Math research 2014

Mathematical Analysis and Its Applications in the
Dept. of Math at Tallinn Uni:
the past and possible future
Andi Kivinukk
Matemaatika osakond, Tallinna Ülikool
IFI seminar, TLÜ
November 19, 2014
A. Kivinukk (Tallinna Ülikool) 1 / 18

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 )
functional analysis, mathematical statistics, data analysis

Staff
Part-time lecturer Anna Šeletski (at TLU since )
functional analysis (summability theory)
Doctoral student Tarmo Metsmägi (defences January, 2015)
theory)
Doctoral student Anna Saksa
theory, Fourier series)

Summability methods, speeds of convergence and ...
Comparison of summability methods, speeds of
convergence and statistical convergence
Topics by: Anne Tali, her former doctoral student Anna Šeletski,
Ulrich Stadtmüller (University of Ulm)
A number sequence x = (n) can be convergent or divergent, but only
convergent sequences are needed in practice.
A divergent sequence x = (n) can be transformed into convergent
sequence y = (n) by some transformation A. Then it is said that
sequence x is A-convergent, i.e., convergent with respect summability
method (transformation) A.
The most common transformations A are matrix transformations
A = (ank ) defined by
n =
1X
k=0
ank k ; n = 0; 1; 2; ::::

Summability methods, speeds of convergence and ...
The following problems are discussed for certain families fAg ( is a
continuous parameter).
1) Methods A are compared by their convergence fields (i.e., by the
sets of all A-convergent sequences) and by speed of convergence.
2) The estimates for speeds of methods A are found.
3) Different types of A-convergence, like ordinary convergence,
strong convergence and statistical convergence, are charaterized and
compared.
4) Transformations A are characterized as bounded operators in
sequence spaces lp:
Anna Šeletski defended her doctoral thesis Comparison of
summability methods by summability fields, speeds of convergence
and statistical convergence in a Riesz-type family in TLU in 2011.

Series and sequences
Topics by
Maria Zeltser
Releasing monotonicity assumption in different tests for convergence
of number series using the WM property. A non-negative tending to
zero sequence fakg weak monotone, written WMS; if for some C 0
it satisfies
ak Can for any k 2 [n; 2n]:
It appears that in Maclaurin-Cauchy integral test, Cauchy condensation
test, Schlömilch and Abel’s k-th term theorem monotonicity can be
replaced by WM. Another generalization of monotonicity allows us to
give conditions for a sequence fakg to satisfy mkak ! 0 for a given
sequence fmkg tending to infinity.

Another research interest is related to describing sequence spaces
with the help of 0-1 sequences which it contains. The aim is to find
conditions in case of different types of sequence spaces E when any
given sequence space F with a good structure containing all 0-1
sequences of the space E contains the space E itself.
Could it be interesting for computer science ??? AK

Approximations, Fourier Analysis, Shannon sampling series
Approximations, Fourier Analysis, Shannon sampling
series
Topics by:
AK and his doctoral students Tarmo Metsmägi and Anna Saksa
Consider a periodic function f 2 C2 as a signal it can be recovered by
its Fourier series or by some generalization
Un(f ; x) :=
Xn
k=n
(
k
n )f ^(k)eikx ;
defined by the window function
2 C[1;1]; (0) = 1; (u) = 0 (juj 1):

Approximations, Fourier Analysis, Shannon sampling series
For non-periodic case the Fourier transform or the Shannon sampling
operators
(SWf )(t) :=
X
k2Z
f (
k
W )s(Wt k)
have used. Here the kernel function is defined by
s(t) :=
Z 1
0
(u) cos(tu)du:
Typical problems are how to characterize the error
kf SWf kC:

