Monthly Archives: August 2017

Notes 2. generalized de Rham theorem

다음의 정리는 미적분학 시간에 배운 정리다.

Proposition. For a smooth vector field F:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}, the following hold:

  1. If \nabla\cdot F=0, there exists G:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3} such that \nabla\times G=F.
  2. If \nabla\times F=0, then there exists a scalar field f:\mathbb{R}^{3}\rightarrow\mathbb{R} such that \nabla f=F.

위의 정리는 어떤 벡터장이 conservative field임을 알려주는 조건이 된다. 이와 관련된 기하학적인 방향으로의 일반화는 다음과 같은 정리가 있다.

Theorem(de Rham). For a smooth manifold M,

    \[ H_{\mathrm{dR}}^{k}\left(M\right)\approx H^{k}\left(M;\mathbb{R}\right). \]

de Rham theorem의 한 결과가 처음에 언급한 정리의 proposition이며, 이 proposition의 한 일반화가 cohomology의 언어로 기술된 de Rham theorem이다. 보통 위와 같은 내용은 미분기하 수업을 들었을 때 많이들 들었을 것이다.

이 절의 관심사는 첫 proposition의 다른 방향의 일반화다. 즉, 더 약한 조건 하에서 conservative vector field임을 판정할 수 있는지를 알아보고자 한다. 잠시 기호를 하나 도입하면, C_{0,\sigma}^{\infty}\left(\Omega\right)을 space of smooth divergence free test function on \Omega이라 하자. 즉,

    \[ C_{0,\sigma}^{\infty}\left(\Omega\right)=\left\{ u\in C_{0}^{\infty}\left(\Omega\right)^{n}:\mathrm{div}u=0\text{ in }\Omega\right\} \]

이다.

만약에 \Omega\mathbb{R}^{3}의 한 simply connected domain이라 하고 uC^{1}-vector field이 \int_{\Omega}u\cdot wdx=0 for all w\in C_{0,\sigma}^{\infty}\left(\Omega\right)라 하면, u=\nabla p를 만족하는 p를 찾을 수 있다. 조건 \int_{\Omega}u\cdot wdx=0에 임의의 h\in C_{0}^{\infty}\left(\Omega\right)에 대하여 w=\nabla\times h
하고

    \[ \nabla\cdot\left(v_{1}\times v_{2}\right)=v_{2}\cdot\nabla\times v_{1}-v_{1}\cdot\nabla\times v_{2} \]

임을 염두하면,

    \[ \int_{\Omega}\nabla\times u\cdot hdx=0\quad\text{for all }h\in C_{0}^{\infty}\left(\Omega\right) \]

을 얻는다. 이로부터 \nabla\times u=0이다. 즉 위에서 언급된 conservative vector field의 조건이다. 이제 \Omega가 simply connected이므로 Stokes’ theorem에 의하여 u=\nabla p
얻는다.

그러나 위와 같은 조건은 Navier-Stokes equation의 수학적인 이론을 전개하기에는 기술적인 문제를 해결해주지는 못한다. 이 글에서는 보다 일반적인 경우에 대하여 de Rham theorem을 증명하도록 한다. 지금부터는 \Omega\subset\mathbb{R}^{n}을 bounded Lipschitz domain이라 하자. 다음과 같은 정리가 잘 알려져있다:

Theorem. Let \Omega be a bounded Lipschitz domain in \mathbb{R}^{n} and \zeta\in C_{0}^{\infty}\left(\Omega\right) a fixed function with \int_{\Omega}\zeta dx=1. Then there exists a linear operator

    \[ \mathcal{B}_{\Omega}:C_{0}^{\infty}\left(\Omega\right)\rightarrow C_{0}^{\infty}\left(\Omega\right)^{n} \]

such that for each g\in C_{0}^{\infty}\left(\Omega\right), the vector field u=\mathcal{B}_{\Omega}\left[g\right] satisfies

    \[ \mathrm{div}u=g-\left(\int_{\Omega}gdx\right)\zeta\quad\text{in }\Omega \]

and

    \[ \left\Vert u\right\Vert _{m+1,q;\Omega}\le C\left(m,q,n,\Omega\right)\left\Vert g\right\Vert _{m,q;\Omega} \]

for every m\in\mathbb{N}\cup\left\{ 0\right\} and 1<q<\infty. Moreover, by continuity, \mathcal{B}_{\Omega} can be extended uniquely to a bounded linear operator from W_{0}^{m,q}\left(\Omega\right) into W_{0}^{m+1,q}\left(\Omega\right)^{n}, called the Bogovskii operator and denoted again by \mathcal{B}_{\Omega}.

