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Interpolation and the Algebras AP

In: Complex Analysis and Special Topics in Harmonic Analysis

Author

Listed:
  • Carlos A. Berenstein

    (University of Maryland, Mathematics Department and Institute for Systems Research)

  • Roger Gay

    (Université de Bordeaux I, Centre de Recherche en Mathématiques)

Abstract

In the first chapter, we have seen how the Leitmotiv of the boundary values of holomorphic functions lead us naturally to introduce several transforms, in particular, the Fourier-Borel and Fourier transforms, and found out that many questions can be posed in equivalent terms in the algebras of entire functions with growth conditions, Exp(Ω) and F(ɛ’(ℝ)), specially problems relating to convolution equations. In the case of distributions, this relation will be come more evident in Chapter 6. The aim of this chapter is to study a more general class of algebras, the Hörmander algebras, A p (Ω). We shall see that the ideal theory of these algebras is intimately related to the study of interpolation varieties. In the previous volume [BG, Chapter 3], we have shown that to be the case for the algebras of holomorphic functions ℋ(Ω), and we found out that one could study interpolation questions with the help of the inhomogeneous Cauchy-Riemann equation. The same will be the case here. This time, though, we shall be obliged to consider the problem of solving the Cauchy-Riemann equation with growth constraints.

Suggested Citation

  • Carlos A. Berenstein & Roger Gay, 1995. "Interpolation and the Algebras AP," Springer Books, in: Complex Analysis and Special Topics in Harmonic Analysis, chapter 0, pages 109-197, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4613-8445-8_2
    DOI: 10.1007/978-1-4613-8445-8_2
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