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- W4287704637 abstract "We are interested in the ``smoothest'' averaging that can be achieved by convolving functions $f in ell^2(mathbb{Z})$ with an averaging function $u$. More precisely, suppose $u:{-n, ldots, n} to mathbb{R}$ is a symmetric function normalized to $sum_{k=-n}^{n}u(k) = 1$. We show that every convolution operator is not-too-smooth, in the sense that $$sup_{f in ell^2(mathbb{Z})} frac{| nabla (f*u)|_{ell^2(mathbb{Z})}}{|f|_{ell^2}}geq frac{2}{2n+1},$$ and we show that equality holds if and only if $u$ is constant on the interval ${-n, ldots, n}$. In the setting where smoothness is measured by the $ell^2$-norm of the discrete second derivative and we further restrict our attention to functions $u$ with nonnegative Fourier transform, we establish the inequality $$sup_{f in ell^2(mathbb{Z})} frac{| Delta (f*u)|_{ell^2(mathbb{Z})}}{|f|_{ell^2(mathbb{Z})}} geq frac{4}{(n+1)^2},$$ with equality if and only if $u$ is the triangle function $u(k)=(n+1-|k|)/(n+1)^2$. We also discuss a continuous analogue and several open problems." @default.
- W4287704637 created "2022-07-26" @default.
- W4287704637 creator A5054778975 @default.
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- W4287704637 date "2020-07-27" @default.
- W4287704637 modified "2023-09-29" @default.
- W4287704637 title "The smoothest average: Dirichlet, Fej'er and Chebyshev" @default.
- W4287704637 doi "https://doi.org/10.48550/arxiv.2007.13700" @default.
- W4287704637 hasPublicationYear "2020" @default.
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