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A simple observation about compactness and fast decay of Fourier coefficients. (English) Zbl 1203.42002
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.
42A16Fourier coefficients, special Fourier series, etc.
46B50Compactness in Banach (or normed) spaces
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