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Completeness of $ \{{\rm sin}\ nx+Ki$ $ {\rm cos}\ nx\}$


Author: Jonathan I. Ginsberg
Journal: Proc. Amer. Math. Soc. 29 (1971), 291-293
MSC: Primary 42.17
DOI: https://doi.org/10.1090/S0002-9939-1971-0276679-6
MathSciNet review: 0276679
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Abstract: Let $ {\mathbf{C}}[ {a,b} ]$ be the space of continuous functions on the interval $ [ {a,b} ]$. It is shown that the set of functions $ \{ {\sin nx + Ki\cos nx} \}_{n = 1}^\infty ,K \ne \pm 1$, is incomplete in $ {\mathbf{C}}[ {0,\pi + a} ],a > 0$.


References [Enhancements On Off] (What's this?)

  • [1] R. Feinerman and D. J. Newman, Completeness of $ \{ A\sin nx + B\cos nx\} $ on $ [0,\pi ]$, Michigan Math. J. 15 (1968), 305-312. MR 38 #3689. MR 0235380 (38:3689)
  • [2] N. Levinson, Gap and density theorems, Amer. Math. Soc. Colloq. Publ., vol. 26, Amer. Math. Soc., Providence, R.I., 1940. MR 2, 180. MR 0003208 (2:180d)
  • [3] A. Zygmund, Trigonometrical series, 2nd ed., Chelsea, New York, 1952. MR 17, 844. MR 0076084 (17:844d)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1971-0276679-6
Keywords: Completeness, $ {\mathbf{C}}[0,\pi + a]$
Article copyright: © Copyright 1971 American Mathematical Society

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