SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Furthermore, the map  $f \mapsto \sigma(f)$  is a continuous linear operator with respect to the  $L^{\infty}$ -norm.
Sketch of proof. By splitting  $f$  into real and imaginary part we may assume that  $f$  is real-valued. Fourier-analytical results (recalled in Appendix C.1) imply that each  $C^k$ -function  $f$  can be well approximated by a linear combination of (multivariate) Chebyshev polynomials. Precisely, we have
$$
\left\| f - P \right\| _ {L ^ {\infty} \left(\Omega_ {n}; \mathbb {R}\right)} \leq \frac {c}{m ^ {k}} \cdot \left\| f \right\| _ {C ^ {k} \left(\Omega_ {n}; \mathbb {R}\right)},
$$
where  $P$  is given via the formula
$$
P (z) = \sum_ {0 \leq \mathbf {k} \leq 2 m - 1} \mathcal {V} _ {\mathbf {k}} ^ {m} (f) \cdot T _ {\mathbf {k}} (z), \quad z \in \Omega_ {n}.
$$
Here, the functions  $T_{\mathbf{k}}$  are multivariate versions of Chebyshev polynomials and  $\nu_{\mathbf{k}}^{m}(f)$  are continuous linear functionals in  $f$ . The constant  $c > 0$  only depends on  $n$  and  $k$ . See Appendix C.1 for a rigorous proof of this approximation property. Approximating the polynomials  $T_{\mathbf{k}}$  by neural networks according to Theorem 3.1 yields the desired claim; see Appendix C.3 for the details.
Remark 3.3. Theorem 3.1 can not only be used to derive approximation rates for the approximation of  $C^k$ -functions but can be applied to any function class that is well approximable by algebraic polynomials. For example, it can be used to prove an order of approximation of  $\nu^{-m^{1/(2n)}/17}$  for the approximation of functions  $f: \Omega_n \to \mathbb{C}$  that can be holomorphically extended onto some polyellipse in  $\mathbb{C}^{2n}$ . The parameter  $\nu > 1$  specifies the size of this polyellipse. We refer the interested reader to Appendix D for detailed definitions, statements and proofs for this fact.
4 Optimality of the derived approximation rate
In this section we discuss the optimality of the approximation rate proven in Theorem 3.2. We first deduce from general results by DeVore et al. [19] that the rate is optimal in the setting that the map which assigns to a function  $f \in C^k(\Omega_n; \mathbb{C})$  the weights of the approximating network is continuous, as is the case in Theorem 3.2. However, it might be possible to achieve a better degree of approximation if this map is not required to be continuous, which is discussed in the second part of this section. Proofs for all the statements in this section are given in Appendices E.1, F.2 and F.3.
Continuous weight selection. We begin by considering the case of continuous weight selection. As mentioned in the introduction, this is a natural assumption, since in classical training algorithms such as (S)GD, continuous operations based on samples  $f(x_{j})$  are used to adjust the weights.
In [19, Theorem 4.2] a lower bound of  $m^{-k / s}$  is established for the rate of approximating functions  $f$  of Sobolev regularity  $W^{k,\infty}$  in the following very general setting: The set of functions that is used for approximation can be parametrized using  $m$  (real) parameters and the map that assigns to  $f$  the parameters of the approximating function is continuous with respect to some norm on  $W^{k,\infty}([-1,1]^s;\mathbb{R})$ . A detailed version of the proof of that statement (for  $C^k$  instead of  $W^{k,\infty}$ ) is contained in Appendix E.1. A careful transfer of this result to the complex-valued setting yields the following theorem (see also Appendix E.1).
