n_0)}(x_0)|}$  and denote  $l = \frac{\ln 2}{2} (1 + \Omega)^{-M}$ . For all  $x$  such that  $|x - x_0| \leq l$  we have
$$
\Re \left(\bar {z} _ {0} f ^ {\left(n _ {0}\right)} (x)\right) \geq \frac {1}{2} | f ^ {\left(n _ {0}\right)} \left(x _ {0}\right) |. \tag {32}
$$
It follows that we can find in [0, 1] a subinterval of length  $l / 4$  on which either
$$
\Re \left(\bar {z} _ {0} f ^ {\left(n _ {0} - 1\right)} (x)\right) \geq \frac {l}{2 \cdot 4} \left| f ^ {\left(n _ {0}\right)} \left(x _ {0}\right) \right| \tag {33}
$$
$$
\Re \left(\bar {z} _ {0} f ^ {\left(n _ {0} - 1\right)} (x)\right) \leq - \frac {l}{2 \cdot 4} \left| f ^ {\left(n _ {0}\right)} \left(x _ {0}\right) \right|. \tag {34}
$$
Making another iteration, we can find in [0, 1] a subinterval of length  $l / 4^2$  on which either
$$
\Re \left(\bar {z} _ {0} f ^ {\left(n _ {0} - 2\right)} (x)\right) \geq \frac {l}{4 ^ {2}} \cdot \frac {l}{2 \cdot 4} | f ^ {\left(n _ {0}\right)} \left(x _ {0}\right) | \tag {35}
$$
$$
\Re \left(\bar {z} _ {0} f ^ {(n _ {0} - 2)} (x)\right) \leq - \frac {l}{4 ^ {2}} \cdot \frac {l}{2 \cdot 4} | f ^ {(n _ {0})} (x _ {0}) |. \tag {36}
$$
Continuing in this way, for any  $n = n_0, n_0 - 1, \ldots, 0$  we find in [0, 1] a subinterval of length  $l / 4^{n_0 - n}$  on which either
$$
\Re \left(\bar {z} _ {0} f ^ {\left(n _ {0} - n\right)} (x)\right) \geq \frac {l ^ {n _ {0} - n}}{2 ^ {\left(n _ {0} - n\right) \left(n _ {0} - n + 1\right) + 1}} \left| f ^ {\left(n _ {0}\right)} \left(x _ {0}\right) \right| \tag {37}
$$
$$
\Re \left(\bar {z} _ {0} f ^ {\left(n _ {0} - n\right)} (x)\right) \leq - \frac {l ^ {n _ {0} - n}}{2 ^ {\left(n _ {0} - n\right) \left(n _ {0} - n + 1\right) + 1}} \left| f ^ {\left(n _ {0}\right)} \left(x _ {0}\right) \right|. \tag {38}
$$
Taking  $n = 0$ , this implies
$$
\left| f ^ {\left(n _ {0}\right)} \left(x _ {0}\right) \right| \leq 2 ^ {n _ {0} \left(n _ {0} + 1\right) + 1} l ^ {- n _ {0}} \max  _ {x \in [ 0, 1 ]} | f (x) | \tag {39}
$$
and so, by definition of  $x_0$  and  $n_0$ , for any  $n = 0,\ldots ,M - 1$
$$
\begin{array}{l} \max  _ {x \in [ 0, 1 ]} | f ^ {(n)} (x) | = | f ^ {\left(n _ {0}\right)} \left(x _ {0}\right) | (40) \\ \leq 2 ^ {n _ {0} \left(n _ {0} + 1\right) + 1} l ^ {- n _ {0}} \max  _ {x \in [ 0, 1 ]} | f (x) | (41) \\ \leq 2 ^ {(M - 1) M + 1} \left(\left(2 / \ln 2\right) (1 + \Omega) ^ {M}\right) ^ {M - 1} \max  _ {x \in [ 0, 1 ]} | f (x) | (42) \\ \leq 2 ^ {M (M + 1)} \left(1 + \Omega\right) ^ {M (M - 1)} \max  _ {x \in [ 0, 1 ]} | f (x) |. (43) \\ \end{array}
$$
This proves the claim for  $n \leq M - 1$ . If  $n \geq M$ , then we use again Eq. (21) and the bound  $\sum_{m} |b_m| \leq (1 + \Omega)^M$ .
