SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Proof. The proof is by induction over  $s$ . The case  $s = 1$  is an application of Faa di Bruno's formula: We can write  $Tf = f \circ g$  with  $g(x) = \cos(x)$ . We then take  $\ell \in \mathbb{N}_0$  with  $\ell \leq k$  and some  $x \in [-\pi, \pi]$ . The set partition version of Faa di Bruno's formula (see for instance [24, p. 219]) then yields
$$
\left| (f \circ g) ^ {(\ell)} (x) \right| \leq \sum_ {\pi \in \Pi_ {\ell}} \left(\left| f ^ {(| \pi |)} (g (x)) \right| \cdot \prod_ {B \in \pi} \left| g ^ {(| B |)} (x) \right|\right).
$$
Here,  $\Pi_{\ell}$  denotes the set of all partitions of the set  $\{1,\dots ,\ell \}$ . Since all derivatives of  $g$  are bounded by 1 in absolute value and  $|\pi |\leq \ell$  for every partition  $\pi \in \Pi_{\ell}$  we get
$$
\left\| \left(f \circ g\right) ^ {(\ell)} \right\| _ {L ^ {\infty} ([ - \pi , \pi ]; \mathbb {C})} \leq \left| \Pi_ {\ell} \right| \cdot \| f \| _ {C ^ {\ell} ([ - 1, 1 ]; \mathbb {C})}.
$$
The number  $|\Pi_{\ell}|$  is the number of possible partitions of the set  $\{1, \dots, \ell\}$  and is the so-called  $\ell$ -th Bell number. It can be bounded from above by  $\ell^{\ell}$  (see [13, Theorem 2.1]). This proves the case  $s = 1$ .
We now assume that the claim holds for an arbitrary but fixed  $s \in \mathbb{N}$ . Take  $\alpha \in \mathbb{N}_0^{s+1}$  with  $|\alpha| \leq k$ . We decompose  $\alpha = (\alpha', \alpha_{s+1})$  with  $\alpha' \in \mathbb{N}_0^s$ . For a fixed variable  $y_{s+1} \in [-1, 1]$ , we define
$$
f _ {y _ {s + 1}} \left(y _ {1}, \dots , y _ {s}\right) := f \left(y _ {1}, \dots , y _ {s}, y _ {s + 1}\right) \quad \text {f o r} \quad \left(y _ {1}, \dots , y _ {s}\right) \in [ - 1, 1 ] ^ {s}.
$$
We denote  $g(x_{1},\ldots ,x_{s + 1})\coloneqq (\cos (x_{1}),\ldots ,\cos (x_{s + 1}))$ ,  $g_{s}(x_{1},\dots,x_{s})\coloneqq (\cos (x_{1}),\dots,\cos (x_{s}))$  and  $\theta (x_{s + 1})\coloneqq \cos (x_{s + 1})$ . For every  $(x_{1},\dots,x_{s + 1})\in [-\pi ,\pi ]^{s + 1}$  it then holds
$$
(f \circ g) (x _ {1}, \dots , x _ {s + 1}) = \left(f _ {\theta \left(x _ {s + 1}\right)} \circ g _ {s}\right) (x _ {1}, \dots , x _ {s}).
$$
We now differentiate  $f \circ g$  with respect to the multiindex  $\alpha$  and get
$$
\begin{array}{l} \left[ \partial^ {\alpha} (f \circ g) \right] \left(x _ {1}, \dots , x _ {s + 1}\right) = \frac {\partial^ {\alpha_ {s + 1}}}{\partial x _ {s + 1} ^ {\alpha_ {s + 1}}} \left[ \partial^ {\alpha^ {\prime}} \left(f _ {\theta \left(x _ {s + 1}\right)} \circ g _ {s}\right) \left(x _ {1}, \dots , x _ {s}\right) \right] \\ = \left(h _ {x _ {1}, \dots , x _ {s}} \circ \theta\right) ^ {\left(\alpha_ {s + 1}\right)} \left(x _ {s + 1}\right) \\ \end{array}
$$
$h_{x_1,\dots,x_s}(y_{s+1}) \coloneqq \partial^{\alpha'}(f_{y_{s+1}} \circ g_s)(x_1, \dots, x_s)$  for  $(x_1, \dots, x_s) \in [-\pi, \pi]^s$  and  $y_{s+1} \in [-1, 1]$ .
