Last modified:

07/12/2003

---

no.

Welcome to

designed by Kayo Arimoto

Akio Arimoto

Now at Electronic and Computer Engineering , Musashi Institute of Technology, Tokyo 158,Japan

All e-mail should be addressed to:

arimoto@iname.com

Postal mail should be addressed to:

Akio Arimoto

Department of Electronics and Computer Engineering

Musashi Institute of Technology

1-28-1 Tamazutumi Setagaya-ku

Tokyo 158, Japan

---

Akio Arimoto’papers

l A simple proof of the classification of normal Toeplitz matrices
Electron. J. Linear Algebra 9 (2002), 108-111.

math.LA/0204276 A Simple Proof of the Classification of Normal Toeplitz Matrices. Akio Arimoto. 5 pages. LA reference Ito http://www.math.cs.musashi-tech.ac.jp/~arimoto/mathematics/ito/ITO1996.PDF Farenick http://www.math.cs.musashi-tech.ac.jp/~arimoto/mathematics/farenick.pdf

l Completeness of Trigonometric System with Integer Indices , 　psfile

Journal of Approximation Theory, vol.112,no.2,Oct 1,2001,pp.311-317

Zbl 0988.42007
Arimoto, Akio
Completeness of trigonometric system with integer indices $\{ e^{inx};x\in\Re\}$. (English)
[J] J. Approximation Theory 112, No.2, 311-317 (2001). [ISSN 0021-9045]

Summary: Necessary and sufficient conditions are given which ensure the completeness of the trigonometric systems with integer indices; $\{e^{inx}; x\in{\germ R}\}^\infty_{n= -\infty}$ or $\{e^{inx}; x\in{\germ R}\}^\infty_{n=1}$ in $L^\alpha(\mu,{\germ R})$, $\alpha\ge 1$. If there exists a support $\Lambda$ of the measure $\mu$ which is a wandering set, that is, $\Lambda+ 2k\pi$, $k= 0,\pm 1,\pm 2,\dots$, are mutually disjoint for different $k$'s, then the linear span of our trigonometric system $\{e^{inx}; x\in{\germ R}\}^\infty_{n=-\infty}$ is dense in $L^\alpha(\mu,{\germ R})$, $\alpha\ge 1$. The converse statement is also true.

MSC 2000:

*42A65 Completeness of sets of functions

MATH Database 1931-1998
a service of the European Mathematical Society
http://www.emis.de/cgi-bin/MATH
in MATH Database, Zentralblatt fur Mathematik / Mathematics Abstracts:
Copyright (c) 1997,1998 European Mathematical Society, FIZ Karlsruhe & Springer-Verlag.
[ZB/w3] Retrieval Software : Copyright (c) 1996 Cellule MathDoc, UJF & CNRS.

-----------------------------------------------------------------------

870.44004

Arimoto, Akio; Ito, Takashi

Singularly positive definite sequences and parametrization of extreme points. (English)

[J] Linear Algebra Appl. 239, 127-149 (1996). [ISSN 0024-3795 ]

Authors' abstract: In the truncated classical moment problems, the set of all solutions constitutes a convex set of positive measures. We are concerned with extreme points of this convex set. It is shown that the extreme points can be characterized in terms of the singularly positive definite extensions of a given positive definite finite sequence.

[ P.Ressel (Eichstaett) ]

Keywords: truncated moment problem; positive definite sequence; singularly positive definite extension; extreme points; convex set

Classification:

*44A60 Moment problems

46A55 Convexity in topological linear spaces

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641.60049

Arimoto, Akio

Approximation of the finite prediction for a weakly stationary process. (English)

[J] Ann. Probab. 16, No.1, 355-360 (1988). [ISSN 0091-1798]

Some estimation problems for the difference between the finite predictor and infinite predictor of stationary stochastic processes are obtained. These estimates have a more precise order than the previous one obtained by the author [Ann. Inst. Stat. Math. 33, 101-113 (1981; Zbl. 484.60031)].

