A class of reproducing kernels
J. R. Hattemer · Illinois Journal of Mathematics · 1968
Let P(x) be a homogeneous positive definite polynomial of order 2m, m > 0 an teer, hang constant coecients.e show that the emel K(x, y) ' exp (2vi P(z) dz, where x e E, y > 0, W 1 0, the imaginary prt of is positive nd the rel prt of is negative, stisfies the follog five properties"(1) K(x, y) e L (E), dependently of y;(2) f g(x, y) dx 1;(3) f>oIg(x,y)qdxOyO,l.q;(4) K(x, y) 0}, where P(D) is the erential operator obtaed from P(x) by replacg each occurrence of x by O/Ox, i 1, n. Lettg xx x', x _x from thehomogeneity of P, we obtain by a simple change of variable the follog identities for K" g(x, y) y-g(xy -, 1) xg(x', y x]-) for all x e E, y > 0.