위 정리의 증명은 Notes 1 을 참고하기 바란다.

이 정리의 응용으로 다음과 같은 형태의 de Rham theorem을 증명하고자 한다. 우선 정리를 쓰기 전에 몇가지 기호를 도입한다. m\ge1이고 1<q<\infty일 때, W^{-m,q}\left(\Omega\right)W_{0}^{m,q^{\prime}}\left(\Omega\right)의 dual 공간이다. 여기서 q'=\frac{q}{q-1} 이다. 그리고 이 공간에 부여되는 norm은

    \[ \left\Vert f\right\Vert _{-m,q;\Omega}=\sup\left\{ \left\langle f,\phi\right\rangle :\phi\in W_{0}^{m,q'}\left(\Omega\right),\left\Vert \phi\right\Vert _{m,q';\Omega}\le1\right\} \]

이다.

이제 정리를 쓰면 다음과 같다.

Theorem. Let \Omega be a bounded Lipschitz domain in \mathbb{R}^{n} and let m\in\mathbb{Z} and 1<q<\infty. If f\in W^{m,q}\left(\Omega\right)^{n} satisfies

    \[ \left\langle f,\Phi\right\rangle =0\quad\text{for all }\Phi\in C_{0,\sigma}^{\infty}\left(\Omega\right), \]

then there exists \psi\in W^{m+1,q}\left(\Omega\right) such that

    \[ f=\nabla\psi\quad\text{in }\Omega \]

and

    \[ \left\Vert \psi\right\Vert _{m+1,q;\Omega}\le C\left\Vert f\right\Vert _{m,q;\Omega} \]

for some constants C=C\left(m,q,\Omega\right).

우선 m<-1이라 하자. 그러면 fW_{0}^{-m,q'}\left(\Omega\right)^{n}위에서의 한 bounded linear functional이다. 그런데 Bogovskii operator는 W_{0}^{-m-1,q'}\left(\Omega\right)^{n}에서 W_{0}^{-m,q'}\left(\Omega\right)^{n} 으로 보내주는 bounded linear operator이므로

    \begin{align*} \left\langle f,\mathcal{B}\left[g\right]\right\rangle & \le\left\Vert f\right\Vert _{m,q;\Omega}\left\Vert \mathcal{B}\left[g\right]\right\Vert _{-m,q';\Omega}\\ & \le C\left\Vert f\right\Vert _{m,q;\Omega}\left\Vert g\right\Vert _{-m-1,q';\Omega} \end{align*}

이 성립한다. 여기서 g\in W_{0}^{-m-1,q'}\left(\Omega\right)이다.

이제 \psi

    \[ \left\langle \psi,g\right\rangle =-\left\langle f,\mathcal{B}\left[g\right]\right\rangle \quad\text{for all }g\in W_{0}^{-m-1,q'}\left(\Omega\right) \]

가 되도록 정의하면, \psi는 bounded linear operator on W^{-m-1,q'}_0(\Omega)이므로(m=-1일 때는 L^{q'} (\Omega)위에서) m<-1일 때는 \psi\in W^{m+1,q}\left(\Omega\right)이 되며 \left\Vert \psi\right\Vert _{m+1,q;\Omega}\le C\left\Vert f\right\Vert _{m,q;\Omega}을 얻는다. m=-1일 경우에는 Riesz representation theorem에 의하여 \psi\in L^{q}\left(\Omega\right)를 얻으며 같은 estimate를 얻는다.

