SlowGuess/NeurIPS_2023_OCR huggingface dataset:
[AA20] Omar Abu Arqub. Numerical simulation of time-fractional partial differential equations arising in fluid flows via reproducing kernel method. International Journal of Numerical Methods for Heat & Fluid Flow, 30(11):4711-4733, 2020.
[AAS20] Omar Abu Arqub and Mohammed Al-Smadi. An adaptive numerical approach for the solutions of fractional advection-diffusion and dispersion equations in singular case under riesz's derivative operator. Physica A: Statistical Mechanics and its Applications, 540:123257, 2020.
[Ame14] William F Ames. Numerical methods for partial differential equations. Academic press, 2014.
[Arq18] Omar Abu Arqub. Numerical solutions for the robin time-fractional partial differential equations of heat and fluid flows based on the reproducing kernel algorithm. International Journal of Numerical Methods for Heat & Fluid Flow, 28(4):828-856, 2018.
[ASA19] Mohammed Al-Smadi and Omar Abu Arqub. Computational algorithm for solving fredholm time-fractional partial integrodifferential equations of dirichlet functions type with error estimates. Applied Mathematics and Computation, 342:280-294, 2019.
[Bar01] E Barkai. Fractional fokker-planck equation, solution, and application. Physical Review E, 63(4):046118, 2001.
$\left[\mathrm{BCE}^{+}22\right]$  Krishna Balasubramanian, Sinho Chewi, Murat A Erdogdu, Adil Salim, and Shunshi Zhang. Towards a theory of non-log-concave sampling: first-order stationarity guarantees for Langevin monte carlo. In Conference on Learning Theory, pages 2896-2923. PMLR, 2022.
[BCRB22] Eloise Berthier, Justin Carpentier, Alessandro Rudi, and Francis Bach. Infinite-dimensional sums-of-squares for optimal control. In 2022 IEEE 61st Conference on Decision and Control (CDC), pages 577-582. IEEE, 2022.
[BDMP17] Nicolas Brosse, Alain Durmus, Éric Moulines, and Marcelo Pereyra. Sampling from a log-concave distribution with compact support with proximal Langevin monte carlo. In *Conference on learning theory*, pages 319–342. PMLR, 2017.
[BEL15] Sebastien Bubeck, Ronen Eldan, and Joseph Lehec. Finite-time analysis of projected Langevin monte carlo. Advances in Neural Information Processing Systems, 28, 2015.
[BH17] Jake Bouvrie and Boumediene Hamzi. Kernel methods for the approximation of some key quantities of nonlinear systems. Journal of Computational Dynamics, 4(1&2):1-19, 2017.
[Zho08] Ding-Xuan Zhou. Derivative reproducing properties for kernel methods in learning theory. Journal of computational and Applied Mathematics, 220(1-2):456-463, 2008.
[ZL22] Yuping Zheng and Andrew Lamperski. Constrained langevin algorithms with 1-mixing external random variables. arXiv preprint arXiv:2205.14192, 2022.
[ZZ22] Xiaolong Zhang and Xicheng Zhang. Ergodicity of supercritical sdes driven by alpha-stable processes and heavy-tailed sampling. arXiv preprint arXiv:2201.10158, 2022.
A Existence of an appropriate PSD matrix  $M_{\varepsilon}$  (Fokker-Planck Equation)
A.1 Useful Results
Consider the mollifier function  $g$  defined as in [RC21].
$$
g (x) = \frac {2 ^ {- d / 2}}{V _ {d}} \| x \| ^ {- d} J _ {d / 2} (2 \pi \| x \|) J _ {d / 2} (4 \pi \| x \|), \tag {13}
$$
where  $J_{d/2}$  is the Bessel function of the first kind of order  $d/2$  and  $V_d = \int_{\|x\| \leq 1} dx = \frac{\pi^{d/2}}{\Gamma(d/2+1)}$ .  $g_v(x) = v^{-d} g(x/v)$ .
Lemma 1 ([RC21], Lemma D.3). The function  $g$  defined above satisfies  $g \in L^{1}(\mathbb{R}^{d}) \cap L^{2}(\mathbb{R}^{d})$  and  $\int g(x) dx = 1$ . Moreover, for any  $\omega \in \mathbb{R}^d$ , we have
$$
\mathbf {1} _ {\{\| \omega \| <   1 \}} (\omega) \leq \mathcal {F} [ g ] (\omega) \leq \mathbf {1} _ {\{\| \omega \| \leq 3 \}} (\omega).
$$
The following proposition provides a useful characterization of the space  $W_2^\beta(\mathbb{R}^d)$ .
