\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}