# The Analysis Seminar in Thessaloniki

2007-2008

ARISTOTLE UNIVERSITY OF THESSALONIKI

DEPARTMENT OF
MATHEMATICS

DIVISION OF
MATHEMATICAL ANALYSIS

Tel. 2310 997906, Fax 2310
997945

Organizing Committee:

D.Betsakos

I.
Gasparis

Seminar Room: 3rd floor in the building of the School of
Sciences

# Programm of Talks

The seminar takes place
every Tuesday 11-12 a.m. (unless otherwise stated)

### Tuesday, October 30, 2007.

**Vasileios Spentzos**, Aristotle
University of Thessaloniki.

Title: *The spectral theorem for self-adjoint
operators on Hilbert spaces.*

(Master thesis).

### Tuesday, November 6, 2007.

**Nina Snigireva**, University of St
Andrews, Scotland.

Title: *Fourier transforms of in-homogeneous
self-similar measures.*

### Tuesday, November 13, 2007.

**Anestis Fotiadis**, University of
Warwick, U.K.

Title: *Harmonic maps between manifolds of negative
curvature.*

### Tuesday, November 20, 2007.

**Ioanna Kirezi**, University of
Crete.

Title: *Monotonicity properties of Darboux sums.*

## Analysis Mini Conference

### Tuesday, November 27, 2007.

organized by M. Marias

10.30-11.15
**Nikos Bournaveas (Univ. of Edinbourgh),**

*Velocity averaging lemmas
for kinetic transport equations.*

11.30-12.00 **Michel Marias
(A.U.Th.),**

*Spectral multipliers and Riesz transforms on
manifolds.*

12.00-13.00. Coffee break

13.15-14.00
**Paschalis Karageorgis (Univ. of Dublin),**

*Existence and behaviour
of solutions for wave equations with potential*.

14.15-14.45
**Dimitrios Betsakos (A.U.Th.),**

*Symmetrization and heat kernels for
the Laplacian.*.

### Tuesday, December 11, 2007.

**Anestis Tsomidis**, Aristotle
University of Thessaloniki.

Title: *Iteration of complex polynomials-Julia
sets.*

(Master thesis).

### Tuesday, December 18, 2007.

**Minos Petrakis**, Technical University
of Crete.

Title: *Radon-Nikodym property on Banach spaces.*

### Tuesday, January 8, 2008.

**Dimitris Koukoulopoulos**, University of
Illinois at Urbana-Champaign.

Title: *A probabilistic approach to number
theory and its application to a problem on localization of factorizations.*

**Abstract:** Probabilistic number theory was born in 1917 in a paper by
Hardy and Ramanujan and was further developed by Turan, Kubilius, Cramer,
Erd\"os, Brun, Selberg and various others. Despite the deterministic nature of
number theory, probabilistic arguments offer very accurate predictions about the
behavior of various arithmetic quantities, which then we have to establish
rigorously. I will present the application of such ideas to the following
problem: how many integers up to $x$ are divisible by a product of $k$ integers
with each of the factors lying in a prescribed interval?

### Tuesday, March 18, 2008.

**Michel Marias**, A.U.Th..

Title:
*The spectrum of the Laplacian and the Riesz transforms on hyperbolic
manifolds.*

### Tuesday, April 8, 2008.

**Vassilios Rothos **, Division of
Mathematics, Faculty of Engineering, A.U.Th..

Title: *Stationary and
Travelling Waves in Nonlinear Lattices with Analytical Methods.*

**Abstract:** We study stationary and travelling waves on a
two-dimensional lattice with linear and nonlinear coupling between nearest
particles and a periodic nonlinear substrate potential. We show the existence of
both uniform sliding states and periodic travelling waves as well in a
two-dimensional sine-Gordon lattice equation using topological and variational
methods. We present some recent results on dicrete Nonlinear Schrodinger
equation with applications. We use variational methods (mountain-pass) and
dynamical systems methods (center manifold, normal forms).

### Tuesday, April 15, 2008.

**Athanasia Baharoglou, **Department of
Mathematics, A.U.Th.

Title: *Extended Universal Taylor series on doubly
connected domains*

**Abstract:** We prove the existence of
universal Taylor series on doubly connected domains which are valid on a part of
the boundary. On another part of the boundary the universal functions and their
derivatives are continuously extendable. Furthermore, we show similar results as
those mentioned above with respect to every center. In conclusion, we mention
that universal functions may vanish at 1.

