| Popis: |
1. Let R be the additive group of all real numbers and t → Tt a representation of this group on an n-dimensional complex vector space X. This means that to each real number t a linear operator Tt : X → X is assigned in such a way that $${\text{T}}_{{\text{t + s}}} = {\text{T}}_{\text{t}} {\text{T}}_{\text{s}} , {\text{T}}_0 = {\text{I}}$$ (1) Since the set {Tt : t ∈ R} consists of operators which commute one to the other the irreducible representations are one dimensional. Hence we are lead to find all functions f: R → C (C is the set of all complex numbers) such that $${\text{f}}\left( {{\text{t}}\,{\text{ + }}\,{\text{s}}} \right) = {\text{f}}\left( {\text{t}} \right)\,\,{\text{f}}\,\,{\text{(s),}}\,\,\,\,\,{\text{f}}\,\,{\text{(0)}}\,{\text{ = }}\,\,{\text{1}}{\text{.}}$$ (2) |
