Abstract
In the criterion for polynomial denseness in the space C 0w established by de Branges in 1959, we replace the requirement of the existence of an entire function by an equivalent requirement of the existence of a polynomial sequence. We introduce the notion of strict compactness of polynomial sets and establish sufficient conditions for a polynomial family to possess this property.
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REFERENCES
Yu. M. Berezanskii, G. F. Us, and Z. G. Sheftel', Functional Analysis [in Russian], Vyshcha Shkola, Kiev (1990).
I. P. Natanson, Theory of Functions of a Real Variable [in Russian], Nauka, Moscow 1974).
S. N. Mergelyan, “Weighted polynomial approximations,” Usp. Mat. Nauk, 11, No.5, 107–152 (1956).
A. G. Bakan, “Criterion of polynomial denseness and general form of a linear continuous functional on the space C ow ,” Ukr. Mat. Zh., 54, No.5, 610–622 (2002).
S. Bernstein, “Le probleme de l'approximation des fonctions continues sur tout l'axe reel at l'une de ses applications,” Bull. Math. France, 52, 399–410 (1924).
P. Koosis, The Logarithmic Integral, I, Cambridge University Press, Cambridge (1988).
C. Berg, “Moment problems and polynomial approximation,” in: Ann. Fac. Sci. Univ. Toulouse, Stieltjes Spec. Issue (1996), pp. 9–32.
A. Borichev and M. Sodin, “The Hamburger moment problem and weighted polynomial approximation on discrete subsets of the real line,” J. Anal. Math., 71, 219–264 (1998).
A. G. Bakan, “Polynomial density in L p (R 1, dµ) and representation of all measures which generate a determinate Hamburger moment problem,” in: M. Lassonde (editor), Approximation, Optimization, and Mathematical Economics (Pointe-aa-Pitre, 1999), Physica, Heidelberg (2001), pp. 37–46.
N. I. Akhiezer, “On weighted approximation of continuous functions on the entire number axis,” Usp. Mat. Nauk, 11, No.4, 3–43 (1956).
L. de Branges, “The Bernstein problem,” Proc. Amer. Math. Soc., 10, 825–832 (1959).
M. Sodin and P. Yuditskii, “Another approach to de Branges' theorem on weighted polynomial approximation,” in: L. Zalcman (editor), Proc. Ashkelon Workshop Compl. Function Theory, Isr. Math. Conf. Proc. (May 1996), Vol. 11, American Mathematical Society, Providence, RI (1997), pp. 221–227.
B. Ya. Levin, Distribution of Roots of Entire Functions [in Russian], Gostekhteorizdat, Moscow (1956).
H. Hamburger, “Hermitian transformations of deficiency index (1, 1), Jacobi matrices and undetermined moment problems,” Amer. J. Math., 66, 489–522 (1944).
N. I. Akhiezer, Classical Moment Problem [in Russian], Fizmatgiz, Moscow (1961).
V. K. Dzyadyk and V. V. Ivanov, “On asymptotics and estimates for the uniform norms of the Lagrange interpolation polynomials corresponding to the Chebyshev nodal points,” Anal. Math., 9, 85–97 (1983).
B. V. Shabat, Introduction to Complex Analysis [in Russian], Part 1, Nauka, Moscow (1985).
A. I. Markushevich, Theory of Analytic Functions [in Russian], Vol. 1, Nauka, Moscow (1967).
I. I. Hirschman and D. V. Widder, The Convolution Transform, Princeton University Press, Princeton, NJ (1955).
A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series [in Russian], Nauka, Moscow (1981).
E. Laguerre, “Sur les fonctions du genre zero et du genre un,” C. R. Acad. Sci. Paris, 98, 828–831 (1982).
G. Polya, “Uber Annaherung durch Polynome mit lauter reellen Wurzein,” Rend. Circ. Math. Palermo, 36, 279–295 (1913).
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Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 3, pp. 305–319, March, 2005.
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Bakan, A.G. Polynomial Form of de Branges Conditions for the Denseness of Algebraic Polynomials in the Space C 0w . Ukr Math J 57, 364–381 (2005). https://doi.org/10.1007/s11253-005-0196-5
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DOI: https://doi.org/10.1007/s11253-005-0196-5