Geodesic
Szeg\H o theorem, Carath\'eodory domains, and boundedness of calculating functionals
Matematičeskie zametki, Tome 77 (2005) no. 1, pp. 3-15.

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Suppose that $G$ is a bounded simply connected domain on the plane with boundary $\Gamma$, $z_0\in G$, $\omega$ is the harmonic measure with respect to $z_0$, on $\Gamma$, $\mu$ is a finite Borel measure with support $\operatorname{supp}(\mu)\subseteq\Gamma$, $\mu_a+\mu_s$ is the decomposition of $\mu$ with respect to $\omega$, and $t$ is a positive real number. We solve the following problem: for what geometry of the domain $G$ is the condition $$ \int\ln\biggl(\frac{d\mu_a}{d\omega}\biggr)\,d\omega=-\infty $$ equivalent to the completeness of the polynomials in$L^t(\mu)$ or to the unboundedness of the calculating functional $p\to p(z_0)$, where $p$ is a polynomial in $L^t(\mu)$? We study the relationship between the densities of the algebras of rational functions in $L^t(\mu)$ and $C(\Gamma)$. For $t=2$, we obtain a sufficient criterion for the unboundedness of the calculating functional in the case of finite Borel measures with support of an arbitrary geometry. 
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F. G. Abdullaev; A. A. Dovgoshey. Szeg\H o theorem, Carath\'eodory domains, and boundedness of calculating functionals. Matematičeskie zametki, Tome 77 (2005) no. 1, pp. 3-15. http://geodesic.mathdoc.fr/item/MZM_2005_77_1_a0/

[1] Szegö G., “Beiträge zur Theorie der Toeplitzschen Formen, I”, Math. Z., 6 (1920), 167–202 | DOI | MR

[2] Kolmogorov A. N., “Interpolirovanie i ekstrapolirovanie statsionarnykh sluchainykh posledovatelnostei”, Izv. AN SSSR. Ser. matem., 5 (1941), 3–14 | MR | Zbl

[3] Krein M. G., “Ob odnom obobschenii issledovanii G. Szegö, V. I. Smirnova i A. N. Kolmogorova”, Dokl. AN SSSR, 46 (1945), 95–98

[4] Thomson J., “Approximation in the mean by polynomials”, Ann. of Math., 133 (1991), 477–507 | DOI | Zbl

[5] Thomson J., “Bounded point evaluation and polynomial approximation”, Proc. Amer. Math. Soc., 123 (1995), 1757–1761 | DOI | MR | Zbl

[6] Akeroyd J., “An extension of Szegö's theorem”, Indiana Univ. Math. J., 43 (1994), 1339–1347 | DOI | MR | Zbl

[7] Akeroyd J., “An extension of Szegö's theorem, II”, Indiana Univ. Math. J., 45 (1996), 241–252 | DOI | MR | Zbl

[8] Akeroyd J., Saleeby E. G., “On polynomials approximation in the mean”, Contemprary Math., 232 (1999), 23–26 | MR | Zbl

[9] Akeroyd J., Saleeby E. G., “A class $P^t(d\mu)$ spaces whole point evaluations vary with $t$”, Proc. Amer. Math. Soc., 127 (1999), 537–542 | DOI | MR | Zbl

[10] Conway J. B., Subnormal Operators, Pitman, Boston, 1981 | MR | Zbl

[11] Volberg A. L., “Srednekvadraticheskaya polnota mnogochlenov za predelami teoremy Segë”, Dokl. AN SSSR, 241 (1978), 521–527

[12] Trent T., “Extension of the theorem of Szegö”, Michigan Math. J., 26 (1979), 373–377 | DOI

[13] Browder A., Introduction to function algebras, W. A. Benjamin, New York–Amsterdam, 1969 | MR | Zbl

[14] Helson H., Lowdenslager D., “Prediction theory and Fourier series in several variables”, Acta Math., 99 (1958), 165–202 | DOI | Zbl

[15] Stahl H., Totik V., General Orthogonal Polynomials, Cambridge University Press, Cambridge, 1992 | Zbl

[16] Gamelin T., Ravnomernye algebry, Mir, M., 1969

[17] Philip J. D., Interpolation and Approximation, Blaisdell Publ. Co., New York–Toronto–London, 1963

[18] Vitushkin A. G., “Analiticheskaya emkost mnozhestv v zadachakh teorii priblizhenii”, UMN, 22:6 (1967), 141–199

[19] Donald L. C., Measure Theory, Birkhäuser, Boston–Basel–Berlin, 1993

[20] Ullman J. L., “On the regular behavior of orthogonal polynomials”, Proc. London Math. Soc., 24 (1972), 119–148 | DOI | Zbl

[21] Rudin W., Functional Analysis, 2nd edition, McGaw-Hill, Singapore, 1991 | Zbl

[22] Kheiman U., Kennedi P., Subgarmonicheskie funktsii, T. 1, Mir, M., 1980