[ M.P.Mokljacuk ]

Citations: Zbl.484.60031

Keywords: prediction; estimation problems

Classification:

*60G25 Prediction theory

60K20 Appl. of Markov renewal processes

60G10 Stationary processes

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484.60031

Arimoto, Akio

Asymptotic behavior of difference between a finite predictor and an infinite predictor for a weakly stationary stochastic process. (English)

[J] Ann. Inst. Stat. Math. 33, 101-113 (1981). [ISSN 0020-3157]

Keywords: weakly stationary; prediction error

Classification:

*60G25 Prediction theory

62M20 Prediction, etc. (statistics)

60G10 Stationary processes

60G15 Gaussian processes

482.60035

Arimoto, Akio

On the order of complete regularity for a weakly stationary random sequence. (English)

[J] Yokohama Math. J. 29, 77-87 (1981). [ISSN 0044-0523]

Citations: Zbl.203.502

Keywords: stationary random sequence; complete linear regularity; Hardy class; strong mixing condition; time series; modulus of continuity

Classification:

*60G10 Stationary processes

60G17 Sample path properties

------------------------------------------------------------

351.60039

Arimoto, Akio

Note on the strong law of large numbers for a weakly stationary stochastic process. (English)

[J] Rep. Stat. Appl. Res., Un. Jap. Sci. Eng. 22, 164-167 (1975). [ISSN 0034-4842]

Classification:

*60G10 Stationary processes

60F15 Strong limit theorems

----------------------------------------------------------------------

321.60029

Arimoto, Akio

On the asymptotic uncorrelatedness of Fourier coefficients of a stationary stochastic process. (English)

[J] Rep. Stat. Appl. Res., Un. Jap. Sci. Eng. 22, 41-46 (1975). [ISSN 0034-4842]

Classification:

*60G10 Stationary processes

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395.62031

Arimoto, Akio

On the estimation of the probability density of a distribution function. (English)

[J] Keio Math. Semin. Rep. 1, 31-35 (1973). [ISSN 0388-3469]

Keywords: ORDERS OF CONSISTENCY; KERNEL TYPE ESTIMATOR; ASYMPTOTIC UNBIASEDNESS; NONPARAMETRIC ESTIMATION OF DENSITIES

------------------------------------------------------------

368.62083

Arimoto, Akio

Some theorems for harmonizable stochastic processes. (English)

[CA] Stoch. Processes, Keio math. Semin. Rep. No.1, 24-30 (1973).

Classification:

*62G05 Nonparametric estimation

------------------------------------------------------

385.60040

Arimoto, Akio

Some theorems for harmonizable stochastic processes. (English)

[J] Keio Math. Semin. Rep. 1, 24-29 (1973). [ISSN 0388-3469]

Classification:

*60G12 General second order processes

----------------------------------------------------

251.60030

Arimoto, Akio

On the degree of approximation of harmonizable stochastic processes by the smoothing procedure. (English)

[J] Rep. Stat. Appl. Res., Un. Jap. Sci. Eng. 19, 42-47 (1972). [ISSN 0034-4842]

Classification:

*60G10 Stationary processes

60G35 Appl. of stochastic processes

We must read the following papers to write more original papers.

Yoshino,Takashi 1999 The conditions that the product of Hankel operators is also a Hankel Operator Arch.Math.73 146-153

Ikramov,Kh.D. 1994 Describing Normal Toeplitz Matrices

Comp.Maths Math.Phys. 34,3,399-404

Gel'fgat,V.I. 1995 A Normality Criterion for Toeplitz matrices

Comp.Maths Math.Phys. 35,9,1147-1150

Ikramov,Kh.D.;Chugunov,V.N. 1996 Normality Condition for a Complex Toeplitz Matrix

Comp.Maths Math.Phys. 36,2,131-137

Gel'fgat,V.I. 1998 Commutation Criterion for Toeplitz Matrices

Comp.Maths Math.Phys. 38,1,7-10

Preprints, Preprints from internet

1/15/2002

Jussi Vaisala, A survey of near isometries 11/Jan/2002

also see his home page :

http://www.helsinki.fi/~jvaisala/preprints.html

Seminary Notes

General Solution for a linear Prediction Problem of Stationar Processes ( V.M.Adamian, D.Z.Arov) pdf

Condition that Toeplitz operator is normal in the Bergman space.( not yet completed)

Numerical Range

Algebra of Quantum Mechanics (pdf)

