On discrete analytic functions: Products, rational functions and reproducing kernels

Daniel Alpay, Palle Jorgensen, Ron Seager, Dan Volok

Research output: Contribution to journalArticlepeer-review

24 Scopus citations

Abstract

We introduce a family of discrete analytic functions, called expandable discrete analytic functions, which includes discrete analytic polynomials, and define two products in this family. The first one is defined in a way similar to the Cauchy-Kovalevskaya product of hyperholomorphic functions, and allows us to define rational discrete analytic functions. To define the second product we need a new space of entire functions which is contractively included in the Fock space. We study in this space some counterparts of Schur analysis.

Original languageEnglish
Pages (from-to)393-426
Number of pages34
JournalJournal of Applied Mathematics and Computing
Volume41
Issue number1-2
DOIs
StatePublished - 1 Mar 2013

Keywords

  • 2D lattice ℤ
  • Cauchy integral representation
  • Cauchy-Kovalevskaya theorem
  • Cauchy-Riemann equations
  • Difference operators
  • Discrete analytic functions
  • Expandable functions
  • Fock space
  • Fourier transform
  • Lie algebra of operators
  • Multipliers
  • Rational functions
  • Realizable linear systems
  • Reproducing kernel Hilbert space
  • Schur analysis

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