Abstract
Let θ be an inner function, let α ∈ C, ¦α¦=1. Then the harmonic function ℜ[(α+θ)]/(α−θ)] is the Poisson integral of a singular measureσ α D. N. Clark's known theorem enables us to identify in a natural manner the space H2 ⊖ θH2 with the space L2 (σ α ).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Literature cited
D. N. Clark, “One dimensional perturbations of restricted shifts,” J. Anal. Math.,25, 169–191 (1972).
N. K. Nikol'skii, Treatise on the Shift Operator. Spectral Function Theory, Springer, Berlin (1986).
A. B. Aleksandrov, “Invariant subspaces of shift operators. An axiomatic approach,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst.,113, 7–26 (1981).
A. B. Aleksandrov, “Invariant subspaces of the backward shift operator in the space Hp (p ∈ (0, 1)),” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst.,92, 7–29 (1979).
A. B. Aleksandrov, “The multiplicity of the boundary values of inner functions,” Izv. Akad. Nauk ArmSSR, Mat.,22, No. 5, 490–503 (1987).
A. L. Volberg, “Thin and thick families of rational fractions,” in: Complex Analysis and Spectral Theory (Leningrad, 1979/1980), Lecture Notes in Math., No. 864, Springer, Berlin (1981), pp. 440–480.
B. Erikke (B. Jöricke) and V. P. Khavin, “Traces of harmonic functions and comparison of Lp-norms of analytic functions,” Math. Nachr.,123, 225–254 (1985).
P. Koosis, “Moyennes quadratiques pondérées de fonctions périodiques et de leurs conjuguées harmoniques,” C. R. Acad. Sci. Paris, Ser. A-B,291, No. 4, A255-A257 (1980).
S. V. Hruscev and S. A. Vinogradov, “Free interpolation in the space of uniformly convergent Taylor series,” in: Complex Analysis and Spectral Theory, Seminar, Leningrad (1979/80), Lecture Notes in Math., No. 864, 171–213 (1981).
M. G. Goluzina, “On the multiplication and division of Cauchy-Stieltjes type integrals,” Vestnik Leningrad. Univ. Mat. Mekh. Astronom., No. 19, vyp. 4, 8–15 (1981).
S. V. Kislyakov, “Classical problems of Fourier analysis,” Itogi Nauki i Tekhniki, Fundam. Napravl.,15, 135–195 (1987).
J. B. Garnett, Bounded Analytic Functions, Academic Press, New York (1981).
J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, Lecture Notes in Math., No. 338, Springer, Berlin (1973).
A. Pietsch, Operator Ideals, North-Holland, Amsterdam (1980).
P. R. Ahern and D. N. Clark, “On inner functions with Hp-derivative,” Michigan Math. J.,21, No. 2, 115–128 (1974).
E. F. Collingwood and A. J. Lohwater, The Theory of Cluster Sets, Cambridge Univ. Press, London (1966).
A. L. Vol'berg and S. R. Treil', “Imbedding theorems for the invariant subspaces of the backward shift operator,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst.,149, 38–51 (1986).
B. Cohn, “Carleson measures for functions orthogonal to invariant subspaces,” Pacific J. Math.,103, No. 2, 347–364 (1982).
Additional information
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 170, pp. 7–33, 1989.
Rights and permissions
About this article
Cite this article
Aleksandrov, A.B. Inner functions and related spaces of pseudocontinuable functions. J Math Sci 63, 115–129 (1993). https://doi.org/10.1007/BF01099304
Issue Date:
DOI: https://doi.org/10.1007/BF01099304