Illinois Journal of Mathematics

Reproducing kernel Hilbert spaces generated by the binomial coefficients

Daniel Alpay and Palle Jorgensen

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Abstract

We study a reproducing kernel Hilbert space of functions defined on the positive integers and associated to the binomial coefficients. We introduce two transforms, which allow us to develop a related harmonic analysis in this Hilbert space. Finally, we mention connections with the theory of discrete analytic functions, statistics, and with the quantum case.

Article information

Source
Illinois J. Math., Volume 58, Number 2 (2014), 471-495.

Dates
Received: 14 November 2013
Revised: 17 December 2014
First available in Project Euclid: 7 July 2015

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1436275494

Digital Object Identifier
doi:10.1215/ijm/1436275494

Mathematical Reviews number (MathSciNet)
MR3367659

Zentralblatt MATH identifier
1332.46035

Subjects
Primary: 46E22: Hilbert spaces with reproducing kernels (= [proper] functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) [See also 47B32] 47H60: Multilinear and polynomial operators [See also 46G25] 11B65: Binomial coefficients; factorials; $q$-identities [See also 05A10, 05A30]

Citation

Alpay, Daniel; Jorgensen, Palle. Reproducing kernel Hilbert spaces generated by the binomial coefficients. Illinois J. Math. 58 (2014), no. 2, 471--495. doi:10.1215/ijm/1436275494. https://projecteuclid.org/euclid.ijm/1436275494


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References

  • D. Alpay, The Schur algorithm, reproducing kernel spaces and system theory, American Mathematical Society, Providence, RI, 2001. Translated from the 1998 French original by Stephen S. Wilson, Panoramas et Synthèses.
  • D. Alpay, P. Jorgensen, R. Seager and D. Volok, On discrete analytic functions: Products, rational functions and reproducing kernels, J. Appl. Math. Comput. 41 (2013), 393–426.
  • N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337–404.
  • L. Carlitz, $q$-Bernoulli numbers and polynomials, Duke Math. J. 15 (1948), 987–1000.
  • L. Carlitz, Expansions of $q$-Bernoulli numbers, Duke Math. J. 25 (1958), 355–364.
  • R. J. Duffin, Basic properties of discrete analytic functions, Duke Math. J. 23 (1956), 335–363.
  • J. Ferrand, Fonctions préharmoniques et fonctions préholomorphes, Bull. Sci. Math. (2) 68 (1944), 152–180.
  • P. E. T. Jorgensen and A. M. Paolucci, States on the Cuntz algebras and $p$-adic random walks, J. Aust. Math. Soc. 90 (2011), no. 2, 197–211.
  • P. E. T. Jorgensen and A. M. Paolucci, Markov measures and extended zeta functions, J. Appl. Math. Comput. 38 (2012), no. 1–2, 305–323.
  • P. E. T. Jorgensen, L. M. Schmitt and R. F. Werner, $q$-relations and stability of $C^\ast$-isomorphism classes, Algebraic methods in operator theory, Birkhäuser Boston, Boston, MA, 1994, pp. 261–271.
  • O. G. Jørsboe, Equivalence or singularity of Gaussian measures on function spaces, Various Publications Series, vol. 4, Matematisk Institut, Aarhus Universitet, Aarhus, 1968.
  • R. Lasser and E. Perreiter, Homomorphisms of $l^1$-algebras on signed polynomial hypergroups, Banach J. Math. Anal. 4 (2010), no. 2, 1–10.
  • L. Mendo, Asymptotically optimum estimation of a probability in inverse binomial sampling under general loss functions, J. Statist. Plann. Inference 142 (2012), no. 10, 2862–2870.
  • L. Mendo and J. M. Hernando, Estimation of a probability with optimum guaranteed confidence in inverse binomial sampling, Bernoulli 16 (2010), no. 2, 493–513.
  • H. Meschkovski, Hilbertsche Räume mit Kernfunktion, Springer, Berlin, 1962.
  • S. Saitoh, Theory of reproducing kernels and its applications, Pitman Research Notes in Mathematics Series, vol. 189, Longman Scientific and Technical, Harlow, 1988.
  • L. Schwartz, Sous espaces hilbertiens d'espaces vectoriels topologiques et noyaux associés (noyaux reproduisants), J. Anal. Math. 13 (1964), 115–256.
  • A. Tucker, Applied combinatorics, 3rd ed., Wiley, New York, 1995.
  • W. Zieliński, The shortest Clopper–Pearson confidence interval for binomial probability, Comm. Statist. Simulation Comput. 39 (2010), no. 1, 188–193.