Banach space problem solved by W.T.Gowers Fields Medalist 1998

1999.2.23

1997.7.29

1997.7.20

1997.6.9

1997.5.22

1997.2.25

:

A letter from Kharkov

---

my favorite Mathematicians:

Shuichi,Maeda
A.E.,Nussbaum
L.D.Pitt
Sodin,Misha

---

Journal home page

Mathematics journals

Real Analysis Exchange 

Printed Journals containing Mathematics and with Internet sites

---

History of Mathematics and more

McTutor

---

Featured Review (American Mathematical Society)

Doma\'nski, Pawe\l(PL-POZNM); Vogt, Dietmar(D-WUPP)
The space of real-analytic functions has no basis. (English. English summary)
Studia Math. 142 (2000), no. 2, 187--200.

References for Moment Problem （２００１．１１．７）

Hamburger and Stieltjes moment problems in several variables Author(s): F.-H. Vasilescu. 

  Trans. Amer. Math. Soc. November 2,2001(pdf)

Preprints of my works

A characterization of N-extremal measures for the Hamburger moment problem,(in preparation)

Hamburger Moment Problem references are Here

Reprints

 
 

 On Canonical Solutions to the Hamburger Moment Problem, Hokkaido 
University Technical Report Series in Mathematics,Ser#.41,pp.71-75(1996)
 

You can find my works in the past by


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Yes, I will appreciate your offered hand (collaboration,jobs,marriage,or any 
other opportunities).

 
We should recognize an importance of "internet" for our research work.
I give here an example for my case.

Discussions

Hamburger moment problem
If you are interested in the following articles, don't hesitate to contact me
via e-mail or other means.

1.1.

I firstly posted the following news to one of the news groups "sci.math.research" the other day at the end of 1994.

 Let
 (1) { s_0,s_1,...}
 be a given positive definite Hamburger moment sequence, in other words
 for some positive measure \mu we can represent moments as
 s_k=\int{x^k d\mu}.
 In case when representing measure \mu is unique, we say (1) is determinate.
 Otherwise we say (1) is indeterminate.
 Now by a first backward extension of (1), we mean a positive definite
 Hamburger moment sequence
 (2) { t_0,t_1,t_2(=s_0),t_3(=s_1),...}.
 And by a second backward extension of (1), we mean a first backward extension
 of (2), etc.
 F.M.Wright[1956 Proc.of AMS,413-422] proved

 Theorem:
 (a)An indeterminate Hamburger moment sequence can be always backward
 extendable once.
 (b)(1) is an indeterminate Hamburger sequence and if (2) is a first
 backward extension of (1), then (2) is an indeterminate Hamburger moment
 sequence if and only if (t_0,t_1) is an interior of some closed parabolic
 region.

 We can easily see that if we take ( t_0,t_1) on the boundary of this
 parabolic region, (2) becomes a determinate moment sequence.

 Our conjecture is that
 " N-extremal measures are always found in such way."

 Here we mean by N-extremal measure \mu such that polynomials are dense in
 L^2(\mu).[N.I.Akiezer, The classical moment problem,Oliver and Boyd,
 Edinburgh 1965, p.43]

 Any commens will be appreciated.

1.2
It is very happy that I can announce here many correspondences to this news.

From hrubin@stat.purdue.edu Wed Dec 28 05:49:31 1994
Subject: Re: Hamburger moment problem
Newsgroups: sci.math.research
Organization: Purdue University Statistics Department
Status: RO
In article <9412250520.AA26763@tansei2.tansei.cc.u-tokyo.ac.jp> you write:
>
> Let
>(1) { s_0,s_1

Your result MUST be SLIGHTLY wrong. I read Shohat and Tamarkin
many years ago.
An extremal solution of an indeterminate moment problem is given
by putting maximal mass at a point; in fact, at all points where
it puts mass. The resulting measure is discrete. So if 0 is not
in the spectrum of the distribution, there must exist moments of
order -1 and -2. If the new problem (the backwards extension) is
not determined, we would have a contradiction to the density of
the polynomials for the original problem in L_2; the nasc for a
moment problem to have a unique solution is that either the 
first forward extension has a unique solution OR 1 is in the
L_2 closure of the polynomials divisible by x.
The case where 0 is a point of the spectrum, however, is not
of this type, although it is a limit.
So the final answer is that N-extremal measures are either
found in this way, or are the limits of such at t_0 -> \infty.

