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1993 | 105 | 2 | 151-158
Tytuł artykułu

Extremal functions of the Nevanlinna-Pick problem and Douglas algebras

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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.
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  • Topzhkovskaya St., D.2, Kor. 1, KV. 76, 197342 St. Petersburg, Russia
Bibliografia
  • [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.
  • [3] C. Guillory, K. Izuchi and D. Sarason, Interpolating Blaschke products and division in Douglas algebras, Proc. Roy. Irish Acad. 84A (1984), 1-7.
  • [4] C. Guillory and D. Sarason, The algebra of quasicontinuous functions, ibid., 57-67.
  • [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] K. Izuchi, QC-level sets and quotients of Douglas algebras, J. Funct. Anal. 65 (1986), 293-308.
  • [7] K. Izuchi, Countably generated Douglas algebras, Trans. Amer. Math. Soc. 299 (1987), 171-192.
  • [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.
  • [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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