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  "body": "\n## Definition\n\nLet $(\\mu_n)_{n \\ge 0}$ be a sequence of real numbers.\nDefine its iterated differences by $\\Delta^0 \\mu_k = \\mu_k$ and $\\Delta^{j+1} \\mu_k = \\Delta^j \\mu_k - \\Delta^j \\mu_{k+1}$, so that\n$$\\Delta^j \\mu_k = \\sum_{i=0}^{j} (-1)^i \\binom{j}{i} \\mu_{k+i}.$$\nThe *Hausdorff matrix* of $(\\mu_n)$ is the lower triangular matrix $H_\\mu = (h_{nk})$ with\n$$h_{nk} = \\binom{n}{k} \\Delta^{n-k} \\mu_k \\quad (0 \\le k \\le n), \\qquad h_{nk} = 0 \\quad (k \u003e n).$$\nA [[sequence|0QE8STGJH2]] $(s_k)$ of complex numbers is *Hausdorff summable* by $H_\\mu$ to $s$ if $t_n = \\sum_{k=0}^{n} h_{nk} s_k \\to s$ as $n \\to \\infty$.\nA series is Hausdorff summable if the sequence of its partial sums is.\n\n## Remarks\n\nExample: for $\\mu_n = \\frac{1}{n+1} = \\int_0^1 t^n \\, dt$ one gets $\\Delta^{n-k} \\mu_k = \\int_0^1 t^k (1 - t)^{n-k} \\, dt = \\frac{k! \\, (n-k)!}{(n+1)!}$.\nSo $h_{nk} = \\frac{1}{n+1}$, and $t_n = \\frac{s_0 + \\cdots + s_n}{n+1}$ is the Cesàro mean.\nSimilarly, $\\mu_n = 2^{-n}$ gives $h_{nk} = \\binom{n}{k} 2^{-n}$, the Euler $(E, 1)$ means.\n\nNon-example of regularity: for $\\mu_0 = 1$ and $\\mu_n = 0$ for $n \\ge 1$, one has $\\Delta^{n-k} \\mu_k = 0$ for $k \\ge 1$ and $\\Delta^n \\mu_0 = 1$.\nSo $t_n = s_0$ for all $n$, and the method sends every sequence to its first term.\n\nHausdorff showed that $H_\\mu$ is regular if and only if $\\mu_n = \\int_0^1 t^n \\, d\\chi(t)$ for a function $\\chi$ of bounded variation on $[0, 1]$ with $\\chi(0) = 0$, $\\chi(1) = 1$ and $\\chi$ continuous at $0$.\nIn the non-example, $\\chi$ jumps from $0$ to $1$ at $0$, which violates the last condition.\n",
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