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Restrictions of analytic functions. III


Authors: Marvin Rosenblum and James Rovnyak
Journal: Proc. Amer. Math. Soc. 52 (1975), 222-226
MSC: Primary 47B35; Secondary 30A78, 42A40
DOI: https://doi.org/10.1090/S0002-9939-1975-0399926-2
MathSciNet review: 0399926
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Abstract: Characterizations are given of all real and all pure imaginary functions on a real Borel set which occur as restrictions of boundary functions of functions holomorphic and having positive imaginary part in the upper half-plane.


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  • [1] N. Aronszajn and W. F. Donoghue, Jr., On exponential representations of analytic functions in the upper half-plane with positive imaginary part, J. Analyse Math. 5 (1956/57), 321-388; Supplement, ibid. 12 (1964), 113-127. MR 29 #6025. MR 0168769 (29:6025)
  • [2] M. Rosenblum and J. Rovnyak, Two theorems on finite Hilbert transforms, J. Math. Anal. Appl. 48 (1974), 708-720. MR 0365006 (51:1259)
  • [3] -, Restrictions of analytic functions. I, Proc. Amer. Math. Soc. 48 (1975), 113-119. MR 0399924 (53:3764a)
  • [4] -, Restrictions of analytic functions, II, Proc. Amer. Math. Soc. 51 (1975), 335-343. MR 0399925 (53:3764b)
  • [5] E. C. Titchmarsh, Introduction to the theory of Fourier integrals, 2nd ed., Clarendon Press, Oxford, 1948.
  • [6] F. G. Tricomi, Integral equations, Interscience, New York, 1957. MR 20 #1177. MR 0094665 (20:1177)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1975-0399926-2
Keywords: Hilbert transform, analytic function with positive imaginary part, exponential representation
Article copyright: © Copyright 1975 American Mathematical Society

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