-- 
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette 
IN47907-1399Phone: (317)494-6054
hrubin@stat.purdue.edu (Internet, bitnet) 
{purdue,pur-ee}!a.stat!hrubin(UUCP)


Message 2:
From nevai@ops.mps.ohio-state.edu Thu Jan 5 05:25:40 1995
Message-Id: <9501042024.AA16992@ops.mps.ohio-state.edu>
Subject: Hamburger moment problem...
To: a80919@tansei.cc.u-tokyo.ac.jp
Date: Wed, 4 Jan 1995 15:24:28 -0500 (EST)
Cc: askey@math.wisc.edu (Dick Askey)
Reply-To: nevai@math.ohio-state.edu (Paul Nevai)
Organization: The Ohio State University
X-Mailer: ELM [version 2.4 PL23]
Content-Type: text
Status: RO

Our conjecture is that 
 " N-extremal measures are always found in such way."
Here we mean by N-extremal measure \mu such that polynomials are dense in
L^2(\mu).[N.I.Akiezer, The classical moment problem,Oliver and Boyd,
Edinburgh 1965, p.43]

Dear Professor Akio (is this your family name?):
Can you be more specific about what you mean by "in such way"? 

Happy new year...Paul N.

Paul Nevai pali+@osu.edu
Department of Mathematics nevai@math.ohio-state.edu
The Ohio State University nevai@ohstpy.bitnet
231 West Eighteenth Avenue 1-614-292-3317 (Office)
Columbus, Ohio 43210-1174 1-614-292-5310 (Answering Machine)
The United States of America 1-614-292-1479 (Math Dept Fax)

1.3.
I sent a mail to Prof. Nevai at 1/6/95.

Dear Professor Nevai:

Thank you for your correspondence. My family name is Arimoto. In Japanese style
I am called Arimoto Akio.

If we have an indeterminate Hamburger moment sequence
(1) { s_0,s_1,s_2,...},
then for any real y
 \sum_{p=0}^{\infty}[X_p(0)+yY_p(0)]^2 
is finite. ( Here Y_p are orthonormal polynomials and X_p are polynomials of 
second kind,i.e. Akiezer's P_k and Q_k.).
Let 
(2) x = \sum_{p=0}^{\infty}[X_p(0)+yY_p(0)]^2.
By virtue of Wright's theory we have a determinate moment sequence
(3) {x,y,s_0,s_1,s_2,...},
which is a first backward extension of (1). 
I guess (3) determines all the N extremal measures "in such way" where
(x,y)'s are taken from the points on the parabola (2).

The other day,
Herman Rubin, Dept. of Statistics, Purdue Univ.,
mailed me

>Your result MUST be SLIGHTLY wrong. I read Shohat and Tamarkin
>many years ago.
>An extremal solution of an indeterminate moment problem is given
>by putting maximal mass at a point; in fact, at all points where
>it puts mass. The resulting measure is discrete. So if 0 is not
>in the spectrum of the distribution, there must exist moments of
>order -1 and -2. If the new problem (the backwards extension) is
>not determined, we would have a contradiction to the density of
>the polynomials for the original problem in L_2; the nasc for a
>moment problem to have a unique solution is that either the 
>first forward extension has a unique solution OR 1 is in the
>L_2 closure of the polynomials divisible by x.
>The case where 0 is a point of the spectrum, however, is not
>of this type, although it is a limit.
>So the final answer is that N-extremal measures are either
>found in this way, or are the limits of such at t_0 -> \infty.