Using the case  $s = 1$ , we get
$$
\left| \left[ \partial^ {\boldsymbol {\alpha}} (f \circ g) \right] \left(x _ {1}, \dots , x _ {s + 1}\right) \right| = \left| \left(h _ {x _ {1}, \dots , x _ {s}} \circ \theta\right) ^ {\left(\boldsymbol {\alpha} _ {s + 1}\right)} \left(x _ {s + 1}\right) \right| \leq \boldsymbol {\alpha} _ {s + 1} ^ {\boldsymbol {\alpha} _ {s + 1}} \cdot \left\| h _ {x _ {1}, \dots , x _ {s}} \right\| _ {C ^ {\boldsymbol {\alpha} _ {s + 1}} ([ - 1, 1 ]; \mathbb {C})}
$$
for any fixed  $(x_{1},\ldots ,x_{s})\in [-\pi ,\pi ]^{s}$
It remains to bound  $\| h_{x_1,\ldots ,x_s}\|_{C^{\alpha_{s + 1}}([-1,1];\mathbb{C})}$ . To this end, we fix  $\ell \in \mathbb{N}_0$  with  $\ell \leq \alpha_{s + 1}$ . We further denote
$$
F _ {y _ {1}, \dots , y _ {s}} (y _ {s + 1}) := f (y _ {1}, \dots , y _ {s}, y _ {s + 1}) \quad \text {f o r} \quad (y _ {1}, \dots , y _ {s + 1}) \in [ - 1, 1 ] ^ {s + 1}.
$$
For arbitrary  $(x_{1},\ldots ,x_{s})\in [-\pi ,\pi ]^{s}$  and  $y_{s + 1}\in [-1,1]$  we then see
$$
h _ {x _ {1}, \ldots , x _ {s}} ^ {(\ell)} (y _ {s + 1}) = \partial^ {\boldsymbol {\alpha} ^ {\prime}} \left[ (x _ {1}, \ldots , x _ {s}) \mapsto F _ {g _ {s} (x _ {1}, \ldots , x _ {s})} ^ {(\ell)} (y _ {s + 1}) \right] = \partial^ {\boldsymbol {\alpha} ^ {\prime}} \left[ H _ {y _ {s + 1}} \circ g _ {s} \right] (x _ {1}, \ldots , x _ {s})
$$
$$
H _ {y _ {s + 1}} \left(y _ {1}, \dots , y _ {s}\right) := F _ {y _ {1}, \dots , y _ {s}} ^ {(\ell)} \left(y _ {s + 1}\right) \quad \text {f o r} \left(y _ {1}, \dots , y _ {s}\right) \in [ - 1, 1 ] ^ {s}.
$$
Hence, we see by induction that
$$
\begin{array}{l} \left| h _ {x _ {1}, \dots , x _ {s}} ^ {(\ell)} \left(y _ {s + 1}\right) \right| = \left| \partial^ {\boldsymbol {\alpha} ^ {\prime}} \left[ H _ {y _ {s + 1}} \circ g _ {s} \right] \left(x _ {1}, \dots , x _ {s}\right) \right| \stackrel {{\mathrm {I H}}} {{\leq}} \prod_ {j = 1} ^ {s} \boldsymbol {\alpha} _ {j} ^ {\boldsymbol {\alpha} _ {j}} \cdot \left\| H _ {y _ {s + 1}} \right\| _ {C ^ {\left| \boldsymbol {\alpha} ^ {\prime} \right|} \left([ - 1, 1 ] ^ {s}; \mathbb {C}\right)} \\ \leq \prod_ {j = 1} ^ {s} \boldsymbol {\alpha} _ {j} ^ {\boldsymbol {\alpha} _ {j}} \cdot \| f \| _ {C ^ {| \boldsymbol {\alpha} |} ([ - 1, 1 ] ^ {s + 1}; \mathbb {C})} \\ \end{array}
$$
Remark C.20. For a multiindex  $\alpha \in \mathbb{N}_0^s$  with  $|\alpha| \leq k$  we see
$$
\prod_ {j = 1} ^ {s} \alpha_ {j} ^ {\boldsymbol {\alpha} _ {j}} \leq k ^ {\sum_ {j = 1} ^ {s} \boldsymbol {\alpha} _ {j}} \leq k ^ {k}.
$$
Hence, the norm of the operator introduced in Lemma C.11 can be bounded from above by  $k^k$ .
$$
T _ {\ell} (\cos (x)) = \cos (\ell x), \quad x \in \mathbb {R}.
$$
For a multi-index  $\mathbf{k} \in \mathbb{N}_0^s$  we define
$$
T _ {\mathbf {k}} (x) := \prod_ {j = 1} ^ {s} T _ {\mathbf {k} _ {j}} \left(x _ {j}\right), \quad x \in [ - 1, 1 ] ^ {s}.
$$
The proof of Theorem 3.2 relies on the fact that  $C^k$ -functions can be approximated at a certain rate using linear combinations of the  $T_{\mathbf{k}}$  (see Theorem C.15). We also refer to Figure 4 for an illustration of the overall proof strategy of Theorem 3.2.
