@article{Dmytryshyn_2021, title={Approximation by interpolation spectral subspaces of operators with discrete spectrum}, volume={55}, url={http://matstud.org.ua/ojs/index.php/matstud/article/view/174}, DOI={10.30970/ms.55.2.162-170}, abstractNote={<p> </p> <p>The paper describes approximation properties of interpolation<br>spectral subspaces of an unbounded operator $A$ with discrete<br>spectrum $\sigma(A)$ in a Banach space $\mathfrak X$, as well as<br>ones corresponding subspaces ${\mathcal E}_{q,p}^{
u}(A)$ of<br>analytic vectors relative to $A$. Some properties of subspaces<br>${\mathcal E}_{q,p}^{
u}(A)$ are established, including the<br>possibility of their identification with the interpolation subspaces<br>obtained by the real method of interpolation. A relation between spectral subspaces and subspaces ${\mathcal<br>E}_{q,p}^{
u}(A)$ of analytic vectors of $A$ is also<br>established.</p> <p>We prove the inequalities that provide a sharp estimate of<br>errors for the best approximations by interpolation spectral<br>subspaces, as well as the subspaces ${\mathcal E}_{q,p}^{
u}(A)$.<br>Such inequalities fully characterize the subspace of elements from<br>$\mathfrak X$ in relation to rapidity of approximations. The<br>obtained estimates of spectral approximation errors are expressed<br>in terms of the quasi-norms of the approximation spaces $\mathcal<br>{B}_{q,p,\tau}^{s}(A)$ associated with the given operator $A$. In<br>this regard, the $E$-functional is used that plays a similar role<br>as the module of smoothness in the function theory.</p> <p>We use the so-called normalization factor to write the constants<br>in the estimates of spectral approximation errors. This normalization<br>factor is determined by the parameters $\tau$ and $s$ of the<br>approximation spaces $\mathcal {B}_{q,p,\tau}^{s}(A)$ and has a<br>special form in the case $\tau(1+s)=2$.</p> <p>Applications to spectral approximations of the regular elliptic<br>operators with variable smooth coefficients in the space<br>$L_q(\Omega)$ over an open bounded set $\Omega\subset\mathbb{R}^n$<br>and some self-adjoint ordinary elliptic differential operators in<br>a bounded interval $\Omega=(a,b)$ are shown.</p> <p> </p>}, number={2}, journal={Matematychni Studii}, author={Dmytryshyn, M.I.}, year={2021}, month={Jun.}, pages={162-170} }