Kravchuk Polynomials and Induced/Reduced Operators on Clifford Algebras

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• 作者：G. Stacey Staples (1)
  
  1. Department of Mathematics and Statistics ; Southern Illinois University Edwardsville ; Edwardsville ; IL聽 ; 62026-1653 ; USA
  
• 关键词：Operator calculus ; Kravchuk polynomials ; Clifford algebras ; Quantum probability ; 15A66 ; 47C05 ; 81R05 ; 60B99
• 刊名：Complex Analysis and Operator Theory
• 出版年：2015
• 出版时间：February 2015
• 年：2015
• 卷：9
• 期：2
• 页码：445-478
• 全文大小：473 KB
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• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Operator Theory
  Analysis
  
• 出版者：Birkh盲user Basel
• ISSN：1661-8262

文摘

Kravchuk polynomials arise as orthogonal polynomials with respect to the binomial distribution and have numerous applications in harmonic analysis, statistics, coding theory, and quantum probability. The relationship between Kravchuk polynomials and Clifford algebras is multifaceted. In this paper, Kravchuk polynomials are discovered as traces of conjugation operators in Clifford algebras, and appear in Clifford Berezin integrals of Clifford polynomials. Regarding Kravchuk matrices as linear operators on a vector space \(V\) , the action induced on the Clifford algebra over \(V\) is equivalent to blade conjugation, i.e., reflections across sets of orthogonal hyperplanes. Such operators also have a natural interpretation in terms of raising and lowering operators on the algebra. On the other hand, beginning with particular linear operators on the Clifford algebra \(\mathcal {C}\ell _Q(V)\) , one obtains Kravchuk matrices as operators on the paravector space \(V_*\) through a process of operator grade-reduction. Symmetric Kravchuk matrices are recovered as representations of grade-reductions of maps induced by negative-definite quadratic forms on \(V\) .