3.5.1 Appendix: Real Analytic Functions

Multi-index notation

The following multi-index notation simplifies many presentations of formulas. Let $x=(x_1,\ldots,x_n)$ and
$$u:\ \Omega\subset\mathbb{R}\mapsto \mathbb{R}^1\ \ (\mbox{or}\ \mathbb{R}^m\ \mbox{for systems}).$$
The n-tuple of non-negative integers (including zero)
$$\alpha=(\alpha_1,\ldots,\alpha_n)$$
is called multi-index. Set
$$\begin{eqnarray*} |\alpha|&=&\alpha_1+\ldots+\alpha_n\\ \alpha!&=&\alpha_1!\alpha_2!\cdot\ldots\cdot\alpha_n!\\ x^\alpha&=&x_1^{\alpha_1}x_2^{\alpha_2}\cdot\ldots\cdot x_n^{\alpha_n}\ \ (\mbox{for a monom})\\ D_k&=&\frac{\partial}{\partial x_k}\\ D&=&(D_1,\ldots,D_n)\\ Du&=&(D_1u,\ldots,D_nu)\equiv\nabla u\equiv \text{grad}\ u\\ D^\alpha&=&D_1^{\alpha_1}D_2^{\alpha_2}\cdot\ldots\cdot D_n^{\alpha_n}\equiv\frac{\partial^{|\alpha|}}{\partial x_1^{\alpha_1}\partial x_2^{\alpha_2}\ldots\partial x_n^{\alpha_n}}. \end{eqnarray*}$$
Define a partial order by
$$\alpha\ge\beta\ \ \mbox{if and only if}\ \ \alpha_i\ge\beta_i\ \ \mbox{for all}\ i.$$
Sometimes we use the notations
$${\bf 0}=(0,0\ldots,0), \ \ \bf{ 1}=(1,1\ldots,1),$$
where
${\bf 0},\ {\bf 1}\in\mathbb{R}.$

Using this multi-index notion, we have

1.
$$(x+y)^\alpha=\sum_{\begin{array}{c}\beta,\gamma\\ \beta+\gamma=\alpha\end{array}}\frac{\alpha!}{\beta!\gamma!}x^\beta y^\gamma,$$
where $x,\ y\in\mathbb{R}$ and $\alpha,\ \beta,\ \gamma$ are multi-indices.

2. Taylor expansion for a polynomial $f(x)$ of degree $m$:
$$f(x)=\sum_{|\alpha|\le m}\frac{1}{\alpha!}\left(D^\alpha f(0)\right) x^\alpha,$$
here is $D^\alpha f(0):=\left(D^\alpha f(x)\right)|_{x=0}$.

3. Let $x=(x_1,\ldots,x_n)$ and $m\ge0$ an integer, then
$$
(x_1+\ldots+x_n)^m=\sum_{|\alpha|=m}\frac{m!}{\alpha!}x^\alpha.
\]

4.
$$
\alpha!\le|\alpha|!\le n^{|\alpha|}\alpha!.
\]

5. Leibniz's rule:
$$
D^\alpha(fg)=
\sum_{ $$\begin{array}{c}\beta,\gamma\\ \beta+\gamma=\alpha\end{array}$$ }\frac{\alpha!}{\beta!\gamma!}(D^\beta f)(D^\gamma g).
\]

6.
$$\begin{eqnarray*} D^\beta x^\alpha&=&\frac{\alpha!}{(\alpha-\beta)!}x^{\alpha-\beta}\ \ \mbox{if}\ \alpha\ge\beta,\\ D^\beta x^\alpha&=&0\ \ \mbox{otherwise}. \end{eqnarray*}$$

7. Directional derivative:
$$\frac{d^m}{dt^m}f(x+ty)=\sum_{|\alpha|=m}\frac{|\alpha|!}{\alpha!}\left(D^\alpha f(x+ty)\right)y^\alpha,$$
where $x,\ y\in\mathbb{R}$ and $t\in\mathbb{R}^1$.

