Διήμερο στην Αρμονική και Μιγαδική Ανάλυση

http://fourier.math.uoc.gr/ mk/ch2002

Τμήμα Μαθηματικών
Πανεπιστήμιο Κρήτης
Λεωφόρος Κνωσού
714 09 Ηράκλειο

E-mail: ch2002@fourier.math.uoc.gr

25-26 Μαΐου 2002


Περιεχόμενα

1 Ανακοίνωση

Το Τμήμα Μαθηματικών του Πανεπιστημίου Κρήτης οργανώνει το Σαββατοκύριακο 25-26 Μαΐου 2002 στο Ηράκλειο ένα διήμερο ομιλιών στην Αρμονική και Μιγαδική ανάλυση.

Καλούνται να συμμετάσχουν, είτε δίνοντας μια ομιλία πάνω στην έρευνά τους είτε απλά ως ακροατές, όσοι εργάζονται ερευνητικά στον ευρύτερο τομέα της Αρμονικής και Μιγαδικής Ανάλυσης.

Θα προσπαθήσουμε να καλύψουμε τουλάχιστον τα τοπικά έξοδα όσο γίνεται περισσοτέρων ομιλητών.

2 Πώς να συμμετάσχετε

Δηλώστε έγκαιρα συμμετοχή στέλνοντας e-mail στο

ch2002@fourier.math.uoc.gr
ή με FAX (υπ' όψιν Μ. Κολουντζάκη στο 0810393881) ή ταχυδρομείο.

3 Οργανωτές

Μανόλης Κατσοπρινάκης, Αναπληρωτής Καθηγητής, Πανεπ. Κρήτης
Μιχάλης Κολουντζάκης, Αναπληρωτής Καθηγητής, Πανεπ. Κρήτης
Βασίλης Νεστορίδης, Καθηγητής, Πανεπ. Αθηνών
Μιχάλης Παπαδημητράκης, Αναπληρωτής Καθηγητής, Πανεπ. Κρήτης

4 Επιβεβαιωμένοι ομιλητές

  1. Γιώργος Αλεξόπουλος, Orsay
    The asymptotic behavior of convolution powers on Lie groups
    We shall explain the method of Bloch wave expansions and how it can be used to study the asymptotic behavior of convolution powers of densities on Lie groups.

  2. Νίκος Ατρέας, Α.Π.Θ.
    Sampling in multiresolution analysis subspaces of periodic sequences
    We establish the notion of multiresolution analysis (MRA) on the space of N-periodic sequences, where $ N=p^M, \; p,M \in \bf Z\rm ^+$ and we give sufficient conditions for the existence of sampling Theorems on MRA subspaces; in particular we present several examples of sampling formulas of Haar and Shannon-type. Each sampling expansion is an MRA projection, so we estimate its Truncation error.

  3. Βάγια Βλάχου, Πανεπιστήμιο Αθηνών
    Identical approximative sequences for various notions of universality (with G. Costakis)
    We will prove that there exist functions which are universal with respect to overconvergence, to derivatives, to antiderivatives and to translations and they realise approximations simultaneously with the same approximation sequence. Our result is generic.

  4. Μαρίζα Ζυμωνοπούλου, University of Missouri-Columbia
    Extremal sections of complex $ \ell_p$-balls, $ 0<p\leq 2$
    Let $ 0< p \leq 2.$ For $ \xi \in {\mathbf C}^{n}, \xi\neq 0,$ denote by $ H_{\xi}=\left\{x \in {\mathbf C}^{n}: (x,\xi )=0\right\}$ the complex hyperplane in $ {\mathbf C}^{n}$ orthogonal to $ \xi.$ The $ (n-1)-$dimensional complex volume of $ B_{p}({\mathbf C}^{n})\cap H_{\xi}$ is minimal if $ \vert\xi^{1}\vert= \ldots = \vert\xi^{n}\vert,$ for $ \xi=(\xi^{1},\xi^{2}, \ldots ,\xi^{n}),$ where $ \xi^{j} \in {\mathbf C}, j=1, \ldots ,n$ and maximal, if $ \xi$ has only one non-zero coordinate, $ (\xi^{1},0, \ldots ,0), \xi^{1} \in {\mathbf C}. $

  5. Μιχάλης Κολουντζάκης, Πανεπιστήμιο Κρήτης
    On a problem of Turan about positive definite functions (with S.G. Revesz)
    We study the following question posed by Turan. Suppose $\Omega$ is a convex body in Euclidean space ${\mathbf R}^d$ which is symmetric with respect to the origin. Of all positive definite functions supported in $\Omega$, and with value $1$ at the origin, which one has the largest integral? It is probably the case that the extremal function is the indicator of the half-body convolved with itself and properly scaled, but this has been proved only for a small class of domains so far. We add to this class of known Turan domains the class of all spectral convex domains. These are all convex domains which have an orthogonal basis of exponentials $e_\lambda(x) = \exp 2\pi i{\langle \lambda, x \rangle}$, $\lambda \in {\mathbf R}^d$. As a corollary we obtain that all convex domains which tile space by translation are Turan domains.

