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  "body": "\n## Definition\n\nLet $N \\ge 0$ be an integer.\nThe *Dirichlet kernel* of order $N$ is the [[trigonometric polynomial|J3NJQ5FZJ5]] $D_N \\colon \\R \\to \\R$ given by\n$$D_N(x) = \\sum_{n=-N}^{N} e^{inx} = 1 + 2 \\sum_{n=1}^{N} \\cos nx.$$\n\n## Remarks\n\nSumming the geometric progression gives $D_N(x) = \\frac{\\sin((N + \\frac{1}{2}) x)}{\\sin(x/2)}$ for $x \\notin 2\\pi\\Z$, and $D_N(x) = 2N + 1$ for $x \\in 2\\pi\\Z$.\nIt is even, $2\\pi$-periodic, and $\\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} D_N(x) \\, dx = 1$, since only the term $n = 0$ contributes.\nFor $f$ integrable and $2\\pi$-periodic, the partial sums of its [[Fourier series|G12KXDHPPS]] are the [[convolutions|2Z29JN3VES]]\n$$S_N f(x) = \\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} f(x - t) D_N(t) \\, dt.$$\nExample: $D_1(x) = 1 + 2 \\cos x$, and indeed $\\frac{\\sin(3x/2)}{\\sin(x/2)} = 3 - 4 \\sin^2(x/2) = 1 + 2 \\cos x$.\nNon-example: $\\sum_{n=0}^{N} e^{inx}$ is not a Dirichlet kernel, since its sum is not symmetric in $n$; it is not real-valued for $N \\ge 1$.\nSome authors use the factor $\\frac{1}{2\\pi}$ in $D_N$ itself, or the period $1$ with $D_N(x) = \\sum_{\\abs{n} \\le N} e^{2\\pi i n x}$.\n",
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