8. Taylor's theorem: Let $u\in C^{m+1}$ in a neighborhood $N(y)$ of $y$, then, if $x\in N(y)$,
$$u(x)=\sum_{|\alpha|\le m}\frac{1}{\alpha!}\left(D^\alpha u(y)\right)(x-y)^\alpha+R_m,$$
where
$$R_m=\sum_{|\alpha|=m+1}\frac{1}{\alpha!}\left(D^\alpha u(y+\delta(x-y))\right)x^\alpha,\ 0<\delta<1,$$
$\delta=\delta(u,m,x,y)$,
or
$$R_m=\frac{1}{m!}\int_0^1\ (1-t)^m\Phi^{(m+1)}(t)\ dt,$$
where $\Phi(t)=u(y+t(x-y))$. It follows from 7. that
$$
R_m=(m+1)\sum_{|\alpha|=m+1}\frac{1}{\alpha!}\left(\int_0^1\ (1-t)D^\alpha u(y+t(x-y))\ dt\right)(x-y)^\alpha.
\]

9. Using multi-index notation, the general linear partial differential equation of order $m$ can be written as
$$
\sum_{|\alpha|\le m}a_\alpha (x)D^\alpha u=f(x)\ \ \mbox{in}\ \Omega\subset\mathbb{R}.
\]

Power series

Here we collect some definitions and results for power series in $\mathbb{R}$.

Definition. Let $c_\alpha\in\mathbb{R}^1\ (\mbox{or}\ \in\mathbb{R}^m)$. The series
$$\sum_\alpha c_\alpha\equiv\sum_{m=0}^\infty\left(\sum_{|\alpha|=m}c_\alpha\right)$$
is said to be convergent if
$$\sum_\alpha |c_\alpha|\equiv\sum_{m=0}^\infty\left(\sum_{|\alpha|=m}|c_\alpha|\right)$$
is convergent.

Remark. According to the above definition, a convergent series is absolutely convergent. It follows that we can rearrange the order of summation.

Using the above multi-index notation and keeping in mind that we can rearrange convergent series, we have

10. Let $x\in\mathbb{R}$, then
$$\begin{eqnarray*} \sum_\alpha x^\alpha&=&\prod_{i=1}^n\left(\sum_{\alpha_i=0}^\infty x_i^{\alpha_i}\right)\\ &=&\frac{1}{(1-x_1)(1-x_2)\cdot\ldots\cdot(1-x_n)}\\ &=&\frac{1}{({\bf 1}-x)^{\bf 1}}, \end{eqnarray*}$$
provided $|x_i|<1$ is satisfied for each $i$.

11. Assume $x\in\mathbb{R}$ and $|x_1|+|x_2|+\ldots+|x_n|<1$, then
$$\begin{eqnarray*} \sum_\alpha\frac{|\alpha|!}{\alpha!}x^\alpha&=&\sum_{j=0}^\infty\sum_{|\alpha|=j}\frac{|\alpha|!}{\alpha!}x^\alpha\\ &=&\sum_{j=0}^\infty(x_1+\ldots+x_n)^j\\ &=&\frac{1}{1-(x_1+\ldots+x_n)}. \end{eqnarray*}$$

12. Let $x\in\mathbb{R}$, $|x_i|<1$ for all $i$, and $\beta$ is a given multi-index. Then
$$\begin{eqnarray*} \sum_{\alpha\ge\beta}\frac{\alpha!}{(\alpha-\beta)!}x^{\alpha-\beta}&=&D^\beta\frac{1}{({\bf 1}-x)^1}\\ &=&\frac{\beta!}{({\bf 1}-x)^{1+\beta}}\ \end{eqnarray*}$$

13. Let $x\in\mathbb{R}$ and $|x_1|+\ldots+|x_n|<1$. Then
$$\begin{eqnarray*} \sum_{\alpha\ge\beta}\frac{|\alpha|!}{(\alpha-\beta)!}x^{\alpha-\beta}&=&D^\beta\frac{1}{1-x_1-\ldots-x_n}\\ &=&\frac{|\beta|!}{(1-x_1-\ldots-x_n)^{1+|\beta|}}\ . \end{eqnarray*}$$

Consider the power series
$$\begin{equation} \label{power}\tag{3.34} \sum_\alpha c_\alpha x^\alpha \end{equation}$$
and assume this series is convergent for a $z\in\mathbb{R}$. Then, by definition,
$$\mu:=\sum_\alpha|c_\alpha||z^\alpha|<\infty$$
and the series ($\ref{power}$) is uniformly convergent for all $x\in Q(z)$, where
$$
Q(z):\ \ |x_i|\le|z_i|\ \ \mbox{for all}\ \ i.
\]

Figure 3.5.1.1: Definition of $D\in Q(z)$

Thus the power series ($\ref{power}$) defines a continuous function defined on $Q(z)$, according to a theorem of Weierstrass.