Proposition 4 (Characterization of the Sobolev space  $W_2^k(\mathbb{R}^d)$ , [Wen04]). Let  $k \in \mathbb{N}$ . The norm of the Sobolev space  $\| \cdot \|_{W_2^k(\mathbb{R}^d)}$  is equivalent to the following norm
$$
\left\| f \right\| _ {W _ {2} ^ {k} (\mathbb {R} ^ {d})} ^ {\prime 2} = \int_ {\mathbb {R} ^ {d}} \left| \mathcal {F} [ f ] (\omega) \right| ^ {2} (1 + \| \omega \| ^ {2}) ^ {k} d \omega , \quad \forall f \in L ^ {2} (\mathbb {R} ^ {d}) \tag {14}
$$
and satisfies
$$
\frac {1}{(2 \pi) ^ {2 k}} \| f \| _ {W _ {2} ^ {k} (\mathbb {R} ^ {d})} ^ {2} \leq \| f \| _ {W _ {2} ^ {k} (\mathbb {R} ^ {d})} ^ {\prime} \leq 2 ^ {2 k} \| f \| _ {W _ {2} ^ {k} (\mathbb {R} ^ {d})} ^ {2}, \quad \forall f \in L ^ {2} (\mathbb {R} ^ {d}). \tag {15}
$$
Moreover, when  $k > d / 2$ , then  $W_2^k (\mathbb{R}^d)$  is a reproducing kernel Hilbert space.
Lemma 2. Consider the definition of  $g_v(x) = v^{-d}g(x / v)$  where  $g$  is defined in equation (13). Let  $\beta > 2, q \in \mathbb{N}$ . Let  $f_1, \ldots, f_q \in W_2^\beta(\mathbb{R}^d) \cap L^\infty(\mathbb{R}^d)$  and the function  $p^\star = \sum_{i=1}^{q} f_i^2$ . Let us assume that the Assumption 2 is also satisfied. Let  $\varepsilon \in (0,1]$  and let  $\eta \in \mathbb{R}_{++}^d$ . Let  $\phi_{X,T}$  be the feature map of the Gaussian kernel with bandwidth  $\eta$  and let  $\mathcal{H}_X \otimes \mathcal{H}_T$  be the associated RKHS. Then there exists  $\mathsf{M}_{\varepsilon} \in \mathbb{S}_+(\mathcal{H}_X \otimes \mathcal{H}_T)$  with  $\mathrm{rank}(\mathsf{M}_{\varepsilon}) \leq q$ , such that for the representation  $\hat{p}(x,t) = \phi_{X,T}^\top \mathsf{M}_{\varepsilon} \phi_{X,T}$ , following holds for  $v > 0$  and  $k \in \{1,\dots,d\}$ ,
$$
\begin{array}{l} \left\| \frac {\partial \hat {p} (x , t)}{\partial t} - \frac {\partial p ^ {\star} (x , t)}{\partial t} \right\| _ {L ^ {1} (\mathbb {R} ^ {d})} \leq 8 \pi (2 v) ^ {(\beta - 1)} \sum_ {i = 1} ^ {q} \| f _ {i} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d})} \| f _ {i} \| _ {L _ {2} (\mathbb {R} ^ {d})} \\ + 2 (2 v) ^ {\beta} \sum_ {i = 1} ^ {q} \| f _ {i} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d})} \left\| \frac {\partial f _ {i}}{\partial t} \right\| _ {L _ {2} (\mathbb {R} ^ {d})} \| g \| _ {L _ {1} (\mathbb {R} ^ {d})}, \tag {16} \\ \end{array}
$$
$$
