Spectrum of Positive Definite Functions on Product Hypergroups

This paper aims to show that the amenability of K sub 1 /sub x K sub 2 /sub is equivalent to the following condition: If φ is a continuous positive definite function defined on K sub 1 /sub x K sub 2 /sub and φ ≥ 0 then the constant function 1 sub K1xK2 /sub belongs to the spectrum of φ , which K sub 1 /sub and K sub 2 /sub are locally compact hypergroups as defined by R. Jewett , with convolutions * sub 1 /sub ; * sub 2 /sub respectively. Our study deals with the cases of exponentially bounded product hypergroups and discrete solvable product hypergroups. And study of conditionally exponential convex functions.