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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

$equation$

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

$equation$

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