Abstract
We discuss the difference between orthogonal polynomials on finite and infinite dimensional vectors spaces. In particular we prove an infinite dimensional extension of Favard lemma.
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L. Accardi, A. Barhoumi and A. Dhahri, Identification of the theory of orthogonal polynomials in d-indeterminates with the theory of 3-diagonal symmetric interacting Fock spaces on \({\mathbb{C}^d}\), Infin. Dimens. Anal. Quantum. Probab. Relat. Top. 20 no. 1 (2017), 1750004–1–55, arXiv:1401.5434v1 [math.FA], 21 Jan 2014.
Accardi L., Bozejko M.: Interacting Fock space and Gaussianization of probability measures. Infin. Dim. Anal. Quantum Probab. Rel. Topics 1, 663–670 (1998)
L. Accardi, H.-H. Kuo and A.I. Stan, Characterization of probability measures through the canonically associated interacting Fock spaces, Inf. Dim. Anal. Quant. Prob. Rel. Top. 7 no. 4 (2004), 485–505.
L. Accardi, H.-H. Kuo and A. Stan, Probability measures in terms of creation, annihilation, and neutral operators, in: Quantum Probability and Infinite Dimensional Analysis: From Foundations to Applications [QP–PQ XVIII], M. Schürmann and U. Franz (eds.), World Scientific, 2005, pp. 1–11.
L. Accardi, H.-H. Kuo and A. Stan, Moments and commutators of probability measures, Infin. Dim. Anal. Quantum Probab. Rel. Topics 10 no. 4 (2007), 591–612.
L. Accardi and M. Nhani, Interacting Fock Spaces and Orthogonal Polynomials in several variables, in: The Crossroad of Non-Commutativity, Infinite-Dimensionality, Obata, Hora, Matsui (eds.) World Scientific, 2002, pp. 192–205. Preprint Volterra, N. 523 (2002).
L. Accardi and M. Skeide, Interacting Fock space versus full Fock module, Commun. Stoch. Anal. 2 no. 3 (2008), 423–444. Volterra Preprint N. 328 (1998).
L. Accardi, Y.G. Lu and I. Volovich, The QED Hilbert module and Interacting Fock spaces, Publications of IIAS (Kyoto), 1997.
D. Alpay, P.E.T. Jorgensen and D.P. Kimsey, Moment problems in an infinite number of variables, Infin. Dim. Anal. Quantum Probab. Rel. Topics 21 no. 4 (2015).
A. Hora and N. Obata, Quantum Probability and Spectral Analysis of Graphs, Springer Theoretical and Mathematical Physics, Springer, 2007.
Acknowledgements
We are grateful to the referee for several appropriate comments, in particular for having detected an imprecision in our previous proof of Lemma 4.23.
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Lecture delivered at the Seminario Matematico e Fisico di Milano on January 27, 2015
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Accardi, L., Dhahri, A. Orthogonal Polynomial Decomposition for Random Fields with All Moments. Milan J. Math. 87, 21–56 (2019). https://doi.org/10.1007/s00032-019-00291-6
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DOI: https://doi.org/10.1007/s00032-019-00291-6