Lemma 2. The  $L^{\infty}$  and  $L^2$  norms are equivalent on the set  $H_{M,\Omega}^{(2)}$ : for any  $f \in H_{M,\Omega}^{(2)}$
$$
\| f \| _ {2} \leq \| f \| _ {\infty} \leq 2 [ 2 (1 + \Omega) ] ^ {M (M + 1) / 2} \| f \| _ {2} \tag {44}
$$
Proof. Only the second inequality is nontrivial. By Lemma 1,
$$
\left\| f ^ {\prime} \right\| _ {\infty} \leq c \| f \| _ {\infty}, \quad c = [ 2 (1 + \Omega) ] ^ {M (M + 1)}. \tag {45}
$$
It  $x_0 = \arg \max_{x\in [0,1]}|f(x)|$  , then for  $|x - x_0|\leq 1 / c$  we have  $|f(x)|\geq (1 - |x - x_0|c)|f(x_0)|$  and so
$$
\left\| f \right\| _ {2} ^ {2} \geq \int_ {0} ^ {1 / c} (1 - t c) ^ {2} | f \left(x _ {0}\right) | ^ {2} d t = \frac {1}{3 c} \| f \| _ {\infty} ^ {2}, \tag {46}
$$
implying the desired result.
Consider now the expansion of  $f_{r}$  over different frequency clusters:
$$
f _ {r} = \sum_ {t = 1} ^ {s} f _ {t, r}, \tag {47}
$$
where  $f_{t,r}$  denotes the  $I_{t}$ -component of  $f_{r}$ . We first show that we can discard the components associated with unbounded frequencies.
Lemma 3. If  $f_{r}$  converge to  $f_{*}$  uniformly on [0,1], then  $f_{t_0,r}$  also converge to  $f_{*}$  uniformly on [0,1].
Proof. Step 1. Let us first prove the statement in the case  $f_{*} = 0$ . Observe that each  $f_{t,r}$  can be represented as
$$
f _ {t, r} = \sum_ {q = \pm 1} e ^ {i q \omega_ {t, r} x} g _ {t, r, q}, \tag {48}
$$
where  $g_{t,r,1} = \overline{g_{t,r, - 1}}$  and  $g_{t,r,q}\in H_{2N,\Omega}^{(2)}$  with some  $\Omega$  independent of  $t$  and  $r$ . We use this representation for all  $t$  except  $t = t_0$  (since  $f_{t_0,r}\in H_{2N,\Omega}^{(2)}$  already):
$$
f _ {r} = f _ {t _ {0}, r} + \sum_ {t \neq t _ {0}} ^ {s} \sum_ {q = \pm 1} e ^ {i q \omega_ {t, r} x} g _ {t, r, q}. \tag {49}
$$
$$
\left\| f _ {r} \right\| _ {2} ^ {2} = \left\langle f _ {r}, f _ {r} \right\rangle = \left\| f _ {t _ {0}, r} \right\| _ {2} ^ {2} + \sum_ {t \neq t _ {0}} ^ {s} \sum_ {q = \pm 1} \left\| g _ {t, r, q} \right\| _ {2} ^ {2} + \text {c r o s s - t e r m s}. \tag {50}
$$
Observe that a product of  $g, h \in H_{M,\Omega}^{(2)}$  belongs to  $H_{M^2,2\Omega}^{(2)}$ . It follows that each cross-term can be written as
$$
\langle e ^ {i \Delta \omega_ {r}}, g _ {r} \rangle , \tag {51}
$$
where  $\Delta \omega_r \stackrel{r \to \infty}{\longrightarrow} \infty$  and  $g_r \in H_{(2N)^2, 2\Omega}^{(2)}$ ; the function  $g_r$  being a product of two functions out of the collection  $F = \{f_{t_0,r}\} \cup \{(g_{t,r,q})_{t \neq t_0, q = \pm 1}\}$ . We have
$$
