SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} \partial_ {\mathrm {w i r t}} ^ {e _ {j}} \left[ w \mapsto \left(\partial_ {\mathrm {w i r t}} ^ {| \mathbf {m} |} \bar {\partial} _ {\mathrm {w i r t}} ^ {| \ell |} \phi\right) \left(w ^ {T} z + b\right) \right] \\ = \left(\partial_ {\mathrm {w i r t}} ^ {| \mathbf {m} | + 1} \overline {{\partial}} _ {\mathrm {w i r t}} ^ {| \boldsymbol {\ell} |} \phi\right) \left(w ^ {T} z + b\right) \cdot \partial_ {\mathrm {w i r t}} ^ {e _ {j}} \left[ w \mapsto w ^ {T} z + b \right] \\ + \left(\partial_ {\mathrm {w i r t}} ^ {| \mathbf {m} |} \bar {\partial} _ {\mathrm {w i r t}} ^ {| \ell | + 1} \phi\right) (w ^ {T} z + b) \cdot \partial_ {\mathrm {w i r t}} ^ {e _ {j}} \left[ w \mapsto \overline {{w ^ {T} z + b}} \right] \\ = z ^ {e _ {j}} \cdot \left(\partial_ {\mathrm {w i r t}} ^ {| \mathbf {m} | + 1} \overline {{\partial}} _ {\mathrm {w i r t}} ^ {| \boldsymbol {\ell} |} \phi\right) (w ^ {T} z + b) + \left(\partial_ {\mathrm {w i r t}} ^ {| \mathbf {m} |} \overline {{\partial}} _ {\mathrm {w i r t}} ^ {| \boldsymbol {\ell} | + 1} \phi\right) (w ^ {T} z + b) \cdot \overline {{\overline {{\partial}} _ {\mathrm {w i r t}} ^ {e _ {j}} [ w \mapsto w ^ {T} z + b ]}} \\ = z ^ {e _ {j}} \cdot \left(\partial_ {\mathrm {w i r t}} ^ {| \mathbf {m} | + 1} \overline {{\partial}} _ {\mathrm {w i r t}} ^ {| \ell |} \phi\right) \left(w ^ {T} z + b\right). \\ \end{array}
$$
By induction, this proves the claim.
As the last preparation for the proof of Theorem 3.1, we need the following lemma.
Lemma B.7. Let  $\phi : \mathbb{C} \to \mathbb{C}$  and  $\delta > 0$ ,  $b \in \mathbb{C}$ ,  $k \in \mathbb{N}_0$ , such that  $\left.\phi\right|_{B_{\delta}(b)} \in C^k(B_{\delta}(b); \mathbb{C})$ . Let  $m, n \in \mathbb{N}$  and  $\varepsilon > 0$ . For  $\pmb{p} \in \mathbb{N}_0^{2n}$ ,  $h > 0$  and  $z \in \Omega_n$  we write
$$
\begin{array}{l} \Phi_ {\boldsymbol {p}, h} (z) := (2 h) ^ {- | \boldsymbol {p} |} \sum_ {0 \leq r \leq \boldsymbol {p}} (- 1) ^ {| \boldsymbol {p} | - | \boldsymbol {r} |} \left( \begin{array}{c} \boldsymbol {p} \\ \boldsymbol {r} \end{array} \right) (\phi_ {z} \circ \varphi_ {n}) \left(h (2 \boldsymbol {r} - \boldsymbol {p})\right) \\ = (2 h) ^ {- | \boldsymbol {p} |} \sum_ {0 \leq \boldsymbol {r} \leq \boldsymbol {p}} (- 1) ^ {| \boldsymbol {p} | - | \boldsymbol {r} |} \left( \begin{array}{c} \boldsymbol {p} \\ \boldsymbol {r} \end{array} \right) \phi \left(\left[ \varphi_ {n} \left(h (\boldsymbol {2 r} - \boldsymbol {p})\right) \right] ^ {T} \cdot z + b\right), \\ \end{array}
$$
where  $\phi_z$  is as introduced in Lemma B.6 and  $\varphi_n$  is as in (A.1). Furthermore, let
$$
\phi_ {\boldsymbol {p}}: \quad \Omega_ {n} \times B _ {\frac {\delta}{\sqrt {2 n}}} (0) \to \mathbb {C}, \quad (z, w) \mapsto \partial^ {\boldsymbol {p}} \phi_ {z} (w).
$$
Then there exists  $h^* > 0$  such that
$$
\left\| \Phi_ {\boldsymbol {p}, h} - \phi_ {\boldsymbol {p}} (\cdot , 0) \right\| _ {L ^ {\infty} \left(\Omega_ {n}; \mathbb {C}\right)} \leq \varepsilon
$$
for all  $\pmb{p} \in \mathbb{N}_0^{2n}$  with  $|\pmb{p}| \leq k$  and  $\pmb{p} \leq m$  and  $h \in (0, h^*)$ .
