2015-12-17 research seminar 2nd part

Mathematical Analysis and Its Applications in the
Dept. of Math at Tallinn Uni:
the past, present and possible future
Andi Kivinukk
Matemaatika osakond, Tallinna Ülikool
DTI seminar, TLÜ
detsember 16, 2015
A. Kivinukk (Tallinna Ülikool) 1 / 20

Staff
Staff
Prof Anne Tali (at TLU since 1973)
Mathematical competencies: mathematical analysis, applications of
functional analysis (summability theory)
Prof AK (at TLU since 1993)
Mathematical competencies: mathematical analysis (approximation
theory and its applications, in particular in signal analysis), Fourier
analysis, mathematical ﬁnance (option theory), optimization
Senior researcher Maria Zeltser (at TLU since 2004 )
Mathematical competencies: mathematical analysis, applications of
functional analysis, mathematical statistics, data analysis

Staff
Lecturer, PhD Anna Šeletski (at TLU since 2014)
Mathematical competencies: mathematical analysis (partial differential
equations), applications of functional analysis (summability theory)
Non-staff member researcher, PhD Tarmo Metsmägi
theory)
Doctoral student Anna Saksa
theory, Fourier series)

Approximations, Fourier Analysis, Shannon sampling series
Approximations, Fourier Analysis, Shannon sampling
series
Topics by:
AK and his doctoral student Anna Saksa and Tarmo Metsmägi
Introduction
Warning: MATH do not exist without numbers or speciﬁc symbols !
A crucial number in Math is
π = 3.1415926535897932384626433832795028...,
here are 3.5 × 101 digits.
Mathematicians and computer scientists discovered new approaches
that, when combined with increasing computational power, extended
the decimal representation of π to, as of 2015, over 13.3 trillion (1013)
digits.

Usually we use an approximation π = 3.14, as you know - probably ?
Using the approximation we did the error π − π < 0.0016.
Some persons may like the approximation π = 3.1416, in that case the
error is
π − π < 0.000075 or π − π > −0.000075
or using the absolute value |π − π| < 0.000075.
Three important things happened:
0) An approximation is simpler as the object itself !
1) It does not matter is an approximation bigger or less from the true
value - the error is error. Therefore we use the absolute value.
2) If our approximation uses more digits the error will be smaller
(0.000075 < 0.0016).

Our topic, AK, Anna Saksa and Tarmo Metsmägi
In our topic the complicated objects are functions or more generally
operators. You may consider these as input-output machines , like a
mincing machine.
For quite arbitrary functions f the Fourier partial sums Snf perform an
universal approximation method.
The error, analogically to the absolute value, is given by ||f − Snf||, and
here the parameter n is a natural number and for bigger n the
approximation will be better.
But (Arbitrary cannot be perfect !) even for the continuous functions
(these have continuous graphs) the Fourier series may fail. In this case
some generalization is used:
Un(f, x) :=
n
k=−n
λ(
k
n
)f∧
(k)eikx
.

In fact, the basis functions eikx = cos kx + i sin kx are 2π-periodic. For
non-periodic case the Fourier transform or the Shannon sampling
operators
(SW f)(t) :=
k∈Z
f(
k
W
)s(Wt − k)
are used. For a ﬁnite interval, e.g. [0, 1], another type of operators
(Bnf)(x) =
n
k=0
f(
k
n
)pk,n(x)
are used.
Again, typical problems are how to characterize the error
f − SW f .

Summability methods, speeds of convergence and ...
Comparison of summability methods, speeds of
convergence and statistical convergence
Topics by: Anne Tali, her former doctoral student, now Lecturer, PhD
Anna Šeletski and co-author Ulrich Stadtmüller (University of Ulm)
A number sequence x = (ξn) can be convergent or divergent, but only
convergent sequences are common in practice.
Example. The sequence (1, 1
2 , 1
3 , 1
4 , ..., 1
100, ...) in infinity seems to be
"equal" to 0, and by definition we call it to be convergent.
But we are not sure what will happen in infinity for (1, 0, 1, 0, 1, 0, 1, ...),
thus, we call it to be divergent.
A divergent sequence x = (ξn) can be transformed into convergent
sequence y = (ηn) by some operator A. Then it is said that sequence
x is A-convergent.

The most common operators A are matrix transformations A = (ank )
deﬁned by
ηn =
∞
k=0
ankξk , n = 0, 1, 2, ....
The following problems are discussed for certain families {Aα} (α is a
continuous parameter).
1) Methods Aα are compared by their sets of all Aα-convergent
sequences and by speed of convergence.
2) The estimates for speeds of methods Aα are found.
3) Different types of Aα-convergence, like ordinary convergence,
strong convergence and statistical convergence, are characterized and
compared.
4) Transformations Aα are characterized as bounded operators in
sequence spaces lp.

Anna Šeletski
is involved in another project with Jaan Janno, Tallinn Uni of
Technology, studying the solitary waves (in channels or in some micro
elements like constructions materials).

Series and sequences
Topics by
Maria Zeltser
Series are sums with inﬁnity number of terms:
∞
k=0
ak = a0 + a1 + a2 + ... + a100 + ...
It is up the terms ak, could the expression above be meaningful or not !
Example. In case ak ≡ 1 we have
∞
k=0
1 = 1 + 1 + 1 + ... + 1 + ...
and it is certainly a huge number or even more - inﬁnity. We call that
this series is divergent.

