A Cauchy-Riemann equation for generalized analytic functions
HTML articles powered by AMS MathViewer
- by John Wermer PDF
- Proc. Amer. Math. Soc. 138 (2010), 1667-1672 Request permission
Abstract:
We denote by $T^{2}$ the torus: $z = \exp i\theta , w = \exp i\phi$, and we fix a positive irrational number $\alpha$. $A_{\alpha }$ denotes the space of continuous functions $f$ on $T^{2}$ whose Fourier coefficient sequence is supported by the lattice half-plane $n + m\alpha \geq 0$. R. Arens and I. Singer introduced and studied the space $A_{\alpha }$, and it turned out to be an interesting generalization of the disk algebra. Here we construct a differential operator $X_{\Sigma }$ on a certain 3-manifold $\Sigma _{0}$ such that $X_{\Sigma }$ characterizes $A_{\alpha }$ in a manner analogous to the characterization of the disk algebra by the Cauchy-Riemann equation in the disk.References
- Richard Arens and I. M. Singer, Generalized analytic functions, Trans. Amer. Math. Soc. 81 (1956), 379–393. MR 78657, DOI 10.1090/S0002-9947-1956-0078657-5
- Theodore W. Gamelin, Uniform algebras, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1969. MR 0410387
- G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, Oxford, at the Clarendon Press, 1954. 3rd ed. MR 0067125
- Henry Helson and David Lowdenslager, Prediction theory and Fourier series in several variables, Acta Math. 99 (1958), 165–202. MR 97688, DOI 10.1007/BF02392425
- Kenneth Hoffman and I. M. Singer, Maximal subalgebras of $C(\Gamma )$, Amer. J. Math. 79 (1957), 295–305. MR 85478, DOI 10.2307/2372683
Additional Information
- John Wermer
- Affiliation: Department of Mathematics, Brown University, 151 Thayer Street, Providence, Rhode Island 02912
- Email: wermer@math.brown.edu
- Received by editor(s): May 8, 2009
- Published electronically: December 18, 2009
- Communicated by: Franc Forstneric
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 138 (2010), 1667-1672
- MSC (2000): Primary 32-XX
- DOI: https://doi.org/10.1090/S0002-9939-09-10228-9
- MathSciNet review: 2587451