\begin{array}{l} \left\| \frac {\partial p ^ {\star} (x , t)}{\partial t} - \frac {\partial \hat {p} (x , t)}{\partial t} \right\| _ {L ^ {2} (\mathbb {R} ^ {d})} \leq 8 \pi (2 v) ^ {(\beta - 1)} \sum_ {i = 1} ^ {q} \| f _ {i} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d})} \| f _ {i} \| _ {L ^ {\infty} (\mathbb {R} ^ {d})} \\ + 2 (2 v) ^ {\beta} \sum_ {i = 1} ^ {q} \| f _ {i} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d})} \left\| \frac {\partial f _ {i}}{\partial t} \right\| _ {L ^ {\infty} (\mathbb {R} ^ {d})} \| g \| _ {L _ {1} (\mathbb {R} ^ {d})} \tag {17} \\ \end{array}
$$
$$
\begin{array}{l} \left\| \frac {\partial p ^ {\star} (x , t)}{\partial x _ {k}} - \frac {\partial \hat {p} (x , t)}{\partial x _ {k}} \right\| _ {L ^ {1} (\mathbb {R} ^ {d})} \leq 8 \pi (2 v) ^ {(\beta - 1)} \sum_ {i = 1} ^ {q} \| f _ {i} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d})} \| f _ {i} \| _ {L _ {2} (\mathbb {R} ^ {d})} \\ + 2 (2 v) ^ {\beta} \sum_ {i = 1} ^ {q} \| f _ {i} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d})} \left\| \frac {\partial f _ {i}}{\partial x _ {k}} \right\| _ {L _ {2} (\mathbb {R} ^ {d})} \| g \| _ {L _ {1} (\mathbb {R} ^ {d})} \tag {18} \\ \end{array}
$$
$$
\left\| \frac {\partial p ^ {\star} (x , t)}{\partial x _ {k}} - \frac {\partial \hat {p} (x , t)}{\partial x _ {k}} \right\| _ {L ^ {2} \left(\mathbb {R} ^ {d}\right)} \leq 8 \pi (2 v) ^ {(\beta - 1)} \sum_ {i = 1} ^ {q} \| f _ {i} \| _ {W _ {2} ^ {\beta} \left(\mathbb {R} ^ {d}\right)} \| f _ {i} \| _ {L ^ {\infty} \left(\mathbb {R} ^ {d}\right)} \tag {19}
$$
$$
+ 2 (2 v) ^ {\beta} \sum_ {i = 1} ^ {q} \| f _ {i} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d})} \left\| \frac {\partial f _ {i}}{\partial x _ {k}} \right\| _ {L ^ {\infty} (\mathbb {R} ^ {d})} \| g \| _ {L _ {1} (\mathbb {R} ^ {d})}. \tag {20}
$$
Proof. We have  $g$  defined as in [RC21].
$$
g _ {v} (x) = \frac {2 ^ {- d / 2}}{V _ {d}} \| x \| ^ {- d} J _ {d / 2} (2 \pi \| x \|) J _ {d / 2} (4 \pi \| x \|), \tag {21}
$$
where  $J_{d/2}$  is the Bessel function of the first kind of order  $d/2$  and  $V_d = \int_{\|x\| \leq 1} dx = \frac{\pi^{d/2}}{\Gamma(d/2+1)}$ .  $g_v(x) = v^{-d} g(x/v)$ . We use the result of lemma 1 to prove the statement. For a function  $f \in W_2^\beta(\mathbb{R}^d)$ ,
$$
\begin{array}{l} \left\| \frac {\partial}{\partial x _ {i}} f - \frac {\partial}{\partial x _ {i}} (f \star g _ {v}) \right\| _ {L _ {2} (\mathbb {R} ^ {d})} ^ {2} = \left\| \mathcal {F} \left[ \frac {\partial}{\partial x _ {i}} f \right] - \mathcal {F} \left[ \frac {\partial}{\partial x _ {i}} (f \star g _ {v}) \right] \right\| _ {L _ {2} (\mathbb {R} ^ {d})} ^ {2} \\ = \left\| 2 \pi \iota \omega_ {i} \mathcal {F} [ f ] - 2 \pi \iota \omega_ {i} \mathcal {F} [ f \star g _ {v} ] \right\| _ {L _ {2} (\mathbb {R} ^ {d})} = \left\| 2 \pi \iota \omega_ {i} \mathcal {F} [ f ] (1 - \mathcal {F} [ g _ {v} ]) \right\| _ {L _ {2} (\mathbb {R} ^ {d})} ^ {2} \\ = 4 \pi^ {2} \int_ {\mathbb {R} ^ {d}} \omega_ {i} ^ {2} | \mathcal {F} [ f ] (\omega) | ^ {2} | 1 - \mathcal {F} [ g ] (v \omega) | ^ {2} d \omega \leq 4 \pi^ {2} \int_ {t \| \omega \| \geq 1} \omega_ {i} ^ {2} | \mathcal {F} [ f ] (\omega) | ^ {2} d \omega \\ = 4 \pi^ {2} \int_ {v \| \omega \| \geq 1} \omega_ {i} ^ {2} (1 + \| \omega \| ^ {2}) ^ {- \beta} (1 + \| \omega \| ^ {2}) ^ {\beta} | \mathcal {F} [ f ] (\omega) | ^ {2} d \omega \\ \leq \sup  _ {v \| \omega \| \geq 1} \frac {4 \pi^ {2} \omega_ {i} ^ {2}}{(1 + \| \omega \| ^ {2}) ^ {\beta}} \int (1 + \| \omega \| ^ {2}) ^ {\beta} | \mathcal {F} [ f ] (\omega) | ^ {2} d \omega \\ \leq \sup _ {v \| \omega \| \geq 1} \frac {4 \pi^ {2} \omega_ {i} ^ {2}}{(1 + \| \omega \| ^ {2}) ^ {\beta}} 2 ^ {2 \beta} \| f \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d})} ^ {2} \leq \sup _ {v \| \omega \| \geq 1} \frac {4 \pi^ {2}}{(1 + \| \omega \| ^ {2}) ^ {\beta - 1}} 2 ^ {2 \beta} \| f \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d})} ^ {2} \\ \leq \frac {1 6 \pi^ {2} (2 v) ^ {2 (\beta - 1)}}{(1 + v ^ {2}) ^ {\beta - 1}} \| f \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d})} ^ {2} \\ \end{array}
$$
$$
\left\| \frac {\partial}{\partial x _ {i}} f - \frac {\partial}{\partial x _ {i}} \left(f \star g _ {v}\right) \right\| _ {L _ {2} \left(\mathbb {R} ^ {d}\right)} ^ {2} \leq \frac {1 6 \pi^ {2} (2 v) ^ {2 (\beta - 1)}}{\left(1 + v ^ {2}\right) ^ {\beta - 1}} \| f \| _ {W _ {2} ^ {\beta} \left(\mathbb {R} ^ {d}\right)} ^ {2} \tag {22}
$$
The same result will also hold for  $\frac{\partial}{\partial t}$  without loss of generality. Hence,
$$
\left\| \frac {\partial}{\partial t} f - \frac {\partial}{\partial t} \left(f \star g _ {v}\right) \right\| _ {L _ {2} \left(\mathbb {R} ^ {d}\right)} ^ {2} \leq \frac {1 6 \pi^ {2} (2 v) ^ {2 (\beta - 1)}}{\left(1 + v ^ {2}\right) ^ {\beta - 1}} \| f \| _ {W _ {2} ^ {\beta} \left(\mathbb {R} ^ {d}\right)} ^ {2} \tag {23}
$$
We now bound  $\left\| \frac{\partial p^{\star}(x,t)}{\partial t} - \frac{\partial \hat{p}(x,t)}{\partial t} \right\|_{L^{r}(\mathbb{R}^{d})}$  and  $\left\| \frac{\partial p^{\star}(x,t)}{\partial x_{i}} - \frac{\partial \hat{p}(x,t)}{\partial x_{i}} \right\|_{L^{r}(\mathbb{R}^{d})}$ .
As in [RC21], let us denote,  $f_{i} \star g_{v}$  as  $f_{i,v}$ . We consider
$$
M _ {\varepsilon} = \sum_ {i = 1} ^ {q} f _ {i, v} f _ {i, v} ^ {\top}.
$$
$$
\hat {p} (x, t) = f (x, t) = \phi_ {X, T} (x, t) ^ {\top} M _ {\varepsilon} \phi_ {X, T} (x, t) = \sum_ {i = 1} ^ {q} f _ {i, v} ^ {2} (x, t).
