(untitled)
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 L_n(ψ) =_k=0^na_kψ^(k), where the constant coefficients a_k R may be adapted to f. We prove that for each f C^(n)(I), there is a selection of coefficients {a_1, …,a_n} and a corresponding linear combinationS_n( f , t) =_k=1^nb_k e^λ_k^t of functions ψ_k(t) = e^λ_kt in the nullity of L which satisfies the following Jackson’s type inequality: \|f^(m)-S_n^(m)(f,t)\|_∞|I|^1/qe^|λ_n||I| |a_n|2^n-m-1/p|λ_n|^n-m-1 \|L_n(f)\|_p where |λ_n| = _k |λ_k|, 0 ≤ m ≤ n−1, p,q ≥ 1, and 1p+1q= 1. For the particular operator M_n( f ) = f +1/(2n)! f ^(2n) the rate of approximation by the eigenvalues of M_n for non-periodic analytic functions on intervals of restricted length is established to be exponential. Applications in algorithms and numerical examples are discussed. , number=2, journal=Analysis in Theory and Applications, year=2011, month=Apr., pages=187–200