Characterization of polyanalytic functions by meromorphic extensions from chains of circles

Mark L. Agranovsky

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Let Ct = {z ∈ ℂ: {pipe}z - c(t){pipe} = r(t), t ∈ (0, 1)} be a C1-family of circles in the plane such that limt→0+Ct = {a}, limt→1-Ct = {b}, a ≠ b, and {pipe}c′(t){pipe}2 + {pipe}r′(t){pipe}2 ≠ 0. The discriminant set S of the family is defined as the closure of the set {c(t) + r(t)w(t), t ∈ [0, 1]}, where w = w(t) is the root of the quadratic equation c̄′(t)w2 + 2r′(t)w + c′(t) = 0 with {pipe}w{pipe} < 1, if such a root exists. Suppose that S does not contain a continuous curve joining a and b. We prove that if a smooth and regular (in a certain sense) function f in Ω = ∪Ct, possesses, for each t ∈ (0, 1), a meromorphic extension inside Ct whose only singular point is a pole at c(t) of order at most ν ∈ {0} ∪ ℕ, then f is polyanalytic of order at most ν, i. e., f(z) = h0(z) + h1(z)z̄+̇̇̇z̄νhν(z), where the h′js are analytic functions in Ω. For ν = 0, the condition on S can be omitted. If {pipe}c′(t){pipe} > {pipe}r′(t){pipe}, t ∈ (0, 1), then S = ∅ and the condition of regularity can be dropped. A hyperbolic version of the result is also given.

Original languageEnglish
Pages (from-to)305-329
Number of pages25
JournalJournal d'Analyse Mathematique
Issue number1
StatePublished - Jan 2011

Bibliographical note

Funding Information:
∗This work was partially supported by ISF (Israel Science Foundation), Grant 688/08. Some of this research was done as a part of European Networking Program HCAA.


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