Proof. Fix  $\mathbf{p} \in \mathbb{N}_0^{2n}$  with  $|\mathbf{p}| \leq k$  and  $\mathbf{p} \leq m$ . The map
$$
B _ {\sqrt {2 n} + 1} (0) \times B _ {\delta / (\sqrt {2 n} + 1)} (0) \to \mathbb {C}, (z, w) \mapsto \phi (w ^ {T} z + b)
$$
is in  $C^k$  since
$$
\left| w ^ {T} z \right| \leq \| w \| \cdot \| z \| <   \frac {\delta}{\sqrt {2 n} + 1} \cdot (\sqrt {2 n} + 1) = \delta .
$$
Therefore, the map
$$
B _ {\sqrt {2 n} + 1} (0) \times B _ {\delta / (\sqrt {2 n} + 1)} (0) \to \mathbb {C}, \quad (z, w) \mapsto \partial^ {\mathbf {p}} \phi_ {z} (w)
$$
is continuous and in particular uniformly continuous on the compact set
$$
\Omega_ {n} \times \overline {{B _ {\delta / (3 n)}}} (0) \subseteq B _ {\sqrt {2 n} + 1} (0) \times B _ {\delta / (\sqrt {2 n} + 1)} (0).
$$
Here, we employed  $\sqrt{2n} + 1 < 3n$  for every  $n \in \mathbb{N}$ . Hence, there exists  $h_{\mathbf{p}} \in (0, \frac{\delta}{3n \cdot \sqrt{2n} \cdot m})$ , such that
$$
| \phi_ {\mathbf {p}} (z, \xi) - \phi_ {\mathbf {p}} (z, 0) | \leq \frac {\varepsilon}{\sqrt {2}}
$$
for all  $\xi \in \varphi_{n}\left(h\cdot [-m,m]^{2n}\right)$ ,  $h\in (0,h_{\mathbf{p}})$  and  $z\in \Omega_{n}$ . Now fix such an  $h\in (0,h_{\mathbf{p}})$  and  $z\in \Omega_{n}$ . Applying Theorem B.5 to both components of  $\left(\varphi_1^{-1}\circ \Phi_{\mathbf{p},h}\right)(z)$  and  $\varphi_1^{-1}\circ \phi_z\circ \varphi_n\big|_{\left(-\frac{\delta}{3n},\frac{\delta}{3n}\right)^{2n}}$  separately yields the existence of two real vectors  $\xi_{1}$ ,  $\xi_{2}\in h\cdot [-m,m]^{2n}$ , such that
$$
\begin{array}{l} \left(\varphi_ {1} ^ {- 1} \circ \Phi_ {\mathbf {p}, h} (z)\right) _ {1} = \left[ \partial^ {\mathbf {p}} \left(\varphi_ {1} ^ {- 1} \circ \phi_ {z} \circ \varphi_ {n}\right) \left(\xi_ {1}\right) \right] _ {1} \\ \text {a n d} \quad \left(\varphi_ {1} ^ {- 1} \circ \Phi_ {\mathbf {p}, h} (z)\right) _ {2} = \left[ \partial^ {\mathbf {p}} \left(\varphi_ {1} ^ {- 1} \circ \phi_ {z} \circ \varphi_ {n}\right) \left(\xi_ {2}\right) \right] _ {2}. \end{array}
$$
Rewriting this yields
$$
\operatorname {R e} \left(\Phi_ {\mathbf {p}, h} (z)\right) = \operatorname {R e} \left(\phi_ {\mathbf {p}} \left(z, \varphi_ {n} \left(\xi_ {1}\right)\right)\right) \quad \text {a n d} \quad \operatorname {I m} \left(\Phi_ {\mathbf {p}, h} (z)\right) = \operatorname {I m} \left(\phi_ {\mathbf {p}} \left(z, \varphi_ {n} \left(\xi_ {2}\right)\right)\right).
$$
Using this property, we deduce
$$
\left| \operatorname {R e} \left(\Phi_ {\mathbf {p}, h} (z) - \phi_ {\mathbf {p}} (z, 0)\right) \right| = \left| \operatorname {R e} \left(\phi_ {\mathbf {p}} \left(z, \varphi_ {n} \left(\xi_ {1}\right)\right) - \phi_ {\mathbf {p}} (z, 0)\right) \right| \leq \left| \phi_ {\mathbf {p}} \left(z, \varphi_ {n} \left(\xi_ {1}\right)\right) - \phi_ {\mathbf {p}} (z, 0) \right| \leq \frac {\varepsilon}{\sqrt {2}}
$$
and analogously for the imaginary part. Since  $z \in \Omega_n$  and  $h \in (0, h_{\mathbf{p}})$  have been chosen arbitrarily we get the claim by choosing
$$
h ^ {*} := \min  \left\{h _ {\mathbf {p}}: \mathbf {p} \in \mathbb {N} _ {0} ^ {2 n} \text {w i t h} | \mathbf {p} | \leq k \text {a n d} \mathbf {p} \leq m \right\}.
$$
Using the previous two lemmas and the results from Appendix B.1 and Appendix B.2, we can now prove Theorem 3.1.
