A hypercomplex system (h.c.s.) 𝐿 1 ( 𝑄 , 𝑚 ) is, roughly speaking, a space which is defined by a structure measure ( 𝑐 ( 𝐴 , 𝐵 , 𝑟 ) , ( 𝐴 , 𝐵 ∈ ℬ ( 𝑄 ) ) ) , such space has been studied by Berezanskii and Krein. Our main result is to define the exponentially convex functions (e.c.f.) on (h.c.s.), and we will study their properties. The definition of such functions is a natural generalization of that defined on semigroup.