Abstract
For each function f, f ε VMO, there exists a unique function f0, analytic in the circle\(\mathbb{D}\) and such that ∥f−f0∥∞=f{∥∞∶gεVMOA}. We define the operator of best approximation (nonlinear) A, Af=f0, fεVMO, In the paper one considers the question of the preservation of a class under the action of the operator i.e. finding the classes X, X ⊂ VMO, AX ⊂ X. One investigates the classes X containing unbounded functions. It is proved that if P_X is the space of the symbols of the Hankel operators from a Banach space E of functions into the Hardy space H2, then AX ⊂ X. For E one can take “almost” any space.
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Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 141, pp. 5–17, 1985.
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Vol'berg, A.L., Tolokonnikov, V.A. Hankel operators and problems of best approximation of unbounded functions. J Math Sci 37, 1269–1275 (1987). https://doi.org/10.1007/BF01327036
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DOI: https://doi.org/10.1007/BF01327036