이제 \Phi\in C_{0}^{\infty}\left(\Omega\right)^{n}이라 하고, g=\mathrm{div}\Phi라 하자. 그러면 divergence theorem에 의하여

    \[ \mathrm{div}\mathcal{B}\left[g\right]=\mathrm{div}\Phi-\left(\int_{\Omega}\mathrm{div}\Phi dx\right)\zeta=\mathrm{div}\Phi \]

를 얻고, linearlity에 의하여 \mathcal{B}\left[g\right]-\Phi\in C_{0,\sigma}^{\infty}\left(\Omega\right)를 얻는다. 따라서

    \[ -\left\langle \psi,\mathrm{div}\Phi\right\rangle =-\left\langle \psi,g\right\rangle =\left\langle f,\mathcal{B}\left[g\right]\right\rangle =\left\langle f,\Phi\right\rangle \]

을 얻는다. 이로부터 원하는 정리가 m\le-1일 때 증명된다. m\ge0일 경우에는 함수라서 횔더 부등식을 사용해서 앞에서 얻었던 유사한 부등식을 얻으면 된다. \Box

사실 \mathbb{R}^{n}은 sequence of bounded Lipschitz domain들의 union으로 표현할 수 있으므로, 위의 정리로부터 다음의 정리를 자명하게 얻는다:

Theorem. Let \Omega be an arbitrary domain in \mathbb{R}^{n} and let 1<q<\infty. If f\in L_{\mathrm{loc}}^{q}\left(\Omega\right) satisfies

    \[ \int_{\Omega}f\cdot\Phi dx=0\quad\text{for all }\Phi\in C_{0,\sigma}^{\infty}\left(\Omega\right), \]

then there exists \psi\in W_{\mathrm{loc}}^{1,q}\left(\Omega\right) such that

    \[ f=\nabla\psi\quad\text{in }\Omega. \]

이 정리는 다양한 곳에서 응용이 있다. 대표적인 응용으로는 벡터필드를 divergence free part와 curl free part로 쪼갤 수 있다는 Helmholtz-Weyl decomposition을 얻을 수 있으며, 이로부터 Stokes equation의 해의 유일성을 얻을 수 있다.

Weierstrass approximation theorem

In this note, we prove the Weierstrass approximation theorem.

1. Bernstein’s approximation

Theorem. Let f:\left[0,1\right]\rightarrow\mathbb{R} be continuous and let \varepsilon>0 be given. Then there exists a polynomial p\left(x\right) such that \norm{p-f}{\infty}<\varepsilon.

We can construct a polynomial p explicitly. For each n, we define

    \[ p_{n}\left(x\right)=\sum_{k=0}^{n}\binom{n}{k}f\left(\frac{k}{n}\right)x^{k}\left(1-x\right)^{n-k}. \]

We call this polynomial as Bernstein polynomials. The polynomial can be interpreted in terms of probabilistic idea.

For 0\le x\le1, we interpret x as a probability of getting a head of the coin(success) and 1-x as a probability of getting a tail of the coin(fail). In n tosses, the probability of k-success is

    \[ \binom{n}{k}x^{k}\left(1-x\right)^{n-k}. \]

In the polynomial, f\left(\frac{k}{n}\right) has the meaning `f\left(\frac{k}{n}\right) dollars is paid out when exactly k heads turn up when n tosses are made). So the average amount paid out when n tosses are made is

    \[ \sum_{k=0}^{n}\binom{n}{k}f\left(\frac{k}{n}\right)x^{k}\left(1-x\right)^{n-k}. \]

As n\rightarrow\infty, then in a typical game, \frac{k}{n}\rightarrow x. So the average payout converges to f\left(x\right).

This is an intuitive idea, not a rigorous proof. Since most of people don’t know the theory of probability, we give a proof which does not contains any probabilistic interpretation.

