古代エジプト王の礼拝堂、カイロの遺跡で発見

古代エジプト王の礼拝堂、カイロの遺跡で発見

ＡＦＰ＝時事 4月15日(水)10時22分配信

古代エジプト王の礼拝堂、カイロの遺跡で発見

エジプトの首都カイロにあるヘリオポリス神殿跡で見つかった、古代エジプト第30王朝のファラオ（王）、ネクタネボ1世が使用していた礼拝堂の下部に使われていた彫刻入りの石材（2015年4月14日提供）。【翻訳編集】 AFPBB News

【AFP＝時事】エジプトの首都カイロ（Cairo）にある古代神殿遺跡で、約2300年前のファラオ（王）が使用していた礼拝堂の一部が見つかった。同国考古省が14日、発表した。

NASA衛星画像から古代エジプトのピラミッド17基見つかる

遺跡は、古代エジプト王朝の首都で、現在はカイロの労働者・中産階級の住宅が立ち並ぶヘリオポリス（Heliopolis）にある。礼拝堂を建立したのは、エジプト第30王朝のファラオ、ネクタネボ1世（Nectanebo I）（在位紀元前380～363年）だ。

考古省の声明によると、より大きな神殿内にあるこの礼拝堂は彫刻が施された玄武岩で造られており、メルエンプタハ（Merneptah）王のカルトゥーシュ（王記）が刻まれた彫像の一部も見つかった。彫像は、第19王朝ラムセス2世（Ramses II）の王子であるメルエンプタハ王が神にささげ物をする姿を表現したものだ。

声明によると、同遺跡を発掘しているエジプトとドイツの共同チームは、今回の発掘シーズン中に、日干しレンガ造りの神殿の残りの部分の発掘を終えたい考えだ。この地域では、12世紀にイスラム地区カイロ（Islamic Cairo、旧市街）の建造のため古代遺跡の石材が多数流用されたため、こうした発見は極めて珍しい。【翻訳編集】 AFPBB Newshttp://headlines.yahoo.co.jp/hl?a=20150415-00000013-jij_afp-sctch

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\begin{document}

\title{\bf Announcement 212: What are reproducing kernels?}

\author{{\it Institute of Reproducing Kernels}\\

Kiryu 376-0041, Japan\\

\date{}

\maketitle

{\bf Abstract: } In this announcement, we shall state simply a general meaning for reproducing kernels. We would like to answer the general and essential question that: what are reproducing kernels?

\bigskip

\section{ Introduction}

\medskipBased on \cite{ss,ss2}, we would like to introduce the concept of reproducing kernels and at the same time, we would like to answer the general and essential question that: what are reproducing kernels?

\bigskip

\section{ What is a reproducing kernel}

\medskip

We shall consider a family of {\bf any complex valued functions} $\{U_n(x)\}_{n=0}^\infty$ defined on an abstract set $E$ that are linearly independent. Then, we consider the form:

\begin{equation}

K_N(x,y) =\sum_{n=0}^N U_n(x) \overline{U_n(y)}.

\end{equation}

Then, $K_N(x,y)$ is a {\bf reproducing kernel} in the following sense:

We shall consider the family of all the functions, for arbitrary complex numbers $\{C_n\}_{n=0}^N$

\begin{equation}

F(x) =\sum_{n=0}^N C_nU_n(x)

\end{equation}

and we introduce the norm

\begin{equation}

\Vert F \Vert^2=\sum_{n=0}^N |C_n|^2.

\end{equation}

The function space forms a Hilbert space $H_{K_N}(E)$ determined by the kernel $K_N(x,y)$ with the inner product induced from the norm (2.3), as usual. Then, we note that, for any $y \in E$

\begin{equation}

K_N(\cdot,y) \in H_{K_N}(E)

\end{equation}

and for any $ F \in H_{K_N}(E)$ and for any $y \in E$

\begin{equation}

F (y) =( F(\cdot), K_N(\cdot,y) )_{ H_{K_N}(E)} = \sum_{n=0}^N C_n U_n(y) .

\end{equation}

The properties (2.4) and (2.5) are called a {\bf reproducing property} of the kernel $K_N(x,y)$ for the Hilbert space $H_{K_N}(E)$, because the functions $F$ in the inner product (2.5) are appeared in the left hand　side. This formula may be considered that the functions $F$ may be represented by the kernel $K_N(x,y)$ and the Hilbert space $H_{K_N}(E)$ is represented by the kernel $K_N(x,y)$.

\bigskip

\section{ A general reproducing kernel}

\medskip

We wish to introduce a preHilbert space by

\[H_{K_\infty}:=

\bigcup_{N \geqq 0}H_{K_N}(E).\]

For any $ F\in H_{K_\infty}$, there exists a space $H_{K_M}(E)$ containing the function $F$ for some $M \geqq 0$. Then, for any

$N$ such that $ M< N$,

$$

H_{K_M}(E) \subset H_{K_N}(E)

$$

and, for the function $ F \in H_{K_M}$,

$$

\Vert F\Vert_{H_{K_M}(E) } = \Vert F\Vert_{H_{K_N}(E)}.

$$

Therefore, there exists the limit:

\[\|F\|_{H_{K_\infty}}:=

\lim_{N \to \infty}\|F\|_{H_{K_N}(E)}.\]

Denote by $H_\infty$ the completion of $H_{K_\infty}$ with respect to this norm.

Note that for any

$ M < N$, and for any $F_M \in H_{K_M}(E)$, $F_M \in H_{K_N}(E)$ and furthermore,

in particular, that

\[\langle f,g \rangle_{H_{K_M(E)}}=

\langle f,g \rangle_{H_{K_N(E)}}\]

for all $N>M $ and $f,g \in H_{K_M}(E)$.

