TY - JOUR
AU - Dmytryshyn, M.I.
AU - Lopushansky, O.V.
PY - 2019/06/30
Y2 - 2024/11/04
TI - Spectral approximations of strongly degenerate elliptic differential operators
JF - Carpathian Mathematical Publications
JA - Carpathian Math. Publ.
VL - 11
IS - 1
SE - Scientific articles
DO - 10.15330/cmp.11.1.48-53
UR - https://journals.pnu.edu.ua/index.php/cmp/article/view/1507
SP - 48-53
AB - <p>We establish analytical estimates of spectral approximations errors for strongly degenerate elliptic differential operators in the Lebesgue space $L_q(\Omega)$ on a bounded domain $\Omega$. Elliptic operators have coefficients with strong degeneration near boundary. Their spectrum consists of isolated eigenvalues of finite multiplicity and the linear span of the associated eigenvectors is dense in $L_q(\Omega)$. The received results are based on an appropriate generalization of Bernstein-Jackson inequalities with explicitly calculated constants for quasi-normalized Besov-type approximation spaces which are associated with the given elliptic operator. The approximation spaces are determined by the functional $E\left(t,u\right)$, which characterizes the shortest distance from an arbitrary function ${u\in L_q(\Omega)}$ to the closed linear span of spectral subspaces of the given operator, corresponding to the eigenvalues such that not larger than fixed ${t>0}$. Such linear span of spectral subspaces coincides with the subspace of entire analytic functions of exponential type not larger than ${t>0}$. The approximation functional $E\left(t,u\right)$ in our cases plays a similar role as the modulus of smoothness in the functions theory.</p>
ER -