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2012.05.30
カテゴリ： カテゴリ未分類
Dear Professors Castro, Minh and Riva:
I wrote down the open question as in the attached way that I stated yesterday at coffee time.
You can forward the manuscript unconditionally.
I know the increasing power of Hanoi and Italy, however, their problems may not be easy.
With best regards,
Sincerely yours,
Saburou Saitoh
2012.5.30.09:10
at Aveiro
\usepackage{latexsym,amsmath,amssymb,amsfonts,amstext,amsthm}
\newcommand{\bm}[1]{\mbox{\boldmath $#1$}}
\numberwithin{equation}{section}
\begin{document}
\title{\bf Open questions on analytic functions\thanks{This work was supported in part by {\em Center for Research and Development in Mathematics and Applications} of University of Aveiro, and
the Portuguese Science Foundation ({\em FCT--Funda\c{c}\~{a}o para a Ci\^{e}ncia
e a Tecnologia}). }}
\author{Saburou Saitoh\\
{\small\it Center for Research and Development in Mathematics and Applications},\\
{\small Department of Mathematics, University of Aveiro,
3810-193 Aveiro, Portugal}\\
\date{ }
\maketitle
\section{Introduction}
\label{sect1}
We would like to propose two open questions. One is a very simple conjecture, however, its proof will be very difficult. The second conjecture may not be solved over the next three hundred years.
Let $D$ be an $N$ ply-connected and bounded regular domain on the complex $z$ plane whose boundary components are $\{ C_j\}_{j=1}^N$, the $C_N$ is the outer boundary and all the boundary components are closed analytic Jordan curves (for simplicity).
The following result is very fundamental: For any given complex numbers $\{\alpha_j\}_{j=1}^{N-1}$, there exist analytic functions on $D\cup \partial D$ satisfying
\begin{equation}
\int_{C_j} f(z) dz = \alpha_j, \quad \quad j = 1,2,...,N-1.
\end{equation}
The question is: How will be when we change $dz$ by $|dz|$? Our conjecture is as follows:
For any given complex numbers $\{\alpha_j\}_{j=1}^{N}$, there exist analytic functions on $D\cup \partial D$ (or, of the analytic Hardy $H_1(D)$) functions satisfying
\begin{equation}
\int_{C_j } f(z) |dz| = \alpha_j, \quad \quad j = 1,2,...,N,
\end{equation}
except annulus domains $D$. Of course, for annulus domain cases, we can give one period only, arbitrarily.
\section{Hardy reproducing kernels }
In \cite{Sa97, Sa10}, in connection with the deep theories of D. A. Hejhal and J. D. Fay, the author proposed the following conjecture:
Meanwhile, for the Hardy reproducing kernel we can give the conjecture in the converse direction, from the case of doubly connected domains
\begin{equation}
K(z,\overline{u}) << K_t(z,\overline{u}) \hat{K}_t(z,\overline{u}).
\end{equation}
From the relation we can derive various results, the conjecture will be, however, very difficult to solve.
The author attacked to this problem several years, it was, however, impossible to solve it and in those days, the author derived the above result applying the result of Hejhal conversely to the general theory of reproducing kernels by Aronszajn. The author thinks for the conjecture we must use the deep theory of J. D. Fay, however, his profound theory seems to be too deep and great for the author. So, from the viewpoints of fundamental results and applicable analysis, the author turned his research interest to the following applications of the general theory of reproducing kernels.
\bibliographystyle{plain}
\begin{thebibliography}{10}
%1
%10
\bibitem{Sa97}
S.~Saitoh. {\it Integral Transforms, Reproducing Kernels and
their Applications}, Pitman Research Notes in Mathematics Series
vol.~369 (London: Addison Wesley Longman), 1997.
\bibitem{Sa10}
S. Saitoh,
{Theory of reproducing kernels: Applications to approximate solutions of bounded linear operator functions on Hilbert spaces},
\newblock AMS Translations, Series 2, {\bf 230} (2010), 107--134.
