Mark Agranovsky, Bar-Ilan University, Ramat Gan, Israel Dan ...

Mark Agranovsky
Bar-Ilan University, Ramat Gan, Israel
e-mail: agranovs@math.biu.ac.il
Boundary Forelli Theorem and Meromorphic Extension from Circles
Characterization of holomorphic functions, of one and several complex variables, and
their boundary values, in integral-geometric spirit (via moment conditions on varieties of
curves) attract attention of complex analysts since ’70s. However, many natural and simply
formulated questions are still not answered. I will present a recent result on characterization
of boundary values of holomorphic functions in the complex unit ball, in terms of onedimensional
holomorphic extension along bundles of complex lines. This result is a boundary
analog of celebrated Forelli theorem about holomorphicity on slices, which in turn is a
variation of the clasical Hartogs’ theorem about separate analyticity. The proof of boundary
Forelli theorem is based on reduction to characterization of polyanalytic functions in planar
domains in terms of meromorphic extendibility from chains of circles.
Dan Mangoubi
Hebrew University of Jerusalem, Jerusalem, Israel
e-mail: mangoubi@gmail.com
Geometry of Eigenfunctions and Harmonic Functions
Let (M, g) be a compact Riemannian manifold of dimension n. Let L be the Laplace-
Beltrami operator on M, and let u be an eigenfunction: Lu = λu. We study the geometry
of nodal sets and the nodal domains as λ becomes arbitrary large. In particular, we address
the following problem: Faber-Krahn inequality shows that one has a lower bound on the
volume of each nodal domain A λ :
Vol(A λ ) > C/λ n/2 .
Can one find a ball inside A λ of radius C ε /λ 1/2+ε inside A λ ? The answer is positive in
dimension two and open in dimension n ≥ 3. We will show that in order to obtain bounds
in dimension n ≥ 3, one is naturally lead to consider harmonic functions f in the unit ball
of R n , with f(0) = 0, and to look for lower bounds on the volume of f > 0. I will describe
what is known about these bounds.
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Alexander M. Olevskii
Tel Aviv University, Tel Aviv, Israel
e-mail: olevskii@yahoo.com
Wiener’s ”Closer of Translates” Problem
and Piatetskii-Shapiro Uniqueness Phenomenon
Wiener characterized cyclic vectors (with respect to translations) in l p (Z) and L p (R) (p =
1, 2) in terms of zero sets of Fourier transform. He conjectured that a similar characterization
should be true for 1 < p < 2. I will discuss this conjecture.
Joint work with Nir Lev.
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Yosef Yomdin
Weizmann Institute of Sciences, Rehovot, Israel
e-mail: yosef.yomdin@weizmann.ac.il
Invisible Sets for Integral Measurements
and Moment vanishing Problem
The problem of reconstruction of semi-algebraic sets and functions from integral measurements,
like moments or Fourier transform, naturally arises in Signal Processing. For
certain (incomplete) measurements there are ”invisible” signals. Their description leads
to the moment vanishing problem: give conditions for identical vanishing of the moments
m k = ∫ P k (x)q(x)dx, for various classes of P and q, and various integration domains. Recently
a serious progress has been achieved in some special cases of this problem, and relations
have been found with the Mathieu conjecture in representations of compact Lie groups,
and (through the recent work of Wenhua Zhao) with certain questions around the Jacobian
conjecture.