$$
This gives,
$$
\begin{array}{l} \frac {\partial}{\partial x _ {i}} p ^ {\star} - \frac {\partial}{\partial x _ {i}} \hat {p} = 2 \sum_ {i = 1} ^ {q} \left[ f _ {i} \frac {\partial f _ {i}}{\partial x _ {i}} - f _ {i, v} \frac {\partial f _ {i , v}}{\partial x _ {i}} \right] = 2 \sum_ {i = 1} ^ {q} \left[ f _ {i} \frac {\partial f _ {i}}{\partial x _ {i}} - f _ {i} \frac {\partial f _ {i , v}}{\partial x _ {i}} + f _ {i} \frac {\partial f _ {i , v}}{\partial x _ {i}} - f _ {i, v} \frac {\partial f _ {i , v}}{\partial x _ {i}} \right] \\ = 2 \sum_ {i = 1} ^ {q} \left[ f _ {i} \left(\frac {\partial f _ {i}}{\partial x _ {i}} - \frac {\partial f _ {i , v}}{\partial x _ {i}}\right) + (f _ {i} - f _ {i, v}) \frac {\partial f _ {i , v}}{\partial x _ {i}} \right] \\ \end{array}
$$
$$
\left\| \frac {\partial}{\partial x _ {i}} p ^ {\star} - \frac {\partial}{\partial x _ {i}} \hat {p} \right\| _ {L _ {1} (\mathbb {R} ^ {d})} \leq 2 \sum_ {i = 1} ^ {q} \left\| \frac {\partial f _ {i}}{\partial x _ {i}} - \frac {\partial f _ {i , v}}{\partial x _ {i}} \right\| _ {L _ {2} (\mathbb {R} ^ {d})} \| f _ {i} \| _ {L _ {2} (\mathbb {R} ^ {d})} + 2 \sum_ {i = 1} ^ {q} \| f _ {i} - f _ {i, v} \| _ {L _ {2} (\mathbb {R} ^ {d})} \left\| \frac {\partial f _ {i , v}}{\partial x _ {i}} \right\| _ {L _ {2} (\mathbb {R} ^ {d})}. \tag {24}
$$
$$
\left\| \frac {\partial}{\partial t} p ^ {\star} - \frac {\partial}{\partial t} \hat {p} \right\| _ {L _ {1} \left(\mathbb {R} ^ {d}\right)} \leq 2 \sum_ {i = 1} ^ {q} \left\| \frac {\partial f _ {i}}{\partial t} - \frac {\partial f _ {i , v}}{\partial t} \right\| _ {L _ {2} \left(\mathbb {R} ^ {d}\right)} \| f _ {i} \| _ {L _ {2} \left(\mathbb {R} ^ {d}\right)} + 2 \sum_ {i = 1} ^ {q} \| f _ {i} - f _ {i, v} \| _ {L _ {2} \left(\mathbb {R} ^ {d}\right)} \left\| \frac {\partial f _ {i , v}}{\partial t} \right\| _ {L _ {2} \left(\mathbb {R} ^ {d}\right)}. \tag {25}
$$
$$
\begin{array}{l} \left\| \frac {\partial p ^ {\star} (x , t)}{\partial t} - \frac {\partial \hat {p} (x , t)}{\partial t} \right\| _ {L ^ {1} (\mathbb {R} ^ {d})} \leq 2 \sum_ {i = 1} ^ {q} \left\| \frac {\partial f _ {i}}{\partial t} - \frac {\partial f _ {i , v}}{\partial t} \right\| _ {L ^ {2} (\mathbb {R} ^ {d})} \| f _ {i} \| _ {L ^ {2} (\mathbb {R} ^ {d})} + 2 \sum_ {i = 1} ^ {q} \| f _ {i} - f _ {i, v} \| _ {L ^ {2} (\mathbb {R} ^ {d})} \left\| \frac {\partial f _ {i , v}}{\partial t} \right\| _ {L ^ {2} (\mathbb {R} ^ {d})} \\ \leq \frac {8 \pi (2 v) ^ {(\beta - 1)}}{(1 + v ^ {2}) ^ {\frac {\beta - 1}{2}}} \sum_ {i = 1} ^ {q} \| f _ {i} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d})} \| f _ {i} \| _ {L _ {2} (\mathbb {R} ^ {d})} + 2 (2 v) ^ {\beta} \sum_ {i = 1} ^ {q} \| f _ {i} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d})} \left\| \frac {\partial f _ {i , v}}{\partial t} \right\| _ {L _ {2} (\mathbb {R} ^ {d})} \\ \leq 8 \pi (2 v) ^ {(\beta - 1)} \sum_ {i = 1} ^ {q} \| f _ {i} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d})} \| f _ {i} \| _ {L _ {2} (\mathbb {R} ^ {d})} + 2 (2 v) ^ {\beta} \sum_ {i = 1} ^ {q} \| f _ {i} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d})} \left\| \frac {\partial f _ {i}}{\partial t} \star g _ {v} \right\| _ {L _ {2} (\mathbb {R} ^ {d})} \\ \leq 8 \pi (2 v) ^ {(\beta - 1)} \sum_ {i = 1} ^ {q} \| f _ {i} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d})} \| f _ {i} \| _ {L _ {2} (\mathbb {R} ^ {d})} + 2 (2 v) ^ {\beta} \sum_ {i = 1} ^ {q} \| f _ {i} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d})} \left\| \frac {\partial f _ {i}}{\partial t} \right\| _ {L _ {2} (\mathbb {R} ^ {d})} \| g _ {v} \| _ {L _ {1} (\mathbb {R} ^ {d})} \\ = 8 \pi (2 v) ^ {(\beta - 1)} \sum_ {i = 1} ^ {q} \| f _ {i} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d})} \| f _ {i} \| _ {L _ {2} (\mathbb {R} ^ {d})} + 2 (2 v) ^ {\beta} \sum_ {i = 1} ^ {q} \| f _ {i} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d})} \left\| \frac {\partial f _ {i}}{\partial t} \right\| _ {L _ {2} (\mathbb {R} ^ {d})} \| g \| _ {L _ {1} (\mathbb {R} ^ {d})}. \\ \end{array}
$$
$$
v = \min  \left(\left(\frac {\varepsilon}{2 ^ {\beta - 1} 1 6 \pi C _ {1}}\right) ^ {\frac {1}{\beta - 1}}, \left(\frac {\varepsilon}{2 ^ {\beta + 1} C _ {2}}\right) ^ {\frac {1}{\beta}}\right) \tag {26}
$$
where  $C_1 = \sum_{i=1}^q \|f_i\|_{W_2^\beta(\mathbb{R}^d)} \|f_i\|_{L_2(\mathbb{R}^d)}$  and  $C_2 = \sum_{i=1}^q \|f_i\|_{W_2^\beta(\mathbb{R}^d)} \left\| \frac{\partial f_i}{\partial t} \right\|_{L_2(\mathbb{R}^d)} \|g\|_{L_1(\mathbb{R}^d)}$ .
$$
\left\| \frac {\partial p ^ {\star} (x , t)}{\partial t} - \frac {\partial \hat {p} (x , t)}{\partial t} \right\| _ {L ^ {1} (\mathbb {R} ^ {d})} \leq \varepsilon .
$$
$$
\left\| \frac {\partial p ^ {\star} (x , t)}{\partial t} - \frac {\partial \hat {p} (x , t)}{\partial t} \right\| _ {L ^ {2} (\mathbb {R} ^ {d})} = 2 \sum_ {i = 1} ^ {q} \left\| \frac {\partial f _ {i}}{\partial t} - \frac {\partial f _ {i , v}}{\partial t} \right\| _ {L ^ {2} (\mathbb {R} ^ {d})} \| f _ {i} \| _ {L ^ {\infty} (\mathbb {R} ^ {d})} + 2 \sum_ {i = 1} ^ {q} \| f _ {i} - f _ {i, v} \| _ {L ^ {2} (\mathbb {R} ^ {d})} \left\| \frac {\partial f _ {i , v}}{\partial t} \right\| _ {L ^ {\infty} (\mathbb {R} ^ {d})}.
$$
$$
\left\| \frac {\partial f _ {i , v}}{\partial t} \right\| _ {L ^ {\infty} (\mathbb {R} ^ {d})} = \left\| \frac {\partial f _ {i}}{\partial t} \star g _ {v} \right\| _ {L ^ {\infty} (\mathbb {R} ^ {d})} \leq \left\| \frac {\partial f _ {i}}{\partial t} \right\| _ {L ^ {\infty} (\mathbb {R} ^ {d})} \| g _ {v} \| _ {L ^ {1} (\mathbb {R} ^ {d})} = \left\| \frac {\partial f _ {i}}{\partial t} \right\| _ {L ^ {\infty} (\mathbb {R} ^ {d})} \| g \| _ {L ^ {1} (\mathbb {R} ^ {d})}.
$$
$$
\begin{array}{l} \left\| \frac {\partial p ^ {\star} (x , t)}{\partial t} - \frac {\partial \hat {p} (x , t)}{\partial t} \right\| _ {L ^ {2} (\mathbb {R} ^ {d})} \leq 8 \pi (2 v) ^ {(\beta - 1)} \sum_ {i = 1} ^ {q} \| f _ {i} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d})} \| f _ {i} \| _ {L ^ {\infty} (\mathbb {R} ^ {d})} \\ + 2 (2 v) ^ {\beta} \sum_ {i = 1} ^ {q} \| f _ {i} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d})} \left\| \frac {\partial f _ {i}}{\partial t} \right\| _ {L ^ {\infty} (\mathbb {R} ^ {d})} \| g \| _ {L _ {1} (\mathbb {R} ^ {d})}. \\ \end{array}
$$
$$
v = \min  \left(\left(\frac {\varepsilon}{2 ^ {\beta - 1} 1 6 \pi C _ {1}}\right) ^ {\frac {1}{\beta - 1}}, \left(\frac {\varepsilon}{2 ^ {\beta + 1} C _ {2}}\right) ^ {\frac {1}{\beta}}\right)
$$
where  $C_1 = \sum_{i=1}^q \|f_i\|_{W_2^\beta(\mathbb{R}^d)} \|f_i\|_{L^\infty(\mathbb{R}^d)}$  and  $C_2 = \sum_{i=1}^q \|f_i\|_{W_2^\beta(\mathbb{R}^d)} \left\| \frac{\partial f_i}{\partial t} \right\|_{L^\infty(\mathbb{R}^d)} \|g\|_{L_1(\mathbb{R}^d)}$ .