The interior of $Q(z)$ is not empty if and only if $z_i\not=0$ for all $i$, see Figure 3.5.1.1.
For given $x$ in a fixed compact subset $D$ of $Q(z)$ there is a $q$, $0<q<1$, such that
$$|x_i|\le q|z_i|\ \ \mbox{for all}\ i.$$
Set
$$
f(x)=\sum_\alpha c_\alpha x^\alpha.
\]

Proposition A1. (i) In every compact subset $D$ of $Q(z)$ one has $f\in C^\infty(D)$ and
the formal differentiate series, that is $\sum_\alpha D^\beta c_\alpha x^\alpha$, is uniformly convergent on the closure of $D$ and is equal to $D^\beta f$.}

(ii)
$$|D^\beta f(x)|\le M|\beta|!r^{-|\beta|}\ \ \mbox{in}\ D,$$
where
$$M=\frac{\mu}{(1-q)^n},\qquad \qquad r=(1-q)\min_i|z_i|.$$

Proof. See F. John [10], p. 64. Or an exercise. Hint: Use formula 12. where $x$ is replaced by $(q,\ldots,q)$.

Remark. From the proposition above it follows
$$
c_\alpha=\frac{1}{\alpha!}D^\alpha f(0).
\]

Definition. Assume $f$ is defined on a domain $\Omega\subset\mathbb{R}$, then $f$ is said to be {\it real analytic in $y\in\Omega$} if there are $c_\alpha\in\mathbb{R}^1$ and if there is a neighborhood $N(y)$ of $y$ such that
$$f(x)=\sum_\alpha c_\alpha(x-y)^\alpha$$
for all $x\in N(y)$, and the series converges (absolutely) for each $x\in N(y)$.
A function $f$ is called {\it real analytic in $\Omega$} if it is real analytic for each $y\in\Omega$.
We will write $f\in C^\omega(\Omega)$ in the case that $f$ is real analytic in the domain $\Omega$.
A vector valued function $f(x)=(f_1(x),\ldots,f_m)$ is called real analytic if each coordinate is real analytic.

Proposition A2. (i) Let $f\in C^\omega(\Omega)$. Then $f\in C^\infty(\Omega)$.}

(ii)
Assume $f\in C^\omega(\Omega)$. Then for each $y\in \Omega$ there exists a neighborhood $N(y)$ and positive constants $M$, $r$ such that
$$f(x)=\sum_\alpha\frac{1}{\alpha!}(D^\alpha f(y))(x-y)^\alpha$$
for all $x\in N(y)$, and the series converges (absolutely) for each $x\in N(y)$, and
$$|D^\beta f(x)|\le M|\beta|!r^{-|\beta|}.$$

The proof follows from Proposition A1.

An open set $\Omega\in\mathbb{R}$ is called connected if $\Omega$ is not a union of two nonempty
open sets with empty intersection. An open set $\Omega\in\mathbb{R}$ is connected if and only if its path connected, see [11], pp. 38, for example. We say that $\Omega$ is path connected if for any $x,y\in\Omega$ there is a continuous curve $\gamma(t)\in\Omega$, $0\le t\le1$, with $\gamma(0)=x$ and $\gamma(1)=y$. From the theory of one complex variable we know that a continuation of an analytic function is uniquely determined. The same is true for real analytic functions.

Proposition A3. Assume $f\in C^\omega(\Omega)$ and $\Omega$ is connected. Then
$f$ is uniquely determined if for one $z\in\Omega$ all $D^\alpha f(z)$ are known.

Proof. See F. John [10], p. 65. Suppose $g, h\in C^\omega(\Omega)$ and
$D^\alpha g(z)=D^\alpha h(z)$ for every $\alpha$. Set $f=g-h$ and
$$\begin{eqnarray*} \Omega_1&=&\{x\in\Omega:\ D^\alpha f(x)=0\ \ \mbox{for all}\ \alpha\},\\ \Omega_2&=&\{x\in\Omega:\ D^\alpha f(x)\not=0\ \ \mbox{for at least one}\ \alpha\}. \end{eqnarray*}$$
The set $\Omega_2$ is open since $D^\alpha f$ are continuous in $\Omega$. The set $\Omega_1$ is also open since $f(x)=0$ in a neighbourhood of $y\in\Omega_1$. This follows from
$$f(x)=\sum_\alpha \frac{1}{\alpha!}(D^\alpha f(y))(x-y)^\alpha.$$
Since $z\in\Omega_1$, i. e., $\Omega_1\not=\emptyset$, it follows $\Omega_2=\emptyset$.

