2015-12-17 research seminar 2nd part

Mathematical Analysis and Its Applications in theDept. of Math at Tallinn Uni:

the past, present and possible future

Andi Kivinukk

Matemaatika osakond, Tallinna Ülikool

DTI seminar, TLÜdetsember 16, 2015

A. Kivinukk (Tallinna Ülikool) 1 / 20

Staff

Staff

Prof Anne Tali (at TLU since 1973)

Mathematical competencies: mathematical analysis, applications offunctional analysis (summability theory)

Prof AK (at TLU since 1993)

Mathematical competencies: mathematical analysis (approximationtheory and its applications, in particular in signal analysis), Fourieranalysis, mathematical finance (option theory), optimization

Senior researcher Maria Zeltser (at TLU since 2004 )

Mathematical competencies: mathematical analysis, applications offunctional analysis, mathematical statistics, data analysis


Staff

Lecturer, PhD Anna Šeletski (at TLU since 2014)

Mathematical competencies: mathematical analysis (partial differentialequations), applications of functional analysis (summability theory)

Non-staff member researcher, PhD Tarmo Metsmägi

Mathematical competencies: mathematical analysis (approximationtheory)

Doctoral student Anna Saksa

Mathematical competencies: mathematical analysis (approximationtheory, Fourier series)


Approximations, Fourier Analysis, Shannon sampling series

Approximations, Fourier Analysis, Shannon samplingseries

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 approachesthat, when combined with increasing computational power, extendedthe decimal representation of π to, as of 2015, over 13.3 trillion (1013)digits.



Usually we use an approximation π̃ = 3.14, as you know - probably ?

Using the approximation we did the error π − π̃ < 0.0016.

Some persons may like the approximation π̃ = 3.1416, in that case theerror isπ̃ − π < 0.000075 or π − π̃ > −0.000075or using the absolute value |π − π̃| < 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 truevalue - 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 generallyoperators . You may consider these as input-output machines , like amincing machine.

For quite arbitrary functions f the Fourier partial sums Snf perform anuniversal approximation method.

The error, analogically to the absolute value, is given by ||f − Snf ||, andhere the parameter n is a natural number and for bigger n theapproximation will be better.

But (Arbitrary cannot be perfect !) even for the continuous functions(these have continuous graphs) the Fourier series may fail. In this casesome generalization is used:

Un(f , x) :=n∑

k=−n

λ(kn)f∧(k)eikx .



In fact, the basis functions eikx = cos kx + i sin kx are 2π-periodic. Fornon-periodic case the Fourier transform or the Shannon samplingoperators

(SW f )(t) :=∑

k∈Z

f (kW

)s(Wt − k)

are used. For a finite interval, e.g. [0,1], another type of operators

(Bnf )(x) =n∑

k=0

f (kn)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 ofconvergence and statistical convergence

Topics by: Anne Tali , her former doctoral student, now Lecturer, PhDAnna Šeletski and co-author Ulrich Stadtmüller (University of Ulm)

A number sequence x = (ξn) can be convergent or divergent, but onlyconvergent sequences are common in practice.

Example. The sequence (1, 12 ,

13 ,

14 , ...,

1100 , ...) 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 convergentsequence y = (ηn) by some operator A. Then it is said that sequencex 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 acontinuous parameter).

1) Methods Aα are compared by their sets of all Aα-convergentsequences 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 andcompared.

4) Transformations Aα are characterized as bounded operators insequence spaces lp.



Anna Šeletski

is involved in another project with Jaan Janno , Tallinn Uni ofTechnology, studying the solitary waves (in channels or in some microelements 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 thatthis series is divergent.



More hopeful seems to be

∞∑

k=0

1(k + 1)2 = 1 +

122 +

132 + ...+

11002 + ... ,

because, although the number of terms is infinity, at the "end" we addvery 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 Zeltserfound 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 NumericalHarmonic Analysis, Springer, 2014, 65–88.

Sampling Theory in Signal and Image Processing, 13 (2014), no2, 189–206, and many other issues

In Proc. of the 9th Intern. Conf. on Sampling Theory andApplications , 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 INTERNATIONALCONFERENCE OF NUMERICAL ANALYSIS AND APPLIEDMATHEMATICS 2013: ICNAAM 2013: Rhodes, Greece, 21-27September 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 MathematicsColloquium and Finnish Mathematical Days 2014, Book ofAbstracts, 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, EstonianMathematical Society, 2013.


Conferences, Work-shops, Seminars, (co-)organized by our working group

Conferences, Work-shops, Seminars, (co-)organizedby our working group

Approximations, Summations and Applications, Laulasmaa, Dec.11, 2015Methods of Analysis and Algebra, Intern. conf. dedicated to thecentennial of professor Gunnar Kangro, Tartu, Estonia, September1-6, 2013International Workshop on Approximations, Harmonic Analysis,Operators and Sequences" , Narva-Joesuu, Oct. 3-5, 2008Finnish-Estonian Mathematics Colloquium = FinEst Math 2002,TallinnNB ! 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 FinnishMathematical 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 andStatistics , Tartu, 2012, 2013.

Conf. Numerical Analysis and Applied Mathematics ICNAAM2013: Rhodes, Greece, 21-27 September 2013.

14.03-21.03.2013, Israel, Bar-Ilan University, talk and scientificwork 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 andApplications, Alba di Canazei, September 9-14, 2012.


Ending Projects

Ending projects

Function and sequence spaces in approximations and theirapplications, ETF 8627, 2011 - 2014 (2015) (M. Zeltser, AK,Tatjana Tamberg, Anna Saksa, Tarmo Metsmägi )

Estonian Center of Excellence Mesosystems Theory andApplications, 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 Estonianpolitics in science)

No students, no money, no staff (Now still the situation better thanin physics, biology, ...)


2015-12-17 research seminar 2nd part

Education