Advertisement

Journal of Fourier Analysis and Applications

, Volume 23, Issue 6, pp 1426–1444 | Cite as

Adaptative Decomposition: The Case of the Drury–Arveson Space

  • Daniel Alpay
  • Fabrizio Colombo
  • Tao Qian
  • Irene Sabadini
Article
  • 252 Downloads

Abstract

The maximum selection principle allows to give expansions, in an adaptive way, of functions in the Hardy space \(\mathbf H_2\) of the disk in terms of Blaschke products. The expansion is specific to the given function. Blaschke factors and products have counterparts in the unit ball of \(\mathbb C^N\), and this fact allows us to extend in the present paper the maximum selection principle to the case of functions in the Drury–Arveson space of functions analytic in the unit ball of \(\mathbb C^N\). This will give rise to an algorithm which is a variation in this higher dimensional case of the greedy algorithm. We also introduce infinite Blaschke products in this setting and study their convergence.

Keywords

Drury–Arveson space Adaptative decomposition Blaschke products 

Mathematics Subject Classification

47A56 41A20 

Notes

Acknowledgments

The authors thank the Macao Government FDCT 098/2012/A3 and D. Alpay thanks the Earl Katz family for endowing the chair which supported his research.

References

  1. 1.
    Agler, J., McCarthy, J.: Complete Nevanlinna–Pick kernels. J. Funct. Anal. 175, 111–124 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Agler, J., McCarthy, J.: Pick interpolation and Hilbert function spaces. In: Graduate Studies in Mathematics, vol. 44. American Mathematical Society, Providence, RI (2002)Google Scholar
  3. 3.
    Alpay, D.: A Complex Analysis Problem Book. Birkhäuser, Basel (2011)CrossRefzbMATHGoogle Scholar
  4. 4.
    Alpay, D., Kaptanoğlu, H.T.: Integral formulas for a sub-Hardy space of the ball with complete Nevanlinna–Pick reproducing kernel. C. R. Acad. Sci. 333, 285–290 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Alpay, D., Kaptanoğlu, H.T.: Gleason’s problem and homogeneous interpolation in Hardy and Dirichlet-type spaces of the ball. J. Math. Anal. Appl. 276(2), 654–672 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Alpay, D., Kaptanoğlu, H.T.: Some finite-dimensional backward shift-invariant subspaces in the ball and a related interpolation problem. Integral Equ. Oper. Theory 42, 1–21 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Alpay, D., Colombo, F., Qian, T., Sabadini, I.: Adaptative decomposition: the matrix-valued case. Proc. Am. Math. Soc (to appear)Google Scholar
  8. 8.
    Arveson, W.: Subalgebras of \(C^*\)-algebras. III. Multivariable operator theory. Acta Math. 181, 159–228 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Ball, J., Bolotnikov, V., Fang, Q.: Schur-class multipliers on the Arveson space: de Branges–Rovnyak reproducing kernel spaces and commutative transfer-function realizations. J. Math. Anal. Appl. 341(1), 519–539 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Ball, J., Trent, T., Vinnikov, V.: Interpolation and commutant lifting for multipliers on reproducing kernel Hilbert spaces. In: Proceedings of Conference in Honor of the 60–th Birthday of M.A. Kaashoek. Operator Theory: Advances and Applications, vol. 122, pp. 89–138. Birkhauser, Basel (2001)Google Scholar
  11. 11.
    Choquet, G.: Cours d’analyse, Tome II: Topologie. Masson, 120 bd Saint–Germain, Paris VI (1973)Google Scholar
  12. 12.
    Drury, S.W.: A generalization of von Neumann’s inequality to the complex ball. Proc. Am. Math. Soc. 68(3), 300–304 (1978)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Favard, J.: Cours d’analyse de l’École Polytechnique. Tome 1. Introduction-Operations. Gauthier-Villars, Paris (1968)Google Scholar
  14. 14.
    Qian, T.: Two-dimensional adaptive Fourier decomposition. Math. Met. Appl. Sci. 39(10), 2431–2448 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Qian, T., Wang, Y.B.: Adaptive decomposition into basic signals of non-negative instantaneous frequencies—a variation and realization of greedy algorithm. Adv. Comput. Math. 34(3), 279–293 (2011)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Rudin, W.: Function theory in the unit ball of \({\mathbb{C}^n}\). Springer, Berlin (1980)CrossRefzbMATHGoogle Scholar
  17. 17.
    Shalit, O.: Operator theory and function theory in Drury–Arveson space and its quotients. In: Alpay, D. (ed.) Handbook of Operator Theory, pp. 1125–1180. Springer, Basel (2015)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Daniel Alpay
    • 1
    • 2
  • Fabrizio Colombo
    • 3
  • Tao Qian
    • 4
  • Irene Sabadini
    • 3
  1. 1.Department of MathematicsBen-Gurion University of the NegevBeer-ShevaIsrael
  2. 2.Department of mathematicsChapman UniversityOrangeUSA
  3. 3.Dipartimento di MatematicaPolitecnico di MilanoMilanItaly
  4. 4.Department of MathematicsUniversity of MacauMacaoChina

Personalised recommendations