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Holomorphic Hermite polynomials in two variables
- Publication Year :
- 2017
- Publisher :
- arXiv, 2017.
-
Abstract
- Generalizations of the Hermite polynomials to many variables and/or to the complex domain have been located in mathematical and physical literature for some decades. Polynomials traditionally called complex Hermite ones are mostly understood as polynomials in z and z ‾ which in fact makes them polynomials in two real variables with complex coefficients. The present paper proposes to investigate for the first time holomorphic Hermite polynomials in two variables. Their algebraic and analytic properties are developed here. While the algebraic properties do not differ too much for those considered so far, their analytic features are based on a kind of non-rotational orthogonality invented by van Eijndhoven and Meyers. Inspired by their invention we merely follow the idea of Bargmann's seminal paper (1961) giving explicit construction of reproducing kernel Hilbert spaces based on those polynomials. “Homotopic” behaviour of our new formation culminates in comparing it to the very classical Bargmann space of two variables on one edge and the aforementioned Hermite polynomials in z and z ‾ on the other. Unlike in the case of Bargmann's basis our Hermite polynomials are not product ones but factorize to it when bonded together with the first case of limit properties leading both to the Bargmann basis and suitable form of the reproducing kernel. Also in the second limit we recover standard results obeyed by Hermite polynomials in z and z ‾ .
- Subjects :
- Pure mathematics
Hermite polynomials
creation and annihilation operators
Applied Mathematics
010102 general mathematics
Hilbert space
Holomorphic function
hermite polynomials in two complex variables
01 natural sciences
Domain (mathematical analysis)
010101 applied mathematics
van Eijndhoven and Meyers type orthogonality
symbols.namesake
Kernel (algebra)
Factorization
Orthogonality
Mathematics - Classical Analysis and ODEs
symbols
Classical Analysis and ODEs (math.CA)
FOS: Mathematics
0101 mathematics
Algebraic number
reproducing kernel Hilbert space
Analysis
Mathematics
Subjects
Details
- Database :
- OpenAIRE
- Accession number :
- edsair.doi.dedup.....ea80ff501f20546269c884ec99109284
- Full Text :
- https://doi.org/10.48550/arxiv.1706.04491