Previous |  Up |  Next

Article

Keywords:
completeness; Banach space; complex Müntz theorem
Summary:
Let $E=\bigcup _{n=1}^{\infty }I_{n}$ be the union of infinitely many disjoint closed intervals where $I_{n}=[a_{n}$, $b_{n}]$, $0<a_{1}<b_{1}<a_{2}<b_{2}<\dots <b_{n}<\dots $, $\lim _{n\rightarrow \infty }b_{n}=\infty .$ Let $\alpha (t)$ be a nonnegative function and $\{\lambda _{n}\}_{n=1}^{\infty }$ a sequence of distinct complex numbers. In this paper, a theorem on the completeness of the system $\{t^{\lambda _{n}}\log ^{m_{n}}t\}$ in $C_{0}(E)$ is obtained where $C_{0}(E)$ is the weighted Banach space consists of complex functions continuous on $E$ with $f(t){\rm e}^{-\alpha (t)}$ vanishing at infinity.
References:
[1] Boivin, A., Zhu, Ch.: On the completeness of the system $\{z^{\tau_{n}}\}$ in $L^{2}$. J. Approximation Theory 118 (2002), 1-19. MR 1928254
[2] Borichev, A. A., Sodin, M.: Krein's entire functions and the Bernstein approximation problem. Ill. J. Math. 45 (2001), 167-185. MR 1849992 | Zbl 0989.41003
[3] Borwein, P., Erdélyi, T.: Polynomials and Polynomial Inequalities. Springer-Verlag, New York (1995). MR 1367960 | Zbl 0840.26002
[4] Branges, L. de: The Bernstein problem. Proc. Am. Math. Soc. 10 (1959), 825-832. DOI 10.1090/S0002-9939-1959-0114080-0 | MR 0114080 | Zbl 0092.06905
[5] Deng, G. T.: Incompleteness and closure of a linear span of exponential system in a weighted Banach space. J. Approximation Theory 125 (2003), 1-9. DOI 10.1016/j.jat.2003.09.004 | MR 2016836 | Zbl 1036.30002
[6] Deng, G. T.: On weighted polynomial approximation with gaps. Nagoya Math. J. 178 (2005), 55-61. MR 2145315 | Zbl 1082.41017
[7] Deng, G. T.: Incompleteness and minimality of complex exponential system. Sci. China, Ser. A 50 (2007), 1467-1476. DOI 10.1007/s11425-007-0093-5 | MR 2390463 | Zbl 1130.30028
[8] Halmos, P. R.: Measure Theory, 2nd printing, Graduate Texts in Mathematics. 18. Springer-Verlag, New York-Heidelberg-Berlin (1974). MR 0453532
[9] Izumi, S.-I., Kawata, T.: Quasi-analytic class and closure of $\{t^{n}\}$ in the interval $(-\infty, \infty)$. Tohoku Math. J. 43 (1937), 267-273.
[10] Levin, B. Y.: Lectures on Entire Functions, Translations of Mathematical Monographs, 150. Providence RI., American Mathematical Society (1996). MR 1400006
[11] Malliavin, P.: Sur quelques procédés d'extrapolation. Acta Math. 83 (1955), 179-255. DOI 10.1007/BF02392523 | MR 0075297 | Zbl 0067.05104
[12] Mergelyan, S. N.: On the completeness of system of analytic functions. Amer. Math. Soc. Transl. Ser. 2 (1962), 109-166. MR 0131561
[13] Markushevich, A. I.: Theory of Functions of a Complex Variable, Selected Russian Publications in the Mathematical Sciences. Prentice-Hall (1965).
[14] Rudin, W.: Real and Complex Analysis, 3rd. ed. McGraw-Hill, New York (1987). MR 0924157 | Zbl 0925.00005
[15] Sedletskij, A. M.: Nonharmonic analysis. J. Math. Sci., New York 116 (2003), 3551-3619. MR 2024093 | Zbl 1051.42018
[16] Shen, X.: On the closure $\{ z^{\tau_{n}}\log^{j} z\} $ in a domain of the complex plane. Acta Math. Sinica 13 (1963), 405-418 Chinese Chinese Math. 4 (1963), 440-453 English.
[17] Shen, X.: On the completeness of $\{ z^{\tau_{n}}\log^{j} z\} $ on an unbounded curve of the complex plane. Acta Math. Sinica 13 (1963), 170-192 Chinese Chinese Math. 12 (1963), 921-950 English.
[18] Shen, X.: On approximation of functions in the complex plane by the system of functions $\{ z^{\tau_{n}}\log^{j} z\} $. Acta Math. Sinica 14 (1964), 406-414 Chinese Chinese Math. 5 (1965), 439-446 English. MR 0179534
[19] Yang, X. D.: Incompleteness of exponential system in the weighted Banach space. J. Approx. Theory 153 (2008), 73-79. DOI 10.1016/j.jat.2008.01.004 | MR 2432554 | Zbl 1149.30025
[20] Zhu, Ch.: Some Results in Complex Approximation with Sequence of Complex Exponents. Thesis of the University of Werstern Ontario, Canada (1999).
[21] Zikkos, E.: On a theorem of normal Levinson and a variation of the Fabry gap theorem. Complex Variables, Theory Appl. 50 (2005), 229-255. DOI 10.1080/02781070500032804 | MR 2125918
Partner of
EuDML logo