Despite of his advice, I believe these troubles will be conqured.
ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
o akio arimoto department of industrial engineering|
o email:a80919@tansei.cc.u-tokyo.ac.jp musashi institute of technology |
o telephone:(03) 3703-3111 setagaya-ku tamazutsumi 1-28-1 | 
o telefax: (03) 5707-2136 tokyo 158, japan |
ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo

1.4
I posted the second news at 1/10/95.

Dear Modelator

 Let \mu be a positive Borel measure on R and its Stieltjes transform 
I_{\mu}(z)=\int{d\mu(t)/(z-t)}, which is a holomorphic function of z in C-R.
A Hamburger's moment sequence is given by s_k=\int{x^k d\mu(x)},k=0,1,2,3,... 
Let P_k(z) be orthonormal polynomials in L^2(\mu) and their associated second
polynomials: Q_k(z)=\int{(P_k(z)-P_k(t))/(z-t)}d\mu(t).
 The Fourier coefficient of 1/(z-t) with respect to P_k(t) is easily seen
to be I_{\mu}(z)P_k(z)-Q_k(z). The density of polynomials in L^2(\mu) is
equivalent to that 1/(z-t) is precisely approximated by polynomials in 
L^2(\mu). These are the contents of the following statement.

 Proposition: \mu is N-extremal iff the Parseval equality holds:
(1) \sum_{k=0}^{\infty}|I_{\mu}(z)P_k(z)-Q_k(z)|^2
 =\int{d\mu(t)/(|z-t|^2)}.
 
For z=0, (1) becomes the Wright's parabola stated in my previous article
 <9412250520.AA26763@tansei2.tansei.cc.u-tokyo.ac.jp> :
 
(2) \sum_{k=0}^{\infty}(s_1P_k(0)+Q_k(0))^2
 =s_2,
where s_1,s_2 are given by the first backward extension of {s_0,s_1,...}.
Our conjecture was the converse of the above deduction,i.e.
 (2)==>(1).

It is pointed out by H.Rubin that the case where 0 is a support point of \mu
causes us some troubles. Can we now prove or disprove our conjecture?

 Acknowledgements for replies from: Herman Rubin and Paul Nevai. 
Thanks for more helps in advance.

oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
o akio arimoto department of industrial engineering o
o email:a80919@tansei.cc.u-tokyo.ac.jp musashi institute of technology o
o telephone:(03) 3703-3111 setagaya-ku tamazutsumi 1-28-1 o 
o telefax: (03) 5707-2136 tokyo 158, japan o
oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo

I have again many correspondences.

From jroach@Starbase.NeoSoft.COM Thu Jan 12 02:52:51 1995
Date: Wed, 11 Jan 1995 11:52:48 -0600
From: Jack Roach 
Message-Id: <199501111752.LAA00604@Starbase.NeoSoft.COM>
X-Provider: NeoSoft, Inc.: Internet Service Provider (713) 684-5969
To: a80919@tansei.cc.u-tokyo.ac.jp (A80919)
Subject: Re: Hamburger moment problem (2)
Newsgroups: sci.math.research
X-Newsreader: TIN [version 1.2 PL2]
Status: RO

You are probabily aware of the connection with continued fractions but if 
not, you may want to see H. S. Wall, "The Analytic Theory of Continued 
Fractions," Van Nostrand (1948), reprinted by Chelsea (1967), LC # 
66-24296. There is quite a bit on various moment problems in this book.

Jack Roach 

--------------------------------------------------------------------------
From berg@math.ku.dk Wed Jan 18 21:52:21 1995
Received: by gelfand id AA05455
Date: Wed, 18 Jan 1995 13:51:42 +0100
From: Christian Berg 
Message-Id: <199501181251.AA05455@gelfand>
To: Walter=Van=Assche%twi%WIS@cc3.kuleuven.ac.be
Subject: Hamburger moment problem...
Cc: nevai@math.ohio-state.edu, a80919@tansei.cc.u-tokyo.ac.jp
X-Char-Esc: 29
Status: RO

Dear Akio Arimoto, Walter Van Assche, Paul Nevai,
Via Paul and Walter I received the following:


 Forwarded to:  email@pub@wis[berg@math.ku.dk]
 cc: 
 Comments by: Walter Van Assche@twi@WIS
 Comments: 


 This is certainly of interest to you, Christian.

 The best for 1995!