Theorem 4.1. Let  $n, k \in \mathbb{N}$ . Then there exists a constant  $c = c(n, k) > 0$  with the following property: For any  $m \in \mathbb{N}$ , any map  $\overline{a}: C^k(\Omega_n; \mathbb{C}) \to \mathbb{C}^m$  that is continuous with respect to some norm on  $C^k(\Omega_n; \mathbb{C})$  and any map  $M: \mathbb{C}^m \to C(\Omega_n; \mathbb{C})$  we have
$$
\sup  _ {f \in C ^ {k} (\Omega_ {n}; \mathbb {C}), \| f \| _ {C ^ {k} (\Omega_ {n}; \mathbb {C})} \leq 1} \| f - M (\overline {{a}} (f)) \| _ {L ^ {\infty} (\Omega_ {n}; \mathbb {C})} \geq c \cdot m ^ {- k / (2 n)}.
$$
The interpretation of Theorem 4.1 is as follows: If one approximates  $C^k$ -functions on  $\Omega_n$  using a set of functions that can be parametrized using  $m$  (complex) parameters and one assumes that the weight assignment is continuous, one cannot achieve a better rate of approximation than  $m^{-k/(2n)}$ . As a special case it can be deduced that the approximation rate is at most  $m^{-k/(2n)}$  when approximating  $C^k$ -functions using shallow CVNNs with  $m$  parameters under continuous weight assignment (see Corollary E.3). One can show that this even holds for deep CVNNs. Hence, for continuous weight selection the rate proven in Theorem 3.2 is optimal, even in the class of (possibly) deep networks.
Discontinuous weight selection. When we drop the continuity assumption on the selection of the weights, the behavior of the optimal approximation rate is more subtle. Precisely, we show that there are activation functions for which the rate of approximation can be improved to  $m^{-k / (2n - 1)}$ . On the other hand, we also show that there are activation functions for which an improvement of the approximation rate (up to logarithmic factors) is not possible.
Theorem 4.2. There exists a function  $\phi \in C^{\infty}(\mathbb{C};\mathbb{C})$  which is non-polyharmonic with the following additional property: For every  $k\in \mathbb{N}$  and  $n\in \mathbb{N}$  there exists a constant  $c = c(n,k) > 0$  such that
Figure 3: Illustration of the construction of the activation function in Theorem 4.2.
for any  $m\in \mathbb{N}$  and  $f\in C^k$ $(\Omega_{n};\mathbb{C})$  there is a first layer  $\Phi \in \mathcal{F}_{n,m}^{\phi}$  and a vector  $\sigma \in \mathbb{C}^{m}$  such that
$$
\left\| f - \sigma^ {T} \Phi \right\| _ {L ^ {\infty} (\Omega_ {n}; \mathbb {C})} \leq c \cdot m ^ {- k / (2 n - 1)} \cdot \| f \| _ {C ^ {k} (\Omega_ {n}; \mathbb {C})}.
$$
Sketch of proof. The function  $\phi$  is constructed in the following way: Take a countable dense subset  $\{u_{\ell}\}_{\ell \in \mathbb{N}}$  of  $C(\Omega_1;\mathbb{C})$ , for instance the set of all polynomials in  $z$  and  $\overline{z}$  with coefficients in  $\mathbb{Q} + i\mathbb{Q}$ . Define  $\phi$  in a way such that
$$
\phi (z + 3 \ell) = u _ {\ell} (z)
$$
for every  $z \in \Omega_1$  and  $\ell \in \mathbb{N}$ . Furthermore, to ensure that  $\phi$  is non-polyharmonic, let  $\phi(z) = e^{\mathrm{Re}(z)}$  for every  $z \in \Omega_1$ . The smoothness of  $\phi$  can be accomplished by multiplying with a smooth bump function; see Lemma F.4 for details. The construction of  $\phi$  is illustrated in Figure 3.
Let then  $f \in C^{k}(\Omega_{n};\mathbb{C})$  be arbitrary. General results from the theory of ridge functions [22] show that there exist  $b_{1},\ldots ,b_{m}\in \mathbb{C}^{n}$  and  $g_{1},\dots,g_{m}\in C(\Omega_{1};\mathbb{C})$  such that
$$
\left\| f (z) - \sum_ {j = 1} ^ {m} g _ {j} \left(b _ {j} ^ {T} \cdot z\right) \right\| _ {L ^ {\infty} \left(\Omega_ {n}; \mathbb {C}\right)} \leq c \cdot m ^ {- k / (2 n - 1)} \cdot \| f \| _ {C ^ {k} \left(\Omega_ {n}; \mathbb {C}\right)};
$$
see Proposition F.3 and Appendix F.1 for a detailed proof of this fact. Approximating the functions  $g_{j}$  by suitable functions  $u_{\ell_j}$  and expressing those functions via  $\phi (\bullet + 3\ell_j)$  yields the claim.