Proof of Theorem 3.2. Choose  $M \in \mathbb{N}$  as the largest integer for which  $(16M - 7)^{2n} \leq m$ , where we assume without loss of generality that  $9^{2n} \leq m$ , which can be done by choosing  $\sigma_{j} = 0$  for all  $j \in \{1, \dots, m\}$  for  $m < 9^{2n}$ , at the cost of possibly enlarging  $c$ . First we note that by the choice of  $M$  the inequality
$$
m \leq (1 6 M + 9) ^ {2 n}
$$
holds true. Since  $16M + 9 \leq 25M$ , we get  $m \leq 25^{2n} \cdot M^{2n}$  or equivalently
$$
\frac {m ^ {1 / 2 n}}{2 5} \leq M. \tag {C.7}
$$
According to Theorem C.15 we choose a constant  $c_{1} = c_{1}(n,k)$  with the property that for any function  $f\in C^{k}\left([ - 1,1 ]^{2n};\mathbb{R}\right)$  there exists a polynomial
$$
P = \sum_ {0 \leq \mathbf {k} \leq 2 M - 1} \mathcal {V} _ {\mathbf {k}} ^ {M} (f) \cdot T _ {\mathbf {k}}
$$
of coordinatewise degree at most  $2M - 1$  satisfying
$$
\| f - P \| _ {L ^ {\infty} ([ - 1, 1 ] ^ {2 n}; \mathbb {R})} \leq \frac {c _ {1}}{M ^ {k}} \cdot \| f \| _ {C ^ {k} ([ - 1, 1 ] ^ {2 n}; \mathbb {R})}.
$$
Furthermore, according to Theorem C.15, we choose a constant  $c_{2} = c_{2}(n)$ , such that the inequality
$$
\sum_ {0 \leq \mathbf {k} \leq 2 M - 1} \left| \mathcal {V} _ {\mathbf {k}} ^ {M} (f) \right| \leq c _ {2} \cdot M ^ {n} \cdot \| f \| _ {L ^ {\infty} ([ - 1, 1 ] ^ {2 n}; \mathbb {R})} \leq c _ {2} \cdot M ^ {n} \cdot \| f \| _ {C ^ {k} ([ - 1, 1 ] ^ {2 n}; \mathbb {R})}
$$
holds for all  $f \in C^{k}$  ([-1, 1]2n; R). The final constant is then defined to be
$$
c = c (n, k) := \sqrt {2} \cdot 2 5 ^ {k} \cdot \left(c _ {1} + c _ {2}\right).
$$
Fix  $\mathbf{k} \leq 2M - 1$ . Since  $T_{\mathbf{k}}$  is a polynomial of componentwise degree less or equal to  $2M - 1$  with  $\varphi_{n}$  as in (A.1), we have a representation
$$
\left(T_{\mathbf{k}}\circ \varphi_{n}^{-1}\right)(z) = \sum_{\substack{\boldsymbol{\ell}^{1},\boldsymbol{\ell}^{2}\in \mathbb{N}_{0}^{n}\\ \boldsymbol{\ell}^{1},\boldsymbol{\ell}^{2}\leq 2M - 1}}a_{\boldsymbol{\ell}^{1},\boldsymbol{\ell}^{2}}^{\mathbf{k}}\prod_{t = 1}^{n}\operatorname {Re}\left(z_{t}\right)^{\boldsymbol{\ell}_{t}^{1}}\operatorname {Im}\left(z_{t}\right)^{\boldsymbol{\ell}_{t}^{2}}
$$
with suitably chosen coefficients  $a_{\ell^1,\ell^2}^{\mathbf{k}} \in \mathbb{C}$ . By using the identities  $\operatorname{Re}(z_t) = \frac{1}{2}(z_t + \overline{z_t})$  and also  $\operatorname{Im}(z_t) = \frac{1}{2i}(z_t - \overline{z_t})$  we can rewrite  $T_{\mathbf{k}} \circ \varphi_n^{-1}$  into a complex polynomial in  $z$  and  $\overline{z}$ , i.e.,
$$
\left(T_{\mathbf{k}}\circ \varphi_{n}^{-1}\right)(z) = \sum_{\substack{\boldsymbol{\ell}^{1},\boldsymbol{\ell}^{2}\in \mathbb{N}_{0}^{n}\\ \boldsymbol{\ell}^{1},\boldsymbol{\ell}^{2}\leq 4M - 2}}b_{\boldsymbol{\ell}^{1},\boldsymbol{\ell}^{2}}^{\mathbf{k}}z^{\boldsymbol{\ell}^{1}}\overline{z}^{\boldsymbol{\ell}^{2}}
$$
with complex coefficients  $b_{\ell^1,\ell^2}^{\mathbf{k}} \in \mathbb{C}$ . Using Theorem 3.1, we choose  $\rho_1, \ldots, \rho_m \in \mathbb{C}^n$  and  $b \in \mathbb{C}$ , such that for any polynomial  $P \in \{T_{\mathbf{k}} \circ \varphi_n^{-1} : \mathbf{k} \leq 2M - 1\} \subseteq \mathcal{P}_{4M - 2}^n$  there are coefficients  $\sigma_1(P), \ldots, \sigma_m(P) \in \mathbb{C}$ , such that
$$
\left\| g _ {P} - P \right\| _ {L ^ {\infty} \left(\Omega_ {n}; \mathbb {C}\right)} \leq M ^ {- k - n}, \tag {C.8}
$$
$$
g _ {P} := \sum_ {t = 1} ^ {m} \sigma_ {t} (P) \phi (\rho_ {t} ^ {T} z + b).
