カリフォルニアでUFO目撃相次ぐ～4K撮影の超光速UFOに上空を浮遊する大群～それぞれ動画が話題に
150313-011
3月に入りカリフォルニアでのUFO目撃が相次いでいます。
3月7日にアップされた、4Kの高解像度で撮影されたという動画には超光速で右から左に移動していくUFOが記録されていたとして海外でも話題になっています。
見やすくした2倍スロー動画の再生数は既に20万回を突破。
そして3月11日には、カリフォルニア上空にかなりの数の白く光るUFOが浮遊している動画が公開され、こちらも注目を集めているようです。
天変地異や大事件の際にUFOが現れていた、という話をよく聞きますが、カリフォルニアで何か起きるのでしょうか。http://www.buzznews.jp/?p=1458302
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\begin{document}
\title{\bf Announcement 212: What are reproducing kernels?}
\author{{\it Institute of Reproducing Kernels}\\
Kawauchi-cho, 5-1648-16,\\
Kiryu 376-0041, Japan\\
E-mail: kbdmm360@yahoo.co.jp\\}
\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}
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