More hopeful seems to be
∞
k=0
1
(k + 1)2
= 1 +
1
22
+
1
32
+ ... +
1
1002
+ ... ,
because, although the number of terms is inﬁnity, at the "end" we add
very small numbers like 0.001, ..., 0.000001, ... etc.
We call that this series is convergent.
The topic of convergent/divergent series is very old, but Maria Zeltser
found a very fresh view to this area.

A selection of Journals, Books, etc. where we have published
A selection of Journals, Books, ...
In: New Perspectives on Approximation and Sampling Theory. A.
I. Zayed and G. Schmeisser (Eds.) Applied and Numerical
Harmonic Analysis, Springer, 2014, 65–88.
Sampling Theory in Signal and Image Processing, 13 (2014), no
2, 189–206, and many other issues
In Proc. of the 9th Intern. Conf. on Sampling Theory and
Applications , Singapore, May 2-6, 2011, A. Khong, F. Oggier
(Eds.), Nanyang Techn. Univ., 2011, 1–4

Math. Nachr., 2009, 282, 2, 288–306.
Filomat, 2015
Proc. Estonian Acad. Sci., many many times
Wave Motion, 52 (2015)
Mathematical Modelling and Analysis , 2010, 15, 103–112, etc.
Journal of Function Spaces and Applications, 1–8, 2013.
Mathematica Slovaca, 63(6), 1333–1346, 2014.

In: AIP Conference Proceedings, 11TH INTERNATIONAL
CONFERENCE OF NUMERICAL ANALYSIS AND APPLIED
MATHEMATICS 2013: ICNAAM 2013: Rhodes, Greece, 21-27
September 2013, 770–773, 2013.
In: Fourier Analysis. Pseudo-differential Operators,
Time-Frequency Analysis and Partial Differential Equations,
Springer, 357– 374, 2014.
In: Abstracts of the International Congress of Mathematicians
(ICM 2014), Seoul, Korea, 272–273, 2014.
In: FINEST MATH 2014 : Fourth Finnish-Estonian Mathematics
Colloquium and Finnish Mathematical Days 2014, Book of
Abstracts, University of Helsinki, 9-10 January 2014, 2014.
In: Kangro-100 : Methods of Analysis and Algebra, Intern. conf.
dedicated to the centennial of professor Gunnar Kangro, Tartu,
Estonia, September 1-6, 2013, Book of Abstracts, Tartu, Estonian
Mathematical Society, 2013.

Conferences, Work-shops, Seminars, (co-)organized by our working group
Conferences, Work-shops, Seminars, (co-)organized
by our working group
Approximations, Summations and Applications, Laulasmaa, Dec.
11, 2015
Methods of Analysis and Algebra, Intern. conf. dedicated to the
centennial of professor Gunnar Kangro, Tartu, Estonia, September
1-6, 2013
International Workshop on Approximations, Harmonic Analysis,
Operators and Sequences" , Narva-Joesuu, Oct. 3-5, 2008
Finnish-Estonian Mathematics Colloquium = FinEst Math 2002,
Tallinn
NB ! Approved by the Intern. Steering Committee of SampTA:
Intern. Conf. SampTA (Sampling Theory and Applications),
Tallinn, July 3 - 7, 2017;
with 150 - 180 foreigners !

Conferences, Lectures, etc.
Fourth Finnish-Estonian Mathematics Colloquium and Finnish
Mathematical Days 2014, Univ. of Helsinki, 9-10 January 2014.
Intern. Conf. on Operator Theory, 28 April - 01 May, 2014,
Hammamet, Tunisia.
Intern. Congress of Mathematicians (ICM 2014), Seoul, Korea,
13-21 August 2014.
ISAAC 9th congress, Krakow, Poland, 5-9 august, 2013.
Lecturers in Estonian Doctoral School of Mathematics and
Statistics , Tartu, 2012, 2013.
Conf. Numerical Analysis and Applied Mathematics ICNAAM
2013: Rhodes, Greece, 21-27 September 2013.
14.03-21.03.2013, Israel, Bar-Ilan University, talk and scientiﬁc
work with a co-author.

11th Intern. Conf. on Sampling Theory and Applications, May 25 -
29, 2015, Washington DC (American Uni)
Modern Time-Frequency Analysis, Strobl, Austria, June 1-7, 2014.
10th Intern. Conf. on Sampling Theory and Applications, July 1st -
July 5th, 2013, Jacobs Univ. Bremen
ERASMUS lecturer at Babes - Bolyai University of Cluj - Napoca,
Romania, March, 2013.
3rd Dolomites Workshop on Constructive Approximation and
Applications, Alba di Canazei, September 9-14, 2012.

Ending Projects
Ending projects
Function and sequence spaces in approximations and their
applications, ETF 8627, 2011 - 2014 (2015) (M. Zeltser, AK,
Tatjana Tamberg, Anna Saksa, Tarmo Metsmägi )
Estonian Center of Excellence Mesosystems Theory and
Applications, AU/8211, 2011 - 2015 (AK, M. Zeltser, A. Saksa)

Future: some 2 - 5 years later
Future: some 2 - 5 years later
Retirements, new colleagues with ??? (unknown) qualiﬁcation
With high probability no pure scientiﬁc projects (due to Estonian
politics in science)
No students, no money, no staff (Now still the situation better than
in physics, biology, ...)

2015-12-17 research seminar 2nd part

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2015-12-17 research seminar 2nd part