Proof of Theorem 3.1. Let  $b \in U$  satisfy
$$
\partial_ {\mathrm {w i r t}} ^ {\ell_ {1}} \overline {{\partial}} _ {\mathrm {w i r t}} ^ {\ell_ {2}} \phi (b) \neq 0 \quad \text {f o r a l l} \ell_ {1}, \ell_ {2} \in \mathbb {N} _ {0} \text {w i t h} \ell_ {1}, \ell_ {2} \leq m n.
$$
Such a point  $b$  exists according to Proposition 2.1. Let  $p \in \mathcal{P}'$  and fix  $\mathbf{m}, \ell \in \mathbb{N}_0^n$  with  $\mathbf{m}, \ell \leq m$ . For each  $z \in \Omega_n$ , using Lemma B.6, we then have
$$
\begin{array}{l} z ^ {\mathbf {m}} \bar {z} ^ {\boldsymbol {\ell}} = \left[ \left(\partial_ {\mathrm {w i r t}} ^ {| \mathbf {m} |} \bar {\partial} _ {\mathrm {w i r t}} ^ {| \boldsymbol {\ell} |} \phi\right) (b) \right] ^ {- 1} \partial_ {\mathrm {w i r t}} ^ {\mathbf {m}} \bar {\partial} _ {\mathrm {w i r t}} ^ {\boldsymbol {\ell}} \phi_ {z} (0) \\ \stackrel {\text {P r o p . A . 1}} {=} \left[ \left(\partial_ {\mathrm {w i r t}} ^ {| \mathbf {m} |} \overline {{\partial}} _ {\mathrm {w i r t}} ^ {| \ell |} \phi\right) (b) \right] ^ {- 1} \cdot \sum_ {\substack {\mathbf {p} = (\mathbf {p} ^ {\prime}, \mathbf {p} ^ {\prime \prime}) \in \mathbb {N} _ {0} ^ {2 n} \\ \mathbf {p} ^ {\prime} + \mathbf {p} ^ {\prime \prime} = \mathbf {m} + \ell}} b _ {\mathbf {p} ^ {\prime}, \mathbf {p} ^ {\prime \prime}} \partial^ {(\mathbf {p} ^ {\prime}, \mathbf {p} ^ {\prime \prime})} \phi_ {z} (0) \tag{B.6} \\ \end{array}
$$
with suitably chosen complex coefficients  $b_{\mathbf{p}^{\prime},\mathbf{p}^{\prime \prime}}\in \mathbb{C}$  depending only on  $\mathbf{p}^{\prime}$ ,  $\mathbf{p}^{\prime \prime}$ ,  $\mathbf{m}$  and  $\ell$ . Here we used that  $|\mathbf{m}|, |\ell|\leq mn$ . Since  $\mathcal{P}'\subseteq \mathcal{P}_m^n$  is bounded and  $p\in \mathcal{P}'$ , we can write
$$
p(z) = \sum_{\substack{\mathbf{m},\boldsymbol {\ell}\in \mathbb{N}_{0}^{n}\\ \mathbf{m},\boldsymbol {\ell}\leq m}}a_{\mathbf{m},\boldsymbol{\ell}}z^{\mathbf{m}}\overline{z}^{\boldsymbol{\ell}}
$$
with  $|a_{\mathbf{m},\boldsymbol{\ell}}| \leq c$  for some constant  $c = c(\mathcal{P}') > 0$ . In combination with (B.6), this easily implies that we can rewrite  $p$  as
$$
p (z) = \sum_ {\substack {\mathbf {p} \in \mathbb {N} _ {0} ^ {2 n} \\ \mathbf {p} \leq 2 m}} c _ {\mathbf {p}} (p) \partial^ {\mathbf {p}} \phi_ {z} (0) \tag{B.7}
$$
with coefficients  $c_{\mathbf{p}}(p) \in \mathbb{C}$  satisfying  $|c_{\mathbf{p}}(p)| \leq c'$  for some constant  $c' = c'(\phi, b, \mathcal{P}', m, n)$ . By Lemma B.7, we choose  $h^* > 0$ , such that
$$
\left| \Phi_ {\mathbf {p}, h ^ {*}} (z) - \partial^ {\mathbf {p}} \phi_ {z} (0) \right| \leq \frac {\varepsilon}{\sum_ {\mathbf {q} \in \mathbb {N} _ {0} ^ {2 n} , \mathbf {q} \leq 2 m} c ^ {\prime}}
$$
