@article{Gorbachuk_Hrushka_Torba_2005, title={Direct and Inverse Theorems in the Theory of Approximation by the Ritz Method}, volume={57}, url={https://umj.imath.kiev.ua/index.php/umj/article/view/3629}, abstractNote={For an arbitrary self-adjoint operator <em class="EmphasisTypeItalic ">B</em> in a Hilbert space <span class="InlineEquation" id="IE1">\(\mathfrak{H}\)</span>, we present direct and inverse theorems establishing the relationship between the degree of smoothness of a vector <span class="InlineEquation" id="IE2">\(x \in \mathfrak{H}\)</span> with respect to the operator <em class="EmphasisTypeItalic ">B</em>, the rate of convergence to zero of its best approximation by exponential-type entire vectors of the operator <em class="EmphasisTypeItalic ">B</em>, and the <em class="EmphasisTypeItalic ">k</em>-modulus of continuity of the vector <em class="EmphasisTypeItalic ">x</em> with respect to the operator <em class="EmphasisTypeItalic ">B</em>. The results are used for finding <em class="EmphasisTypeItalic ">a priori</em> estimates for the Ritz approximate solutions of operator equations in a Hilbert space.}, number={5}, journal={Ukrains’kyi Matematychnyi Zhurnal}, author={Gorbachuk, M. L. and Hrushka, Ya. I. and Torba, S. M.}, year={2005}, month={May}, pages={633–643} }