Recall

    \[ \left(x+y\right)^{n}=\sum_{k=0}^{n}\binom{n}{k}x^{k}y^{n-k}. \]

Differentiating the identity with respect to x. Then

    \[ nx\left(x+y\right)^{n-1}=\sum_{k=0}^{n}k\binom{n}{k}x^{k}y^{n-k}. \]

Again differentiating the identity with respect to x. Then

    \[ n\left(n-1\right)x^{2}\left(x+y\right)^{n-2}=\sum_{k=0}^{n}k\left(k-1\right)\binom{n}{k}x^{k}y^{n-k}. \]

Let r_{k}\left(x\right)=\binom{n}{k}x^{k}\left(1-x\right)^{n-k}.
Put y=1-x. Then

    \[ \sum_{k=0}^{n}r_{k}\left(x\right)=1,\quad\sum_{k=0}^{n}kr_{k}\left(x\right)=nx,\quad\sum_{k=0}^{n}k\left(k-1\right)r_{k}\left(x\right)=n\left(n-1\right)x^{2}. \]

Hence by computation,

(1)   \begin{equation*} \sum_{k=0}^{n}\left(k-nx\right)^{2}r_{k}\left(x\right)=nx\left(1-x\right). \end{equation*}

Choose M so that \left|f\left(x\right)\right|\le M on \left[0,1\right]. Since f is uniformly continuous, for given \varepsilon, there exists \delta>0 such that \left|x-y\right|<\delta implies \left|f\left(x\right)-f\left(y\right)\right|<\varepsilon.
Now

    \begin{align*} \left|f\left(x\right)-p_{n}\left(x\right)\right| & =\left|f\left(x\right)-\sum_{k=0}^{n}f\left(\frac{k}{n}\right)r_{k}\left(x\right)\right|\\ & \le\sum_{k=0}^{n}\left|f\left(x\right)-f\left(\frac{k}{n}\right)\right|r_{k}\left(x\right). \end{align*}

We divide \left\{ 0,\dots,n\right\} into two parts: (i) \left|k-nx\right|<\delta n (ii) \left|k-nx\right|\ge\delta n.

If \left|k-nx\right|<\delta n, then \left|f\left(x\right)-f\left(\frac{k}{n}\right)\right|<\varepsilon. So in this case, the summation is bounded by \varepsilon since r_{k}\left(x\right)\ge0 and \sum r_{k}=1. For the second type, the corresponding summation is bounded by

    \[ 2M\sum_{k:\left|k-nx\right|\ge\delta n}r_{k}\left(x\right)\le\frac{2M}{n^{2}\delta^{2}}\sum_{k=0}^{n}\left(k-nx\right)^{2}r_{k}\left(x\right) \]

since \left|\left(k-nx\right)/n\delta\right|\ge1. By (1), this is bounded by

    \[ \frac{2M}{n\delta^{2}}\frac{1}{4}\le\frac{M}{2\delta^{2}n} \]

since x\left(1-x\right)\le\frac{1}{4} on \left[0,1\right]. Thus, for every \varepsilon>0, there is a \delta>0 such that

    \[ \norm{f-p_{n}}{\infty}\le\varepsilon+\frac{M}{2\delta^{2}n}. \]

Choose n so that \frac{M}{2\delta^{2}n}<\varepsilon. Then for any k\ge n,

    \[ \norm{f-p_{k}}{\infty}<2\varepsilon. \]

This completes the proof.

2. Landau’s approximation

There is another proof using special `approximation to the identity’. The definition of approximation to the identity is the following:

Definition. We say a family \left\{ K_{n}\right\} _{n=1}^{\infty} is said to be an approximation to the identity if

  • For all n\ge1, \int_{-\infty}^{\infty}K_{n}\left(x\right)dx=1
  • There exists M>0 such that for all n\ge1,

        \[ \int_{-\infty}^{\infty}\left|K_{n}\left(x\right)\right|dx\le M. \]

  • For every \delta>0,

        \[ \int_{\left|x\right|\ge\delta}\left|K_{n}\left(x\right)\right|dx\rightarrow0 \]

    as n\rightarrow\infty.

Roughly speaking, this concept concentrate and localize a function. One motivation of this concept is a dirac delta function. We leave the following theorem as an exercise.

