Direct and inverse theorems on approximation of solutions of operator equations (Q2735872)

The authors continue to develop theoretical and applied aspects of the conception of smooth, analytical and entire elements from the definition domain \(D(A)\) of a selfajoint positive operator \(A\) in a Hilbert space \(H\). They annouce results on the rate of the convergence to zero of \(\|Au_n- f\|_H\), where \(u_n\) is some Ritz's approximation of a solution \(u\) of the equation \(Au= f\). This approximation is build on the eigenelements basis of some selfadjoint positive definite operator \(B\) with \(D(B)= D(A)\), and the estimate of the convergence rate depends on the smoothness of \(u\) in a sense of spaces constructed by \(B\). An application to operators \(Au= -u''+ q(x)u\), \(Bu= u''\) with \(D(A)= D(B) = H^2 (0,\pi)\cap H^1(0,\pi)\) gives a connection between the convergence degree to zero of \(\|u-u_n\|\) and the usual smoothness of \(f\) with some boundary value conditions.