For a real valued function f defined on a finite interval I we consider the problem of approximating f from null spaces of differential operators of the form Ln(ψ) =n ∑k=0 akψ(k), where the constant coefficients ak ∈ R may be adapted to f. We prove that for each f ∈ C(n) (I), there is a selection of coefficients {a1,…, an } and a corresponding linear combinationSn (f, t) = n∑k=l bkeλkt of functions ψk(t) = eλkt in the nullity of L which satisfies the following Jackson's type inequality: ||f(m) - Sn(m)(f ,t)||∞ ≤|I|1/qe|λn||I|/|an|2n-m-1/p|λn|n-m-l ||Ln(f)||p,where |λn| = max|λk|, 0 ≤ m ≤ n- 1, p,q ≥ 1, and 1/p +1/q = 1. For the particular operator Mn(f) = f + 1/(2n) !f(2n) the rate of approximation by the eigenvalues of Mn for non-periodic analytic functions on intervals of restricted length is established to be exponential. Applications in algorithms and numerical examples are discussed.