A structure theorem for sum form functional equations

Summary It is proved that, iff _ij:]0, 1 C (i = 1, ,k;j = 1, ,l) are measurable, satisfy the equation (1) (with some functionsg _it, h_jt:]0, 1 C), then eachf _ij is in a linear space (called Euler space) spanned by the functionsx x ^j(logx) ^k (x ]0, 1;j = 1, ,M;k = 0, ,m _j – 1), where ₁, , _M are distinct complex numbers andm ₁, , m_M natural numbers. The dimension of this linear space is bounded by a linear function ofN.