Making the case for Cauchy transforms
Making the case for Cauchy transforms
Making the case for Cauchy transforms
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2π<br />
f (z) =<br />
0<br />
f (eiθ )<br />
1 − e−iθ dθ<br />
z 2π<br />
Privalov, Morera, Plemelj, and Sokhotski considered<br />
(Kµ)(z) =<br />
2π<br />
0<br />
1<br />
1 − e −iθ z dµ(θ),<br />
where µ is a measure on [0, 2π]. Actually, <strong>the</strong>y considered<br />
(Kµ)(z) =<br />
2π<br />
0<br />
1<br />
1 − e−iθ dF (θ), F ∈ BV [0, 2π].<br />
z<br />
K = {Kµ : µ is a measure on [0, 2π]}<br />
William T. Ross (University of Richmond) <strong>Making</strong> <strong>the</strong> <strong>case</strong> <strong>for</strong> <strong>Cauchy</strong> trans<strong>for</strong>ms 5 / 42