 Walter Van Assche

 -------------------------- [Original Message] ------------------------- 
 

 This is interesting! Take care...Paul

 #############################################################################

 >From a80919@tansei.cc.u-tokyo.ac.jp Tue Dec 27 15:57:40 EST 1994
 Newsgroups: sci.math.research
 Subject: Hamburger moment problem
 Date: Sun, 25 Dec 94 14:20:40 JST

 Let
 (1) { s_0,s_1,...}
 be a given positive definite Hamburger moment sequence, in other words
 for some positive measure \mu we can represent moments as
 s_k=\int{x^k d\mu}.
 In case when representing measure \mu is unique, we say (1) is determinate.
 Otherwise we say (1) is indeterminate.
 Now by a first backward extension of (1), we mean a positive definite
 Hamburger moment sequence
 (2) { t_0,t_1,t_2(=s_0),t_3(=s_1),...}.
 And by a second backward extension of (1), we mean a first backward extension
 of (2), etc.
 F.M.Wright[1956 Proc.of AMS,413-422] proved

 Theorem:
 (a)An indeterminate Hamburger moment sequence can be always backward
 extendable once.
 (b)(1) is an indeterminate Hamburger sequence and if (2) is a first
 backward extension of (1), then (2) is an indeterminate Hamburger moment
 sequence if and only if (t_0,t_1) is an interior of some closed parabolic
 region.

 We can easily see that if we take ( t_0,t_1) on the boundary of this
 parabolic region, (2) becomes a determinate moment sequence.

 Our conjecture is that
 " N-extremal measures are always found in such way."

 Here we mean by N-extremal measure \mu such that polynomials are dense in
 L^2(\mu).[N.I.Akiezer, The classical moment problem,Oliver and Boyd,
 Edinburgh 1965, p.43]

 Any commens will be appreciated.

Here are the comments of Christian Berg:
Let $\mu$ be an indeterminate probability. I write $\mu\sim\nu$ to
denote that the measure $\nu$ has the same moments as $\mu$ and I put
$V_\mu=\{\nu\mid \mu\sim\nu\}.$ For the parabolic region $P\mu$ associated to
$\mu$ it is convenient to consider the set of measures
 $M_\mu=\{\sigma\mid t^2\sigma\sim\mu\}$ because then 
$$P_\mu=\{(\int d\sigma(t),\int td\sigma(t)) \mid \sigma\inM\mu\}$$.
It is easy to see that
$$M_\mu=\{t^{-2}d\nu(t)+a\delta_0\mid a\geq 0, \nu\sim\mu,\int
t^{-2}d\nu(t)<\infty\}$$. For an interior point $(x,y)$ of the parabolic region there are infinitely many measures $\sigma\in M\mu$ for which $$(x,y)=(\int d\sigma(t),\int td\sigma(t))$$. For a boundary point $(x,y)$ of the parabolic region there is exactly one measure $\sigma\in M_\mu$ with this property and it is given as $$\sigma=t^{-2}d\nu_c(t)$$, where $c$ is a special parameter value and $\nu_s$ is the family of Nevanlinna extremal measures in $V_\mu$, where the parameter $s$ ranges over the one-compactification of the real line. The special parameter value $c$ is the unique number for which $c=-1/y$. In other words, the boundary curve of the parabolic region has the parametric representation $$(\int t^{-2}d\nu_{-1/s}(t),s)$$, where $s$ runs over the real line. The unique measure $\sigma$ given above is a determinate measure, and the measures arising in this way from the boundary of parabolic regions associated with indeterminate measures $\mu$ can be described within the class of determinate measures using the index of determinacy introduced and used in recent manuscripts of Duran and myself, namely as the measures $\sigma$ for which $\ind_0(\sigma)=0$. All the results above except the last one with the index is in unpublished notes of mine from 1980 when I read the paper by Wright. If you think it is of interest I could perhaps publish it together with the index result. The last statement on Arimotos email: "Our conjecture is that
N-extremal measures are always found is such way" seems a little mysterious to me since the measures are determinate (and of index 0 at 0). However I realize that Akhiezer uses the word N-extremal in a double sense: Either a determinate measure or an indeterminate measure for which the polynomials are dense in $L^2$. (In the first case polynomials are also dense in $L^2$ by the Riesz theorem). I usually prefer to reserve the word N-extremal for the indeterminate case. In any case I thank Arimoto for having aroused my interest again in backwards extensions of Hamburger sequences and the parabolic regions. I look forward to hearing your comments. Sincerely yours Christian Berg |----------------------------------------------------------------------| | Christian Berg |  | | Matematisk Institut | e-mail: berg@math.ku.dk | | Universitetsparken 5 | phone: +45 35320728 | | DK-2100 Copenhagen | fax: +45 35320704 | | Denmark | secretary: +45 35320722 | |----------------------------------------------------------------------| |----------------------------------------------------------------------| | Christian Berg |  | | Matematisk Institut | e-mail: berg@math.ku.dk | | Universitetsparken 5 | phone: +45 35320728 | | DK-2100 Copenhagen | fax: +45 35320704 | | Denmark | secretary: +45 35320722 | |----------------------------------------------------------------------| Date: Mon, 13 Feb 1995 17:15:39 +0100 From: Christian Berg >
Message-Id: <199502131615.AA17665@gelfand>
To: a80919@tansei.cc.u-tokyo.ac.jp
Subject: moment sequences 
X-Charset: ASCII
X-Char-Esc: 29
Status: RO

