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

Daniel Alpay; Palle Jorgensen; Ron Seager; Dan Volok

doi:10.1007/s12190-012-0608-2

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

Daniel Alpay, Palle Jorgensen, Ron Seager, Dan Volok

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

32 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 language	English
Pages (from-to)	393-426
Number of pages	34
Journal	Journal of Applied Mathematics and Computing
Volume	41
Issue number	1-2
DOIs	https://doi.org/10.1007/s12190-012-0608-2
State	Published - 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

ASJC Scopus subject areas

Computational Mathematics
Applied Mathematics

Access to Document

10.1007/s12190-012-0608-2

Cite this

@article{0225bf83cc0d4ee9b2737b9141fe5bc7,

title = "On discrete analytic functions: Products, rational functions and reproducing kernels",

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.",

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",

author = "Daniel Alpay and Palle Jorgensen and Ron Seager and Dan Volok",

year = "2013",

month = mar,

day = "1",

doi = "10.1007/s12190-012-0608-2",

language = "English",

volume = "41",

pages = "393--426",

journal = "Journal of Applied Mathematics and Computing",

issn = "1598-5865",

publisher = "Springer Nature",

number = "1-2",

}

TY - JOUR

T1 - On discrete analytic functions

T2 - Products, rational functions and reproducing kernels

AU - Alpay, Daniel

AU - Jorgensen, Palle

AU - Seager, Ron

AU - Volok, Dan

PY - 2013/3/1

Y1 - 2013/3/1

N2 - 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.

AB - 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.

KW - 2D lattice ℤ

KW - Cauchy integral representation

KW - Cauchy-Kovalevskaya theorem

KW - Cauchy-Riemann equations

KW - Difference operators

KW - Discrete analytic functions

KW - Expandable functions

KW - Fock space

KW - Fourier transform

KW - Lie algebra of operators

KW - Multipliers

KW - Rational functions

KW - Realizable linear systems

KW - Reproducing kernel Hilbert space

KW - Schur analysis

UR - http://www.scopus.com/inward/record.url?scp=84873339370&partnerID=8YFLogxK

U2 - 10.1007/s12190-012-0608-2

DO - 10.1007/s12190-012-0608-2

M3 - Article

AN - SCOPUS:84873339370

SN - 1598-5865

VL - 41

SP - 393

EP - 426

JO - Journal of Applied Mathematics and Computing

JF - Journal of Applied Mathematics and Computing

IS - 1-2

ER -

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

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this