\begin{array}{l} \left| \left\langle e ^ {i \Delta \omega_ {r}}, g _ {r} \right\rangle \right| = \left| \int_ {0} ^ {1} e ^ {- i \Delta \omega_ {r}} g _ {r} (x) d x \right| (52) \\ = \left| \Delta \omega_ {r} \right| ^ {- 1} \left| \int_ {0} ^ {1} g _ {r} (x) d e ^ {- i \Delta \omega_ {r}} \right| (53) \\ \leq \left| \Delta \omega_ {r} \right| ^ {- 1} \left(2 \| g _ {r} \| _ {\infty} + \left| \int_ {0} ^ {1} e ^ {- i \Delta \omega_ {r}} g _ {r} ^ {\prime} (x) d x \right|\right) (54) \\ \leq \left| \Delta \omega_ {r} \right| ^ {- 1} (2 + c) \| g _ {r} \| _ {\infty}, (55) \\ \end{array}
$$
where we used the inequality  $\| g_r' \|_\infty \leq c \| g_r \|_\infty$  with a constants  $c = c(N, \Omega)$  as established in Lemma 1. Observe that if  $g_r$  is a product of two functions from the collection  $F$ , say  $g, h$ , then we have
$$
\left\| g _ {r} \right\| _ {\infty} \leq \left\| g \right\| _ {\infty} \left\| h \right\| _ {\infty} \leq C \left\| g \right\| _ {2} \left\| h \right\| _ {2}, \tag {56}
$$
with another suitable constant  $C = C(N, \Omega)$ , by Lemma 2. It follows that the cross-terms in Eq. (50) are dominated by the explicitly written terms:
$$
| \text {c r o s s - t e r m s} | = o \left(\left\| f _ {t _ {0}, r} \right\| _ {2} ^ {2} + \sum_ {t \neq t _ {0}} ^ {s} \sum_ {q = \pm 1} \| g _ {t, r, q} \| _ {2} ^ {2}\right), \quad r \rightarrow \infty . \tag {57}
$$
Therefore, if  $\| f_r\|_{\infty} = o(1)$  as  $r\to \infty$ , then we also have
$$
\left\| f _ {t _ {0}, r} \right\| _ {\infty} = O \left(\left\| f _ {t _ {0}, r} \right\| _ {2}\right) = O \left(\left\| f _ {r} \right\| _ {2}\right) = O \left(\left\| f _ {r} \right\| _ {\infty}\right) = o (1) \tag {58}
$$
and similarly all
$$
\left\| g _ {t, r, q} \right\| _ {\infty} = o (1). \tag {59}
$$
Step 2. Now we prove the result for a general  $f_{*}$ . If  $\| f_r - f_*\| _2\to 0$ , then the sequence  $f_{r}$  is Cauchy:
$$
\lim  _ {r _ {1} \rightarrow \infty} \lim  _ {r _ {2} \rightarrow \infty} \| f _ {r _ {1}} - f _ {r _ {2}} \| _ {2} = 0. \tag {60}
$$
$$
f _ {r _ {1}} - f _ {r _ {2}} = \left(f _ {t _ {0}, r _ {1}} - f _ {t _ {0}, r _ {2}}\right) + \sum_ {t \neq t _ {0}} ^ {s} \sum_ {q = \pm 1} e ^ {i q \omega_ {t, r _ {1}} x} g _ {t, r _ {1}, q} - \sum_ {t \neq t _ {0}} ^ {s} \sum_ {q = \pm 1} e ^ {i q \omega_ {t, r _ {2}} x} g _ {t, r _ {2}, q}. \tag {61}
$$
Assuming that  $r_2 = r_2(r_1)$  and grows sufficiently fast, the frequencies  $\omega_{t,r_1}, \omega_{t,r_2}$  and 0 (corresponding to the first term) will form disjoint clusters. We can then use the argument from Step 1 and conclude that all  $\| g_{t,r,q} \|_{\infty} = o(1)$ , implying the desired result.
