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数学代写|偏微分方程代考Partial Differential Equations代写|The Weierstraß approximation theorem

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数学代写|偏微分方程代考Partial Differential Equations代写|The Weierstraß approximation theorem

Let $\Omega \subset \mathbb{R}^n$ with $n \in \mathbb{N}$ denote an open set and $f(x) \in C^k(\Omega)$ with $k \in$ $\mathbb{N} \cup{0}=: \mathbb{N}0$ a $k$-times continuously differentiable function. We intend to prove the following statement: There exists a sequence of polynomials $p_m(x), x \in \mathbb{R}^n$ for $m=1,2, \ldots$ which converges on each compact subset $C \subset \Omega$ uniformly towards the function $f(x)$. Furthermore, all partial derivatives up to the order $k$ of the polynomials $p_m$ converge uniformly on $C$ towards the corresponding derivatives of the function $f$. The coefficients of the polynomials $p_m$ depend on the approximation, in general. If this were not the case, the function $$f(x)=\left{\begin{array}{cl} \exp \left(-\frac{1}{x^2}\right), & x>0 \ 0, & x \leq 0 \end{array}\right.$$ could be expanded into a power series. However, this leads to the evident contradiction: $$0 \equiv \sum{k=0}^{\infty} \frac{f^{(k)}(0)}{k !} x^k$$
In the following Proposition, we introduce a ‘mollifier’ which enables us to smooth functions.

Proposition 1. We consider the following function to each $\varepsilon>0$, namely
\begin{aligned} K_{\varepsilon}(z) & :=\frac{1}{\sqrt{\pi \varepsilon}^n} \exp \left(-\frac{|z|^2}{\varepsilon}\right) \ & =\frac{1}{\sqrt{\pi \varepsilon}^n} \exp \left(-\frac{1}{\varepsilon}\left(z_1^2+\ldots+z_n^2\right)\right), \quad z \in \mathbb{R}^n . \end{aligned}
Then this function $K_{\varepsilon}=K_{\varepsilon}(z)$ possesses the following properties:

1. We have $K_{\varepsilon}(z)>0$ for all $z \in \mathbb{R}^n$;
2. The condition $\int_{\mathbb{R}^n} K_{\varepsilon}(z) d z=1$ holds true;
3. For each $\delta>0$ we observe: $\lim {\varepsilon \rightarrow 0+} \int{|z| \geq \delta} K_{\varepsilon}(z) d z=0$.

数学代写|偏微分方程代考Partial Differential Equations代写|Parameter-invariant integrals and differential forms

In the basic lectures of Analysis the following fundamental result is established.
Theorem 1. (Transformation formula for multiple integrals)
Let $\Omega, \Theta \subset \mathbb{R}^n$ denote two open sets, where we take $n \in \mathbb{N}$. Furthermore, let $y=\left(y_1\left(x_1, \ldots, x_n\right), \ldots, y_n\left(x_1, \ldots, x_n\right)\right): \Omega \rightarrow \Theta$ denote a bijective mapping of the class $C^1\left(\Omega, \mathbb{R}^n\right)$ satisfying
$$J_y(x):=\operatorname{det}\left(\frac{\partial y_i(x)}{\partial x_j}\right){i, j=1, \ldots, n} \neq 0 \quad \text { for all } \quad x \in \Omega .$$ Let the function $f=f(y): \Theta \rightarrow \mathbb{R} \in C^0(\Theta)$ be given with the property $$\int{\Theta}|f(y)| d y<+\infty$$
for the improper Riemannian integral of $|f|$. Then we have the transformation formula
$$\int_{\Theta} f(y) d y=\int_{\Omega} f(y(x))\left|J_y(x)\right| d x .$$
In the sequel, we shall integrate differential forms over $m$-dimensional surfaces in $\mathbb{R}^n$.

