On the completeness of the system \(\{t^{\lambda _{n}}\log ^{m_{n}}t\}\) in \(C_{0}(E)\). (English) Zbl 1265.30177
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 consisting of complex functions continuous on \(E\) with \(f(t)\text{e}^{-\alpha (t)}\) vanishing at infinity.
References:
| [1] | A. Boivin, Ch. Zhu: On the completeness of the system $$\(\backslash\){ {z\^{{T_n}}}\(\backslash\)} $$ in L 2. J. Approximation Theory 118 (2002), 1–19. · Zbl 1014.30004 · doi:10.1006/jath.2002.3693 |
| [2] | A.A. Borichev, M. Sodin: Krein’s entire functions and the Bernstein approximation problem. Ill. J. Math. 45 (2001), 167–185. · Zbl 0989.41003 |
| [3] | P. Borwein, T. Erdélyi: tPolynomials and Polynomial Inequalities. Springer-Verlag, New York, 1995. |
| [4] | L. de Branges: The Bernstein problem. Proc. Am. Math. Soc. 10 (1959), 825–832. · Zbl 0092.06905 |
| [5] | G.T. Deng: Incompleteness and closure of a linear span of exponential system in a weighted Banach space. J. Approximation Theory 125 (2003), 1–9. · Zbl 1036.30002 · doi:10.1016/j.jat.2003.09.004 |
| [6] | G.T. Deng: On weighted polynomial approximation with gaps. Nagoya Math. J. 178 (2005), 55–61. · Zbl 1082.41017 |
| [7] | G.T. Deng: Incompleteness and minimality of complex exponential system. Sci. China, Ser. A 50 (2007), 1467–1476. · Zbl 1130.30028 · doi:10.1007/s11425-007-0093-5 |
| [8] | P.R. Halmos: Measure Theory, 2nd printing, Graduate Texts in Mathematics. 18. Springer-Verlag, New York-Heidelberg-Berlin, 1974. |
| [9] | S.-I. Izumi, T. Kawata: Quasi-analytic class and closure of {t n} in the interval (,. Tohoku Math. J. 43 (1937), 267–273. · JFM 63.0197.03 |
| [10] | B.Y. Levin: Lectures on Entire Functions, Translations of Mathematical Monographs, 150. Providence RI., American Mathematical Society, 1996. |
| [11] | P. Malliavin: Sur quelques procédés d’extrapolation. Acta Math. 83 (1955), 179–255. · Zbl 0067.05104 · doi:10.1007/BF02392523 |
| [12] | S.N. Mergelyan: On the completeness of system of analytic functions. Amer. Math. Soc. Transl. Ser. 2 (1962), 109–166. · Zbl 0122.31601 |
| [13] | A. I. Markushevich: Theory of Functions of a Complex Variable, Selected Russian Publications in the Mathematical Sciences. Prentice-Hall, 1965. |
| [14] | W. Rudin: Real and Complex Analysis, 3rd. ed. McGraw-Hill, New York, 1987. · Zbl 0925.00005 |
| [15] | A.M. Sedletskij: Nonharmonic analysis. J. Math. Sci., New York 116 (2003), 3551–3619. · Zbl 1051.42018 · doi:10.1023/A:1024107924340 |
| [16] | X. Shen: On the closure $$\(\backslash\){ {z\^{{T_n}}}{\(\backslash\)log ĵ}z\(\backslash\)} $$ in a domain of the complex plane. Acta Math. Sinica 13 (1963), 405–418 (In Chinese.); Chinese Math. 4 (1963), 440–453. (In English.) |
| [17] | X. Shen: On the completeness of $$\(\backslash\){ {z\^{{T_n}}}{\(\backslash\)log ĵ}z\(\backslash\)} $$ on an unbounded curve of the complex plane. Acta Math. Sinica 13 (1963), 170–192 (In Chinese.); Chinese Math. 12 (1963), 921–950. (In English.) |
| [18] | X. Shen: On approximation of functions in the complex plane by the system of functions $$\(\backslash\){ {z\^{{T_n}}}{\(\backslash\)log ĵ}z\(\backslash\)} $$ . Acta Math. Sinica 14 (1964), 406–414 (In Chinese.); Chinese Math. 5 (1965), 439–446. (In English.) |
| [19] | X.D. Yang: Incompleteness of exponential system in the weighted Banach space. J. Approx. Theory 153 (2008), 73–79. · Zbl 1149.30025 · doi:10.1016/j.jat.2008.01.004 |
| [20] | Ch. Zhu: Some Results in Complex Approximation with Sequence of Complex Exponents. Thesis of the University of Werstern Ontario, Canada, 1999. |
| [21] | E. Zikkos: On a theorem of normal Levinson and a variation of the Fabry gap theorem. Complex Variables, Theory Appl. 50 (2005), 229–255. · Zbl 1092.30004 · doi:10.1080/02781070500032804 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