The last lemma shows that any uniform limit of functions  $f_{r} \in H_{N}^{(2)}$  is also a uniform limit of band-limited functions  $f_{r} \in H_{2M,\Omega}^{(2)}$  for some  $\Omega < \infty$ . Any function  $f_{r} \in H_{N}^{(2)}$  is a solution of the differential equation
$$
D _ {r} f _ {r} = 0, \tag {62}
$$
$$
D _ {r} = \prod_ {n = 1} ^ {N} \left(\frac {d}{d x} - i \omega_ {n r}\right) \left(\frac {d}{d x} + i \omega_ {n r}\right). \tag {63}
$$
If the frequencies  $\omega_{nr}$  are bounded, by compactness we can find their limit points  $\omega_{n}$ . We argue now that the limiting function  $f_{*}$  is a solution of the corresponding limiting equation  $Df = 0$  with
$$
D = \prod_ {n = 1} ^ {N} \left(\frac {d}{d x} - i \omega_ {n}\right) \left(\frac {d}{d x} + i \omega_ {n}\right). \tag {64}
$$
Indeed, such solutions have the form
$$
\mathbf {f} (x) = e ^ {A (x - x _ {0})} \mathbf {f} \left(x _ {0}\right), \tag {65}
$$
(cf. Eq. (24)), while the functions  $f_{r}$  can be written as
$$
\mathbf {f} _ {r} (x) = e ^ {A _ {r} \left(x - x _ {0}\right)} \mathbf {f} _ {r} \left(x _ {0}\right), \tag {66}
$$
with respective matrices  $A_r$ . As  $r \to \infty$ , the matrices  $A_r$  converge (by taking a subsequence, if necessary) to  $A$ , and, by Lemma 1,  $\mathbf{f}_r(x_0) \to \mathbf{f}(x_0)$ . It follows that the function  $f_*$  indeed is representable in the form (65), i.e. is a solution of equation  $Df = 0$ . This immediately implies the claim of the theorem (with isolated zero limiting frequencies  $\omega_n = 0$ ).
Proof 2 ("discretization-based"). Our second proof essentially relies on discretizations of the function  $f_{*}$  and will consist of two steps. In the first step we introduce the discretizations, show that they satisfy discrete difference equations, and describe general solutions of these equations. In the second step we analyze how discretizations at different length scales agree with each other and show that the continuity of  $f_{*}$  enforces representation (6) in the zero spacing limit.
Step 1: Discretization. Suppose that the function  $f_{*}$  is the uniform limit of functions  $f_{r} \in H_{N}^{(2)}, r = 1,2,\ldots$ . Fix some  $\Delta x > 0$  and consider the sequences  $\mathbf{u}^{*} = (u_{s}^{*})_{s=0}^{\lfloor 1/\Delta x \rfloor}$  and  $\mathbf{u}^{(r)} = (u_{s}^{(r)})_{s=0}^{\lfloor 1/\Delta x \rfloor}$  obtained by restricting these functions to the points  $s\Delta x$ :
$$
u _ {s} ^ {*} = f _ {*} (s \Delta x), \quad u _ {s} ^ {(r)} = f _ {r} (s \Delta x), \quad s = 0, 1, \dots , \lfloor 1 / \Delta x \rfloor . \tag {67}
$$
Since  $f_r \in H_N^{(2)}$ , each sequence  $\mathbf{u}^{(r)}$  can be written as a linear combination
$$
u _ {s} ^ {(r)} = \sum_ {n = 1} ^ {N} b _ {n r} e ^ {i \omega_ {n r} s \Delta x} + \overline {{b _ {n r}}} e ^ {- i \omega_ {n r} s \Delta x}, \quad s = 0, 1, \dots , \lfloor 1 / \Delta x \rfloor \tag {68}
$$
with some complex amplitudes  $b_{nr} \in \mathbb{C}$  and frequencies  $\omega_{nr} \in \mathbb{R}$  (the horizontal bar denotes complex conjugation).
