AbstractWe study the decay of the Fourier‐coefficients of vector‐valued functionsF:T→X, Xa Banach space. Differentiable functionsfgenerally have absolutely summable Fourier‐coefficients,f(n)<, iffXisK‐convex. More precise statements on the decay off(n) for regular functionsfcan be given ifXhas Fourier‐typep. Iffbelongs to the Besov space, the sequence (f(n)) belongs to the Lorentz sequence space lt,vwith 1/t= λ + 1/max (u′, p′). This result is the best possible in the vector‐valued case and generalizes the well‐
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