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Reproducing Kernel Hilbert Spaces of Polyanalytic Functions of Infinite Order

Integral Equations and Operator Theory volume 94, Article number: 35 (2022) Cite this article

Abstract

In this paper we introduce reproducing kernel Hilbert spaces of polyanalytic functions of infinite order. First we study in details the counterpart of the Fock space and related results in this framework. In this case the kernel function is given by \(\displaystyle e^{z\overline{w}+\overline{z}w}\) which can be connected to kernels of polyanalytic Fock spaces of finite order. Segal–Bargmann and Berezin type transforms are also considered in this setting. Then, we study the reproducing kernel Hilbert spaces of complex-valued functions with reproducing kernel \(\displaystyle \frac{1}{(1-z\overline{w})(1-\overline{z}w)}\) and \(\displaystyle \frac{1}{1-2\mathrm{Re}\, z\overline{w}}\). The corresponding backward shift operators are introduced and investigated.

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References

  1. Abreu, L.D.: Sampling and interpolation in Bargmann–Fock spaces of polyanalytic functions. Appl. Comput. Harmon. Anal. 29, 287–302 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Abreu, L.D.: Super-wavelets versus poly-Bergman spaces. Int. Equ. Oper. Theory 73, 177–193 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Abreu, L.D., Balazs, P., De Gosson, M., Mouayn, Z.: Discrete coherent states for higher Landau levels. Ann. Phys. 363, 337–353 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Abreu, L.D., Feichtinger, H.G.: Function Spaces of Polyanalytic Functions. Harmonic and Complex Analysis and its Applications, Trends Math., pp. 1–38. Birkhäuser Springer, Cham (2014)

  5. Agler, J.: Nevanlinna–Pick Interpolation on Sobolev Space, Proceedings of the American Mathematica Society. 108(2) (1990)

  6. Agler, J.: On the representation of certain holomorphic functions on a polydisc. Oper. Theory: Adv. Appl. 48, 47–66 (1990)

    MathSciNet  MATH  Google Scholar 

  7. Agranovsky, M.L.: Characterization of polyanalytic functions by meromorphic extensions into chains of circles. J. d’Analyse Math. 113, 305–329 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Alpay, D.: An Advanced Complex Analysis Problem Book. Topological Vector Spaces, Functional Analysis, and Hilbert Spaces of Analytic Functions. Birkhäuser, Basel (2015)

  9. Alpay, D., Bolotnikov, V., Dijksma, A., Sadosky, C.: Hilbert spaces contractively included in the Hardy space of the bidisk. Positivity 5(1), 25–50 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. Alpay, D., Cerejeiras, P., Kaehler, U., Kling, T.: Commutators on Fock spaces. Preprint (2021) arXiv:2208.03394

  11. Alpay, D., Kaptanoglu, T.: Some finite-dimensional backward-shift-invariant subspaces in the ball and a related interpolation problem. Integral Equ. Oper. Theory 42(1), 1–21 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Aronszajn, N.: Theory of reproducing kernels. Trans. Am. Math. Soc. 68, 337–404 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  13. Askour, N., Intissar, A., Mouayn, Z.: Espaces de Bargmann généralisés et formules explicites pour leurs noyaux reproduisants. Comptes Rendus de l’Acad. des Sci.-Ser. I-Math. 325(7), 707–712 (1997)

    MATH  Google Scholar 

  14. Balk, M.B.: Polyanalytic Functions. Akademie-Verlag, Berlin (1991)

    MATH  Google Scholar 

  15. Balk, M.B.: Polyanalytic Functions and their Generalizations, in Complex Analysis, I, 195–253, Encyclopaedia Math. Sci., 85, Springer, Berlin (1997)

  16. Ball, J.A., Bolotnikov, V.: Canonical Transfer-Function Realization for Schur–Agler-Class Functions on Domains with Matrix Polynomial Defining Function in \({\mathbb{C}}^n\). Recent Progress in Operator Theory and Its Applications, 23–55, Operator Theory Advanced Application, 220, Birkhäuser/Springer Basel AG, Basel (2012)

  17. Ball, J.A., Kaliuzhnyi-Verbovetskyi, D.S.: Schur–Agler and Herglotz–Agler classes of functions: positive-kernel decompositions and transfer-function realizations. Adv. Math. 280, 121–187 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ball, J.A., Trent, T.: Unitary colligations, reproducing kernel Hilbert spaces and Nevanlinna–Pick interpolation in several variables. J. Funct. Anal. 157, 1–61 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  19. Bargman, V.: On a Hilbert space of analytic functions and an associated integral transform. Commun. Pure Appl. Math. 14, 187–214 (1961)

