Abstract
Basic result: let {zn} be a sequence of points of the unit disc and {kn} be a sequence of natural numbers, satisfying the conditions:
Then for any bounded sequence of complex numbers
there exists a sequence
such that the function
interpolates ω: where BΛ is the Blaschke product with zeros at the points λn (k)}, M is a constant,
. if N=1 this theorem is proved by Earl (RZhMat, 1972, 1B 163). The idea of the proof, as in Earl, is that if the zeros {λn (k)} run through neighborhoods of the points zn, then the Blaschke products with these zeros interpolate sequences ω, filling some neighborhood of zero in the space Z∞. The theorem formulated is used to get interpolation theorems in classes narrower than H∞.
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Literature cited
L. Carleson, Ann. Math.,76, No. 3, 547–559 (1962).
J. P. Earl, J. London Math. Soc,2, No. 2, 544–548 (1970).
V. E. Katsnel'son, Punkts. Anal. Prim.,1, No. 2, 39–51 (1967).
V. I. Vasyunin, Dokl. Akad. Nauk SSSR,227, No. 1, 11–14 (1976).
N. K. Nikol'skii, Tr. Mat. Inst. Akad. Nauk SSSR,130, Nauka, Leningrad (1977).
P. S. Aleksandrov and B. A. Pasynkov, Introduction to Dimension Theory [in Russian], Moscow (1973).
S. A. Vinogradov and V. P. Khavin, J. Sov. Math.,9, No. 2 (1978).
S. A. Vinogradov and V. P. Khavin, J. Sov. Math.,14, No. 2 (1980).
Additional information
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 73, pp. 195–202, 1977.
I thank S. A. Vinogradov for helpful discussions and interest in the work.
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Videnskii, I.V. Multiple interpolation by Blaschke products. J Math Sci 34, 2139–2143 (1986). https://doi.org/10.1007/BF01741588
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DOI: https://doi.org/10.1007/BF01741588