\bigskip

{\bf Theorem} 　　Under the above conditions,

for any function $F \in H_\infty$ and for $F_N^*$

defined

by

\[

F_N^*(x)=\langle F,K_N(\cdot,x) \rangle_{ H_\infty},

\]

$F_N^* \in H_{K_N}(E)$ for all $N>0$,

and

as $N \to \infty$,

$F_N^* \to F$

in the topology of $H_\infty$.

\medskip

{\bf Proof.}

Just observe that

$$

|F_N^*(x)|^2 \le \Vert F\Vert_{H_\infty}^2 \Vert K_N(\cdot,x) \Vert_{H_\infty}^2

$$

$$

\le \Vert F \Vert_{H_\infty}^2 \Vert K_N(\cdot,x) \Vert_{H_{K_N}(E)}^2

$$

$$

= \Vert F \Vert_{H_\infty}^2 K_N(x,x).

$$

Therefore, we see that

$F_N^* \in H_{K_N}(E)$

and that

$\|F_N^*\|_{H_{K_N}(E)} \le \|F\|_{H_\infty}$.

The mapping

$F \mapsto F^*_N$

being uniformly bounded, and so,

we can assume that

$F \in H_{K_L}(E)$ for any fixed $L $.

However,

in this case,

the result is clear, since, $F \in H_{K_N}(E)$ for $ L< N$

$$

\lim_{N \to \infty}　F_N^*(x) = \lim_{N \to \infty} \langle F,K_N(\cdot,x) \rangle_{ H_\infty}= \lim_{N \to \infty}　\langle F,K_N(\cdot,x) \rangle_{H_{K_N}(E) } =F(x).

$$

\medskip

The Theorem may be looked as a reproducing kernel in the natural topology and by the sense of the Theorem, the reproducing property may be written as follows:

\[F(x)=\langle F,K_\infty(\cdot,x) \rangle_{ H_\infty},\]

with

\begin{equation}

K_\infty(\cdot,x) \equiv \lim_{N \to \infty}K_N(\cdot,x) = \sum_{n=0}^\infty U_n(\cdot) \overline{U_n(x)}.

\end{equation}

Here {\bf the limit does, in general, not need to exist}, however, the series are non-decreasing, in the sense: for any $N>M$, $K_N(y,x) - K_M(y,x)$ is a poisitive definite quadratic form function.

\bigskip

\section{Conclusion}

Any reproducing kernel (separable case) may be considered as the form (3.1) by arbitrary linear independent functions $\{U_n(x)\}$ on an abstract set $E$, here, the sum does not need to converge. Furthermore, the property of linear independent is not essential.

Recall the {\bf double helix structure of gene} for the form (3.1).

The completion $H_\infty$ may be found, in concrete cases, from the realization of the spaces

$H_{K_N}(E)$.

The typical case is that the family $\{U_n(x)\}_{n=0}^\infty$ is a complete orthonormal system in a Hilbert space with the norm

\begin{equation}

\Vert F　\Vert^2 = \int _E |F(x)|^2 dm(x)

\end{equation}

with a $dm$ measurable set $E$ in the usual form $L_2(E,dm)$. Then, the functions (2.2) and the norm (2.3) are realized by this norm and the completion of the space $H_{K_\infty}(E)$ is given by this Hilbert space with the norm (4.1).

The complete version of the contents, see \cite{ss} and the fundamental application to initial value problems using eigenfunctions and reproducing kernels, see \cite{ss2}.

\bigskip

\section{Remarks}

The common fundamental definitions and results on reproducing kernels are given as follows:

\medskip

{\bf Definition:}

Let $E$ be an arbitrary abstract (non-void) set.

Denote by ${\mathcal F}(E)$ \index{${\mathcal F}(E)$}

the set of all complex-valued functions on $E$.

A reproducing kernel Hilbert spaces \index{reproducing kernel Hilbert space}

on the set $E$

is a Hilbert space ${\mathcal H} \subset {\mathcal F}(E)$

coming with a function $K:E \times E \to {\mathcal H}$,

which is called the reproducing kernel, \index{reproducing kernel}

having {\bf the reproducing property} that \index{reproducing property}

\begin{equation}\label{eq:110213-14011}

K_p\equiv K(\cdot,p) \in {\mathcal H}\mbox{ for all }p \in E

\end{equation}

and that

\begin{equation}\label{eq:110213-140}

f(p)=\langle f,K_p \rangle_{\mathcal H}

\end{equation}

holds for all $p \in E$ and all $f \in {\mathcal H}$.

\medskip

{\bf Definition:}

A complex-valued function $k:E \times E \to {\mathbb C}$

is called a

{\bf positive definite quadratic form function}

\index{positive definite quadratic form function}

on the set $E$,

or shortly,

{\bf positive definite function},

\index{positive definite function}

when it satisfies the property that,

for an arbitrary function $X:E \to {\mathbb C}$ and for any finite

subset $F$ of $E$,

\begin{equation}\label{eq:101124-26100}

\sum_{p,q \in F} \overline{X(p)} X(q) k(p,q) \geq 0.

\end{equation}

\medskip

Then, the fundamental result is given by: {\bf a reproducing kernel and a positive definite quadratic form function are the same and are one to one correspondence} with the reproducing kernel Hilbert space.

\bibliographystyle{plain}

\begin{thebibliography}{10}

\bibitem{ss}

Saburou Saitoh and Yoshihiro Sawano,

Generalized delta functions as generalized reproducing kernels.

\bibitem{ss2}

Saburou Saitoh and Yoshihiro Sawano,

General initial value problems using eigenfunctions and reproducing kernels.

\bigskip

(S. Saitoh + Y. Sawano, at the Insititute of Reproducing Kernels, 2015.2.25)

\end{thebibliography}

\end{document}