$$
Note that here we implicitly use the bound  $(4\cdot (4M - 2) + 1)^{2n}\leq m$ . We are now going to show that the chosen constant and the chosen vectors  $\rho_{t}$  have the desired property.
Let  $f \in C^k(\Omega_n; \mathbb{C})$ . By splitting  $f$  into real and imaginary part, we write  $f = f_1 + i \cdot f_2$  with  $f_1, f_2 \in C^k(\Omega_n; \mathbb{R})$ . For the following, fix  $j \in \{1, 2\}$  and note that  $f_j \circ \varphi_n \in C^k([-1, 1]^{2n}; \mathbb{R})$ . By choice of  $c_1$ , there exists a polynomial  $P$  with the property
$$
\left\| f _ {j} \circ \varphi_ {n} - P \right\| _ {L ^ {\infty} ([ - 1, 1 ] ^ {2 n}; \mathbb {R})} \leq \frac {c _ {1}}{M ^ {k}} \cdot \left\| f _ {j} \circ \varphi_ {n} \right\| _ {C ^ {k} ([ - 1, 1 ] ^ {2 n}; \mathbb {R})}
$$
$$
\left\| f _ {j} - P \circ \varphi_ {n} ^ {- 1} \right\| _ {L ^ {\infty} \left(\Omega_ {n}; \mathbb {R}\right)} \leq \frac {c _ {1}}{M ^ {k}} \cdot \| f _ {j} \| _ {C ^ {k} \left(\Omega_ {n}; \mathbb {R}\right)}, \tag {C.9}
$$
where  $P\circ \varphi_n^{-1}$  can be written in the form
$$
\left(P \circ \varphi_ {n} ^ {- 1}\right) (z) = \sum_ {0 \leq \mathbf {k} \leq 2 M - 1} \mathcal {V} _ {\mathbf {k}} ^ {M} \left(f _ {j} \circ \varphi_ {n}\right) \cdot \left(T _ {\mathbf {k}} \circ \varphi_ {n} ^ {- 1}\right) (z).
$$
We choose the function  $g_{T_{\mathbf{k}} \circ \varphi_{n}^{-1}}$  according to (C.8). Thus, writing
$$
g _ {j} := \sum_ {0 \leq \mathbf {k} \leq 2 M - 1} \mathcal {V} _ {\mathbf {k}} ^ {M} \left(f _ {j} \circ \varphi_ {n}\right) \cdot g _ {T _ {\mathbf {k}} \circ \varphi_ {n} ^ {- 1}},
$$
$$
\begin{array}{l} \left\| P \circ \varphi_ {n} ^ {- 1} - g _ {j} \right\| _ {L ^ {\infty} \left(\Omega_ {n}; \mathbb {R}\right)} \leq \sum_ {0 \leq \mathbf {k} \leq 2 M - 1} \left| \mathcal {V} _ {\mathbf {k}} ^ {M} \left(f _ {j} \circ \varphi_ {n}\right) \right| \cdot \underbrace {\left\| T _ {\mathbf {k}} \circ \varphi_ {n} ^ {- 1} - g _ {T _ {\mathbf {k}} \circ \varphi_ {n} ^ {- 1}} \right\| _ {L ^ {\infty} \left(\Omega_ {n}; \mathbb {R}\right)}} _ {\leq M ^ {- k - n}} \\ \leq M ^ {- k - n} \cdot \sum_ {0 \leq \mathbf {k} \leq 2 M - 1} \left| \mathcal {V} _ {\mathbf {k}} ^ {M} \left(f _ {j} \circ \varphi_ {n}\right) \right| \\ \leq \frac {c _ {2}}{M ^ {k}} \| f _ {j} \circ \varphi_ {n} \| _ {C ^ {k} ([ - 1, 1 ] ^ {2 n}; \mathbb {R})} = \frac {c _ {2}}{M ^ {k}} \| f _ {j} \| _ {C ^ {k} \left(\Omega_ {n}; \mathbb {R}\right)}. \tag {C.10} \\ \end{array}
$$
Combining (C.9) and (C.10), we see
$$
\left\| f _ {j} - g _ {j} \right\| _ {L ^ {\infty} \left(\Omega_ {n}; \mathbb {R}\right)} \leq \frac {c _ {1} + c _ {2}}{M ^ {k}} \cdot \left\| f _ {j} \right\| _ {C ^ {k} \left(\Omega_ {n}; \mathbb {R}\right)} \leq \frac {c _ {1} + c _ {2}}{M ^ {k}} \cdot \left\| f \right\| _ {C ^ {k} \left(\Omega_ {n}; \mathbb {C}\right)}.
