Simultaneous Lp-approximations of polynomials and derivatives on the whole space

We have obtained a sufficient condition that a linear sum of an activation function can simultaneously approximate polynomials and their derivatives in the sense of L^p(R^d, μ). If the probability measure μ is rapidly decreasing, a wide variety of differentiable functions satisfy this condition. For the Gaussian measure, rapidly increasing functions such as e^t, exp(t² /2) and others can be activation functions. The proof is constructive and elementary, which enables the approximation formulas to be written explicitly