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        "quote": "> [!example] <a class=\"object-permalink\" href=\"#example-12-4\">例 12.4</a> \n>\n> 设$\\theta ( x )$为<u class=\"concept-term\" data-concept-identity=\"urn:uuid:c7a623f9-79cd-45e9-ad58-94b359b2bff2\" data-concept-route=\"/concept-identities#concept-c7a623f9-79cd-45e9-ad58-94b359b2bff2\" id=\"first-e32fef5c881d81e4aca4-zh\">亥维赛德阶跃函数(Heaviside step function)</u>\n>\n> $$\n> \\theta (x) = \\left\\{ \\begin{array}{l l} 1 & \\text { if } x \\geq 0, \\\\ 0 & \\text { if } x <   0. \\end{array} \\right.\n> $$\n>\n> 这显然是一个局部可积函数，并生成一个正则分布$T _ { \\theta }$。对于任意试验函数$\\varphi \\in { \\mathcal { D } } ^ { 1 } ( \\mathbb { R } )$\n>\n> $$\n> \\begin{aligned}\n> T _ {\\theta^ {\\prime}} (\\varphi) &= T _ {\\theta} (- \\varphi^ {\\prime}) = - \\int_ {- \\infty} ^ {\\infty} \\varphi^ {\\prime} (x) \\theta (x)   \\mathrm{d} x \\\\[1em]\n>  &= - \\int_ {0} ^ {\\infty} \\frac {\\mathrm{d} \\varphi}{\\mathrm{d} x}   \\mathrm{d} x \\\\[1em]\n>  &= \\varphi (0) \\quad \\text { since } \\varphi (\\infty) = 0 \\\\[0.4em]\n>  &= \\delta (\\varphi).\n> \\end{aligned}\n> $$\n>\n> 因此我们得到分布方程，仅当 ${ \\mathcal { D } } ^ { 1 } ( \\mathbb { R } )$ 时成立，\n>\n> $$\n> T _ {\\theta} ^ {\\prime} = \\delta .\n> $$\n>\n> 这通常用“函数”来表述为\n>\n> $$\n> \\delta (x) = \\theta^ {\\prime} (x) = \\frac {\\mathrm{d} \\theta (x)}{\\mathrm{d} x}.\n> $$\n>\n> 直观上，$x = 0$ 处的台阶是“无限陡峭”的。",
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