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We identify which choice of $u$ most effectively smooths the second derivative in the following sense. For each $u$, basic Fourier analysis implies there is a constant $C(u)$ so $\\|\\Delta(u \\ast f)\\|_{\\ell^2(\\mathbb{Z})} \\leq C(u)\\|f\\|_{\\ell^2(\\mathbb{Z})}$ for all $f: \\mathbb{Z} \\to \\mathbb{R}$. 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We identify which choice of $u$ most effectively smooths the second derivative in the following sense. For each $u$, basic Fourier analysis implies there is a constant $C(u)$ so $\\|\\Delta(u \\ast f)\\|_{\\ell^2(\\mathbb{Z})} \\leq C(u)\\|f\\|_{\\ell^2(\\mathbb{Z})}$ for all $f: \\mathbb{Z} \\to \\mathbb{R}$. 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