We want to perform now the limit  $r\to \infty$  and derive a similar representation for  $\mathbf{u}^*$ . To this end observe that, up to the boundary values, a sequence of the form  $u_{s} = be^{i\omega s\Delta x}$  satisfies the finite difference equation
$$
\left(T - e ^ {i \omega \Delta x}\right) \mathbf {u} = 0, \tag {69}
$$
where  $T$  is the left shift:
$$
\left(T u\right) _ {s} = u _ {s + 1}. \tag {70}
$$
Since each sequence  $\mathbf{u}^{(r)}$  has the form of linear combination (68) and the operators  $T - e^{i\omega \Delta x}$  commute, it follows that  $\mathbf{u}^{(r)}$  satisfies the finite difference equation
$$
D ^ {(r)} \mathbf {u} ^ {(r)} \stackrel {\text {d e f}} {=} \prod_ {n = 1} ^ {N} [ (T - e ^ {i \omega_ {n r} \Delta x}) (T - e ^ {- i \omega_ {n r} \Delta x}) ] \mathbf {u} ^ {(r)} = 0. \tag {71}
$$
This equation holds up to the  $2N$  boundary components  $u_{s}^{(r)}$  that are moved by the shifts  $T$  contained in  $D^{(r)}$  outside the domain  $[0,1]$  of the functions  $f_{r}$ .
We would like to perform the limit  $r \to \infty$  in this difference equation, but we cannot directly perform it in the exponents  $i\omega_{nr}\Delta x$ , since the frequencies  $\omega_{nr}$  may be unbounded and not have any limit points. However, the values  $e^{i\omega_{nr}\Delta x}$  belong to the unit circle in  $\mathbb{C}$ , which is a compact set. Therefore, we can choose a subsequence  $r_t$  such that  $e^{i\omega_{nrt}\Delta x} \to z_n^*$  as  $t \to \infty$ , with some  $z_n^* \in \mathbb{C}, |z_n^*| = 1$ .
Performing this limit in the difference equation, we then see that the limiting sequence  $\mathbf{u}^*$  satisfies a difference equation
$$
D \mathbf {u} ^ {*} \stackrel {\text {d e f}} {=} \prod_ {n = 1} ^ {N} \left[ \left(T - z _ {n} ^ {*}\right) \left(T - \overline {{z _ {n} ^ {*}}}\right) \right] \mathbf {u} ^ {*} = 0. \tag {72}
$$
A general solution of this equation is again a linear combination of sine waves, but with possible resonances due to repeated values  $z_{n}^{*}$ . Suppose that the numbers  $z_{n}^{*}$  contain  $K$  distinct nonreal values  $z_{1},\ldots ,z_{K}$  with multiplicities  $M_{k}\geq 1$ . Additionally, let  $M_{\pm}\geq 0$  be the multiplicities of the real values  $z_{\pm} = \pm 1$  among the numbers  $z_{n}^{*}$ . We can then write the general real solution of equation (72) as
$$
u _ {s} ^ {*} = \sum_ {m = 0} ^ {2 M _ {-} - 1} b _ {-, m} s ^ {m} (- 1) ^ {s} + \sum_ {m = 0} ^ {2 M _ {+} - 1} b _ {+, m} s ^ {m} + \sum_ {k = 1} ^ {K} \sum_ {m = 0} ^ {M _ {k} - 1} s ^ {m} \left(b _ {k m} z _ {k} ^ {s} + \overline {{b _ {k m}}} \bar {z _ {k}} ^ {s}\right), \tag {73}
