Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Euler spaces of analytic functions

Author: James Rovnyak
Journal: Trans. Amer. Math. Soc. 308 (1988), 197-208
MSC: Primary 30H05; Secondary 44A15, 46E20
MathSciNet review: 929669
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A formula due to Euler and Legendre is used to construct finite-difference counterparts to the Dirichlet space. The spaces have integral representations and characterizations in terms of area integrals. Their reproducing kernels are logarithms of the reproducing kernels of the Newton spaces, which are counterparts to the Hardy class. A Hilbert space with reproducing kernel

$\displaystyle \log [(1/\overline w z)\,\log \;1/(1 - \overline w z)]$

is also shown to exist and to be related to Bernoulli numbers and combinatorial theory.

References [Enhancements On Off] (What's this?)

  • [1] N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337-404. MR 0051437 (14:479c)
  • [2] L. de Branges, Square summable power series, Springer-Verlag (in preparation).
  • [3] A. Erdélyi, et al., Higher transcendental functions, vol. 1, McGraw-Hill, New York, 1953.
  • [4] C. Jordan, Calculus of finite differences, 3rd ed., Chelsea, New York, 1979.
  • [5] C. Markett, M. Rosenblum, and J. Rovnyak, A Plancherel theory for Newton spaces, Integral Equations and Operator Theory 9 (1986), 831-862. MR 866967 (89a:30038)
  • [6] N. Nielsen, Handbuch der Theorie der Gammafunktion, Chelsea, New York, 1965.
  • [7] N. E. Nörlund, Vorlesungen über Differenzenrechnung, Chelsea, New York, 1954.
  • [8] R. E. A. C. Paley and N. Wiener, Fourier transforms in the complex domain, Amer. Math. Soc. Colloq. Publ., vol. 19, Amer. Math. Soc., Providence, R.I., 1934. MR 1451142 (98a:01023)
  • [9] J. A. Shohat and J. D. Tamarkin, The problem of moments, Math. Surveys, no. 1, Amer. Math. Soc., Providence, R.I., 1943. MR 0008438 (5:5c)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 30H05, 44A15, 46E20

Retrieve articles in all journals with MSC: 30H05, 44A15, 46E20

Additional Information

Article copyright: © Copyright 1988 American Mathematical Society

American Mathematical Society