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  "body": "\n## Definition\n\nLet $n \\ge 1$ be an integer.\nThe *Jackson kernel* of order $n$ is the function\n$$J_n(t) = \\lambda_n \\left(\\frac{\\sin(nt/2)}{\\sin(t/2)}\\right)^4, \\qquad t \\in \\R,$$\nwith the value $\\lambda_n n^4$ at the multiples of $2\\pi$.\nThe constant $\\lambda_n \u003e 0$ is chosen so that $\\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} J_n(t) \\, dt = 1$.\n\n## Remarks\n\nSince $\\left(\\frac{\\sin(nt/2)}{\\sin(t/2)}\\right)^2 = n F_{n-1}(t)$, where $F_{n-1}$ is the [[Fejér kernel|DCCA2T9GKH]], $J_n$ is a constant times $F_{n-1}^2$.\nSo $J_n$ is an even, nonnegative [[trigonometric polynomial|J3NJQ5FZJ5]] of degree $2n - 2$.\nIts key property is $\\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} \\abs{t}^k J_n(t) \\, dt \\le \\frac{C}{n^k}$ for $k = 1, 2$, with $C$ independent of $n$.\n\nExample: for $n = 1$, the quotient is $1$, so $J_1 = 1$ and $\\lambda_1 = 1$.\n\nNon-example: the Fejér kernel $F_n$ is nonnegative with mean $1$, but it fails the key property for $k = 1$.\nIts first moment $\\frac{1}{2\\pi} \\int \\abs{t} F_n(t) \\, dt$ is of order $\\frac{\\log n}{n}$, not $\\frac{1}{n}$.\n\nTexts differ in the indexing; some call $J_m$ with $m \\approx n/2$ the kernel of degree $n$.\nThe convolution $f \\mapsto f * J_n$ is the Jackson operator.\n",
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