I recently learned about the theory of distributions. I'm wondering if it's possible to approximate some compactly supported continuous function $f$ on $\mathbb{R^n}$ using the multivariate Taylor expansion and distributional derivatives in the following fashion: $$P_j= \sum_{\vert\alpha\vert\leqslant j}\frac{1}{j!}D^\alpha\Lambda_{f(\varphi_j*\chi_{\text{supp}(f)})}(\varphi_j)\psi_\alpha(x)$$ where $$\Lambda_g:\mathscr{D}(\mathbb{R}^n)\to\mathbb{C}:\varphi\mapsto\int g\varphi,$$ $$\psi_\alpha(x)=\prod_{i\in\{\alpha_k\}}(x_i-c_i),$$ $c\in\text{Int}(\text{supp}(f))$, and $(\varphi_j)$ is an approximation to the identity around $c$. Does $P_j\to f$ uniformly on $\text{supp}(f)$? Any help is appreciated. Thank you!
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- $\begingroup$ What is your $\varphi_j$ stands for. $\endgroup$Liding Yao– Liding Yao2025-11-06 19:02:11 +00:00Commented Nov 6 at 19:02
- $\begingroup$ @LidingYao $\varphi_j$ is a sequence of smooth, compactly supported functions such that $\int_{\mathbb{R}^n}\varphi_jdx=1$ for each $j$ and whose support lies in $\overline{B(c,\frac{1}{j})}$ for each $j$. $\endgroup$Ray– Ray2025-11-06 19:33:20 +00:00Commented Nov 6 at 19:33
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