Extremal functions of the Nevanlinna-Pick problem and Douglas algebras

V. Tolokonnikov

Studia Mathematica (1993)

  • Volume: 105, Issue: 2, page 151-158
  • ISSN: 0039-3223

Abstract

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The Nevanlinna-Pick problem at the zeros of a Blaschke product B having a solution of norm smaller than one is studied. All its extremal solutions are invertible in the Douglas algebra D generated by B. If B is a finite product of sparse Blaschke products (Newman Blaschke products, Frostman Blaschke products) then so are all the extremal solutions. For a Blaschke product B a formula is given for the number C(B) such that if the NP-problem has a solution of norm smaller than C(B) then all its extremal solutions are Carleson Blaschke products, i.e. can be represented as finite products of interpolating Blaschke products.

How to cite

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Tolokonnikov, V.. "Extremal functions of the Nevanlinna-Pick problem and Douglas algebras." Studia Mathematica 105.2 (1993): 151-158. <http://eudml.org/doc/215991>.

@article{Tolokonnikov1993,
abstract = {The Nevanlinna-Pick problem at the zeros of a Blaschke product B having a solution of norm smaller than one is studied. All its extremal solutions are invertible in the Douglas algebra D generated by B. If B is a finite product of sparse Blaschke products (Newman Blaschke products, Frostman Blaschke products) then so are all the extremal solutions. For a Blaschke product B a formula is given for the number C(B) such that if the NP-problem has a solution of norm smaller than C(B) then all its extremal solutions are Carleson Blaschke products, i.e. can be represented as finite products of interpolating Blaschke products.},
author = {Tolokonnikov, V.},
journal = {Studia Mathematica},
keywords = {Nevanlinna-Pick problem},
language = {eng},
number = {2},
pages = {151-158},
title = {Extremal functions of the Nevanlinna-Pick problem and Douglas algebras},
url = {http://eudml.org/doc/215991},
volume = {105},
year = {1993},
}

TY - JOUR
AU - Tolokonnikov, V.
TI - Extremal functions of the Nevanlinna-Pick problem and Douglas algebras
JO - Studia Mathematica
PY - 1993
VL - 105
IS - 2
SP - 151
EP - 158
AB - The Nevanlinna-Pick problem at the zeros of a Blaschke product B having a solution of norm smaller than one is studied. All its extremal solutions are invertible in the Douglas algebra D generated by B. If B is a finite product of sparse Blaschke products (Newman Blaschke products, Frostman Blaschke products) then so are all the extremal solutions. For a Blaschke product B a formula is given for the number C(B) such that if the NP-problem has a solution of norm smaller than C(B) then all its extremal solutions are Carleson Blaschke products, i.e. can be represented as finite products of interpolating Blaschke products.
LA - eng
KW - Nevanlinna-Pick problem
UR - http://eudml.org/doc/215991
ER -

References

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  1. [1] S.-Y. Chang and D. E. Marshall, Some algebras of bounded analytic functions containing the disc algebra, in: Lecture Notes in Math. 604, Springer, 1977, 12-20. 
  2. [2] J. B. Garnett, Bounded Analytic Functions, Academic Press, New York 1981. Zbl0469.30024
  3. [3] C. Guillory, K. Izuchi and D. Sarason, Interpolating Blaschke products and division in Douglas algebras, Proc. Roy. Irish Acad. 84A (1984), 1-7. Zbl0559.46022
  4. [4] C. Guillory and D. Sarason, The algebra of quasicontinuous functions, ibid., 57-67. Zbl0559.46021
  5. [5] V. P. Havin and S. A. Vinogradov, Free interpolation in H ∞ and some other function classes, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), part 1: 47 (1974), 15-54; part 2: 56 (1976), 12-58 (in Russian). 
  6. [6] K. Izuchi, QC-level sets and quotients of Douglas algebras, J. Funct. Anal. 65 (1986), 293-308. Zbl0579.46037
  7. [7] K. Izuchi, Countably generated Douglas algebras, Trans. Amer. Math. Soc. 299 (1987), 171-192. Zbl0608.46028
  8. [8] A. Nicolau, Finite products of interpolating Blaschke products, preprint, 1992, 16 pp. 
  9. [9] N. K. Nikol'skiĭ, Treatise on the Shift Operator, Springer, Berlin 1986. 
  10. [10] A. Stray, Interpolating sequences and the Nevanlinna-Pick problem, Publ. Math. (Barcelona) 35 (1991), 507-516. Zbl0773.30038
  11. [11] V. A. Tolokonnikov, Blaschke products satisfying the Carleson-Newman conditions and Douglas algebras, Algebra i Analiz 3 (4) (1991), 185-196 (in Russian). 

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