$\Box$

It was shown in Proposition A2 that derivatives of a real analytic function satisfy estimates.
On the other hand it follows, see the next proposition, that a function $f\in C^\infty$ is real analytic if these estimates are satisfied.

Definition. Let $y\in\Omega$ and $M,\ r$ positive constants. Then $f$ is said to be in the class $C_{M,r}(y)$ if $f\in C^\infty$ in a neighbourhood of $y$ and if
$$|D^\beta f(y)|\le M|\beta|!r^{-|\beta|}$$
for all $\beta$.

Proposition A4. $f\in C^\omega(\Omega)$ if and only if $f\in C^\infty(\Omega)$ and for every compact subset $S\subset\Omega$ there are positive constants $M,\;r$ such that
$$f\in C_{M,r}(y)\ \ \mbox{for all}\ y\in S.$$

Proof. See F. John [10], pp. 65-66. We will prove the local version of the proposition, that is, we show it for each fixed $y\in\Omega$. The general version follows from Heine-Borel theorem. Because of Proposition A3 it remains to show that the Taylor series
$$\sum_\alpha\frac{1}{\alpha!}D^\alpha f(y)(x-y)^\alpha$$
converges (absolutely) in a neighborhood of $y$ and that this series is equal to $f(x)$.

Define a neighborhood of $y$ by
$$N_d(y)=\{x\in\Omega:\ \ |x_1-y_1|+\ldots+|x_n-y_n|<d\},$$
where $d$ is a sufficiently small positive constant. Set $\Phi(t)=f(y+t(x-y))$. The one-dimensional Taylor theorem says
$$f(x)=\Phi(1)=\sum_{k=0}^{j-1}\frac{1}{k!}\Phi^{(k)}(0)+r_j,$$
where
$$r_j=\frac{1}{(j-1)!}\int_0^1\ (1-t)^{j-1}\Phi^{(j)}(t)\ dt.$$
From formula 7. for directional derivatives it follows for $x\in N_d(y)$ that
$$\frac{1}{j!}\frac{d^j}{dt^j}\Phi(t)=\sum_{|\alpha|=j}\frac{1}{\alpha!}D^\alpha f(y+t(x-y))(x-y)^\alpha.$$
From the assumption and the multinomial formula 3. we get for $0\le t\le 1$
$$\begin{eqnarray*} \left|\frac{1}{j!}\frac{d^j}{dt^j}\Phi(t)\right|&\le&M\sum_{|\alpha|=j}\frac{|\alpha|!}{\alpha!}r^{-|\alpha|}\left|(x-y)^\alpha\right|\\ &=& Mr^{-j}\left(|x_1-y_1|+\ldots +|x_n-y_n|\right)^j\\ &\le&M\left(\frac{d}{r}\right)^j. \end{eqnarray*}$$
Choose $d>0$ such that $d<r$, then the Taylor series converges (absolutely) in $N_d(y)$ and it is equal to $f(x)$ since the remainder satisfies, see the above estimate,
$$
|r_j|=\left|\frac{1}{(j-1)!}\int_0^1\ (1-t)^{j-1}\Phi^j(t)\ dt\right|\le M\left(\frac{d}{r}\right)^j.
\]

$\Box$

We recall that the notation $f<<F$ ($f$ is majorized by $F$) was defined in the previous section.

Proposition A5. (i) $f=(f_1,\ldots,f_m)\in C_{M,r}(0)$ if and only if $f<<(\Phi,\ldots,\Phi)$, where
$$\Phi(x)=\frac{Mr}{r-x_1-\ldots-x_n}\ .$$ }

(ii) \(f\in C_{M,r}(0)\) and $f(0)=0$ if and only if
$$f<<(\Phi-M,\ldots,\Phi-M),$$
where
$$\Phi(x)=\frac{M(x_1+\ldots+x_n)}{r-x_1-\ldots-x_n}\ .$$

Proof.
$$
D^\alpha\Phi(0)=M|\alpha|!r^{-|\alpha|}.
\]

$\Box$

Remark. The definition of $f<<F$ implies, trivially, that $D^\alpha f<<D^\alpha F$.