The preceding theorem showed that there exists an activation function for which the rate in Theorem 3.2 can be strictly improved, if one allows a discontinuous weight selection. In contrast, the following theorem shows for a certain (quite natural) activation function that the rate  $m^{-k / (2n)}$  cannot be improved (up to logarithmic factors), even if one allows a discontinuous weight assignment.
Theorem 4.3. Let  $n, k \in \mathbb{N}$  and
$$
\phi : \quad \mathbb {C} \to \mathbb {C}, \quad \phi (z) := \frac {1}{1 + e ^ {- \operatorname {R e} (z)}}.
$$
Then  $\phi$  is smooth but non-polyharmonic. Furthermore, there exists a constant  $c = c(n,k) > 0$  with the following property: For any  $m\in \mathbb{N}_{\geq 2}$  there exists a function  $f\in C^{k}(\Omega_{n};\mathbb{C})$  with  $\| f\|_{C^k (\Omega_n;\mathbb{C})}\leq 1$ , such that for every  $\Phi \in \mathcal{F}_{n,m}^{\phi}$  and  $\sigma \in \mathbb{C}^m$  we have
$$
\left\| f - \sigma^ {T} \Phi \right\| _ {L ^ {\infty} (\Omega_ {n}; \mathbb {C})} \geq c \cdot (m \cdot \ln (m)) ^ {- k / (2 n)}.
$$
Sketch of proof. The idea of the proof is based on that of [46, Theorem 4] but instead of the bound for the VC dimension of ReLU networks used in [46], we will employ a bound for the VC dimension stated in [4, Theorem 8.11] using the real sigmoid function. For a detailed introduction to the concept of the VC dimension and related topics, see for instance [39, Chapter 6].
A technical reduction from the complex to the real case (see Appendix F.3) shows that it suffices to show the following: If  $\varepsilon \in (0, \frac{1}{2})$  and  $m \in \mathbb{N}$  are such that for every  $f \in C^k([-1,1]^n; \mathbb{R})$  with  $\| f \|_{C^k([-1,1]^n; \mathbb{R})} \leq 1$  there exists a real-valued shallow network  $\mathcal{N}$  with  $\gamma(x) = \frac{1}{1 + e^{-x}}$  as activation function satisfying  $\| f - \mathcal{N} \|_{L^\infty([-1,1]^n; \mathbb{R})} \leq \varepsilon$ , then necessarily
$$
m \geq c \cdot \frac {\varepsilon^ {- n / k}}{\ln (1 / \varepsilon)},
$$
where the constant  $c$  only depends on  $n$  and  $k$ .
To show that the latter claim holds, we assume that  $\varepsilon$  and  $m$  have the property stated above. Take  $N\in \mathbb{N}$  such that  $N^{-k}\asymp \varepsilon$  and consider the grid
$$
\mathcal {G} := \frac {1}{N} \{- N, \dots , N \} ^ {n} \subseteq [ - 1, 1 ] ^ {n}.
