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), where₁, , _M are distinct complex numbers andm₁, , m_M natural numbers. The dimension of this linear space is bounded by a linear function ofN.