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  "body": "\n## Statement\n\nFor an integer $N \\ge 1$, let $D_N$ be the [[Dirichlet kernel|3JY4BVE4DW]] of order $N$ and let\n$$L_N = \\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} \\abs{D_N(x)} \\, dx$$\nbe its $N$-th *Lebesgue constant*.\nThen\n$$L_N = \\frac{4}{\\pi^2} \\log N + O(1) \\qquad \\text{as } N \\to \\infty.$$\nIn particular $L_N \\to \\infty$.\nLet $C(\\mathbb{T})$ be the space of [[continuous|0X3WGX95FE]] $2\\pi$-periodic functions with the supremum norm.\nThen $L_N$ is the operator norm of the partial sum map $f \\mapsto S_N f$ on $C(\\mathbb{T})$.\nIt is also the norm of the linear functional $f \\mapsto S_N f(0)$ on $C(\\mathbb{T})$.\n\n## Remarks\n\nFor the estimate, use $\\abs{\\sin(x/2)} \\le \\abs{x}/2$ and compare $\\abs{\\sin((N + \\frac{1}{2}) x)}$ on each interval between consecutive zeros with its mean value $\\frac{2}{\\pi}$.\nSumming over about $N$ intervals gives a harmonic sum, hence the $\\log N$.\nThe norm of $f \\mapsto S_N f(0) = \\frac{1}{2\\pi} \\int f(-t) D_N(t) \\, dt$ is at most $L_N$, and it equals $L_N$ by testing on continuous functions that approximate $\\operatorname{sgn} D_N$.\nExample: $L_0 = 1$ and $L_N \u003e 1$ for $N \\ge 1$; the values grow very slowly, like $0.405 \\log N$.\n\n## Explanations\n\n### L1 human\n\nThe Dirichlet kernel has integral $1$, but the integral of its absolute value is much larger and grows like a constant times $\\log N$.\nThis number measures how much a partial Fourier sum can amplify a bounded function.\nSince the amplification is unbounded, partial Fourier sums of bounded continuous functions can become unbounded.\n\n### L2 human\n\nBy the uniform boundedness principle, the unboundedness of $L_N$ implies that some continuous function has a Fourier series that diverges at $0$.\nIt also implies that $S_N f$ does not converge to $f$ in $L^1(\\mathbb{T})$ for every $f \\in L^1(\\mathbb{T})$, because the norm of $S_N$ on $L^1$ is also $L_N$.\nOn the positive side, $\\abs{S_N f - f} \\le (1 + L_N) E_N(f)$, where $E_N(f)$ is the distance from $f$ to trigonometric polynomials of degree at most $N$; combined with Jackson's theorem this gives the Dini–Lipschitz criterion.\nThe Fejér kernel, which has $L^1$ norm $1$, avoids the problem and gives Fejér's theorem.\n",
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