On Positive‐Definite Functions

(ii) r(x) = 0 for almost all x, (iii) both q(x) and r(x) are positive-definite. That r(x) is positive-definite is our principal new result. The theorem is proved in §§ 3-5. In § 6 we sharpen (ii), and show that if#>(a;) is 'isotropic', and m > 1, then r(x) = 0 for x ^ 0. In § 7 we prove the corresponding result for functions positive-definite on the surface of a sphere; there is presumably no difficulty in extending this result to functions positivedefinite on a hypersphere.