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  "body": "\n## Summary\n\nThe Bochner–Riesz means of a function $f$ on $\\R^n$ are the operators $S^\\delta_R f$ that multiply the [[Fourier transform|DPVM34ADZV]] of $f$ by $(1 - \\abs{\\xi}^2/R^2)_+^\\delta$.\nThey are a smoothed version of the spherical partial sums of the Fourier integral, with smoothing order $\\delta \\ge 0$.\nThe central question is for which $\\delta$ and $p$ they are bounded on $L^p(\\R^n)$, which decides whether $S^\\delta_R f \\to f$ in $L^p$ as $R \\to \\infty$.\nThis question is tied to the restriction and Kakeya problems.\n\n## Explanations\n\n### L1 human\n\nTo rebuild a function from its frequencies, one may add up all frequencies of size less than $R$ and let $R$ grow.\nA sharp cut at $\\abs{\\xi} = R$ behaves badly in higher dimensions, so one fades the frequencies out gently near the edge of the ball instead.\nThe exponent $\\delta$ says how gentle the fade is: $\\delta = 0$ is the sharp cut, and larger $\\delta$ gives a smoother fade.\nIn one dimension with $\\delta = 1$ and the factor $1 - \\abs{\\xi}/R$ instead, one gets the familiar Fejér means; the Bochner–Riesz means are the round analogue in $\\R^n$.\n",
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