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  "body": "\n## Definition\n\nA *summability kernel* is a sequence $(k_n)_{n \\ge 1}$ of [[continuous|0X3WGX95FE]] $2\\pi$-[[periodic functions|0ATF6Q0633]] $k_n \\colon \\R \\to \\C$ such that:\n\n1. $\\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} k_n(t) \\, dt = 1$ for every $n$.\n2. There is a constant $C$ with $\\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} \\abs{k_n(t)} \\, dt \\le C$ for every $n$.\n3. For every $\\delta$ with $0 \u003c \\delta \u003c \\pi$, $\\int_{\\delta \\le \\abs{t} \\le \\pi} \\abs{k_n(t)} \\, dt \\to 0$ as $n \\to \\infty$.\n\n## Remarks\n\nFor an integrable $2\\pi$-periodic $f$, the convolutions $k_n * f(x) = \\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} k_n(t) f(x - t) \\, dt$ tend to $f$ in $L^1$.\nIf $f$ is continuous, they tend to $f$ uniformly.\nA kernel with $k_n \\ge 0$ is called positive; then clause 2 follows from clause 1 with $C = 1$.\n\nExample: the [[Fejér kernel|DCCA2T9GKH]] $F_N(t) = \\sum_{\\abs{j} \u003c N} \\left(1 - \\frac{\\abs{j}}{N}\\right) e^{ijt} = \\frac{1}{N} \\left( \\frac{\\sin(Nt/2)}{\\sin(t/2)} \\right)^2$.\nIts mean is the coefficient of $e^{0}$, which is $1$.\nIt is nonnegative, so clause 2 holds with $C = 1$.\nFor $\\delta \\le \\abs{t} \\le \\pi$ it is at most $\\frac{1}{N \\sin^2(\\delta/2)}$, which gives clause 3.\n\nNon-example: the [[Dirichlet kernel|3JY4BVE4DW]] $D_n(t) = \\sum_{\\abs{j} \\le n} e^{ijt}$ satisfies clause 1, but $\\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} \\abs{D_n}$ grows like a constant times $\\log n$, so clause 2 fails.\n\nKernels indexed by a continuous parameter are treated the same way; the [[Poisson kernel|TCYSWCTYYM]] $P_r$ with $r \\to 1^-$ is the main example.\nA summability kernel is a bounded [[approximate identity|CDYXTW3PJF]] of the convolution algebra $L^1(\\mathbb{T})$.\n",
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