\end{thebibliography}
\end{document}
I wrote down the open question as in the attached way that I stated yesterday at coffee time.
You can forward the manuscript unconditionally.
I know the increasing power of Hanoi and Italy, however, their problems may not be easy.
With best regards,
Sincerely yours,
Saburou Saitoh
2012.5.30.09:10
at Aveiro
\usepackage{latexsym,amsmath,amssymb,amsfonts,amstext,amsthm}
\newcommand{\bm}[1]{\mbox{\boldmath $#1$}}
\numberwithin{equation}{section}
\begin{document}
\title{\bf Open questions on analytic functions\thanks{This work was supported in part by {\em Center for Research and Development in Mathematics and Applications} of University of Aveiro, and
the Portuguese Science Foundation ({\em FCT--Funda\c{c}\~{a}o para a Ci\^{e}ncia
e a Tecnologia}). }}
\author{Saburou Saitoh\\
{\small\it Center for Research and Development in Mathematics and Applications},\\
{\small Department of Mathematics, University of Aveiro,
3810-193 Aveiro, Portugal}\\
\date{ }
\maketitle
\section{Introduction}
\label{sect1}
We would like to propose two open questions. One is a very simple conjecture, however, its proof will be very difficult. The second conjecture may not be solved over the next three hundred years.
Let $D$ be an $N$ ply-connected and bounded regular domain on the complex $z$ plane whose boundary components are $\{ C_j\}_{j=1}^N$, the $C_N$ is the outer boundary and all the boundary components are closed analytic Jordan curves (for simplicity).
The following result is very fundamental: For any given complex numbers $\{\alpha_j\}_{j=1}^{N-1}$, there exist analytic functions on $D\cup \partial D$ satisfying
\begin{equation}
\int_{C_j} f(z) dz = \alpha_j, \quad \quad j = 1,2,...,N-1.
\end{equation}
The question is: How will be when we change $dz$ by $|dz|$? Our conjecture is as follows:
For any given complex numbers $\{\alpha_j\}_{j=1}^{N}$, there exist analytic functions on $D\cup \partial D$ (or, of the analytic Hardy $H_1(D)$) functions satisfying
\begin{equation}
\int_{C_j } f(z) |dz| = \alpha_j, \quad \quad j = 1,2,...,N,
\end{equation}
except annulus domains $D$. Of course, for annulus domain cases, we can give one period only, arbitrarily.
\section{Hardy reproducing kernels }
In \cite{Sa97, Sa10}, in connection with the deep theories of D. A. Hejhal and J. D. Fay, the author proposed the following conjecture:
Meanwhile, for the Hardy reproducing kernel we can give the conjecture in the converse direction, from the case of doubly connected domains
\begin{equation}
K(z,\overline{u}) << K_t(z,\overline{u}) \hat{K}_t(z,\overline{u}).
\end{equation}
From the relation we can derive various results, the conjecture will be, however, very difficult to solve.
The author attacked to this problem several years, it was, however, impossible to solve it and in those days, the author derived the above result applying the result of Hejhal conversely to the general theory of reproducing kernels by Aronszajn. The author thinks for the conjecture we must use the deep theory of J. D. Fay, however, his profound theory seems to be too deep and great for the author. So, from the viewpoints of fundamental results and applicable analysis, the author turned his research interest to the following applications of the general theory of reproducing kernels.
\bibliographystyle{plain}
\begin{thebibliography}{10}
%1
%10
\bibitem{Sa97}
S.~Saitoh. {\it Integral Transforms, Reproducing Kernels and
their Applications}, Pitman Research Notes in Mathematics Series
vol.~369 (London: Addison Wesley Longman), 1997.
\bibitem{Sa10}
S. Saitoh,
{Theory of reproducing kernels: Applications to approximate solutions of bounded linear operator functions on Hilbert spaces},
\newblock AMS Translations, Series 2, {\bf 230} (2010), 107--134.
\end{thebibliography}
\end{document}
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