another example of Dirac sequence

Let $\\displaystyle A_{n}=\\left[\\frac{-1}{2^{n}},\\frac{1}{2^{n}}\\right]$ and $\\delta_{n}=2^{n-1}\\chi_{A_{n}}$ for every positive integer $n$ , where $\\chi_{S}$ denotes the characteristic function of the set $S$ . Then $\\{\\delta_{n}\\}$ is a Dirac sequence:

1.
$\\delta_{n}(t)\\geq 0$ for every positive integer $n$ and every $t\\in\\mathbb{R}$ .
2.
Let $n$ be a positive integer. Then $\\displaystyle\\int\\limits_{-\\infty}^{\\infty}\\delta_{n}(t)\\,dt=\\int\\limits_{%\\frac{-1}{2^{n}}}^{\\frac{1}{2^{n}}}2^{n-1}\\,dt=1$ .
3.
Let $r>0$ . Then there exists a positive integer $N$ such that, for every integer $k>N$ , we have $\\displaystyle\\frac{1}{2^{k}}<r$ . Thus, for every integer $k>N$ , we have $\\displaystyle\\int\\limits_{\\mathbb{R}\\setminus\\left[-r,r\\right]}d_{k}(t)\\,dt=0$ .

单词	AnotherExampleOfDiracSequence
释义	another example of Dirac sequence Let $\\displaystyle A_{n}=\\left[\\frac{-1}{2^{n}},\\frac{1}{2^{n}}\\right]$ and $\\delta_{n}=2^{n-1}\\chi_{A_{n}}$ for every positive integer $n$ , where $\\chi_{S}$ denotes the characteristic function of the set $S$ . Then $\\{\\delta_{n}\\}$ is a Dirac sequence: 1. $\\delta_{n}(t)\\geq 0$ for every positive integer $n$ and every $t\\in\\mathbb{R}$ . 2. Let $n$ be a positive integer. Then $\\displaystyle\\int\\limits_{-\\infty}^{\\infty}\\delta_{n}(t)\\,dt=\\int\\limits_{%\\frac{-1}{2^{n}}}^{\\frac{1}{2^{n}}}2^{n-1}\\,dt=1$ . 3. Let $r>0$ . Then there exists a positive integer $N$ such that, for every integer $k>N$ , we have $\\displaystyle\\frac{1}{2^{k}}<r$ . Thus, for every integer $k>N$ , we have $\\displaystyle\\int\\limits_{\\mathbb{R}\\setminus\\left[-r,r\\right]}d_{k}(t)\\,dt=0$ .
随便看	FirstAxiomOfCountability first axiom of countability FirstCountable first countable FirstCountableImpliesCompactlyGenerated first countable implies compactly generated FirstFundamentalForm first fundamental form FirstIsomorphismTheorem first isomorphism theorem first isomorphism theorem FirstIsomorphismTheorem1 FirstIsomorphismTheoremForQuivers First Isomorphism Theorem for quivers FirstOrderLanguage first order language FirstOrderLogic first order logic FirstOrderOperatorsInRiemannianGeometry first order operators in Riemannian geometry FirstorderTheory first-order theory FirstPrimitivePythagoreanTriplets first primitive Pythagorean triplets FisherInformationMatrix