2015年6月30日 奈良新聞
復元された赤田9号墓の大小二つの円筒形陶棺=29日、奈良市大安寺西2丁目の奈良市埋蔵文化財調査センター
円筒形陶棺の出土状況。写真の奥は斜めの状態で見つかった小型で、右側は倒れた大型。手前は同時に出土した小型亀甲形陶棺=奈良市西大寺赤田町(奈良市教育委員会埋蔵文化財センター提供)
奈良市埋蔵文化財調査センターは29日、奈良市西大寺赤田町1丁目の赤田横穴墓群9号墓(7世紀中ごろ)から出土した陶棺(素焼きの棺)が、円筒形だったことが分かったと発表した。低い温度で焼いた軟質の「土師(はじ)質」で、平底のために縦置きしたと推定。同様の陶棺の出土は全国で初めて。白骨化させた遺体を納めた再葬用の棺とみられ、古墳時代末期の埋葬法の変化を知る資料になるという。
平成23年9月に9号墓の墓室(長さ5・7メートル、奥壁幅2・5メートル)から出土した陶棺片を約3年かけて復元。
円筒形陶棺は大小二つあり、大型が口径約30センチ、高さ約93センチ、小型が口径約26センチ、高さ約67センチ。いずれも半球形の蓋(ふた)が付き、本体上部の蓋受け部分には蓋を紐(ひも)で固定するための穴が大型で7カ所、小型で9カ所あった。
底が平らなために自立が可能だった。小型は墓室の西壁沿いで斜めになった状態で出土。大型は奥壁沿いに倒れていたが、いずれも当初は縦置きで埋葬されたと考えられるという。
円筒形陶棺は大阪府と兵庫、滋賀、岐阜の各県で、より高温で焼いた「須恵質」で底が丸い物が出土しているが、土師質で平底の物は出土例がないという。
また、今回見つかった陶棺は、大人の遺体には小さすぎるために再葬用の棺と推定。遺体を木棺などに納めて白骨化させたあと、骨だけ埋葬したと考えられる。
復元を担当した同センターの鐘方正樹主任は「周辺で陶棺を制作していた集団が、陶棺をまねて円筒棺を縦置きで上から蓋をする構造に作り変えたのでは。7世紀は古墳や埋葬施設が小さくなるが、陶棺も小型化が進み機能も変わったのだろう」としている。
赤田横穴墓群(6世紀後半～7世紀中ごろ)は、調査が行われた9基のうち7基で「亀甲形」と呼ばれる陶棺を確認。陶棺と埴輪（はにわ）は製作方法が似ていることから、埴輪の工人集団だった土師氏との関わりが指摘されている。
復元された陶棺は7月1日から、同市大安寺西2丁目の同センターで開かれる夏季発掘調査速報展で展示。8月28日まで。http://www.nara-np.co.jp/20150630091201.html
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\begin{document}
\title{\bf Announcement 212: What are reproducing kernels?}
\author{{\it Institute of Reproducing Kernels}\\
\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}
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