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  "body": "\n## Definition\n\nFor an integer $n \\ge 0$, the Dirichlet kernel is the function $D_n(x) = \\sum_{k=-n}^{n} e^{ikx}$ for real $x$.\nFor an integer $N \\ge 1$, the *Fejér kernel* $F_N$ is the [[Cesàro mean|0P0DJHBGBB]] of $D_0, \\dots, D_{N-1}$:\n$$F_N(x) = \\frac{1}{N} \\sum_{n=0}^{N-1} D_n(x) = \\sum_{k=-(N-1)}^{N-1} \\left(1 - \\frac{\\abs{k}}{N}\\right) e^{ikx}.$$\n\n## Remarks\n\nFor $x$ not a multiple of $2\\pi$, the closed form is\n$$F_N(x) = \\frac{1}{N} \\left( \\frac{\\sin(Nx/2)}{\\sin(x/2)} \\right)^2,$$\nand $F_N(2\\pi m) = N$ for integers $m$.\nSo $F_N \\ge 0$, $F_N$ is $2\\pi$-periodic, and $\\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} F_N(x) \\, dx = 1$, because only the term $k = 0$ has a nonzero integral.\nFor $\\delta \\le \\abs{x} \\le \\pi$ one has $F_N(x) \\le \\frac{1}{N \\sin^2(\\delta/2)}$, which tends to $0$.\n\nExample: $F_1 = D_0 = 1$, and $F_2(x) = \\frac12 (1 + (1 + 2\\cos x)) = 1 + \\cos x$, which agrees with $\\frac12 \\left(\\frac{\\sin x}{\\sin(x/2)}\\right)^2 = 2\\cos^2(x/2)$.\n\nNon-example: the Dirichlet kernel $D_1(x) = 1 + 2\\cos x$ is not a Fejér kernel, and it takes the negative value $-1$ at $x = \\pi$.\n\nFor a $2\\pi$-periodic integrable function $f$ with [[Fourier series|G12KXDHPPS]] partial sums $S_n f$, one has $\\frac{1}{N} \\sum_{n=0}^{N-1} S_n f(x) = \\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} F_N(x - t) f(t) \\, dt$.\n",
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