| Title:
|
On the completeness of the system $\{t^{\lambda _{n}}\log ^{m_{n}}t\}$ in $C_{0}(E)$ (English) |
| Author:
|
Yang, Xiangdong |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
62 |
| Issue:
|
2 |
| Year:
|
2012 |
| Pages:
|
361-379 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
completeness |
| Keyword:
|
Banach space |
| Keyword:
|
complex Müntz theorem |
| MSC:
|
30B60 |
| MSC:
|
30E10 |
| MSC:
|
41A10 |
| idZBL:
|
Zbl 1265.30177 |
| idMR:
|
MR2990182 |
| DOI:
|
10.1007/s10587-012-0035-4 |
| . |
| Date available:
|
2012-06-08T09:40:10Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/142834 |
| . |
| Reference:
|
[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 |
| Reference:
|
[2] Borichev, A. A., Sodin, M.: Krein's entire functions and the Bernstein approximation problem.Ill. J. Math. 45 (2001), 167-185. Zbl 0989.41003, MR 1849992, 10.1215/ijm/1258138261 |
| Reference:
|
[3] Borwein, P., Erdélyi, T.: Polynomials and Polynomial Inequalities.Springer-Verlag, New York (1995). Zbl 0840.26002, MR 1367960 |
| Reference:
|
[4] Branges, L. de: The Bernstein problem.Proc. Am. Math. Soc. 10 (1959), 825-832. Zbl 0092.06905, MR 0114080, 10.1090/S0002-9939-1959-0114080-0 |
| Reference:
|
[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. Zbl 1036.30002, MR 2016836, 10.1016/j.jat.2003.09.004 |
| Reference:
|
[6] Deng, G. T.: On weighted polynomial approximation with gaps.Nagoya Math. J. 178 (2005), 55-61. Zbl 1082.41017, MR 2145315, 10.1017/S0027763000009119 |
| Reference:
|
[7] Deng, G. T.: Incompleteness and minimality of complex exponential system.Sci. China, Ser. A 50 (2007), 1467-1476. Zbl 1130.30028, MR 2390463, 10.1007/s11425-007-0093-5 |
| Reference:
|
[8] Halmos, P. R.: Measure Theory, 2nd printing, Graduate Texts in Mathematics. 18.Springer-Verlag, New York-Heidelberg-Berlin (1974). MR 0453532 |
| Reference:
|
[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. |
| Reference:
|
[10] Levin, B. Y.: Lectures on Entire Functions, Translations of Mathematical Monographs, 150.Providence RI., American Mathematical Society (1996). MR 1400006, 10.1090/mmono/150/28 |
| Reference:
|
[11] Malliavin, P.: Sur quelques procédés d'extrapolation.Acta Math. 83 (1955), 179-255. Zbl 0067.05104, MR 0075297, 10.1007/BF02392523 |
| Reference:
|
[12] Mergelyan, S. N.: On the completeness of system of analytic functions.Amer. Math. Soc. Transl. Ser. 2 (1962), 109-166. MR 0131561 |
| Reference:
|
[13] Markushevich, A. I.: Theory of Functions of a Complex Variable, Selected Russian Publications in the Mathematical Sciences.Prentice-Hall (1965). |
| Reference:
|
[14] Rudin, W.: Real and Complex Analysis, 3rd. ed.McGraw-Hill, New York (1987). Zbl 0925.00005, MR 0924157 |
| Reference:
|
[15] Sedletskij, A. M.: Nonharmonic analysis.J. Math. Sci., New York 116 (2003), 3551-3619. Zbl 1051.42018, MR 2024093 |
| Reference:
|
[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. |
| Reference:
|
[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. |
| Reference:
|
[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 |
| Reference:
|
[19] Yang, X. D.: Incompleteness of exponential system in the weighted Banach space.J. Approx. Theory 153 (2008), 73-79. Zbl 1149.30025, MR 2432554, 10.1016/j.jat.2008.01.004 |
| Reference:
|
[20] Zhu, Ch.: Some Results in Complex Approximation with Sequence of Complex Exponents.Thesis of the University of Werstern Ontario, Canada (1999). |
| Reference:
|
[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. MR 2125918, 10.1080/02781070500032804 |
| . |
