TY - JOUR
T1 - On Approximation of Smooth Functions from Null Spaces of Optimal Linear Differential Operators with Constant Coefficients
AU - Vesselin Vatchev
JO - Analysis in Theory and Applications
VL - 2
SP - 187
EP - 200
PY - 2011
DA - 2011/04
SN - 27
DO - http://doi.org/10.1007/s10496-011-0187-3
UR - https://global-sci.org/intro/article_detail/ata/4592.html
KW - approximation of analytic function, differential operator, fundamental set of solutions.
AB - <p style="text-align: justify;">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 $L_n(\psi) =\sum\limits_{k=0}^{n}a_k\psi^{(k)}$, where the constant coefficients $a_k \in R$ may be adapted to $f$.<br/>We prove that for each $f \in C^{(n)}(I)$, there is a selection of coefficients $\{a_1, \cdots,a_n\}$ and a corresponding linear combination$$S_n( f , t) =\sum_{k=1}^nb_k e^{\lambda_{k^t}}$$of functions $\psi_k(t) = e^{\lambda_kt}$ in the nullity of $L$ which satisfies the following Jackson’s type inequality: $$\|f^{(m)}-S_n^{(m)}(f,t)\|_{\infty}\le \frac{|I|^{1/q}e^{|\lambda_n||I|}}{|a_n|2^{n-m-1/p}|\lambda_n|^{n-m-1}}\|L_n(f)\|_p$$ where $|\lambda_n| = \max\limits_k |\lambda_k|$, $0 \leq m \leq n−1,$ $p,q \geq 1$, and $\frac{1}{p}+\frac{1}{q}= 1.$<br/>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.</p>