Abstract
We consider the well-known Szegö — Kolmogorov — Krein theorems on weighted approximations by functions with semibounded spectra defined on a circle or on a line and suggest an efficient construction that realizes these approximations. This construction is based on relations similar to the Cárleman formula for reconstructing analytic functions in terms of their traces on the boundary of their domains of definition.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. G. Krein, “A generalization of the results of G. Szegö, V. I. Smirnov, and A. N. Kolmogorov,”Dokl. Akad. Nauk SSSR,46, No. 3, 1378–1409 (1945).
M. G. Krein, “On an extrapolational problem of A. N. Kolmogorov,”Dokl. Akad. Nauk SSSR,46, No. 8, 339–341 (1945).
G. Szegö, “Über orthogonale Polynome, die zu einer gegebenen Kurve der komplexen Eben gehören,”Math. Z.,9, 218–247 (1921).
G. Szegö, “Über die Entwickelung einer analytischen Funktion nach den Polynomen eines Orthogonalsystems,”Math. Ann.,82, 188–212 (1921).
A. N. Kolmogorov, “Stationary sequences in Hilbert spaces,”Bull. Mosk. Univ., Math.,2, Issue 6, 1–40 (1941).
N. I. Akhiezer, “On assumptions of A. N. Kolmogorov and M. G. Krein,”Dokl. Akad. Nauk SSSR,50, No. 1, 35–39 (1945).
G. Ts. Tumarkin, “Mean approximations of functions on rectifiable curves,”Mat. Sb.,42, 79–128 (1957).
N. K. Nikol'skii, “Selected problems of weight approximation and spectral analysis,”Trudy Mat. Inst. Akad. Nauk SSSR,70, 1–270 (1974).
N. K. Nikol'skii,Lectures on Shift Operators [in Russian], Nauka, Moscow (1980).
P. Koosis,The Logarithmic Integral, Vol. 1, University Press, Cambridge (1988).
V. I. Smirnov and N. A. Lebedev,Constructive Theory of Functions of Complex Variables [in Russian], Nauka, Moscow-Leningrad (1964).
T. Cárleman,Les Fonctions Quasianalytiques, Hermann, Paris (1926).
I. I. Privalov,Boundary Properties of Analytic Functions [in Russian], Gostekhteoretizdat, Moscow-Leningrad (1950).
L. A. Aizenberg,Cárleman Formulas in Complex Analysis [in Russian], Nauka, Novosibirsk (1990).
V. P. Havin and B. Jöricke,The Uncertainty Principle in Harmonic Analysis (to appear).
I. V. Videnskii. E. M. Gavurina, and V. P. Khavin, “An analog of the Cárleman-Goluzin-Krylov interpolational formula,” in:Theory of Operators and Theory of Functions [in Russian], Issue 1, (1983) pp. 21–31.
P. Koosis,Introduction to H p Spaces Cambridge Univ. Press, New York (1980).
H. P. McKean, “Some questions about Hardy functions,”Springer Lect. Notes Math., 85–86 (1943).
V. P. Khavin, “Impossibility of the Cárleman approximation of the functions continuous on the unit circle by the boundary values of the functions analytic and uniformly continuous in the unit disk,”Investigations in the Theory of Linear Operators and Theory of Functions,2, 161–170 (1971).
D. I. Patil, “Representation ofH p functions,”Bull. Am. Math. Soc.,78, No. 5, 617–620 (1972).
Author information
Authors and Affiliations
Additional information
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, Nos. 1–2, pp. 100–127, January–February. 1994.
Rights and permissions
About this article
Cite this article
Khavin, V.P., Bart, V.A. Szegö-Kolmogorov-Krein theorems on weighted trigonometrical approximation and Cárleman-type relations. Ukr Math J 46, 101–132 (1994). https://doi.org/10.1007/BF01057004
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01057004