$$
In the end, define
$$
g := g _ {1} + i \cdot g _ {2}.
$$
Since the vectors  $\rho_t$  have been chosen fixed, it is clear that, after rearranging,  $g$  has the desired form, i.e.,  $g = \sigma^T\Phi$  where  $\Phi(z) = (\phi(\rho_t z + b))_{t=1}^m$ . Furthermore, one obtains the bound
$$
\begin{array}{l} \left\| f - g \right\| _ {L ^ {\infty} (\Omega_ {n}; \mathbb {C})} \leq \sqrt {\left\| f _ {1} - g _ {1} \right\| _ {L ^ {\infty} (\Omega_ {n} ; \mathbb {R})} ^ {2} + \left\| f _ {2} - g _ {2} \right\| _ {L ^ {\infty} (\Omega_ {n} ; \mathbb {R})} ^ {2}} \\ \leq \frac {c _ {1} + c _ {2}}{M ^ {k}} \cdot \sqrt {\| f \| _ {C ^ {k} (\Omega_ {n} ; \mathbb {C})} ^ {2} + \| f \| _ {C ^ {k} (\Omega_ {n} ; \mathbb {C})} ^ {2}} \\ \leq \frac {\sqrt {2} \cdot \left(c _ {1} + c _ {2}\right)}{M ^ {k}} \cdot \| f \| _ {C ^ {k} \left(\Omega_ {n}; \mathbb {C}\right)}. \\ \end{array}
$$
Using (C.7), we see
$$
\left\| f - g \right\| _ {L ^ {\infty} \left(\Omega_ {n}; \mathbb {C}\right)} \leq \frac {\sqrt {2} \cdot 2 5 ^ {k} \cdot \left(c _ {1} + c _ {2}\right)}{m ^ {k / 2 n}} \cdot \left\| f \right\| _ {C ^ {k} \left(\Omega_ {n}; \mathbb {C}\right)},
$$
The linearity and continuity of the maps  $f \mapsto \sigma_j(f)$  (with respect to the  $\| \cdot \|_{L^{\infty}}$ -norm) follow easily from the fact that the map  $f \mapsto \mathcal{V}_{\mathbf{k}}^{M}(f)$  is a continuous linear functional for every multiindex  $0 \leq \mathbf{k} \leq 2M - 1$ .
Figure 4: Schematic for the proof of the main result (Theorem 3.2). The first row shows the different steps of the proof and the second row indicates the main tools used.
D Approximation of holomorphically extendable functions
In this appendix we provide the proofs for the statements contained in Remark 3.3. The proofs mainly rely on results about sparse polynomial approximation [2].
Definition D.1 (cf. [2, Assumption 2.3]). Let  $s \in \mathbb{N}$  and  $\pmb{\nu} \in (1,\infty)^s$ . For every  $j \in \{1,\dots,s\}$  let
$$
\mathcal {E} _ {\nu_ {j}} := \left\{\frac {z + z ^ {- 1}}{2}: z \in \mathbb {C}, 1 \leq | z | \leq \nu_ {j} \right\}.
$$
We then define the (filled-in) Bernstein polyellipse of parameter  $\pmb{\nu}$  as
$$
\mathcal {E} _ {\boldsymbol {\nu}} := \mathcal {E} _ {\boldsymbol {\nu} _ {1}} \times \dots \times \mathcal {E} _ {\boldsymbol {\nu} _ {s}} \subseteq \mathbb {C} ^ {s}
$$
and observe that  $[-1,1]^s \subseteq \mathcal{E}_\nu$  when we interpret  $[-1,1]^s$  as a subset of  $\mathbb{C}^s$ . We further define
$$
\mathcal {V} _ {s} (\boldsymbol {\nu}) := \left\{f: [ - 1, 1 ] ^ {s} \to \mathbb {C}: \exists \text {o p e n} U \supseteq \mathcal {E} _ {\boldsymbol {\nu}} \text {a n d} \tilde {f}: U \to \mathbb {C} \text {h o l o m o r p h i c w i t h} \tilde {f} \Big | _ {[ - 1, 1 ] ^ {s}} = f \right\}.