$$
For every  $\alpha \in \mathcal{G}$  we pick a number  $z_{\alpha} \in \{0,1\}$  arbitrarily and construct a map  $f \in C^{\infty}([-1,1]^{n};\mathbb{R})$  satisfying  $f(\alpha) = z_{\alpha}$  for every  $\alpha \in \mathcal{G}$ . Scaling of  $f$  to  $\tilde{f}$  ensures  $\| \tilde{f} \|_{C^k([-1,1]^n;\mathbb{R})} \leq 1$ , but then  $\tilde{f}(\alpha) = c_0 \cdot z_{\alpha} \cdot N^{-k}$  where  $c_0 = c_0(n,k) > 0$ . By assumption, we can infer the existence of a shallow real-valued neural network  $\mathcal{N}$  with  $\gamma$  as activation function and  $m$  hidden neurons satisfying  $\| \tilde{f} - \mathcal{N} \|_{L^{\infty}([-1,1]^{n};\mathbb{R})} \leq \varepsilon$ . But this shows
$$
\mathcal {N} (\alpha) \left\{ \begin{array}{l l} > \tilde {c} N ^ {- k}, & \text {i f} z _ {\alpha} = 1, \\ <   \tilde {c} N ^ {- k}, & \text {i f} z _ {\alpha} = 0 \end{array} \right. \quad \text {f o r a l l} \alpha \in \mathcal {G}
$$
with a constant  $\tilde{c} = \tilde{c}(n, k) > 0$ . Since the  $z_{\alpha}$  are arbitrary, it follows that the set
$$
H := \left\{\left. \mathbb {1} \left(\mathcal {N} > \tilde {c} N ^ {- k}\right) \right| _ {\mathcal {G}}: \mathcal {N} \text {s h a l l o w N N w i t h a c t i v a t i o n} \gamma \text {a n d} m \text {h i d d e n n e u r o n s} \right\}
$$
shatters the whole grid  $\mathcal{G}$ . This yields  $\mathrm{VC}(H) \geq |\mathcal{G}| = (2N + 1)^n$ . On the other hand, the bound provided by [4, Theorem 8.11] for linear threshold networks yields  $\mathrm{VC}(H) \lesssim m \cdot \ln(N)$ . Combining the two bounds and using  $N^{-k} \asymp \varepsilon$  yields the claim.
5 Tractability of the considered problem in terms of the input dimension
In this section we discuss the tractability (in terms of the input dimension  $n$ ) of the considered problem, i.e., the dependence of the approximation error on  $n$ . We show a novel result stating that, assuming a continuous weight selection, the problem of approximating  $C^k$ -functions is intractable, i.e., that the number of neurons that is required to achieve a non-trivial approximation error is necessarily exponential in  $n$ . In the literature this is referred to as the curse of dimensionality. The proof of the theorem combines ideas from [19] and [35] and is contained in Appendix E.2.
Theorem 5.1. Let  $s \in \mathbb{N}$ . With  $\| \cdot \|_{C^k ([-1,1]^s;\mathbb{R})}$  defined similarly to (2.1), we write
$$
\| f \| _ {C ^ {\infty} ([ - 1, 1 ] ^ {s}; \mathbb {R})} := \sup  _ {k \in \mathbb {N}} \| f \| _ {C ^ {k} ([ - 1, 1 ] ^ {s}; \mathbb {R})} \in [ 0, \infty ]
$$
for any function  $f \in C^{\infty}([-1,1]^s;\mathbb{R})$  and denote by  $C^{\infty,*s}$  the set of all  $f \in C^{\infty}([-1,1]^s;\mathbb{R})$  for which this expression is finite. Let  $\overline{a}: C^{\infty,*s} \to \mathbb{R}^{2^s - 1}$  be continuous with respect to some norm on  $C^{\infty,*s}$  and moreover, let  $M: \mathbb{R}^{2^s - 1} \to C([-1,1]^s;\mathbb{R})$  be an arbitrary map. Then it holds
$$
\sup_{\substack{f\in C^{\infty ,*,s}\\ \| f\|_{C^{\infty}([-1,1]^{s};\mathbb{R})}\leq 1}}\| f - M(\overline{a} (f))\|_{L^{\infty}([-1,1]^{s};\mathbb{R})}\geq 1.
$$
Note that Theorem 5.1 is formulated for real-valued functions but can be transferred to the complex-valued setting (see Corollary E.6). We decided to include the real-valued statement because it is expected to be of greater interest in the community than the complex-valued analog. Moreover, we stress that Theorem 5.1 is not limited to the class of shallow neural networks but refers to any function class that is parametrizable using finitely many parameters (in particular, e.g., the class of neural networks with possibly more than one hidden layer).
