Vol. 10 No 7 - Pi Mu Epsilon
Vol. 10 No 7 - Pi Mu Epsilon
Vol. 10 No 7 - Pi Mu Epsilon
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566 PI MU EPSILON JOURNAL<br />
the vector ( (, 1 ). that is,<br />
V(() = {(w(. w) I w E C}. (4)<br />
Lemma 5. Let V( () be described above. Then V( () is positive, null, or<br />
negative when (satisfies, respectively, Kl > 1, Kl = 1, or Kl < 1.<br />
Proof. Fix ( E C, and consider the point ( w(. w) E V( (). Then<br />
h((w(, w), (w(, w)) = lw(l 2 -l~f = lwl 2 (1(1 2 - 1).<br />
Thus the restriction of h to V( () is now positive definite exactly when<br />
1(1 2 - I > 0, identically zero exactly when Kl 2 - 1 = 0, and negative definite<br />
exactly when Kf- 1 < 0. I<br />
We are interested particularly in the space M_ of negative lines in CP 1 •<br />
Lemma 5 implies that the space M _ is parameterized by the set<br />
D = { ( E Cl Kl < 1}. This is exactly the unit disk in the complex plane.<br />
4. Construction of the Integral Transform In this section we<br />
construct the integral transform W m on ~· First we choose f E Hm and restrict<br />
fto a line V(() eM~ where (ED. From (4) we may think off!~~'> as a function<br />
ofw E
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