Approximation of functions of finite variation by superpositions of a Sigmoidal function Original Research Article

The aim of this note is to generalize a result of Barron [1] concerning the approximation of functions, which can be expressed in terms of the Fourier transform, by superpositions of a fixed sigmoidal function. In particular, we consider functions of the type h(x) = ∫ℝd ƒ (〈t, x〉)dμ(t), where μ is a finite Radon measure on ℝd and ƒ : ℝ → ℂ is a continuous function with bounded variation in ℝ We show (Theorem 2.6) that these functions can be approximated in L2-norm by elements of the set Gn = {Σi=0staggeredn cig(〈ai, x〉 + bi) : aiℝd, bi, ciℝ}, where g is a fixed sigmoidal function, with the error estimated by C/n1/2, where C is a positive constant depending only on f. The same result holds true (Theorem 2.9) for f : ℝ → ℂ satisfying the Lipschitz condition under an additional assumption that ∫ℝd‖t‖ed|u(t)| > ∞