$$
with  $M_{-} + M_{+} + \sum_{k=1}^{K} M_{k} = N$ . Equivalently, we can write
$$
\mathbf {u} ^ {*} = \sum_ {m = 0} ^ {2 M _ {-} - 1} b _ {-, m} \mathbf {v} _ {-, m} + \sum_ {m = 0} ^ {2 M _ {+} - 1} b _ {+, m} \mathbf {v} _ {+, m} + \sum_ {k = 1} ^ {K} \sum_ {m = 0} ^ {M _ {k} - 1} \left(b _ {k m} \mathbf {v} _ {k m} + \overline {{b _ {k m}}} \overline {{\mathbf {v} _ {k m}}}\right), \tag {74}
$$
where we have introduced the vectors  $\mathbf{v}_{\pm ,m}$ ,  $\mathbf{v}_{km}$ ,  $\overline{\mathbf{v}_{km}} \in \mathbb{C}^{[1 / \Delta x] + 1}$  with components
$$
\left(v _ {\pm , m}\right) _ {s} = s ^ {m} (\pm 1) ^ {s}, \quad \left(v _ {k m}\right) _ {s} = s ^ {m} z _ {k} ^ {s}, \quad \left(\overline {{v _ {k m}}}\right) _ {s} = s ^ {m} \overline {{z _ {k}}} ^ {s}. \tag {75}
$$
The coefficients  $b_{km}$  and  $b_{\pm,m}$  in expansions (73), (74) are determined uniquely as long as  $\Delta x$  is sufficiently small:
Lemma 4. Suppose that  $\Delta x \leq 1/(2N-1)$  so that  $\dim \mathbf{u}^{*} = \lfloor 1 / \Delta x \rfloor + 1 \geq 2N$ . Then the  $2N$  vectors  $\mathbf{v}_{\pm,m}$ ,  $\mathbf{v}_{km}$  and  $\overline{\mathbf{v}_{km}}$  are linearly independent and the coefficients  $b_{\pm,m}, b_{km}$  are uniquely determined.
Proof. For  $1 \leq q \leq M_k$ , consider the finite difference operators  $D_{k,q}$  obtained by removing the factor  $(T - z_k)^q$  from the operator  $D$  that appears in Eq. (72):
$$
D _ {k, q} = \frac {D}{(T - z _ {k}) ^ {q}}. \tag {76}
$$
When applied to  $\mathbf{u}^*$ , this operator kills all the components of the expansions (73), (74) except those associated with  $z_{k}$  and having degree  $m\geq M_k - q$ . In particular,
$$
\left(D _ {k, 1} u ^ {*}\right) _ {s} = a _ {k} b _ {k, M _ {k} - 1} z _ {k} ^ {s} \tag {77}
$$
with the explicit coefficient
$$
a _ {k} = \left(M _ {k} - 1\right)! \left(z _ {k} - \overline {{z _ {k}}}\right) ^ {M _ {k}} \prod_ {k ^ {\prime} \in \{\pm \} \cup \{1, \dots , K \} \backslash \{k \}} \left(\left(z _ {k} - z _ {k ^ {\prime}}\right) \left(z _ {k} - \overline {{z _ {k ^ {\prime}}}}\right)\right) ^ {M _ {k ^ {\prime}}} \neq 0. \tag {78}
$$
We can then find the coefficient  $b_{k,M_k - 1}$  from Eq. (77) if there is at least one  $s$  for which it holds (recall that it is not applicable to a few boundary values of  $s$ ). The highest power of the shift  $T$  appearing in  $D_{k,1}$  is  $T^{2N - 1}$ , so such  $s$  exists as long as  $\dim \mathbf{u}^* \geq 2N$ .