We now examine in what way the constant  $c$  appearing in Theorem 3.2 suffers from the curse of dimensionality. To this end, it is convenient to rewrite the result from Theorem 3.2 as
$$
\sup  _ {\| f \| _ {C ^ {k} \left(\Omega_ {n}; \mathbb {C}\right)} \leq 1} \inf  _ {\Phi \in \mathcal {F} _ {n, m} ^ {\phi}, \sigma \in \mathbb {C} ^ {m}} \| f - \sigma^ {T} \Phi \| _ {L ^ {\infty} \left(\Omega_ {n}; \mathbb {C}\right)} \leq (\tilde {c} \cdot m) ^ {- k / (2 n)}
$$
where the constant  $\tilde{c} = \tilde{c}(n, k) > 0$  is independent of  $m$ . Writing the result in that way, one sees immediately that, if one seeks to have a worst-case approximation error of less than  $\varepsilon > 0$ , it is
sufficient to take  $m = \left\lceil \frac{1}{\tilde{c}} \cdot \varepsilon^{-(2n) / k} \right\rceil$  neurons in the hidden layer of the network. Corollary E.7 shows that it necessarily holds  $\tilde{c} \leq 16 \cdot 2^{-n}$  and therefore, the constant  $\tilde{c}$  unavoidably suffers from the curse of dimensionality. An analysis of the constant (where we refer to Appendices C.1 to C.3 for the details) shows that in our case we can establish the bound  $\tilde{c}(n,k) \geq \exp(-C \cdot n^2) \cdot k^{-4n}$  with an absolute constant  $C > 0$ . We remark that, since the constant suffers from the curse of dimensionality in any case, we have not put much effort into optimizing the constant; there is therefore probably ample room for improvement.
To conduct a comprehensive evaluation of machine learning algorithms, one must analyze the questions of approximation, generalization, and optimization through training algorithms. The present paper, however, only focuses on the aspect of approximation. Analyzing if the proven approximation rate can be attained with learning algorithms such as (stochastic) gradient descent falls outside the scope of this paper. Furthermore, the examination of approximation rates under possibly discontinuous weight assignment is not yet fully resolved by our results. It is an open question which rate is optimally achievable in that case, depending on the choice of the activation function, and specifically in distinguishing between shallow and deep networks. We want to mention the two following points which indicate that this is a quite subtle question:
1. For deep NNs (with more than two hidden layers) with general smooth activation function, it is not possible to derive any non-trivial lower bounds in the setting of unrestricted weight assignment, since there exists an activation function with the property that NNs of constant size using this activation function can approximate any continuous function to arbitrary precision (see [31, Theorem 4]). Note that [31] considers real-valued NNs, but the results can be transferred to CVNNs with a suitable choice of the activation function.
2. In the real-valued case, fully general lower bounds for the approximation capabilities of shallow NNs have been derived by using results from [22] regarding the approximation properties of so-called ridge functions, i.e., functions of the form  $\sum_{j=1}^{m} \phi_j(\langle a_j, x \rangle)$  with  $a_j \in \mathbb{R}^d$  and each  $\phi_j : \mathbb{R} \to \mathbb{R}$ . It is an interesting problem to generalize these results to higher-dimensional ridge functions of the form  $\sum_{j=1}^{m} \phi_j(A_jx)$ , where each  $\phi_j : \mathbb{R}^s \to \mathbb{R}$  and  $A_j \in \mathbb{R}^{s \times d}$ . This would imply lower bounds for shallow CVNNs. However, such a generalization seems to be highly non-trivial and is outside the scope of the present paper.
This paper analyzes error bounds for the approximation of complex-valued  $C^k$ -functions by means of complex-valued neural networks with smooth and non-polyharmonic activation functions. It is demonstrated that complex-valued neural networks with these activation functions achieve the identical approximation rate as real-valued networks that employ smooth and non-polynomial activation functions. This is an important theoretical finding, since CVNNs are on the one hand more restrictive than real-valued neural networks (since the mappings between layers should be  $\mathbb{C}$ -linear and not just  $\mathbb{R}$ -linear), but on the other hand more versatile, since the activation function is a mapping from  $\mathbb{C}$  to  $\mathbb{C}$  (i.e., from  $\mathbb{R}^2 \to \mathbb{R}^2$ ) rather than from  $\mathbb{R}$  to  $\mathbb{R}$  as in the real case. Additionally, it is established that the proven approximation rate is optimal if one assumes a continuous weight selection. In summary, if one focuses on the approximation rate for  $C^k$ -functions, CVNNs have the same excellent approximation properties as real-valued networks.
