再生核研究所

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\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}

\medskip

Based 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 $H_\infty$.