Having found  $b_{k,M_k - 1}$ , we can subtract the corresponding component  $b_{k,M_k - 1}\mathbf{v}_{k,M_k - 1}$  from  $\mathbf{u}^*$  and apply to the remainder the operator  $D_{k,2}$ , which will kill all terms except  $b_{k,M_k - 2}\mathbf{v}_{k,M_k - 2}$ . This allows to determine the next coefficient  $b_{k,M_k - 2}$ . Continuing this process and extending it to other  $z_k$  and  $z_{\pm}$ , we uniquely recover all the coefficients  $b_{\pm,m}, b_{km}$ . The linear independence of the vectors  $\mathbf{v}_{\pm,m}, \mathbf{v}_{km}$  and  $\overline{\mathbf{v}_{km}}$  follows from this uniqueness.
Step 2: The limit  $\Delta x\to 0$  . We consider now discretizations with variable spacing  $\Delta x$  . Our goal is to show that expansions (73) lead to the desired representation (6) in the limit  $\Delta x\rightarrow 0$  . In the notation, we will occasionally add  $\Delta x$  to the subscripts to reflect the dependence of various quantities on the discretization spacing.
It is convenient to rewrite expansion (73) by rescaling the coefficients via  $b_{km} = c_{km}(\Delta x)^m$  and introducing frequencies  $\omega_{k}$  and  $\omega_{-}$  such that  $z_{k} = e^{i\omega_{k}\Delta x}$  and  $-1 = e^{i\omega_{-}\Delta x}$ , so as to make the expansion less  $\Delta x$ -dependent:
$$
\begin{array}{l} u _ {s} ^ {*} = f _ {*} (s \Delta x) (79) \\ = \sum_ {m = 0} ^ {2 M _ {-} - 1} c _ {-, m} (s \Delta x) ^ {m} e ^ {i \omega_ {-} s \Delta x} + \sum_ {m = 0} ^ {2 M _ {+} - 1} c _ {+, m} (s \Delta x) ^ {m} (80) \\ + \sum_ {k = 1} ^ {K} \sum_ {m = 0} ^ {M _ {k} - 1} (s \Delta x) ^ {m} \left(c _ {k m} e ^ {i \omega_ {k} s \Delta x} + \overline {{c _ {k m}}} e ^ {- i \omega_ {k} s \Delta x}\right). (81) \\ \end{array}
$$
The frequencies  $\omega_{k},\omega_{-}$  are determined from the values  $z_{k,\Delta x}$  only up to integer multiples of  $2\pi /\Delta x$  ..
$$
\omega_ {k} \in W _ {k, \Delta w} \stackrel {\text {d e f}} {=} \left\{\frac {- i \ln \left(z _ {k , \Delta x}\right) + 2 \pi l}{\Delta x} \right\} _ {l \in \mathbb {Z}}, \tag {82}
$$
$$
\omega_ {-} \in W _ {-, \Delta w} \stackrel {\text {d e f}} {=} \left\{\frac {\pi + 2 \pi l}{\Delta x} \right\} _ {l \in \mathbb {Z}}. \tag {83}
$$
Observe that expansion (79)-(81) is invariant under downscaling, in the following sense. Since  $s$  appears in it only through the combination  $s\Delta x$ , if this expansion holds for some  $\Delta x$ , then it also holds for any integer multiple  $\Delta x' = l\Delta x$ , with the same coefficients  $c_{\pm,m}, c_{km}$  and frequencies  $\omega_{-}, \omega_{k}$ .
To discuss this effect in more detail, we argue now that we can establish a correspondence between different values  $z_{k}$  across different scales. In general, downscaling is accompanied by aliasing: if the difference equation (72) holds with the values  $z_{n,\Delta x}^{*}$  on the length scale  $\Delta x$ , then on the integer multiple length scale  $l\Delta x$  it holds with the values
$$
z _ {n, l \Delta x} ^ {*} = \left(z _ {n, \Delta x} ^ {*}\right) ^ {l}. \tag {84}
$$