# Torba S. M.

### Characterization of the rate of convergence of one approximate method for the solution of an abstract Cauchy problem

Kashpirovskii A. I., Torba S. M.

Ukr. Mat. Zh. - 2008. - 60, № 4. - pp. 557–563

We consider an approximate method for the solution of the Cauchy problem for an operator differential equation. This method is based on the expansion of an exponential in orthogonal Laguerre polynomials. We prove that the fact that an initial value belongs to a certain space of smooth elements of the operator *A* is equivalent to the convergence of a certain weighted sum of integral residuals. As a corollary, we obtain direct and inverse theorems of the theory of approximation in the mean.

### Direct and inverse theorems of approximate methods for the solution of an abstract Cauchy problem

Ukr. Mat. Zh. - 2007. - 59, № 6. - pp. 838–852

We consider an approximate method for the solution of the Cauchy problem for an operator differential equation based on the expansion of the exponential function in orthogonal Laguerre polynomials. For an initial value of finite smoothness with respect to the operator *A*, we prove direct and inverse theorems of the theory of approximation in the mean and give examples of the unimprovability of the corresponding estimates in these theorems. We establish that the rate of convergence is exponential for entire vectors of exponential type and subexponential for Gevrey classes and characterize the corresponding classes in terms of the rate of convergence of approximation in the mean.

### Direct and Inverse Theorems in the Theory of Approximation by the Ritz Method

Gorbachuk M. L., Hrushka Ya. I., Torba S. M.

Ukr. Mat. Zh. - 2005. - 57, № 5. - pp. 633–643

For an arbitrary self-adjoint operator *B* in a Hilbert space \(\mathfrak{H}\), we present direct and inverse theorems establishing the relationship between the degree of smoothness of a vector \(x \in \mathfrak{H}\) with respect to the operator *B*, the rate of convergence to zero of its best approximation by exponential-type entire vectors of the operator *B*, and the *k*-modulus of continuity of the vector *x* with respect to the operator *B*. The results are used for finding *a priori* estimates for the Ritz approximate solutions of operator equations in a Hilbert space.