Definition 1. Let the open set $T \subset \mathbb{R}^m$ with $m \in \mathbb{N}$ constitute the parameter domain. Furthermore, the symbol
$$X(t)=\left(\begin{array}{c} x_1\left(t_1, \ldots, t_m\right) \ \vdots \ x_n\left(t_1, \ldots, t_m\right) \end{array}\right): T \longrightarrow \mathbb{R}^n \in C^k\left(T, \mathbb{R}^n\right)$$
represents a mapping – with $k, n \in \mathbb{N}$ and $m \leq n$ – whose functional matrix
$$\partial X(t)=\left(X_{t_1}(t), \ldots, X_{t_m}(t)\right), \quad t \in T$$
has the rank $m$ for all $t \in T$. Then we call $X$ a parametrized regular surface with the parametric representation $X(t): T \rightarrow \mathbb{R}^n$.
When $X: T \rightarrow \mathbb{R}^n$ and $\widetilde{X}: \widetilde{T} \rightarrow \mathbb{R}^n$ are two parametric representations, we call them equivalent if there exists a topological mapping
$$t=t(s)=\left(t_1\left(s_1, \ldots, s_m\right), \ldots, t_m\left(s_1, \ldots, s_m\right)\right): \widetilde{T} \longrightarrow T \in C^k(\widetilde{T}, T)$$
with the following properties:

1. $J(s):=\frac{\partial\left(t_1, \ldots, t_m\right)}{\partial\left(s_1, \ldots, s_m\right)}(s)=\left|\begin{array}{ccc}\frac{\partial t_1}{\partial s_1}(s) & \ldots & \frac{\partial t_1}{\partial s_m}(s) \ \vdots & \vdots \ \frac{\partial t_m}{\partial s_1}(s) & \ldots & \frac{\partial t_m}{\partial s_m}(s)\end{array}\right|>0 \quad$ for all $s \in \tilde{T}$;
2. $\tilde{X}(s)=X(t(s))$ for all $s \in \widetilde{T}$.

偏微分方程代写

数学代写|偏微分方程代考Partial Differential Equations代写|The Weierstraß approximation theorem

\begin{aligned} K_{\varepsilon}(z) & :=\frac{1}{\sqrt{\pi \varepsilon}^n} \exp \left(-\frac{|z|^2}{\varepsilon}\right) \ & =\frac{1}{\sqrt{\pi \varepsilon}^n} \exp \left(-\frac{1}{\varepsilon}\left(z_1^2+\ldots+z_n^2\right)\right), \quad z \in \mathbb{R}^n . \end{aligned}

数学代写|偏微分方程代考Partial Differential Equations代写|Parameter-invariant integrals and differential forms

$$J_y(x):=\operatorname{det}\left(\frac{\partial y_i(x)}{\partial x_j}\right){i, j=1, \ldots, n} \neq 0 \quad \text { for all } \quad x \in \Omega .$$函数$f=f(y): \Theta \rightarrow \mathbb{R} \in C^0(\Theta)$的属性为$$\int{\Theta}|f(y)| d y<+\infty$$

$$\int_{\Theta} f(y) d y=\int_{\Omega} f(y(x))\left|J_y(x)\right| d x .$$

$$X(t)=\left(\begin{array}{c} x_1\left(t_1, \ldots, t_m\right) \ \vdots \ x_n\left(t_1, \ldots, t_m\right) \end{array}\right): T \longrightarrow \mathbb{R}^n \in C^k\left(T, \mathbb{R}^n\right)$$

$$\partial X(t)=\left(X_{t_1}(t), \ldots, X_{t_m}(t)\right), \quad t \in T$$

$$t=t(s)=\left(t_1\left(s_1, \ldots, s_m\right), \ldots, t_m\left(s_1, \ldots, s_m\right)\right): \widetilde{T} \longrightarrow T \in C^k(\widetilde{T}, T)$$

$J(s):=\frac{\partial\left(t_1, \ldots, t_m\right)}{\partial\left(s_1, \ldots, s_m\right)}(s)=\left|\begin{array}{ccc}\frac{\partial t_1}{\partial s_1}(s) & \ldots & \frac{\partial t_1}{\partial s_m}(s) \ \vdots & \vdots \ \frac{\partial t_m}{\partial s_1}(s) & \ldots & \frac{\partial t_m}{\partial s_m}(s)\end{array}\right|>0 \quad$ 对于所有$s \in \tilde{T}$;

$\tilde{X}(s)=X(t(s))$ 对于所有$s \in \widetilde{T}$。

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