Approximation of infinitely differentiable multivariate functions is intractable

We prove that L∞L∞-approximation of C∞C∞-functions defined on [0,1]d[0,1]d is intractable and suffers from the curse of dimensionality. This is done by showing that the minimal number of linear functionals needed to obtain an algorithm with worst case error at most ε∈(0,1)ε∈(0,1) is exponential in dd. This holds despite the fact that the rate of convergence is infinite.