$$
\left\| \frac {\partial p ^ {\star} (x , t)}{\partial t} - \frac {\partial \hat {p} (x , t)}{\partial t} \right\| _ {L ^ {2} (\mathbb {R} ^ {d})} \leq \varepsilon .
$$
$$
\begin{array}{l} \left\| \frac {\partial p ^ {\star} (x , t)}{\partial x _ {k}} - \frac {\partial \hat {p} (x , t)}{\partial x _ {k}} \right\| _ {L ^ {1} \left(\mathbb {R} ^ {d}\right)} \leq 8 \pi (2 v) ^ {(\beta - 1)} \sum_ {i = 1} ^ {q} \| f _ {i} \| _ {W _ {2} ^ {\beta} \left(\mathbb {R} ^ {d}\right)} \| f _ {i} \| _ {L _ {2} \left(\mathbb {R} ^ {d}\right)} \tag {27} \\ + 2 (2 v) ^ {\beta} \sum_ {i = 1} ^ {q} \| f _ {i} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d})} \left\| \frac {\partial f _ {i}}{\partial x _ {k}} \right\| _ {L _ {2} (\mathbb {R} ^ {d})} \| g \| _ {L _ {1} (\mathbb {R} ^ {d})} \\ \end{array}
$$
$$
\begin{array}{l} \left\| \frac {\partial p ^ {\star} (x , t)}{\partial x _ {k}} - \frac {\partial \hat {p} (x , t)}{\partial x _ {k}} \right\| _ {L ^ {2} (\mathbb {R} ^ {d})} \leq 8 \pi (2 v) ^ {(\beta - 1)} \sum_ {i = 1} ^ {q} \| f _ {i} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d})} \| f _ {i} \| _ {L ^ {\infty} (\mathbb {R} ^ {d})} \\ + 2 (2 v) ^ {\beta} \sum_ {i = 1} ^ {q} \| f _ {i} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d})} \left\| \frac {\partial f _ {i}}{\partial x _ {k}} \right\| _ {L ^ {\infty} (\mathbb {R} ^ {d})} \| g \| _ {L _ {1} (\mathbb {R} ^ {d})}. \\ \end{array}
$$
Lemma 3. Consider the definition of  $g_v(x) = v^{-d}g(x / v)$  where  $g$  is defined in equation (13). Let  $\beta > 2, q \in \mathbb{N}$ . Let  $f_1, \ldots, f_q \in W_2^\beta(\mathbb{R}^d) \cap L^\infty(\mathbb{R}^d)$  and the function  $p^\star = \sum_{i=1}^q f_i^2$ . Let us assume that the Assumption 2 is also satisfied. Let  $\varepsilon \in (0,1]$  and let  $\eta \in \mathbb{R}_{++}^d$ . Let  $\phi_{X,T}$  be the feature map of the Gaussian kernel with bandwidth  $\eta$  and let  $\mathcal{H}_X \otimes \mathcal{H}_T$  be the associated RKHS. Then there exists  $\mathsf{M}_{\varepsilon} \in \mathbb{S}_+(\mathcal{H}_X \otimes \mathcal{H}_T)$  with  $\mathrm{rank}(\mathsf{M}_{\varepsilon}) \leq q$ , such that for the representation  $\hat{p}(x,t) = f(x,t) = \phi_{X,T}^\top \mathsf{M}_{\varepsilon} \phi_{X,T}$ , following holds for all  $i, j \in \{1, \dots, d\}$ ,
$$
\begin{array}{l} \left\| \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} p ^ {\star} (x, t) - \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} \hat {p} (x, t) \right\| _ {L ^ {1} (\mathbb {R} ^ {d})} \leq \frac {1 6 \pi (2 v) ^ {(\beta - 1)}}{(1 + v ^ {2}) ^ {\frac {\beta - 1}{2}}} (1 + \| g \| _ {L ^ {1} (\mathbb {R} ^ {d})}) \sum_ {m = 1} ^ {q} \| f \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d})} \left\| \frac {\partial f _ {m}}{\partial x _ {j}} \right\| _ {L ^ {2} (\mathbb {R} ^ {d})} \\ + \frac {3 2 \pi^ {2} (2 v) ^ {(\beta - 2)}}{(1 + v ^ {2}) ^ {\frac {\beta - 2}{2}}} \sum_ {m = 1} ^ {q} \| f \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d})} \| f _ {m} \| _ {L ^ {2} (\mathbb {R} ^ {d})} + 2 (2 v) ^ {\beta} \sum_ {m = 1} ^ {q} \| f _ {m} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d})} \left\| \frac {\partial^ {2} f _ {m}}{\partial x _ {i} \partial x _ {j}} \right\| _ {L ^ {2} (\mathbb {R} ^ {d})} \| g \| _ {L ^ {1} (\mathbb {R} ^ {d})}, \\ \end{array}
$$
$$
\begin{array}{l} \left\| \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} p ^ {\star} (x, t) - \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} \hat {p} (x, t) \right\| _ {L ^ {2} (\mathbb {R} ^ {d})} \leq \frac {1 6 \pi (2 v) ^ {(\beta - 1)}}{(1 + v ^ {2}) ^ {\frac {\beta - 1}{2}}} (1 + \| g \| _ {L ^ {1} (\mathbb {R} ^ {d})}) \sum_ {m = 1} ^ {q} \| f \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d})} \left\| \frac {\partial f _ {m}}{\partial x _ {j}} \right\| _ {L ^ {\infty} (\mathbb {R} ^ {d})} \\ + \frac {3 2 \pi^ {2} (2 v) ^ {(\beta - 2)}}{(1 + v ^ {2}) ^ {\frac {\beta - 2}{2}}} \sum_ {m = 1} ^ {q} \| f \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d})} \| f _ {m} \| _ {L ^ {\infty} (\mathbb {R} ^ {d})} + 2 (2 v) ^ {\beta} \sum_ {m = 1} ^ {q} \| f _ {m} \| _ {W _ {2} ^ {\beta} (\mathbb {R} ^ {d})} \left\| \frac {\partial^ {2} f _ {m}}{\partial x _ {i} \partial x _ {j}} \right\| _ {L ^ {\infty} (\mathbb {R} ^ {d})} \| g \| _ {L ^ {1} (\mathbb {R} ^ {d})}. \\ \end{array}
$$
$$
g _ {v} (x) = \frac {2 ^ {- d / 2}}{V _ {d}} \| x \| ^ {- d} J _ {d / 2} (2 \pi \| x \|) J _ {d / 2} (4 \pi \| x \|), \tag {28}
$$
where  $J_{d / 2}$  is the Bessel function of the first kind of order  $d / 2$  and  $V_{d} = \int_{\| x\| \leq 1}dx = \frac{\pi^{d / 2}}{\Gamma(d / 2 + 1)}$ .  $g_{v}(x) = v^{-d}g(x / v)$ . We use result of lemma 1 to prove the statement. We have, for a function  $f\in W_2^\beta (\mathbb{R}^d)$
$$
\begin{array}{l} \left\| \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} f - \frac {\partial^ {2}}{\partial x _ {i} x _ {j}} (f \star g _ {v}) \right\| _ {L _ {2} (\mathbb {R} ^ {d})} ^ {2} = \left\| \mathcal {F} \left[ \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} f \right] - \mathcal {F} \left[ \frac {\partial^ {2}}{\partial x _ {i} \partial x _ {j}} (f \star g _ {v}) \right] \right\| _ {L _ {2} (\mathbb {R} ^ {d})} ^ {2} \\ = \left\| 4 \pi^ {2} \omega_ {i} \omega_ {j} \mathcal {F} [ f ] - 4 \pi^ {2} \omega_ {i} \omega_ {j} \mathcal {F} [ f \star g _ {v} ] \right\| _ {L _ {2} (\mathbb {R} ^ {d})} \\ = \left\| 4 \pi^ {2} \omega_ {i} \omega_ {j} \mathcal {F} [ f ] (1 - \mathcal {F} [ g _ {v} ]) \right\| _ {L _ {2} (\mathbb {R} ^ {d})} ^ {2} \\ \end{array}
$$