Dear Akio Arimoto,
Some time ago I sent you an email concerning backward extensions of
moment sequences. This was a reply to a question which reached me via
Paul Nevai and VanAssche. I wonder whether you have received my email
and if so I should like to hear your opinion. There was something in
your problem that was not so clear to me. Sincerely yours Christian
Berg


Remark: Professor Christian Berg is a very famous mathematician. 
 I am very proud of his acquaintance. Viva internet! 

1.5

I mailed Prof.Berg at 2/14/95

Dear Christian Berg,
 Sorry I'm late in answering your e-mail. I received it on Jan 18th.
I am grateful for your good comments to my question. 
 If we take a point (s_{-2},s_{-1}) on the parabola P_{\mu}, we get the
backward extension which uniquely determines the measure \mu. This measure
is N-extremal. 
My points were:
 (1) there is one to one correspondence between such \mu's and N-extreme 
 measures derived from an indeterminate moment sequence (s_{0},s_{1},...).
 (2)The parabola is the limit case of the Parseval relation which is 
 equivalent to the N-extreme property.

 Now I expect that m canonical measures will also appear from m-th backward
extensions. Can you explain the equation (3.6) in Wright's paper[1956]
without using determinant's arguments?
  Sincerely yours, Akio Arimoto

ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
o akio arimoto department of industrial engineering|
o email:a80919@tansei.cc.u-tokyo.ac.jp musashi institute of technology |
o telephone:(03) 3703-3111 setagaya-ku tamazutsumi 1-28-1 | 
o telefax: (03) 5707-2136 tokyo 158, japan |
ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo

1.6
I again posted to sci.math.research.

Subject: Hamburger moment problem(3)

Dear moderator

\documentstyle{jarticle}
\begin{document}
 Ch.Berg and J.P.R.Christensen ( Ann.Inst.Fourier 31(1981),p.107)
have stated that if the measure $\mu$ arising from a Hamburger moment problem
is indeterminate, then for N-extremal $\mu$,$$ h(x_n,x_m)=0,n\not=m,$$
where $h(x,y)$ is the reproducing kernel associated with $\mu$ and $x_n$
are the supporting points of $\mu$. When $\mu$ has the form $$\mu=\sum_{n=1}^{\infty}a_n\varepsilon_{x_n},a_n>0,$$ we easily see that for any $z$$$
 inf_{p\in polynomials}\int \vert \frac{1}{x-z}-p\vert^2d\mu(x)$$$$ =\sum_{n=1}^{\infty}\vert\frac{1}{x_n-z}-\sum_{m=1}^{\infty}
\frac{a_m}{x_m-z}h(x_n,x_m)\vert^2a_n.$$
From this formula, we can see that $\mu$ is N-extremal iff
$$h(x_n,x_m)=\delta _{n,m}/a_n.$$
Is this well known?  
\end{document}

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o akio arimoto department of industrial engineering|
o email:a80919@tansei.cc.u-tokyo.ac.jp musashi institute of technology |
o telephone:(03) 3703-3111 setagaya-ku tamazutsumi 1-28-1 | 
o telefax: (03) 5707-2136 tokyo 158, japan |
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1.7
I asked a question to Prof.Berg by e-mail at 4/3/95.

