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cumulative_distribution_function [2015/04/09 15:40] nikolaj |
cumulative_distribution_function [2015/04/09 16:16] nikolaj |
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| >So we can use such $S$ to normalize functions. | >So we can use such $S$ to normalize functions. | ||
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| - | >Any linear function of evaluated points are examples for $S$. | + | >For ${\mathbb D}={\mathbb N}$ the general case is $Sf:=\sum_{n=0}^\infty (L_nf)(n)$, where $(L_n)$ is a suitable sequence of linear operations (e.g. differential operators). For $L_n={\mathrm{id}}$ we get the standard sum (see below). |
| - | >So for ${\mathbb D}={\mathbb N}$ the general case is $Sf:=\sum_{n=0}^\infty b_n\cdot f(n)$, where $(b_n)$ is some suitable sequence. | + | >For ${\mathbb D}\subseteq{\mathbb R}^m$ we have integrals. |
| - | >For ${\mathbb D}={\mathbb R}^m$ we have integrals. | + | |
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| >has | >has | ||
| >$\sum_{n=0}^\infty \bar{a}(n)=1$ | >$\sum_{n=0}^\infty \bar{a}(n)=1$ | ||
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| - | >Same with $f:[d_1,d_2]\to{\mathbb R}_{\ge 0}$ a function over a real interval and integration | ||
| >The "monomial bump" on $[-d,d]$, which goes against the constant probability $\frac{1}{2d}$ for large $n$: | >The "monomial bump" on $[-d,d]$, which goes against the constant probability $\frac{1}{2d}$ for large $n$: | ||