The next proposition shows that compositions majorize if the involved functions majorize. More precisely, we have

Proposition A6. Let $f,\ F:\ \mathbb{R}\mapsto\mathbb{R}^m$ and $g,\ G$ maps a neighborhood of $0\in\mathbb{R}^m$ into ${\mathbb R}^p$. Assume all functions $f(x),\ F(x),\ g(u),\ G(u)$ are in $C^\infty$, $f(0)=F(0)=0$, $f<<F$ and $g<<G$. Then
$g(f(x))<<G(F(x))$.}

Proof. See F. John [10], p. 68. Set
$$h(x)=g(f(x)),\ \ \ H(x)=G(F(x)).$$
For each coordinate $h_k$ of $h$ we have, according to the chain rule,
$$D^\alpha h_k(0)=P_\alpha(\delta^\beta g_l(0),D^\gamma f_j(0)),$$
where $P_\alpha$ are polynomials with non-negative integers as coefficients, $P_\alpha$
are independent on $g$ or $f$ and $\delta:=(\partial/\partial u_1,\ldots,\partial/\partial u_m)$. Thus,
$$\begin{eqnarray*} |D^\alpha h_k(0)|&\le&P_\alpha(|\delta^\beta g_l(0)|,|D^\gamma f_j(0)|)\\ &\le&P_\alpha(\delta^\beta G_l(0),D^\gamma F_j(0))\\ &=&D^\alpha H_k(0). \end{eqnarray*}$$
$\Box$

Using this result and Proposition A4, which characterizes real analytic functions, it follows that compositions of real analytic functions are real analytic functions again.

Proposition A7. Assume $f(x)$ and $g(u)$ are real analytic, then $g(f(x))$ is real analytic if $f(x)$ is in the domain of definition of $g$.

Proof. See F. John [10], p. 68. Assume that $f$ maps a neighborhood of $y\in\mathbb{R}$ in $\mathbb{R}^m$ and $g$ maps a neighborhood of $v=f(y)$ in ${\mathbb R}^m$. Then $f\in C_{M,r}(y)$ and $g\in C_{\mu,\rho}(v)$ implies
$$h(x):=g(f(x))\in C_{\mu,\rho r/(mM+\rho)}(y).$$
Once one has shown this inclusion, the proposition follows from Proposition~A4. To show the inclusion, we set
$$h(y+x):=g(f(y+x))\equiv g(v+f(y+x)-f(x))=:g^*(f^*(x)),$$
where $v=f(y)$ and
$$\begin{eqnarray*} g^*(u):&=&g(v+u)\in C_{\mu,\rho}(0)\\ f^*(x):&=&f(y+x)-f(y)\in C_{M,r}(0). \end{eqnarray*}$$
In the above formulas $v,\ y$ are considered as fixed parameters. From Proposition~A5 it follows
$$\begin{eqnarray*} f^*(x)&<<&(\Phi-M,\ldots,\Phi-M)=:F\\ g^*(u)&<<&(\Psi,\ldots,\Psi)=:G, \end{eqnarray*}$$
where
$$\begin{eqnarray*} \Phi(x)&=&\frac{Mr}{r-x_1-x_2-\ldots-x_n}\\ \Psi(u)&=&\frac{\mu\rho}{\rho-x_1-x_2-\ldots-x_n}. \end{eqnarray*}$$
From Proposition A6 we get
$$h(y+x)<<(\chi(x),\ldots,\chi(x))\equiv G(F),$$
where
$$\begin{eqnarray*} \chi(x)&=&\frac{\mu\rho}{\rho-m(\Phi(x)-M)}\\ &=&\frac{\mu\rho(r-x_1-\ldots-x_n)}{\rho r-(\rho+mM)(x_1+\ldots+x_n)}\\ &<<&\frac{\mu\rho r}{\rho r-(\rho+mM)(x_1+\ldots+x_n)}\\ &=&\frac{\mu\rho r/(\rho+mM)}{\rho r/(\rho+mM)-(x_1+\ldots x_n)}. \end{eqnarray*}$$
See an exercise for the ''$<<$''-inequality.

$\Box$

Contributors:

• Prof. Dr. Erich Miersemann (Universität Leipzig)

• Integrated by Justin Marshall.