Dear Professor Berg,

I have a question.
 In p.171 of the book [N.I.Akiezer, The classical moment problem], he
has stated Corollary 2 which can be rewritten compactly as: 
 "The necessary and sufficient condition for an N-extremal measure to
belong to the moment sequence ${s_k}$ is that the set of all zeros {$\lambda_j$}of some entire function $q(z)$ and positive numbers $\mu_j$ satisfy
$$[4.31]: \sum_1^{\infty}\frac{1}{\mu_j(1+\lambda_j^2)[q'(\lambda_j)]^2}<\infty$$ $$[4.34]: \sum_1^{\infty}\frac{1}{\mu_j[q'(\lambda_j)]^2}=\infty$$
and
$$[4.32]: s_m=\sum_1^{\infty}\mu_j\lambda_j^m.$$"

For necessity these condition O.K. But for sufficiency I cannot follow his
ideas. Especially I can't understand his statement in line 16,p.171 that ... self-adjoint and therefore the set of all polynomials is dense in $L_{\sigma}^2$. I think there is a situation such that in $L_{\sigma}^2$ the codimension of the set of all polynomials is positive. The condition [4.31] guarantees us the indeterminancy of the measure. However I can't show whether the condition [4.34] implies the denseness of the 
polynomials in $L_{\sigma}^2$ or not. 
Do I miss something?

 I should be grateful for any conmments on this question.
 Regards,
 Akio Arimoto
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o akio arimoto department of industrial engineering|
o email:a80919@tansei.cc.u-tokyo.ac.jp musashi institute of technology |
o telephone:(03) 3703-3111 setagaya-ku tamazutsumi 1-28-1 | 
o telefax: (03) 5707-2136 tokyo 158, japan |
ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo

Professor Berg kindly replied me:

>N 1 berg@math.ku.dk Mon Apr 3 17:56 32/1692 "question"
Message 1:
From berg@math.ku.dk Mon Apr 3 17:56:07 1995
Received: by gelfand id AA02688
Date: Mon, 3 Apr 1995 10:55:27 +0200
From: Christian Berg 
Message-Id: <199504030855.AA02688@gelfand>
To: a80919@tansei.cc.u-tokyo.ac.jp

Dear Akio Akimoto, Thank you for the email with the question about
Akhiezer. Some years ago I was puzzled by the same question and my
student Henrik Pedersen and I tried to read Hamburgers original paper
from Amer J. Math 1944. We realized that his proof as well as
Akhiezers proof is wrong, but we could not settle if the theorem was
true anyway. We were in correspondance with Paul Koosis from McGill
who finally constructed a counterexample, published in a note in C.R.
Acad. Science Paris 311 (1990), 503-506. I shall send you the
manuscript to my talk at the Stieltjes meeting in Delft in November
1994, where I talked about the connection between entire functions and
the moment problem and it contains an outline of the result of Koosis.
For your convieience I include an old letter I wrote to Krein in 1989
about the mistake in Akhiezer. Unfortunately Krein had died a few
weeks before so I never got any answer.
I regret that I have not yet found time to answer your question from
Wrights paper. I shall return to that very soon. Sincerely yours
Christian Berg


From berg@math.ku.dk Tue Apr 4 21:14:06 1995
 id AA10230; Tue, 4 Apr 95 21:13:55 JST
Received: by gelfand id AA07555
Date: Tue, 4 Apr 1995 14:13:28 +0200
From: Christian Berg 
Message-Id: <199504041213.AA07555@gelfand>
To: a80919@tansei.cc.u-tokyo.ac.jp
Subject: Wrights paper 
X-Charset: ASCII
X-Char-Esc: 29
Status: R

Dear Mr. Arimoto, I looked at Wrights equation 3.6. It is the Parseval
formula for the function $1/t^2$. If the measure \mu has a second
backwards extension then $\int 1/t^4d\mu(t)<\infty$, and Parsevals
formula shows that the series is finite. I cannot see the converse in
the same easy way. Sincerely yours Christian Berg

I should be grateful for any comments on this field. Thanks in advance.

Akio Arimoto

(arimoto@cs.musashi-tech.ac.jp)

有本彰雄