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Zbl 1203.42002
Almira, J.M.
A simple observation about compactness and fast decay of Fourier coefficients.
(English)
[J] Ann. Funct. Anal. AFA 1, No. 1, 41-43, electronic only (2010). ISSN 2008-8752/e

Summary: Let $X$ be a Banach space and suppose $Y\subseteq X$ is a Banach space compactly embedded into $X$, and $(a_k)$ is a weakly null sequence of functionals in $X^*$ . Then there exists a sequence $\{\varepsilon_n\}\searrow 0$ such that $|a_n(y)|\le \varepsilon_n\|y\|_Y$ for every $n\in\Bbb N$ and every $y\in Y$. We prove this result and we use it for the study of fast decay of Fourier coefficients in $L^p(\Bbb T)$ and frame coefficients in the Hilbert setting.
MSC 2000:
*42A16 Fourier coefficients, etc.
46B50 Compactness in normed spaces

Keywords: Fourier coefficient; frame coefficient; continuous linear functional; compactness in Banach spaces

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