The behavior of the approximation rate for unrestricted weight selection is more subtle. It is shown that a rate of  $m^{-k / (2n - 1)}$  can be achieved for certain activation functions (Theorem 4.2) but in general, one cannot improve on the rate that is attainable for continuous weight selection (Theorem 4.3).
While the proven approximation rate is optimal under the assumption of continuous weight selection, the involved constants suffer from the curse of dimensionality. Section 5, however, shows that this is inevitable in the given setting.
Such theoretical approximation results contribute to the mathematical understanding of Deep Learning. The remarkable approximation-theoretical properties of neural networks can be seen as one reason why neural networks provide outstanding results in many applications.
Acknowledgments. PG and FV acknowledge support by the German Science Foundation (DFG) in the context of the Emmy Noether junior research group VO 2594/1-1. FV acknowledges support by the Hightech Agenda Bavaria.
[47] D. Yarotsky and A. Zhevnerchuk. "The phase diagram of approximation rates for deep neural networks". In: Advances in neural information processing systems 33 (2020), pp. 13005-13015. URL: https://proceedings.neurips.cc/paper/2020/file/979a3f14bae523dc5101c52120c535e9-Paper.pdf.
[48] Z. Zhang, H. Wang, F. Xu, and Y.-Q. Jin. “Complex-Valued Convolutional Neural Network and Its Application in Polarimetric SAR Image Classification”. In: IEEE Transactions on Geoscience and Remote Sensing 55.12 (2017), pp. 7177–7188. DOI: 10.1109/TGRS.2017.2743222.
A Notation, Wirtinger derivatives and special activation functions
A.1 Notation and Wirtinger derivatives
Throughout the paper, multi-indices (i.e., elements of  $\mathbb{N}_0^n$ ) are denoted using boldface. For  $\mathbf{m} \in \mathbb{N}_0^n$  we have the usual notation  $|\mathbf{m}| = \sum_{j=1}^{n} \mathbf{m}_j$ . For a number  $m \in \mathbb{N}_0$  and another multi-index  $\mathbf{p} \in \mathbb{N}_0^n$  we write  $\mathbf{m} \leq m$  iff  $\mathbf{m}_j \leq m$  for every  $j \in \{1, \dots, n\}$  and  $\mathbf{m} \leq \mathbf{p}$  iff  $\mathbf{m}_j \leq \mathbf{p}_j$  for every  $j \in \{1, \dots, n\}$ . Furthermore we write
$$
\left( \begin{array}{c} \mathbf {p} \\ \mathbf {r} \end{array} \right) := \prod_ {j = 1} ^ {n} \left( \begin{array}{c} \mathbf {p} _ {j} \\ \mathbf {r} _ {j} \end{array} \right)
$$
for two multi-indices  $\mathbf{p},\mathbf{r}\in \mathbb{N}_0^n$  with  $\mathbf{r}\leq \mathbf{p}$ . For a complex vector  $z\in \mathbb{C}^n$  we write
$$
z ^ {\mathbf {m}} := \prod_ {j = 1} ^ {n} z _ {j} ^ {\mathbf {m} _ {j}} \quad \text {a n d} \quad \overline {{z}} ^ {\mathbf {m}} := \prod_ {j = 1} ^ {n} \overline {{z}} _ {j} ^ {\mathbf {m} _ {j}}.
$$
For a point  $x\in \mathbb{R}^n$  and  $r > 0$  we define
$$
B _ {r} (x) := \left\{y \in \mathbb {R} ^ {n}: \| x - y \| <   r \right\}
$$
with  $\| \cdot \|$  denoting the usual Euclidean distance. This definition is analogously transferred to  $\mathbb{C}^n$ .