for all  $z \in \Omega_n$  and  $\mathbf{p} \in \mathbb{N}_0^{2n}$  with  $\mathbf{p} \leq 2m$ . Furthermore, we can rewrite each function  $\Phi_{\mathbf{p},h^*}$  as
$$
\Phi_{\mathbf{p},h^{*}}(z) = \sum_{\substack{\boldsymbol {\alpha}\in \mathbb{Z}^{2n}\\ |\boldsymbol{\alpha}_{j}|\leq 2m  \forall j}}\lambda_{\boldsymbol {\alpha},\mathbf{p}}\phi \big([ \varphi_{n}  (h^{*}\boldsymbol {\alpha})]^{T}\cdot z + b\big)
$$
with suitable coefficients  $\lambda_{\alpha, \mathbf{p}} \in \mathbb{C}$ . Since the cardinality of the set
$$
\left\{\boldsymbol {\alpha} \in \mathbb {Z} ^ {2 n}: | \boldsymbol {\alpha} _ {j} | \leq 2 m \forall j \right\}
$$
is  $(4m + 1)^{2n}$ , this can be converted to
$$
\Phi_ {\mathbf {p}, h ^ {*}} (z) = \sum_ {j = 1} ^ {N} \lambda_ {j, \mathbf {p}} \phi (\rho_ {j} ^ {T} \cdot z + b).
$$
For  $p$  as in (B.7), we then define
$$
\theta (z):= \sum_{\substack{\mathbf{p}\in \mathbb{N}_{0}^{2n}\\ \mathbf{p}\leq 2m}}c_{\mathbf{p}}(p)\cdot \Phi_{\mathbf{p},h^{*}}(z) = \sum_{j = 1}^{N}\left[\left(\sum_{\substack{\mathbf{p}\in \mathbb{N}_{0}^{2n}\\ \mathbf{p}\leq 2m}}c_{\mathbf{p}}(p)\lambda_{j,\mathbf{p}}\right)\phi (\rho_{j}^{T}\cdot z + b)\right]
$$
and note
$$
| \theta (z) - p (z) | \leq \sum_ {\mathbf {p} \leq 2 m} | c _ {\mathbf {p}} (p) | \cdot | \Phi_ {\mathbf {p}, h ^ {*}} (z) - \partial^ {\mathbf {p}} \phi_ {z} (0) | \leq \varepsilon .
$$
Since the coefficients  $\rho_{j}$  have been chosen independently of the polynomial  $p$ , we can rewrite  $\theta$  in the desired form.
C Postponed proofs for the approximation of  $C^k$ -functions
C.1 Prerequisites from Fourier Analysis
This section is dedicated to reviewing some notations and results from Fourier Analysis. In the end, a quantitative result for the approximation of  $C^k$ $([-1,1]^s; \mathbb{R})$ -functions using linear combinations of multivariate Chebyshev polynomials is derived; see Theorem C.15.
We start by recalling several notations and concepts from Fourier Analysis.
Definition C.1. Let  $s \in \mathbb{N}$  and  $k \in \mathbb{N}_0$ . We define
$$
C _ {2 \pi} ^ {k} \left(\mathbb {R} ^ {s}; \mathbb {C}\right) := \left\{f \in C ^ {k} \left(\mathbb {R} ^ {s}; \mathbb {C}\right): \forall \mathbf {p} \in \mathbb {Z} ^ {s} \forall x \in \mathbb {R} ^ {s}: f (x + 2 \pi \mathbf {p}) = f (x) \right\}.
$$
and  $C_{2\pi}(\mathbb{R}^s;\mathbb{C}):= C_{2\pi}^0 (\mathbb{R}^s;\mathbb{C})$  . For a function  $f\in C_{2\pi}^{k}(\mathbb{R}^{s};\mathbb{C})$  we write
$$
\| f\|_{C^{k}([-\pi ,\pi ]^{s};\mathbb{C})}:= \max_{\substack{\mathbf{k}\in \mathbb{N}_{0}^{s}\\ |\mathbf{k}|\leq k}}\big\| \partial^{\mathbf{k}}f\big\|_{L^{\infty}([-\pi ,\pi ]^{s};\mathbb{C})}\text{and}
$$
$$
\| f \| _ {L ^ {p} \left([ - \pi , \pi ] ^ {s}; \mathbb {C}\right)} := \left(\frac {1}{(2 \pi) ^ {s}} \cdot \int_ {[ - \pi , \pi ] ^ {s}} | f (x) | ^ {p} d x\right) ^ {1 / p} \text {f o r} p \in [ 1, \infty).