Theorem. Let \left\{ K_{n}\right\} _{n=1}^{\infty} be a family of approximation to the identity. If f is continuous everywhere, then

    \[ \lim_{n\rightarrow\infty}\int_{-\infty}^{\infty}K_{n}\left(x-y\right)f\left(y\right)dy \]

converges uniformly to f\left(x\right).

Hence the above theorem gives an uniform approximation to continuous function. From now on, we denote \left(f*g\right)\left(x\right)=\int_{-\infty}^{\infty}f\left(x-y\right)g\left(y\right)dy.

Define the Landau kernels by

    \[ L_{n}\left(x\right)=\begin{cases} \frac{\left(1-x^{2}\right)^{n}}{c_{n}} & \text{if }-1\le x\le1\\ 0 & \text{if }\left|x\right|\ge1, \end{cases} \]

where c_{n} is chosen so that \int_{-\infty}^{\infty}L_{n}\left(x\right)dx=1.

Then one can check \left\{ L_{n}\right\} is a family of approximation to the identity.

Now we are ready to prove the Weierstrass approximation theorem again. By considering the translation, it suffices to consider a continuous function f supported in \left[-\frac{1}{2},\frac{1}{2}\right].

Since \left\{ L_{n}\right\} is a family of approximation to the identity, \left(L_{n}*f\right)\rightarrow f uniformly. Note that \left(L_{n}*f\right) is a sequence of polynomials on \left[-\frac{1}{2},\frac{1}{2}\right]. This completes the proof of Weierstrass approximation theorem.

Some remarks

First of all, with some a slight modification, Bernstein polynomial can be generalized to multivariable setting. Moreover, the following theorem holds.

Theorem. Let u\left(x\right) be a real-valued function defined on \mathbb{R}^{n} with continuous derivatives u_{x_{i}}, u_{x_{i}x_{j}} on \mathbb{R}^{n} for all i,j=1,\dots,n. Then there exists a sequence of polynomials u^{k}\left(x\right) in x_{1},\dots,x_{n} such that u^{k},u_{x_{i}}^{k},u_{x_{i}x_{j}}^{k} converges to u,u_{x_{i}},u_{x_{i}x_{j}}, respectively, for all i,j=1,\dots,n, uniformly on any compact set.

The proof of this theorem is quite technical. So we omit the proof. One can see the proof e.g. Krylov’s introduction to the theory of diffusion. But this book is not appropriate to those who are taking undergraduate analysis course. Here the proof uses some theory of probability theory, the weak law of large numbers. One may ask why such theorem is needed. This theorem is a quite good lemma for proving some fundamental theorem in the theory of stochastic integrals.

Another way to generalize the theorem in the following sense. Let \mathcal{P} denotes the space of polynomials. The Weierstrass theorem  gives \mathcal{P} is dense in C\left(\left[a,b\right]\right)
with the uniform norm. Actually, the class \mathcal{P} can be regarded as a special case. This was proved by Stone.

A family of \mathscr{A} of real functions defined on a set E is said to be an algebra if (i) f+g\in\mathscr{A}, (ii) fg\in\mathscr{A} and (iii) cf\in\mathscr{A} for all f,g\in\mathscr{A} and for all real constants c.

If \mathscr{A} has the property that f\in\mathscr{A} whenever f_{n}\in\mathscr{A} and f_{n}\rightarrow f uniformly on E, we say \mathscr{A} uniformly closed.

We say \mathscr{A} separate points on E if to every pair of distinct points x_{1},x_{2}\in E, there exists a function f\in\mathscr{A} such that f\left(x_{1}\right)\neq f\left(x_{2}\right).

Note that the class \mathcal{P} clearly satisfies separate points on \mathbb{R}.

Theorem (Stone). Let \mathscr{A} be an algebra of real continuous function on a compact set K. If \mathscr{A} separates points on K and if \mathscr{A} vanishes at no point of K, then the set of all functions which is the limits of uniformly convergent sequence of members of \mathscr{A} consists of all real continuous functions
on K.

In this sense, the theorem holds not only in \mathbb{R} but also \mathbb{R}^{n}. Moreover, the theorem does not depend on Euclidean structure. Those who are interested in its proof, see Rudin’s PMA.