$$
Here,  $[-1, 1]^s$  is again interpreted as a subset of  $\mathbb{C}^s$ . Moreover, note that such an extension  $\tilde{f}$  is, if existent, unique, as follows from the identity theorem for holomorphic functions. Hence, the expression
$$
\| f \| _ {\mathcal {V} _ {s} (\boldsymbol {\nu})} := \| \tilde {f} \| _ {L ^ {\infty} \left(\mathcal {E} _ {\boldsymbol {\nu}; \mathbb {C}}\right)}
$$
Definition D.2 (cf. [2, pp. 25, 28 ff.]). Let  $s \in \mathbb{N}$ . We define a probability measure on  $[-1, 1]^s$  via
$$
d \mu_ {s} := \prod_ {j = 1} ^ {s} \frac {1}{\pi \sqrt {1 - x _ {j} ^ {2}}} d x.
$$
We define the normalized Chebyshev polynomials for  $\mathbf{k} \in \mathbb{N}_0^s$  as
$$
\widetilde {T _ {\mathbf {k}}} (x) := 2 ^ {\| \mathbf {k} \| _ {0} / 2} \prod_ {j = 1} ^ {s} \cos (\mathbf {k} _ {j} \operatorname {a r c c o s} (x _ {j})),
$$
where  $\| \mathbf{k}\| _0\coloneqq \# \{1\leq j\leq s:\mathbf{k}_j\neq 0\}$ . Note that this definition differs slightly from the notion used in Appendices C.1 and C.3.
The following lemma is crucial for deriving the approximation rate in Theorem D.5. The proofs can be found in [2].
Lemma D.3 (cf. [2, Remark 2.15 and Theorem 3.2]). Let  $s \in \mathbb{N}$ ,  $k \in \mathbb{N}_0^s$ ,  $\pmb{\nu} \in (1,\infty)^s$  and  $f \in \mathcal{V}_s(\pmb{\nu})$ . Then it holds
1.  $\| \widetilde{T_k}\|_{L^\infty ([-1,1]^s;\mathbb{R})} = 2^{\| \pmb {k}\| _0 / 2},$  where  $\| \pmb {k}\| _0 = \# \{1\leq j\leq s:\pmb {k}_j\neq 0\}$
2.  $|\langle \widetilde{T_k},f\rangle_{\mu_s}|\leq \pmb{\nu}^{-k}\cdot 2^{\| \pmb{k}\| _0 / 2}\cdot \| f\|_{\mathcal{V}_s(\pmb {\nu})}.$
It is a well-known fact that the Chebyshev polynomials  $\widetilde{T_{\mathbf{k}}}$  form an orthonormal basis of  $L_{\mu_s}^2 ([-1,1]^s;\mathbb{C})$ . The following proposition states that functions from  $\nu_{s}(\nu)$  can even be approximated uniformly by linear combinations of the  $\widetilde{T_{\mathbf{k}}}$  at a certain rate. The proof follows essentially by applying Lemma D.3.
Proposition D.4. Let  $s, m \in \mathbb{N}$  and  $\pmb{\nu} \in (1,\infty)^s$ . Let  $\nu := \min_{j=1,\dots,s} \pmb{\nu}_j$ . Then there exists a constant  $c = c(s,\pmb{\nu}) > 0$  with the following property: For every  $f \in \mathcal{V}_s(\pmb{\nu})$ , defining
$$
P_{m}:= \sum_{\substack{\boldsymbol {k}\in \mathbb{N}_{0}^{s}\\ \boldsymbol {k}\leq m}}\langle f,\widetilde{T}_{\boldsymbol{k}}\rangle_{\mu_{s}}\cdot \widetilde{T}_{\boldsymbol{k}},
$$
it holds
$$
\| f - P _ {m} \| _ {L ^ {\infty} ([ - 1, 1 ] ^ {s}; \mathbb {C})} \leq c \cdot \nu^ {- m} \cdot \| f \| _ {\mathcal {V} _ {s} (\nu)}.
