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Biomedical Imaging by Sebastian Aeffner, Timo Aspelmeier, Tim Salditt

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The representation of sgn(t) can be proved by complex analysis and is used here to study the action of the Riesz operator with α = 1 − n on g

where we have used the definition of the Hilbert transform in the last step. The Hilbert transform has the following property

(s)n1(Hg(z))(s)=(H(z)n1g(z))(s).(4.53)

Plugging the last expression for (I1−ng)(s) as well as the last mentioned property of the Hilbert transform into the general inversion formula (4.40) and with s = xnθ we find that

f(x_)=i2(2π)n+1dn_θ1πdz1sz(s)n1(Rf)(n_θ,z)sz|s=x_n_θ=i2(2π)n+1(i)n1(R#H[ z ]n1(Rf)(z))(x_)

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