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  "body": "\n## Statement\n\nLet $\\mu$ be a positive Borel measure on $\\R$ with finite moments of all orders and infinite [[support|FQTXV6BQZG]].\nLet $(p_n)_{n \\ge 0}$ be a sequence of real [[orthogonal polynomials|8TXASARQN1]] for $\\mu$, and let $I$ be the smallest closed interval that contains the support of $\\mu$.\n\n1. For every $n \\ge 1$, the polynomial $p_n$ has $n$ distinct real zeros, and all of them lie in the interior of $I$.\n2. Let $x_{n,1} \u003c \\cdots \u003c x_{n,n}$ be the zeros of $p_n$. Then the zeros of $p_n$ and $p_{n+1}$ strictly interlace:\n$$x_{n+1,k} \u003c x_{n,k} \u003c x_{n+1,k+1} \\quad \\text{for } k = 1, \\dots, n.$$\n\n## Remarks\n\nFor part 1, let $y_1, \\dots, y_m$ be the zeros of odd multiplicity of $p_n$ in the interior of $I$, the points where $p_n$ changes sign.\nSuppose $m \u003c n$.\nThen $p_n(x) \\prod_j (x - y_j)$ does not change sign on $I$ and is not zero $\\mu$-almost everywhere, so its integral against $\\mu$ is not zero.\nBut $\\prod_j (x - y_j)$ has degree $m \u003c n$, and $p_n$ is orthogonal to all polynomials of lower degree, a contradiction.\nPart 2 follows from the three-term recurrence through the Christoffel–Darboux formula.\nIf the leading coefficients are positive, it gives $p_{n+1}'(x) p_n(x) - p_{n+1}(x) p_n'(x) \u003e 0$ for all real $x$.\nEvaluating at consecutive zeros of $p_{n+1}$ shows that $p_n$ changes sign between them.\n\nExample: for the Legendre polynomials, $p_1 = x$ has the zero $0$, and $p_2 = \\frac12(3x^2 - 1)$ has the zeros $\\pm 1/\\sqrt{3}$.\nIndeed $-1/\\sqrt{3} \u003c 0 \u003c 1/\\sqrt{3}$, and all lie in $(-1, 1)$.\n\n## Explanations\n\n### L1 human\n\nThe $n$-th orthogonal polynomial crosses the axis exactly $n$ times, inside the region where the measure lives.\nMoreover, the crossings of consecutive polynomials alternate: between two crossings of one there is exactly one crossing of the other.\nThis regular pattern is a fingerprint of orthogonality.\n\n### L2 human\n\nThe zeros of $p_n$ are the eigenvalues of the $n \\times n$ truncated Jacobi matrix, the symmetric [[tridiagonal matrix|DARX1KZRHZ]] built from the recurrence coefficients.\nIn this language, interlacing is the [[Cauchy interlacing theorem|8QQPVNR80F]] for a matrix and its principal submatrix.\nThe zeros of $p_n$ are the nodes of Gaussian quadrature for $\\mu$, and the fact that they are real, simple and inside $I$ is what makes that quadrature work.\nAs $n \\to \\infty$, the counting measures of the zeros, divided by $n$, describe the asymptotic distribution of the zeros, a central topic of the theory.\n",
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