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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 YX 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 {ε n }0 such that |a n (y)|ε n y Y for every n and every yY. We prove this result and we use it for the study of fast decay of Fourier coefficients in L p (𝕋) and frame coefficients in the Hilbert setting.
MSC:
42A16Fourier coefficients, special Fourier series, etc.
46B50Compactness in Banach (or normed) spaces