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$L^2$ uniform integrability in terms of Fourier coefficients
Given a bounded sequence $(f_n)_n$ in $L^2(\mathbf{T})$ where $\mathbf{T}:=\mathbf{R}/\mathbf{Z}$, the strong compactness of $(f_n)_n$ is equivalent to $$\lim_N \sup_n \sum_{|k|\geq N} |c_k(f_n)|^2=0,$...
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