    Article  MathSciNet  Google Scholar 

  20. Begehr, H.: Orthogonal decompositions of the function space \(L_2(\overline{D};\mathbb{C})\). J. Reine Angew. Math. 549, 191–219 (2002)

    MathSciNet  MATH  Google Scholar 

  21. Benahmadi, A., Ghanmi, A.: Non-trivial 1d and 2d Segal–Bargmann transforms. Integral Transforms Spec. Funct. 30(7), 547–563 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  22. Berezin, F.A.: General concept of quantization. Commun. Math. Phys. 40, 153–174 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  23. de Branges, L.: Some Hilbert spaces of analytic functions I. Trans. Am. Math. Soc. 106, 445–468 (1963)

    MathSciNet  MATH  Google Scholar 

  24. de Branges, L.: Espaces Hilbertiens de fonctions entières. Masson, Paris (1972)

    Google Scholar 

  25. Ismail, M.E.H., Simeonov, P.: Complex Hermite polynomials: their combinatorics and integral operators. Proc. Am. Math. Soc. 143, 1397–1410 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  26. Fricain, E., Mashreghi, J.: The Theory of \(\cal{H}\)(\(b\)) Spaces. Vol. 1, vol. 20 of New Mathematical Monographs. Cambridge University Press, Cambridge (2016)

  27. Fricain, E., Mashreghi, J.: The Theory of \(\cal{H}(b)\) Spaces. Vol. 2, volume 21 of New Mathematical Monographs. Cambridge University Press, Cambridge (2016)

  28. Ito, K.: Complex multiple Wiener integral. Jpn. J. Math. 22, 63–86 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  29. Kolossov, G.V.: Sur les problèmes d’lasticité à deux dimensions. C. R. Acad. Sci. 146, 522–525 (1908)

    Google Scholar 

  30. Lebedev, N.N.: Special Functions and Their Applications. Physico-Technical Institute, Academy of Sciences, U.S.S.R (1972)

  31. Mouayn, Z.: Coherent state transforms attached to generalized Bargmann spaces on the complex plane. Math. Nachr. 284(14–15), 1948–1954 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  32. Quiggin, P.: For which reproducing kernel Hilbert spaces is Pick’s theorem true?. Integral Equation and Operator Theory, 16 (1993)

  33. Saitoh, S.: Theory of Reproducing Kernels and Its Applications, vo. 189. Longman Scientific and Technical (1988)

  34. Shigekawa, I.: Eigenvalue problems for the Schrödinger operator with the magnetic field on a compact Riemannian manifold. J. Funct. Anal. 75, 92–127 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  35. Vasilevski, N.L.: On the structure of Bergman and poly-Bergman spaces. Integr. Equ. Oper. Theory 33, 471–488 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  36. Vasilevski, N.L.: Poly-Fock spaces. In: Differential Operators and Related Topics (pp. 371–386). Birkhäuser, Basel (2000)

  37. Zhu, K.: Analysis on Fock Spaces. Springer, New York (2012)

    Book  MATH  Google Scholar 

  38. Zhu, K.: Singular integral operators on the Fock space. Integral Equ. Oper. Theory 81(4), 451–454 (2015)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Daniel Alpay thanks the Foster G. and Mary McGaw Professorship in Mathematical Sciences, which supported this research. Kamal Diki thanks the Grand Challenges Initiative (GCI) at Chapman University for supporting this research. The authors are grateful to the referee for suggestions considered in Remark 5.17 leading to the equivalence Landau levels operator.

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Authors and Affiliations

  1. Schmid College of Science and Technology, Chapman University, One University Drive, Orange, CA, 92866, USA

    Daniel Alpay & Kamal Diki

  2. Dipartimento di Matematica, Politecnico di Milano, Via E. Bonardi, 9, 20133, Milan, Italy

    Fabrizio Colombo & Irene Sabadini

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  1. Daniel Alpay
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  2. Fabrizio Colombo
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  3. Kamal Diki
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  4. Irene Sabadini
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Correspondence to Daniel Alpay.

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Alpay, D., Colombo, F., Diki, K. et al. Reproducing Kernel Hilbert Spaces of Polyanalytic Functions of Infinite Order. Integr. Equ. Oper. Theory 94, 35 (2022). https://doi.org/10.1007/s00020-022-02713-4

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  • DOI: https://doi.org/10.1007/s00020-022-02713-4

Keywords

  • Polyanalytic Fock space of infinite order
  • Polyanalytic Hardy space of infinite order
  • Backward shift operators
  • Segal–Bargmann transform
  • Berezin transform

Mathematics Subject Classification

  • 30H20
  • 44A15
  • 64E22