# Extremal functions of the Nevanlinna-Pick problem and Douglas algebras

Studia Mathematica (1993)

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

## Access Full Article

top## Abstract

top## How to cite

topTolokonnikov, 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

top- [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] J. B. Garnett, Bounded Analytic Functions, Academic Press, New York 1981. Zbl0469.30024
- [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] C. Guillory and D. Sarason, The algebra of quasicontinuous functions, ibid., 57-67. Zbl0559.46021
- [5] V. P. Havin and S. A. Vinogradov, Free interpolation in ${H}^{\infty}$ 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] K. Izuchi, QC-level sets and quotients of Douglas algebras, J. Funct. Anal. 65 (1986), 293-308. Zbl0579.46037
- [7] K. Izuchi, Countably generated Douglas algebras, Trans. Amer. Math. Soc. 299 (1987), 171-192. Zbl0608.46028
- [8] A. Nicolau, Finite products of interpolating Blaschke products, preprint, 1992, 16 pp.
- [9] N. K. Nikol'skiĭ, Treatise on the Shift Operator, Springer, Berlin 1986.
- [10] A. Stray, Interpolating sequences and the Nevanlinna-Pick problem, Publ. Math. (Barcelona) 35 (1991), 507-516. Zbl0773.30038
- [11] V. A. Tolokonnikov, Blaschke products satisfying the Carleson-Newman conditions and Douglas algebras, Algebra i Analiz 3 (4) (1991), 185-196 (in Russian).

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.