27,433 research outputs found
Poncelet's Theorem, Paraorthogonal Polynomials and the Numerical Range of Compressed Multiplication Operators
There has been considerable recent literature connecting Poncelet's theorem
to ellipses, Blaschke products and numerical ranges, summarized, for example,
in the recent book [11]. We show how those results can be understood using
ideas from the theory of orthogonal polynomials on the unit circle (OPUC) and,
in turn, can provide new insights to the theory of OPUC.Comment: 46 pages, 4 figures; minor revisions from v1; accepted for
publication in Adv. Mat
Integral representations and Liouville theorems for solutions of periodic elliptic equations
The paper contains integral representations for certain classes of
exponentially growing solutions of second order periodic elliptic equations.
These representations are the analogs of those previously obtained by S. Agmon,
S. Helgason, and other authors for solutions of the Helmholtz equation. When
one restricts the class of solutions further, requiring their growth to be
polynomial, one arrives to Liouville type theorems, which describe the
structure and dimension of the spaces of such solutions. The Liouville type
theorems previously proved by M. Avellaneda and F.-H. Lin, and J. Moser and M.
Struwe for periodic second order elliptic equations in divergence form are
significantly extended. Relations of these theorems with the analytic structure
of the Fermi and Bloch surfaces are explained.Comment: 48 page
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