f^(ω)=(2π)2n1Rn∫f(x)e−i(x,ω)dx
f^(ω)=(2π)2n1Rn∫xαe−i(x,ω)dx=(2π)2n1Rn∫k=1∏nxkαke−ixkωkdx=
=k=1∏n(2π1R∫xkαke−ixkωkdxk)
Consider 2π1R∫xαe−ixωdx. Prove by induction that 2π1R∫xαe−ixωdx=iα2πδ(α)(ω).
Since δ(ω)=δ(ω) , for α=0 we have 2π1R∫e−ixωdx=2π1R∫eixωdx=
=2π⋅δ(ω)=2πδ(ω).
Suppose that 2π1R∫xke−ixωdx=ik2πδ(k)(ω), where k≥0.
Then ik+12πδ(k+1)(ω)=idωd(ik2πδ(k)(ω)). By the induction hypothesis we obtain ik+12πδ(k+1)(ω)=idωd(2π1R∫xke−ixωdx))=
=2π1R∫ixkdωde−ixωdx=2π1R∫xk+1e−ixωdx
So we prove this statement for α=k+1.
By the principle of mathematical induction we obtain that 2π1R∫xαe−ixωdx=iα2πδ(α)(ω) for all α≥0.
So f^(ω)=k=1∏n(2π1R∫xkαke−ixkωkdxk)=
=k=1∏n(iαk2πδ(αk)(ωk))=i∣α∣(2π)2nk=1∏nδ(αk)(ωk)
Answer: i∣α∣(2π)2nk=1∏nδ(αk)(ωk)
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