$$
Moreover, we set  $\| f\|_{L^{\infty}([-\pi ,\pi ]^{s};\mathbb{R})}\coloneqq \| f\|_{C^{0}([-\pi ,\pi ]^{s};\mathbb{C})}$
Definition C.2. For any  $s \in \mathbb{N}$  and  $\mathbf{k} \in \mathbb{Z}^s$ , we write
$$
e _ {\mathbf {k}}: \quad \mathbb {R} ^ {s} \to \mathbb {C}, \quad e _ {\mathbf {k}} (x) = e ^ {i \langle \mathbf {k}, x \rangle}
$$
where  $\langle \cdot, \cdot \rangle$  denotes the usual inner product of two vectors. For any  $f \in C_{2\pi}(\mathbb{R}^s; \mathbb{C})$  we define the  $\mathbf{k}$ -th Fourier coefficient of  $f$  to be
$$
\hat {f} (\mathbf {k}) := \frac {1}{(2 \pi) ^ {s}} \int_ {[ - \pi , \pi ] ^ {s}} f (x) \overline {{e _ {\mathbf {k}} (x)}} d x.
$$
Definition C.3. For two functions  $f, g \in C_{2\pi}(\mathbb{R}^s; \mathbb{C})$ , we define their convolution as
$$
f * g: \mathbb {R} ^ {s} \to \mathbb {C}, (f * g) (x) := \frac {1}{(2 \pi) ^ {s}} \int_ {[ - \pi , \pi ] ^ {s}} f (t) g (x - t) d t.
$$
In the following we define several so-called kernels.
Definition C.4. Let  $m \in \mathbb{N}_0$  be arbitrary.
1. The  $m$ -th one-dimensional Dirichlet-kernel is defined as
$$
D _ {m} := \sum_ {h = - m} ^ {m} e _ {h}.
$$
2. The  $m$ -th one-dimensional Fejér-kernel is defined as
$$
F _ {m} := \frac {1}{m} \sum_ {h = 0} ^ {m - 1} D _ {h}.
$$
3. The  $m$ -th one-dimensional de-la-Vallee-Poussin-kernel is defined as
$$
V _ {m} := \left(1 + e _ {m} + e _ {- m}\right) \cdot F _ {m}.
$$
4. Let  $s \in \mathbb{N}$ . We extend the above definitions to dimension  $s$  by letting
$$
D _ {m} ^ {s} \left(x _ {1}, \dots , x _ {s}\right) := \prod_ {p = 1} ^ {s} D _ {m} \left(x _ {p}\right),
$$
$$
F _ {m} ^ {s} \left(x _ {1},..., x _ {s}\right) := \prod_ {p = 1} ^ {s} F _ {m} \left(x _ {p}\right),
$$
$$
V _ {m} ^ {s} \left(x _ {1}, \ldots , x _ {s}\right) := \prod_ {p = 1} ^ {s} V _ {m} \left(x _ {p}\right).
$$
We need the following property of the multivariate extension of the de-la-Vallee-Poussin-kernel.
Proposition C.5. Let  $m, s \in \mathbb{N}$ . Then one has  $\| V_m^s\|_{L^1([-\pi, \pi]^s; \mathbb{C})} \leq 3^s$ .
Proof. From [34, Exercise 1.3 and Lemma 1.4] it follows  $\| F_m\|_{L^1 ([-\pi ,\pi ];\mathbb{C})} = 1$  and hence using the triangle inequality  $\| V_{m}\|_{L^{1}([-\pi ,\pi ];\mathbb{C})}\leq 3$ . The claim then follows using Tonelli's theorem.
The following definition introduces the term of trigonometric polynomial.
Definition C.6. For any  $s \in \mathbb{N}$  and  $m \in \mathbb{N}_0$  we call a function of the form
$$
\mathbb{R}^{s}\to \mathbb{C},\quad x\mapsto \sum_{\substack{\mathbf{k}\in \mathbb{Z}_{0}^{s}\\ -m\leq \mathbf{k}\leq m}}a_{\mathbf{k}}e^{i\langle \mathbf{k},x\rangle}
$$
with coefficients  $a_{\mathbf{k}} \in \mathbb{C}$  a trigonometric polynomial of coordinatewise degree at most  $m$  and denote the space of all those functions with  $\mathbb{H}_m^s$ . Here, we consider the sum over all  $\mathbf{k} \in \mathbb{Z}^s$  with  $-m \leq \mathbf{k}_j \leq m$  for all  $j \in \{1, \dots, s\}$ . We then write
$$
E _ {m} ^ {s} (f) := \min  _ {T \in \mathbb {H} _ {m} ^ {s}} \| f - T \| _ {L ^ {\infty} \left(\mathbb {R} ^ {s}; \mathbb {C}\right)} \tag {C.1}
$$
for any function  $f\in C_{2\pi}(\mathbb{R}^s;\mathbb{C})$
The following proposition shows that convolving with the Fejer kernel produces a trigonometric polynomial of degree at most  $2m - 1$ , while reproducing trigonometric polynomials of degree  $m$ . Furthermore, the norm of the convolution operator is bounded uniformly in  $m$ . These properties will be useful for our proof of Theorem C.15.
