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Question Number 1055 by 123456 last updated on 24/May/15

 f:[0,1]→R  g:[0,1]×N→R  g_n (x)=f[g_(n−1) (x)]  g_0 (x)=x  A{f}(n)=∫_0 ^1 f(t)g_n (t)dt  A{f+g}=^? A{f}+A{g}  A{kf}=^? kA{f}

$$\:{f}:\left[\mathrm{0},\mathrm{1}\right]\rightarrow\mathbb{R} \\ $$$${g}:\left[\mathrm{0},\mathrm{1}\right]×\mathbb{N}\rightarrow\mathbb{R} \\ $$$${g}_{{n}} \left({x}\right)={f}\left[{g}_{{n}−\mathrm{1}} \left({x}\right)\right] \\ $$$${g}_{\mathrm{0}} \left({x}\right)={x} \\ $$$$\mathscr{A}\left\{{f}\right\}\left({n}\right)=\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}{f}\left({t}\right){g}_{{n}} \left({t}\right){dt} \\ $$$$\mathscr{A}\left\{{f}+{g}\right\}\overset{?} {=}\mathscr{A}\left\{{f}\right\}+\mathscr{A}\left\{{g}\right\} \\ $$$$\mathscr{A}\left\{{kf}\right\}\overset{?} {=}{k}\mathscr{A}\left\{{f}\right\} \\ $$

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