A selection of Publications
A. Kivinukk, G. Tamberg, On window methods in generalized
Shannon sampling operators. In: New Perspectives on
Approximation and Sampling Theory. A. I. Zayed and G.
Schmeisser (Eds.) Applied and Numerical Harmonic Analysis,
Springer, 2014, 65–88.
Kivinukk, A. and Metsmägi, T. The variation detracting property of
some Shannon sampling series and their derivatives. Sampl.
Theory Signal Image Process., 13 (2014), no 2, 189–206.
A. Kivinukk, On some Shannon sampling series with the variation
detracting property. 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

Kivinukk, A. and Metsmägi, T. Approximation in variation by the
Meyer-König and Zeller operators. Proc. Estonian Acad. Sci.,
2011, 60, 2, 88-97.
Kivinukk, A. and Metsmägi, T. Approximation in variation by the
Kantorovich operators. Proc. Estonian Acad. Sci., 2011, 60, 4,
201-209.
A. Kivinukk, G. Tamberg, Interpolating generalized Shannon
sampling operators, their norms and approximation properties.
Sampl. Theory Signal Image Process. 8 (2009) 77–95.

U. Stadtmüller, A. Tali, A family of generalized Nörlund methods
and related power series methods applied to double sequences,
Math. Nachr., 2009, 282, 2, 288–306.
U. Stadtmüller, A. Tali, A note on families of generalized Nörlund
matrices as bounded operators on lp. Proc. Estonian Acad. Sci.,
2009, 58, 3, 137–145.
A. Šeletski, A. Tali, Comparison of speeds of convergence in
Riesz-type families of summability methods. II, Math. Model.
Anal., 2010, 15, 103–112.
A. Šeletski, A. Tali, Strong summability methods in a Riesz-type
family, Proc. Estonian Acad. Sci., 2011, 60, 4, 238–250.
A. Šeletski, A. Tali, Comparison of strong and statistical
convergences in some families of summability methods, Filomat
(to appear).

(1.1) M. Zeltser, Bounded domains of generalized Riesz methods
with the Hahn property, Journal of Function Spaces and
Applications, 1–8, 2013.
(1.1) M. Zeltser, On equiconvergence of number series,
Mathematica Slovaca, 63(6), 1333–1346, 2014.
(3.1) M. Zeltser, The Hahn property of bounded domains of some
matrix methods, AIP Conference Proceedings, 11TH
INTERNATIONAL CONFERENCE OF NUMERICAL ANALYSIS
AND APPLIED MATHEMATICS 2013: ICNAAM 2013: Rhodes,
Greece, 21-27 Septmeber 2013, 770–773, 2013.
(5.2) M. Zeltser, On the Hahn property of bounded domains of
special matrix methods, 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, 143–143, 2013.

(1.1) M. Zeltser, Factorable Matrices and their associated Riesz
matrices, Proceedings of the Estonian Academy of Sciences.
Physics. Mathematics, 63 (4), 1–7, 2014 [to apppear].
(3.1) S. Tikhonov, M. Zeltser, Weak Monotonicity Concept and Its
Applications, Fourier Analysis . Pseudo-differential Operators,
Time-Frequency Analysis and Partial Differential Equations,
Springer, 357– 374, 2014.
(5.2) M. Zeltser, Application of weak monotonicity in number series
and Hardy inequalities. Abstracts of the International Congress of
Mathematicians (ICM 2014), Seoul, Korea, 272–273, 2014.
(5.2) M. Zeltser, Weak monotonicity concept and its applications,
FINEST MATH 2014 : Fourth Finnish-Estonian Mathematics
Colloquium and Finnish Mathematical Days 2014, Book of
Abstracts, University of Helsinki, 9-10 January 2014, 2014.

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.

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.

Current Projects
Current projects
Function and sequence spaces in approximations and their
applications, ETF 8627, 2011 - 2014 (M. Zeltser, AK, Tatjana
Tamberg, Anna Saksa, Tarmo Metsmägi )
Stochastic processes in nano- and meso-systems : Theory and
applications in material sciences and bio-chemistry, 2012 - 2014
(M. Zeltser)

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, ...)

Math research 2014

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