Proposition C.7. Let  $s, m \in \mathbb{N}$  and  $k \in \mathbb{N}_0$ . The map
$$
v _ {m}: \quad C _ {2 \pi} \left(\mathbb {R} ^ {s}; \mathbb {C}\right) \to \mathbb {H} _ {2 m - 1} ^ {s}, \quad f \mapsto f * V _ {m} ^ {s}
$$
is well-defined and satisfies
$$
v _ {m} (T) = T \quad f o r a l l \quad T \in \mathbb {H} _ {m} ^ {s}. \tag {C.2}
$$
Furthermore, there exists a constant  $c = c(s) > 0$  (independent of  $m$ ), such that
$$
\left\| v _ {m} (f) \right\| _ {C ^ {k} ([ - \pi , \pi ] ^ {s}; \mathbb {C})} \leq c \cdot \left\| f \right\| _ {C ^ {k} ([ - \pi , \pi ] ^ {s}; \mathbb {C})} \forall f \in C _ {2 \pi} ^ {k} (\mathbb {R} ^ {s}; \mathbb {C}),
$$
$$
\left\| v _ {m} (f) \right\| _ {L ^ {\infty} \left([ \pi , \pi ] ^ {s}; \mathbb {C}\right)} \leq c \cdot \| f \| _ {L ^ {\infty} \left([ - \pi , \pi ] ^ {s}; \mathbb {C}\right)} \forall f \in C _ {2 \pi} \left(\mathbb {R} ^ {s}; \mathbb {C}\right). \tag {C.3}
$$
In fact, it holds  $c(s) \leq \exp(C \cdot s)$  with an absolute constant  $C > 0$ .
Proof. A direct computation shows that  $f * e_{\mathbf{k}} = \hat{f}(\mathbf{k}) \cdot e_{\mathbf{k}}$ . This implies that  $v_{m}$  is well-defined since  $V_{m}^{s}$  is a trigonometric polynomial of coordinatewise degree at most  $2m - 1$ .
The operator is bounded on  $C_{2\pi}^{k}(\mathbb{R}^{s};\mathbb{C})$  and  $C_{2\pi}(\mathbb{R}^s;\mathbb{C})$  with norm at most  $c = 3^s$ , as follows from Young's inequality [34, Lemma 1.1 (ii)], Proposition C.5, and the fact that one has for all  $\mathbf{k}\in \mathbb{N}_0^s$  with  $|\mathbf{k}|\leq k$  the identity
$$
\partial^ {\mathbf {k}} \left(f * V _ {m} ^ {s}\right) = \left(\partial^ {\mathbf {k}} f\right) * V _ {m} ^ {s} \quad \text {f o r} f \in C _ {2 \pi} ^ {k} \left(\mathbb {R} ^ {s}; \mathbb {C}\right).
$$
It remains to show that  $v_{m}$  is the identity on  $\mathbb{H}_m^s$ . We first prove that
$$
e _ {k} * V _ {m} = e _ {k} \tag {C.4}
$$
holds for all  $k\in \mathbb{Z}$  with  $|k|\leq m$  .First note that
$$
e _ {k} * V _ {m} = e _ {k} * F _ {m} + e _ {k} * \left(e _ {m} \cdot F _ {m}\right) + e _ {k} * \left(e _ {- m} \cdot F _ {m}\right).
$$
We then compute
$$
e _ {k} * F _ {m} = \frac {1}{m} \sum_ {\ell = 0} ^ {m - 1} D _ {\ell} * e _ {k} = \frac {1}{m} \sum_ {\ell = 0} ^ {m - 1} \sum_ {h = - \ell} ^ {\ell} \underbrace {e _ {h} * e _ {k}} _ {= \delta_ {k, h} \cdot e _ {k}} = \frac {1}{m} \sum_ {\ell = | k |} ^ {m - 1} e _ {k} = \frac {m - | k |}{m} \cdot e _ {k}.