$$
Proof. Let  $f \in \mathcal{V}_s(\nu)$ . Since the  $\widetilde{T_{\mathbf{k}}}$  form an orthonormal basis of  $L_{\mu_s}^2 ([-1,1]^s;\mathbb{C})$  and since  $\mu_{s}$  is a probability measure, so that  $f \in C([ - 1,1 ]^{s};\mathbb{C}) \subseteq L_{\mu_2}^2 ([ - 1,1 ]^s;\mathbb{C})$ , it follows that
$$
f = \sum_ {\mathbf {k} \in \mathbb {N} _ {0} ^ {s}} \langle f, \widetilde {T _ {\mathbf {k}}} \rangle_ {\mu_ {s}} \cdot \widetilde {T _ {\mathbf {k}}}
$$
with unconditional convergence in  $L_{\mu_s}^2$ . For  $x \in [-1, 1]^s$ , note that
$$
\sum_ {\mathbf {k} \in \mathbb {N} _ {0} ^ {s}} | \langle f, \widetilde {T _ {\mathbf {k}}} \rangle_ {\mu_ {s}} | \cdot | \widetilde {T _ {\mathbf {k}}} (x) | \leq \sum_ {\mathbf {k} \in \mathbb {N} _ {0} ^ {s}} | \langle f, \widetilde {T _ {\mathbf {k}}} \rangle_ {\mu_ {s}} | \cdot \| \widetilde {T _ {\mathbf {k}}} \| _ {L ^ {\infty} ([ - 1, 1 ] ^ {s}; \mathbb {C})} \leq 2 ^ {s} \| f \| _ {\nu_ {s} (\boldsymbol {\nu})} \cdot \sum_ {\mathbf {k} \in \mathbb {N} _ {0} ^ {s}} \boldsymbol {\nu} ^ {- \mathbf {k}}.
$$
Here, we employed Lemma D.3 at the last inequality. From  $\nu > 1$  it follows that
$$
\sum_ {\mathbf {k} \in \mathbb {N} _ {0} ^ {s}} \langle f, \widetilde {T _ {\mathbf {k}}} \rangle_ {\mu_ {s}} \cdot \widetilde {T _ {\mathbf {k}}}
$$
converges pointwise (even uniformly) and, since all the involved functions are continuous, this pointwise limit then has to coincide with the  $L_{\mu_s}^2$  -limit  $f$ . Hence, it holds
$$
\begin{array}{l} \| f - P _ {m} \| _ {L ^ {\infty} \left([ - 1, 1 ] ^ {s}; \mathbb {C}\right)} \leq \sum_ {\mathbf {k} \in \mathbb {N} _ {0} ^ {s}} | \langle f, \widetilde {T _ {\mathbf {k}}} \rangle_ {\mu_ {s}} | \cdot \| \widetilde {T _ {\mathbf {k}}} \| _ {L ^ {\infty} \left([ - 1, 1 ] ^ {s}; \mathbb {C}\right)} \\ \mathbf {k} \not \leq m \\ \leq 2 ^ {s} \| f \| _ {\mathcal {V} _ {s} (\nu)} \cdot \sum_ {\mathbf {k} \in \mathbb {N} _ {0} ^ {s}} \nu^ {- \mathbf {k}}, \\ | \mathbf {k} | \not \leq m \\ \end{array}
$$
where we again used Lemma D.3. To complete the proof we compute
$$
\begin{array}{l} \sum_{\substack{\mathbf{k}\in \mathbb{N}_{0}^{s}\\ |\mathbf{k}|\not\leq m}}\boldsymbol{\nu}^{-\mathbf{k}}\leq \sum_{j = 1}^{s}\sum_{\substack{\mathbf{k}\in \mathbb{N}_{0}^{s}\\ \mathbf{k}_{j} > m}}\boldsymbol{\nu}^{-\mathbf{k}} = \sum_{j = 1}^{s}\left(\prod_{\substack{1\leq \ell \leq s\\ l\neq j}}\left(\sum_{k = 0}^{\infty}\boldsymbol{\nu}_{\ell}^{-k}\right)\cdot \sum_{k = m + 1}^{\infty}\boldsymbol{\nu}_{j}^{-k}\right) \\ \leq \sum_ {j = 1} ^ {s} \left(\nu_ {j} ^ {- (m + 1)} \prod_ {1 \leq \ell \leq s} \left(\sum_ {k = 0} ^ {\infty} \nu_ {\ell} ^ {- k}\right)\right) \\ = \prod_ {1 \leq \ell \leq s} \left(\sum_ {k = 0} ^ {\infty} \nu_ {\ell} ^ {- k}\right) \cdot \sum_ {j = 1} ^ {s} \nu_ {j} ^ {- (m + 1)} \\ = \prod_ {1 \leq \ell \leq s} \left(\frac {\nu_ {\ell}}{\nu_ {\ell} - 1}\right) \cdot s \cdot \nu^ {- (m + 1)} \\ \end{array}
$$
and define  $c(s,\pmb {\nu}):= 2^s\cdot s\cdot \prod_{\ell = 1}^s\frac{\pmb{\nu}_j}{\pmb{\nu}_j - 1}\cdot \pmb{\nu}^{-1}$
To formulate the result for the approximation of holomorphically extendable functions using CVNNs we need to transfer the definition of  $\mathcal{V}_s(\pmb{\nu})$  to the complex setting. For  $\pmb{\nu} \in (1, \infty)^{2n}$  and with  $\varphi_n$  as in Equation (A.1), we hence write
$$
\mathcal {W} _ {n} (\boldsymbol {\nu}) := \left\{f: \Omega_ {n} \to \mathbb {C}: f \circ \varphi_ {n} \big | _ {[ - 1, 1 ] ^ {2 n}} \in \mathcal {V} _ {2 n} (\boldsymbol {\nu}) \right\}
$$
For  $f\in \mathcal{W}_n(\nu)$  we define
$$
\| f \| _ {\mathcal {W} _ {n} (\nu)} := \| f \circ \varphi_ {n} \| _ {\mathcal {V} _ {2 n} (\nu)}.