    We also give a new proof that the Euclidean ball is a Turan domain.

  6. Μιχάλης Μαριάς, Α.Π.Θ.
    $ H^{1}$-boundedness of imaginary powers of the Laplacian on Riemannian manifolds
    Let $ M$ be a complete, noncompact Riemannian manifold. We assume that $ M$ satisfies the doubling volume property i.e.

    $\displaystyle V(x,2r)\leq CV(x,r), $

    for all $ x\in M$ and $ r>0$, and that the Poincare inequality

    $\displaystyle \int_{B(x,r)}\left\vert f(x)-f_{B(x,r)}\right\vert ^{2}dx\leq Cr^{2}\int _{B(x,2r)}\left\Vert \nabla f(x)\right\Vert ^{2}dx $

    holds on $ M$.

    If $ \Delta$ is the Laplace-Beltrami operator, we show that its imaginary powers $ \left( -\Delta\right) ^{i\gamma}$, $ \gamma\in\mathbb{R}$, are bounded on the Hardy space $ H^{1}\left( M\right) $.

  7. Δημήτρης Μπετσάκος, Πανεπιστήμιο Κρήτης
    Inequalities for harmonic measure in the unit disk
    Let $ K$ be a compact subset of the unit disk $ D$ and let $ K^*$ be the circular projection of $ K$ on the radius $ (0,1)$. We will prove the following inequalities for the harmonic measure:


    1. If $ 0\leq r<1$ and $ \pi/2\leq \theta\leq \pi$, then

    $\displaystyle \omega(re^{i\theta},K,D)+\omega(re^{-i\theta},K,D)\geq \omega(re^{i\theta},K^*,D)+\omega(re^{-i\theta},K^*,D).$    


    2. If $ 0\leq r<1,\;0<\theta<\pi/2$ and $ K\subset D \cap \{z:\Re z\geq 0\}$, then

    $\displaystyle \omega(re^{i\theta},K,D)+\omega(re^{-i\theta},K,D)\geq \omega(ir,K^*,D)+\omega(-ir,K^*,D).$    

    These results extend the classical Beurling-Nevanlinna projection theorem.

  8. Βασίλης Νεστορίδης, Πανπιστήμιο Αθηνών
    Generic approximation of all orders derivatives by Taylor series
    Let $ \Omega\subset{\mathbf C}$ be a simply connected domain, $ \zeta\in\Omega$ and $f$ holomorphic in $\Omega$. We denote by $ S_N(z) = \sum_{n=0}^N{f^{(n)}(\zeta) \over n!} (z-\zeta)^n$. Let $ K\subset{\mathbf C}$ be a compact set such that $ K^c$ is connected and $ K \subset \overline{\Omega}^c$ (or $ K\subset\Omega^c$, respectively). Let $ h$ be a polynomial. We are looking for a sequence $ \lambda_n \in {\left\{{0,1,2,\ldots}\right\}}$ such that:
    (i) For every compact set $ L \subset {\mathbf C}$ with $ L^c$ connected such that $ L \subset \overline{\Omega}$ (or $ L\subset \Omega$, respectively) and every $ \ell \in {\left\{{0,1,2,\ldots}\right\}}$ we have:

    $\displaystyle \sup_{z\in L}{\left\vert{S_{\lambda_n}^{(\ell)}(z) - f^{(\ell)}(z)}\right\vert} \to 0 $

    and
    (ii)

    \begin{displaymath} \sup_{z\in K}{\left\vert{S_{\lambda_n}^{(\ell)}(z) - h^{(\el... ..., \ \text{for all $\ell\in{\left\{{0,1,2,\ldots}\right\}}$}. \end{displaymath}

    In the case $ K\subset\Omega^c$, $ L\subset \Omega$, we have a generic property in $ H(\Omega)$. In the case $ K \subset \overline{\Omega}^c$, $ L \subset \overline{\Omega}$ we have a generic property in a natural subspace of $ A^\infty(\Omega)$ provided that $ {\left\{{\infty}\right\}} \cup ({\mathbf C}\setminus\overline{\Omega})$ is connected. It is not possible to have $ K\subset\Omega^c$ and $ L \subset \overline{\Omega}$ simultaneously.