### Tuesday, May 27, 2008.

11:00-12:00

**Anna Savvopoulou,
**S.U.N.Y. at Albany

Title: *Almost everywhere convergence of a case of
weighted averages*

**Abstract:** Given a probability measure $\ds
\mu$ on $\ds\mathbb{Z}$, Calder\'on and Bellow proved a weak type inequality for
the maximal operator of $\ds
\mu_{n}\left(f(x)\right)=\sum_{k\in\mathbb{Z}}\mu^{n}(k)f\left(\sigma^{k}(x)\right)$
where $\ds \mu^{n}$ denotes the convolution product. This talk will focus on the
case of a sequence of probability measures on $\ds \mathbb{Z}$, denoted by $\ds
\left\{\mu_{n}\right\}$, obtained inductively in the following way, $\ds
\mu_{1}=\nu_{1}$, $\ds \mu_{2}=\nu_{1}\ast\nu_{2}$, $\ds\ldots$, $\ds
\mu_{n}=\nu_{1}\ast\nu_{2}\ast\cdots\ast\nu_{n}$, where each one of the $\ds
\nu_{i}$ is in turn a strictly aperiodic probability measure on $\ds \mathbb{Z}$
with expectation $0$ and finite second moment. We will discuss the almost
everywhere convergence of the operators $\ds
\mu_{n}f(x)=\sum_{k\in\mathbb{Z}}\mu_{n}(k)f(\sigma^{k}x)$ for $\ds f\in
L^{1}(X)$ and $\ds x\in X$. Throughout the talk $\ds \sigma $ will stand for a
measure preserving transformation of a probability measure space $\ds X$.

12:00-13:00

**Mihalis Kolountzakis, **University of Crete

Title: *The discrepancy of a needle on a checkerboard
*

**Abstract:**Consider the plane as a checkerboard, with each unit
square colored black or white in an arbitrary manner. We show that for any such
coloring there are straight line segments, of arbitrarily large length, such
that the difference of their white length minus their black length, in absolute
value, is at least the square root of their length, up to a miltiplicative
constant. For the corresponding "finite" problem (NXN checkerboard) we also
prove that we can color it in such a way that the above quantity is at most C
\sqrt{N \log N}, for any placement of the line segment.

### Tuesday, June 3, 2008.

**Christopher Wedrychowicz, **S.U.N.Y. at
Albany

Title: *Discrepancy in the behavior of Ergodic averages along
subsequences*

**Abstract:**The classical theorem of Birkhoff
states that the $\ds
T^{N}f(x)=\frac{1}{N}\sum_{k=0}^{N-1}f\left(\sigma^{k}x\right)$ converges almost
everywhere for $\ds x\in X$ and $\ds f\in L^{1}(X)$, where $\ds \sigma$ is a
measure preserving transformation of a probability measure space $\ds X$. It was
shown that there are operators of the form $\ds
T^{N}f(x)=\frac{1}{N}\sum_{k=0}^{N-1}f\left(\sigma^{n_{k}}x\right)$ for a
subsequence $\ds \{n_{k}\}$ of the positive integers that converge in some $\ds
L^{p}$ spaces while diverging in others. The topic of this talk will examine
this phenomenon in the class of Orlisz spaces $\ds
\left\{L\mbox{Log}^{\beta}L:\beta>0\right\}$

### Friday, June 13, 2008.

11:00-12:00

**Vassili Nestoridis,
**University of Athens

Title: *Some recent results on universal
series*

**Abstract:**We do not know explicitly any universal
Taylor series, but we show that it is possible to build an efficient algorithm
constructing such a series in the disk, half-plane, a polygon or an angle. This
does not seem to be possible when the boundary is complicated. Using the
recently published “ Abstract theory of universal series ” we give a sufficient
condition assuring existence of universal series with small (but not summable)
coefficients. As applications we have universal approximation by translations of
meromorphic functions, as -function, by translations of fundamental solutions
of elliptic operators, by translations of approximate identities, as normal
distributions, as well as, the existence of non periodic universal trigonometric
series.

### Tuesday, June 24, 2008.

**Nadia Gal, **University of Missouri,
Columbia

Title: *Examples of Isometric Equivalence Problem*

**Abstract:**Two operators A and B defined from a Banach space X into
a Banach space Y are isometrically equivalent if there exist surjective
isometries U_X, U_Y of X, respectively of Y, such that U_Y A = B U_X . My focus
is the isometric equivalence problem for a spectrum of operators on a variety of
Banach spaces. I establish conditions of the intertwining surjective isometries
U_X and U_Y for matrix operators on sequence spaces, differentiated composition
operators, integrated composition operators on spaces of analytic functions and
integral operators on a space of absolutely continuous vector-valued functions.
I prove that the Cesaro operator Cf(x)= 1/x \int_0^x f(t) dt, defined on a
rearrangement-invariant Banach function space X commutes with an invertible
isometry if and only if the isometry is modulus one multiple of the identity.

The
Analysis Seminar in 2006-2007.

The
Analysis Seminar in 2005-2006.

The
Analysis Seminar in 2004-2005.

The
Analysis Seminar in 2003-2004.

The
Analysis Seminar in 2002-2003.(in Greek)

The
Analysis Seminar in 1992-2002. (in Greek)