On Approximation of Smooth Functions from Null Spaces of Optimal Linear Differential Operators with Constant Coefficients (Analysis in Theory and Applications, 2011)
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 $L_n(ψ) =\sum\limits_{k=0}^{n}a_kψ^{(k)}$, where the constant coefficients $a_k \in R$ may be adapted to $f$.We prove that for each $f \in C^{(n)}(I)$, there is a selection of coefficients $\{a_1, …,a_n\}$ and a corresponding linear combination$$S_n( f , t) =\sum_{k=1}^nb_k e^{λ_{k^t}}$$of functions $ψ_k(t) = e^{λ_kt}$ in the nullity of $L$ which satisfies the following Jackson’s type inequality: $$\|f^{(m)}-S_n^{(m)}(f,t)\|_{∞}\le \frac{|I|^{1/q}e^{|λ_n||I|}}{|a_n|2^{n-m-1/p}|λ_n|^{n-m-1}}\|L_n(f)\|_p$$ where $|λ_n| = \max\limits_k |λ_k|$, $0 ≤ m ≤ n−1,$ $p,q ≥ 1$, and $\frac{1}{p}+\frac{1}{q}= 1.$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.