Uniqueness of Linear Combinations of Ridge Functions

Ridge functions are multivariate functions of the form g(a ldr x), where g is a univariate function, and a ldr x is the inner product of a isin R^d\{0} and x isin R^d. We are concerned with the uniqueness of representation of a given function as some sum of ridge functions. We prove that if f(x) = Sigma_i=1 ^m g_i(aⁱldr x) = 0 for some aⁱ = (a₁ ⁱ, hellip , a_d ⁱ) isin R^d\{0}, and if g_i isin L_loc ^p(R) (or g_i isin D' (R) and g_i(aⁱ ldr x) isin D' (R^d)), then, each g_i is a polynomial of degree at most m - 2. We also prove a theorem on the smoothness of linear combinations of ridge functions. These results improve the existing results.