$\mathbb{C}^n$  is canonically isomorphic to  $\mathbb{R}^{2n}$  by virtue of the isomorphism
$$
\varphi_ {n}: \mathbb {R} ^ {2 n} \rightarrow \mathbb {C} ^ {n}, (x _ {1}, \dots , x _ {n}, y _ {1}, \dots , y _ {n}) \mapsto (x _ {1} + i y _ {1}, \dots , x _ {n} + i y _ {n}). \tag {A.1}
$$
The Wirtinger derivatives defined in Section 2 have the following properties that we are going to use, which can be found for example in [26, E.1a]. Here, we assume that  $U \subseteq \mathbb{C}$  is open and  $f \in C^{1}(U; \mathbb{C})$ .
1.  $\partial_{\mathrm{wirt}}$  and  $\overline{\partial}_{\mathrm{wirt}}$  are both  $\mathbb{C}$ -linear operators on the set  $C^1 (U;\mathbb{C})$ .
2.  $f$  is complex-differentiable in  $z \in U$  iff  $\overline{\partial}_{\mathrm{wirt}} f(z) = 0$  and in this case the equality
$$
\partial_ {\mathrm {w i r t}} f (z) = f ^ {\prime} (z)
$$
holds true, with  $f'(z)$  denoting the complex derivative of  $f$  at  $z$ .
3. We have the conjugation rules
$$
\overline {{\partial_ {\mathrm {w i r t}} f}} = \overline {{\partial}} _ {\mathrm {w i r t}} \overline {{f}} \quad \text {a n d} \quad \overline {{\overline {{\partial}} _ {\mathrm {w i r t}} f}} = \partial_ {\mathrm {w i r t}} \overline {{f}}.
$$
4. If  $g \in C^1(U; \mathbb{C})$ , the following product rules for Wirtinger derivatives hold for every  $z \in U$ :
$$
\partial_ {\operatorname {w i r t}} (f \cdot g) (z) = \partial_ {\operatorname {w i r t}} f (z) \cdot g (z) + f (z) \cdot \partial_ {\operatorname {w i r t}} g (z),
$$
$$
\bar {\partial} _ {\mathrm {w i r t}} (f \cdot g) (z) = \bar {\partial} _ {\mathrm {w i r t}} f (z) \cdot g (z) + f (z) \cdot \bar {\partial} _ {\mathrm {w i r t}} g (z).
$$
This product rule is not explicitly stated in [26] but follows easily from the product rule for  $\frac{\partial}{\partial x}$  and  $\frac{\partial}{\partial y}$ .
5. If  $V \subseteq \mathbb{C}$  is an open set and  $g \in C^{1}(V; \mathbb{C})$  with  $g(V) \subseteq U$ , then the following chain rules for Wirtinger derivatives hold true:
$$
\partial_ {\mathrm {w i r t}} (f \circ g) = \left[ \left(\partial_ {\mathrm {w i r t}} f\right) \circ g \right] \cdot \partial_ {\mathrm {w i r t}} g + \left[ \left(\bar {\partial} _ {\mathrm {w i r t}} f\right) \circ g \right] \cdot \partial_ {\mathrm {w i r t}} \bar {g},
$$
$$
\bar {\partial} _ {\mathrm {w i r t}} (f \circ g) = \left[ \left(\partial_ {\mathrm {w i r t}} f\right) \circ g \right] \cdot \bar {\partial} _ {\mathrm {w i r t}} g + \left[ \left(\bar {\partial} _ {\mathrm {w i r t}} f\right) \circ g \right] \cdot \bar {\partial} _ {\mathrm {w i r t}} \bar {g}.
$$
6. If  $f \in C^2(U; \mathbb{C})$  then we have
$$
\Delta f (z) = 4 \left(\partial_ {\mathrm {w i r t}} \overline {{\partial}} _ {\mathrm {w i r t}} f\right) (z)
$$
for every  $z \in U$ , with  $\Delta$  denoting the usual Laplace-Operator  $\Delta = \partial^{(2,0)} + \partial^{(0,2)}$  (cf. [7, equation (1.7)]).