$$
Thus,  $\mathcal{W}_n(\nu)$  consists of all complex-valued functions defined on  $\Omega_{n}$  that can be holomorphically extended onto some polyellipse in  $\mathbb{C}^{2n}$ , where  $\Omega_{n}\subseteq \mathbb{C}^{n}$  is interpreted as a subset of  $\mathbb{R}^{2n}$  and then as a subset of  $\mathbb{C}^{2n}$ . The final approximation result then reads as follows.
Theorem D.5. Let  $n\in \mathbb{N}$  and  $\nu \in (1,\infty)^{2n}$ . Set
$$
\nu := \min  _ {1 \leq j \leq 2 n} \nu_ {j}.
$$
Then there exists a constant  $c = c(n,\nu) > 0$  with the following property: For every function  $\phi :\mathbb{C}\to \mathbb{C}$  that is smooth and non-polyharmonic on some open set  $\varnothing \neq U\subset \mathbb{C}$  and for every  $m\in \mathbb{N}$  there exists a first layer  $\Phi \in \mathcal{F}_{n,m}^{\phi}$  with the property that for every  $f\in \mathcal{W}_n(\nu)$  there exist coefficients  $\sigma = \sigma (f)\in \mathbb{C}^{m}$  such that
$$
\left\| f - \sigma^ {T} \Phi \right\| _ {L ^ {\infty} \left(\Omega_ {n}; \mathbb {C}\right)} \leq c \cdot \nu^ {- m ^ {1 / (2 n)} / 1 7} \cdot \left\| f \right\| \mathcal {W} _ {n} (\nu).
$$
Moreover, the map  $f \mapsto \sigma(f)$  is a continuous linear functional with respect to the  $L^{\infty}$ -norm.
Proof. Choose  $M \in \mathbb{N}$  as the largest integer satisfying  $(8M + 1)^{2n} \leq m$ , where we assume without loss of generality that  $9^{2n} \leq m$ . This can be done by choosing  $\sigma = 0$  for  $m < 9^{2n}$  at the cost of possibly enlarging  $c$ . Note that the maximality of  $M$  implies  $(8M + 9)^{2n} > m$ . By using  $8M + 9 \leq 17M$ , this gives us
$$
M > \frac {1}{1 7} \cdot m ^ {1 / (2 n)}. \tag {D.1}
$$
Choose the constant  $c_{1} = c_{1}(2n,\nu)$  according to Proposition D.4.
Fix  $\mathbf{k} \in \mathbb{N}_0^{2n}$  with  $\mathbf{k} \leq M$ . Since  $\widetilde{T_{\mathbf{k}}}$  is a polynomial of componentwise degree at most  $M$ , we have a representation
$$
\left(\widetilde{T_{\mathbf{k}}}\circ \varphi_{n}^{-1}\right)(z) = \sum_{\substack{\ell^{1},\ell^{2}\in \mathrm{N}_{0}^{n}\\ \ell^{1},\ell^{2}\leq M}}a_{\ell^{1},\ell^{2}}^{\mathbf{k}}\prod_{t = 1}^{n}\operatorname {Re}\left(z_{t}\right)^{\ell_{t}^{1}}\operatorname {Im}\left(z_{t}\right)^{\ell_{t}^{2}}
$$
with suitably chosen coefficients  $a_{\ell^1,\ell^2}^{\mathbf{k}} \in \mathbb{C}$ . By using the identities  $\operatorname{Re}(z_t) = \frac{1}{2}(z_t + \overline{z_t})$  and also  $\operatorname{Im}(z_t) = \frac{1}{2i}(z_t - \overline{z_t})$ , we can rewrite  $\widetilde{T_{\mathbf{k}}} \circ \varphi_n^{-1}$  into a complex polynomial in  $z$  and  $\overline{z}$ , i.e.,