$$
$$
\begin{array}{l} e _ {k} * (e _ {m} \cdot F _ {m}) = \frac {1}{m} \sum_ {\ell = 0} ^ {m - 1} (e _ {m} D _ {\ell}) * e _ {k} = \frac {1}{m} \sum_ {\ell = 0} ^ {m - 1} \sum_ {h = - \ell} ^ {\ell} \underbrace {e _ {h + m} * e _ {k}} _ {= \delta_ {k, h + m} \cdot e _ {k}} \\ = \frac{1}{m}\sum_{\substack{0\leq \ell \leq m - 1\\ \ell \geq m - k}}e_{k} = \delta_{k\geq 1}\cdot \frac{k}{m}\cdot e_{k} \\ \end{array}
$$
$$
\begin{array}{l} e _ {k} * (e _ {- m} \cdot F _ {m}) = \frac {1}{m} \sum_ {\ell = 0} ^ {m - 1} (e _ {- m} D _ {\ell}) * e _ {k} = \frac {1}{m} \sum_ {\ell = 0} ^ {m - 1} \sum_ {h = - \ell} ^ {\ell} \underbrace {e _ {h - m} * e _ {k}} _ {= \delta_ {k, h - m} \cdot e _ {k}} \\ = \frac{1}{m}\sum_{\substack{0\leq \ell \leq m - 1\\ \ell \geq k + m}}e_{k} = \delta_{k\leq -1}\cdot \frac{-k}{m}\cdot e_{k}. \\ \end{array}
$$
Adding up those three identities yields (C.4).
To finally prove (C.2), it clearly suffices to show that
$$
e _ {\mathbf {k}} * V _ {m} ^ {s} = e _ {\mathbf {k}}
$$
for all  $\mathbf{k} \in \mathbb{Z}^s$  with  $-m \leq \mathbf{k} \leq m$ . But for such  $\mathbf{k}$ , using  $e_{\mathbf{k}}(x) = \prod_{j=1}^{s} e_{\mathbf{k}_j}(x_j)$ , one obtains
$$
\begin{array}{l} (e _ {\mathbf {k}} * V _ {m} ^ {s}) (x) = \frac {1}{(2 \pi) ^ {s}} \int_ {[ - \pi , \pi ] ^ {s}} \prod_ {j = 1} ^ {s} e _ {\mathbf {k} _ {j}} (t _ {j}) \cdot V _ {m} (x _ {j} - t _ {j}) d t \\ \stackrel {\mathrm {F u b i n i}} {=} \prod_ {j = 1} ^ {s} \left(e _ {\mathbf {k} _ {j}} * V _ {m}\right) (x _ {j}) \stackrel {(\mathrm {C . 4})} {=} \prod_ {j = 1} ^ {s} e _ {\mathbf {k} _ {j}} (x _ {j}) = e _ {\mathbf {k}} (x) \\ \end{array}
$$
for any  $x\in \mathbb{R}^s$  , as was to be shown.
The following result follows from a theorem in [29].
Proposition C.8. Let  $s, k \in \mathbb{N}$ . Then there exists a constant  $c = c(s, k) > 0$ , such that, for  $E_m^s$  as defined in (C.1),
$$
E _ {m} ^ {s} (f) \leq \frac {c}{m ^ {k}} \cdot \| f \| _ {C ^ {k} ([ - \pi , \pi ] ^ {s}; \mathbb {R})}
$$
for all  $m\in \mathbb{N}$  and  $f\in C_{2\pi}^{k}(\mathbb{R}^{s};\mathbb{R})$
In fact, it holds  $c(s,k) \leq \exp(C \cdot ks) \cdot k^k$  with an absolute constant  $C > 0$ .
Proof. We apply [29, Theorem 6.6] with  $n_i = m$  and  $p_i = k$ , which yields the existence of a constant  $c_1 = c_1(s,k) > 0$ , such that
$$
E _ {m} ^ {s} (f) \leq c _ {1} \cdot \sum_ {\ell = 1} ^ {s} \frac {1}{m ^ {k}} \cdot \omega_ {\ell} \left(f, \frac {1}{m}\right)
$$
for all  $m \in \mathbb{N}$  and  $f \in C_{2\pi}^{k}(\mathbb{R}^{s};\mathbb{R})$ , where  $\omega_{\ell}(f,\bullet)$  denotes the modulus of continuity of  $\frac{\partial^k f}{\partial x_\ell^k}$  with respect to  $x_{\ell}$ , where we have the trivial bound
$$
\omega_ {\ell} \left(f, \frac {1}{m}\right) \leq 2 \cdot \| f \| _ {C ^ {k} ([ - \pi , \pi ] ^ {s}; \mathbb {R})}.
$$
Hence, we get
$$
E _ {m} ^ {s} (f) \leq c _ {1} \cdot s \cdot 2 \cdot \| f \| _ {C ^ {k} ([ - \pi , \pi ] ^ {s}; \mathbb {R})} \frac {1}{m ^ {k}},
$$
so the claim follows by choosing  $c \coloneqq 2s \cdot c_1$ .
