On Norm-Dependent Positive Definite Functions
Yasuo Yamasaki · Publications of the Research Institute for Mathematical Sciences · 1990
Any norm-dependent positive definite function on an infinite dimensional normed space can be written as a superposition of exp(—c • ||). Conversely, for a Hilbert space, any superposition of exp(—c -1|) is positive definite. A norm-dependent positive definite function exists only if the norm is of cotype 2. If exp(—||| | ) is positive definite for some c>0, such a form an interval (0, a0] where «0^2. If «0 = 2, then || • || is a Hilbertian norm. For (/*), Q<p^2, we have a0=p. (Though \\x = (E l*n l p ) 1 / p is not n a norm for 0<£<1, the last statement remains valid).