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Information-theoretic-based spreading measures of orthogonal polynomials. (English) Zbl 1276.33011
Summary: The macroscopic properties of a quantum system strongly depend on the spreading of the physical eigenfunctions (wavefunctions) of its Hamiltonian operator over its confined domain. The wavefunctions are often controlled by classical or hypergeometric-type orthogonal polynomials (Hermite, Laguerre and Jacobi).
Here, we discuss the spreading of these polynomials over its orthogonality interval by means of various information-theoretic quantities which grasp some facets of the polynomial distribution not yet analyzed. We consider the information-theoretic lengths closely related to the Fisher information and Rényi and Shannon entropies, which quantify the polynomial spreading far beyond the celebrated standard deviation.

MSC:
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
94A17 Measures of information, entropy
62B10 Statistical aspects of information-theoretic topics
65C60 Computational problems in statistics (MSC2010)
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