A Uniqueness Theorem for Functions with Positive Imaginary Part

  • B. S. Pavlov
Part of the Topics in Mathematical Physics book series (TOMP, volume 4)


In studying the structure of the spectrum of a one-dimensional perturbation of the operator of multiplication by the independent variable in L2(-∞, ∞)the following problem of function theory arises: describe the set of all roots of a function f which is represented in terms of a Cauchy-type integral with positive density which satisfies a certain smoothness condition:
$$\begin{array}{l} f(z) = \int\limits_{ - \infty }^\infty {\frac{{{{\left| {\varphi (t)} \right|}^2}}}{{t - z}}dt - 1,{\mathop{\rm Im}\nolimits} z \ge 0,} \\ \varphi (t) \in Lip\alpha ,\alpha > 0;\left| {\varphi (t)} \right|{t^\alpha } \to 0,t \to \infty \\ \end{array} $$


Unit Disk Half Plane Uniqueness Theorem Regular Function Lower Half Plane 


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Literature Cited

  1. 1.
    A. Zygmund, Trigonometric Series, Vol. 1, Ch. IV, Izd. Mir (1965).Google Scholar
  2. 2.
    L. Carleson, “Sets of uniqueness for functions regular in the unit circle,” Acta Math., Vol. 87, No. 3–4 (1952).Google Scholar

Copyright information

© Consultants Bureau, New York 1971

Authors and Affiliations

  • B. S. Pavlov

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