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.
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D. Alpay thanks the Earl Katz family for endowing the chair which supported his research. The research of the authors was supported in part by the Binational Science Foundation grant 2010117.
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Alpay, D., Jorgensen, P., Seager, R. et al. On discrete analytic functions: Products, rational functions and reproducing kernels. J. Appl. Math. Comput. 41, 393–426 (2013). https://doi.org/10.1007/s12190-012-0608-2
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DOI: https://doi.org/10.1007/s12190-012-0608-2
Keywords
- Discrete analytic functions
- 2D lattice ℤ2
- Reproducing kernel Hilbert space
- Multipliers
- Cauchy integral representation
- Difference operators
- Lie algebra of operators
- Fourier transform
- Realizable linear systems
- Expandable functions
- Rational functions
- Cauchy-Riemann equations
- Cauchy-Kovalevskaya theorem
- Schur analysis
- Fock space