A Markov–Bernstein inequality for Gaussian networks

Let s ≥ 1 be an integer. A Gaussian network is a function on R of the form g(x) = ∑N k=1 ak exp(−‖x − xk‖ ). The minimal separation among the centers, defined by min1≤j 6=k≤N ‖xj − xk‖, is an important characteristic of the network that determines the stability of interpolation by Gaussian networks, the degree of approximation by such networks, etc. We prove that if g(x) = ∑N k=1 ak exp(−‖x − xk‖ ), the minimal separation of g exceeds 1/m, and logN = O(m) then for any integer r ≥ 1, any partial derivative Dg of order r of g satisfies ‖Dg‖p,Rs ≤ cm‖g‖p,Rs .