Let E be a bounded measurable subset of the real line. The finite Hilbert transform is the operator HE defined on one of the spaces LP(E) (1 1, the Hilbert transform of f at x is defined by (1) Hf(x) = li (rmi) f f(tXt -x) 1dt, provided this latter limit exists. It is known that if f is in LP(R), for p > 1, then Hf exists a.e. Moreover, when p > 1, the map f -Hf defines a bounded linear operator on LP(R) (see, e.g., Garsia [2] ). Obviously, it is possible to consider the reduced Hilbert transform HE defmed on LP(E) (p > 1) by HEf(x) = (ri) 1 fE f(t)(t X)-1 dt; here, the integral is clearly singular and must be interpreted as the Cauchy principal value defined in equation (1). The map f -+ HEf is a bounded operator on LP(E) (p > 1). If the set E is bounded, then the operator HE will be referred to as a finite Hilbert transform. Widom [8] described the spectra of the finite Hilbert transforms on all of the LP spaces (1 1, define Received by the editors August 26, 1974. AMS (MOS) subject classifications (1970). Primary 45EO5; Secondary 44A15. (1) This work was supported by the National Science Foundation. Copyright i) 1975, Anierican Mathematical Society 347 This content downloaded from 207.46.13.150 on Thu, 28 Jul 2016 05:11:13 UTC All use subject to http://about.jstor.org/terms