  9. Αρίστος Συσκάκης, Α.Π.Θ. ΑΚΥΡΩΘΗΚΕ
    Classical matrices and composition operators
    We point out that certain classical matrices when acting on spaces of analytic functions, can be written in terms of composition operators. These are the Cesàro and Hilbert matrices,

    \begin{displaymath} {\textbf{Ces}}= \left(%% \begin{array}{ccccc} 1 & 0 & 0 & .... ...frac{1}{5} & . \\ . & . & . & . \\ \end{array}%% \right), \end{displaymath}

    and certain Hausdorff of the form

    $\displaystyle {\textbf{Hau}}= \left( \begin{matrix}c_{0,0}& 0 & 0& . c_{1,0... ... &0& . c_{2,0} & c_{2,1} & c_{2,2} & . . & . &. &. \end{matrix} \right)$    

    where

    $\displaystyle c_{n,k}=\binom{n}{k}\Delta^{n-k}\mu_k, \quad k\leq n,$    

    with $ \Delta$ the forward difference operator $ \Delta\mu_n=\mu_n-\mu_{n+1}$ on scalar sequences, and $ \mu_n$ a moment sequence i.e.

    $\displaystyle \mu_n=\int_0^1t^n d\mu(t), \qquad n=0,1,\cdots ,$    

    where $ \mu$ is a positive finite Borel measure on $ (0,1]$.

  10. Νίκος Στυλιανόπουλος, Πανεπιστήμιο Κύπρου
    Zero Distribution of Bergman Orthogonal Polynomials for Certain Planar Domains (with A.L. Levin and E.B. Saff)
    Let $G$ be a simply-connected domain in the complex plane bounded by a closed Jordan curve $L$, and let $P_n$, $n\geq 0$, be polynomials of respective degrees $n=0,1,\ldots$ that are orthonormal in $G$ with respect to the area measure (the so-called Bergman polynomials). Let $f$ be a conformal map of $G$ onto the unit disk. We characterize in terms of the asymptotic behavior of the zeros of $P_n$'s the case when $f$ has a singularity on $L$. To investigate the opposite case we consider a special class of lens-shaped domains $G$ that are bounded by two orthogonal circular arcs. Utilizing the theory of logarithmic potentials with external fields, we show that the limiting distribution of the zeros of the $P_n$'s for such lens domains is supported on a Jordan arc joining the two vertices of $G$. We determine this arc along with the distribution function.

  11. Στάθης Φίλιππας, Πανεπιστήμιο Κρήτης
    Improved Hardy inequalities
    Let $\Omega$ be a bounded domain in $ {\mathbf R}^N$, $ N \geq 3$, containing the origin. We consider the Hardy inequality

    $\displaystyle \int_{\Omega} \vert\nabla u(x)\vert^2 dx \geq \left(\frac{N-2}{2} \right)^2 \int_{\Omega} \frac{u^2(x)}{\vert x\vert^2} dx. $

    We show that this can be repeatedly improved by adding specific lower order terms in the right hand side, thus obtaining a ``series expansion'' of Hardy's inequality. We also give various extensions.

5 Άλλες συμμετοχές (εκτός Πανεπιστημίου Κρήτης)

  1. Δημήτρης Γατζούρας, Γεωπονικό Πανεπιστήμιο
  2. Σταύρος Παπαδόπουλος

6 Πρόγραμμα ομιλιών

Οι ομιλίες θα γίνουν στην αίθουσα Λ 221 του προκατασκευσμένου κτιρίου του Πανεπιστημίου Κρήτης στο συγκρότημα της Λεωφ. Κνωσού. Η αίθουσα είναι στη βορειότερη πτέρυγα του κτιρίου, πάνω από το κυλικείο.

Ώρα Ομιλητής Τίτλος
Σάββατο    
10:00-10:45 Νίκος Ατρέας Sampling in multiresolution analysis subspaces of periodic sequences
11:00-11:45 Μαρίζα Ζυμωνοπούλου Extremal sections of complex $ \ell_p$-balls, $ 0<p\leq 2$
12:00-12:45 Νίκος Στυλιανόπυλος Zero Distribution of Bergman Orthogonal Polynomials for Certain Planar Domains
     
16:00-16:45 Βάγια Βλάχου Identical approximative sequences for various notions of universality
17:00-17:45 Βασίλης Νεστορίδης Generic approximation of all orders derivatives by Taylor series
     
18:15-19:00 Δημήτρης Μπετσάκος Inequalities for harmonic measure in the unit disk
19:15-20:00 Στάθης Φίλιππας Improved Hardy inequalities
Κυριακή    
10:00-10:45 Γιώργος Αλεξόπουλος The asymptotic behavior of convolution powers on Lie groups
11:00-11:45 Μιχάλης Μαριάς $ H^{1}$-boundedness of imaginary powers of the Laplacian on Riemannian manifolds
     
12:15-13:00 Μιχάλης Κολουντζάκης On a problem of Turan about positive definite functions


Mihalis Kolountzakis 2002-05-24