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On complete biorthogonal systems

Author: Robert M. Young
Journal: Proc. Amer. Math. Soc. 83 (1981), 537-540
MSC: Primary 42C30; Secondary 30D99
MathSciNet review: 627686
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Abstract: Fundamental to the study of bases in a separable Hilbert space $ H$ is the notion of a biorthogonal system. Two sequences $ \left\{ {{f_n}} \right\}$ and $ \left\{ {{g_n}} \right\}$ of elements from $ H$ are said to be biorthogonal if $ ({f_n},{g_m}) = {\delta _{nm}}$. A complete sequence that possesses a biorthogonal sequence is called exact. Despite the symmetry of the definition of biorthogonality, simple examples show that $ \{ {f_n}\} $ may be exact while $ \{ {g_n}\} $ fails to be exact. For sequences of complex exponentials in $ {L^2}( - \pi ,\pi )$, the situation is dramatically different--if the sequence $ \{ {e^{i{\lambda _n}t}}\} $ is exact, then its biorthogonal sequence is also exact.

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

  • [1] B. Ja. Levin, Distribution of zeros of entire functions, American Mathematical Society, Providence, R.I., 1964. MR 0156975
  • [2] Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Springer-Verlag, Berlin-New York, 1977. Sequence spaces; Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 92. MR 0500056
  • [3] Ivan Singer, Bases in Banach spaces. I, Springer-Verlag, New York-Berlin, 1970. Die Grundlehren der mathematischen Wissenschaften, Band 154. MR 0298399
  • [4] Robert M. Young, An introduction to nonharmonic Fourier series, Pure and Applied Mathematics, vol. 93, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 591684

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Keywords: Biorthogonal system, exact sequence, Paley-Wiener space
Article copyright: © Copyright 1981 American Mathematical Society

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