Extremal functions of the Nevanlinna-Pick problem and Douglas algebras
Tolokonnikov, V.
Studia Mathematica, Tome 104 (1993), p. 151-158 / Harvested from The Polish Digital Mathematics Library
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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.

Publié le : 1993-01-01
EUDML-ID : urn:eudml:doc:215991 
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Tolokonnikov, V. Extremal functions of the Nevanlinna-Pick problem and Douglas algebras. Studia Mathematica, Tome 104 (1993) pp. 151-158. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv105i2p151bwm/

[00000] [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.

[00001] [2] J. B. Garnett, Bounded Analytic Functions, Academic Press, New York 1981. | Zbl 0469.30024

[00002] [3] C. Guillory, K. Izuchi and D. Sarason, Interpolating Blaschke products and division in Douglas algebras, Proc. Roy. Irish Acad. 84A (1984), 1-7. | Zbl 0559.46022

[00003] [4] C. Guillory and D. Sarason, The algebra of quasicontinuous functions, ibid., 57-67. | Zbl 0559.46021

[00004] [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).

[00005] [6] K. Izuchi, QC-level sets and quotients of Douglas algebras, J. Funct. Anal. 65 (1986), 293-308. | Zbl 0579.46037

[00006] [7] K. Izuchi, Countably generated Douglas algebras, Trans. Amer. Math. Soc. 299 (1987), 171-192. | Zbl 0608.46028

[00007] [8] A. Nicolau, Finite products of interpolating Blaschke products, preprint, 1992, 16 pp.

[00008] [9] N. K. Nikol'skiĭ, Treatise on the Shift Operator, Springer, Berlin 1986.

[00009] [10] A. Stray, Interpolating sequences and the Nevanlinna-Pick problem, Publ. Math. (Barcelona) 35 (1991), 507-516. | Zbl 0773.30038

[00010] [11] V. A. Tolokonnikov, Blaschke products satisfying the Carleson-Newman conditions and Douglas algebras, Algebra i Analiz 3 (4) (1991), 185-196 (in Russian).