We refer to Appendix C.2 (see Theorem C.18) for a proof of the claimed bound on the constant  $c(s, k)$ .
The above proposition bounds the best possible error of approximating  $f$  by trigonometric polynomials of coordinatewise degree at most  $m$ , but this is in general non-constructive. Our next result shows that a similar bound holds for approximating  $f$  by  $v_{m}(f)$ .
Theorem C.9. Let  $s \in \mathbb{N}$ . Then there exists a constant  $c = c(s) > 0$ , such that the operator  $v_{m}$  from Proposition C.7 satisfies
$$
\left\| f - v _ {m} (f) \right\| _ {L ^ {\infty} \left(\mathbb {R} ^ {s}\right)} \leq c \cdot E _ {m} ^ {s} (f)
$$
for any  $m\in \mathbb{N}$  and  $f\in C_{2\pi}(\mathbb{R}^s;\mathbb{C})$
Proof. For any  $T \in \mathbb{H}_m^s$  one has
$$
\| f - v _ {m} (f) \| _ {L ^ {\infty} \left(\mathbb {R} ^ {s}\right)} \stackrel {{\text {(C . 2)}}} {\leq} \| f - T \| _ {L ^ {\infty} \left(\mathbb {R} ^ {s}\right)} + \| v _ {m} (T) - v _ {m} (f) \| _ {L ^ {\infty} \left(\mathbb {R} ^ {s}\right)} \stackrel {{\text {(C . 3)}}} {\leq} (c + 1) \| f - T \| _ {L ^ {\infty} \left(\mathbb {R} ^ {s}\right)}.
$$
Taking the infimum over all  $T \in \mathbb{H}_m^s$  yields the claim.
By combining Proposition C.8 and Theorem C.9, we immediately get the following bound.
Corollary C.10. Let  $s, k \in \mathbb{N}_0$ . Then there exists a constant  $c = c(s, k) > 0$ , such that
$$
\left\| f - v _ {m} (f) \right\| _ {L ^ {\infty} (\mathbb {R} ^ {s})} \leq \frac {c}{m ^ {k}} \cdot \left\| f \right\| _ {C ^ {k} ([ - \pi , \pi ] ^ {s}; \mathbb {R})}
$$
for every  $m\in \mathbb{N}$  and  $f\in C_{2\pi}^{k}(\mathbb{R}^{s};\mathbb{R})$
In fact, we have  $c(s,k) \leq \exp(C \cdot ks) \cdot k^k$  with an absolute constant  $C > 0$ .
Up to now, we have studied the approximation of periodic functions by trigonometric polynomials, but our actual goal is to approximate non-periodic functions by algebraic polynomials. The next lemma establishes a connection between the two settings.
Lemma C.11. Let  $k \in \mathbb{N}_0$  and  $s \in \mathbb{N}$ . For any function  $f \in C^k([-1,1]^s; \mathbb{C})$ , we define the corresponding periodic function via
$$
f ^ {*}: \mathbb {R} ^ {s} \to \mathbb {C}, f ^ {*} (x _ {1},..., x _ {s}) = f (\cos (x _ {1}),..., \cos (x _ {s}))
$$
and note  $f^{*}\in C_{2\pi}^{k}(\mathbb{R}^{s};\mathbb{C})$  . The map
$$
C ^ {k} \left([ - 1, 1 ] ^ {s}; \mathbb {C}\right)\rightarrow C _ {2 \pi} ^ {k} \left(\mathbb {R} ^ {s}; \mathbb {C}\right), \quad f \mapsto f ^ {*}
$$
is a continuous linear operator with respect to the  $C^k$ -norms on  $C^k ([-1,1]^s;\mathbb{C})$  and  $C_{2\pi}^{k}(\mathbb{R}^{s};\mathbb{C})$ . The operator norm can be bounded from above by  $k^k$ .
Proof. The map is well-defined since  $\cos$  is a smooth function and  $2\pi$ -periodic. The linearity of the operator is obvious, so it remains to show its continuity.
The goal is to apply the closed graph theorem [21, Theorem 5.12]. By definition of  $f^{*}$ , and since  $\cos : [-\pi, \pi] \to [-1, 1]$  is surjective, we have the equality  $\|f\|_{L^{\infty}([-\pi, 1]^{s}; \mathbb{C})} = \|f^{*}\|_{L^{\infty}([-\pi, \pi]^{s}; \mathbb{C})}$ .