TY - JOUR
AU - Dmytryshyn, M.I.
PY - 2021/06/22
Y2 - 2022/01/20
TI - Approximation by interpolation spectral subspaces of operators with discrete spectrum
JF - Matematychni Studii
JA - Mat. Stud.
VL - 55
IS - 2
SE - Articles
DO - 10.30970/ms.55.2.162-170
UR - http://www.matstud.org.ua/ojs/index.php/matstud/article/view/174
SP - 162-170
AB - The paper describes approximation properties of interpolationspectral subspaces of an unbounded operator $A$ with discretespectrum $\sigma(A)$ in a Banach space $\mathfrak X$, as well asones corresponding subspaces ${\mathcal E}_{q,p}^{
u}(A)$ ofanalytic vectors relative to $A$. Some properties of subspaces${\mathcal E}_{q,p}^{
u}(A)$ are established, including thepossibility of their identification with the interpolation subspacesobtained by the real method of interpolation. A relation between spectral subspaces and subspaces ${\mathcalE}_{q,p}^{
u}(A)$ of analytic vectors of $A$ is alsoestablished.We prove the inequalities that provide a sharp estimate oferrors for the best approximations by interpolation spectralsubspaces, as well as the subspaces ${\mathcal E}_{q,p}^{
u}(A)$.Such inequalities fully characterize the subspace of elements from$\mathfrak X$ in relation to rapidity of approximations. Theobtained estimates of spectral approximation errors are expressedin terms of the quasi-norms of the approximation spaces $\mathcal{B}_{q,p,\tau}^{s}(A)$ associated with the given operator $A$. Inthis regard, the $E$-functional is used that plays a similar roleas the module of smoothness in the function theory.We use the so-called normalization factor to write the constantsin the estimates of spectral approximation errors. This normalizationfactor is determined by the parameters $\tau$ and $s$ of theapproximation spaces $\mathcal {B}_{q,p,\tau}^{s}(A)$ and has aspecial form in the case $\tau(1+s)=2$.Applications to spectral approximations of the regular ellipticoperators with variable smooth coefficients in the space$L_q(\Omega)$ over an open bounded set $\Omega\subset\mathbb{R}^n$and some self-adjoint ordinary elliptic differential operators ina bounded interval $\Omega=(a,b)$ are shown.
ER -