A review of the results will be presented:

1. Direct and converse theorems of approximation theory in different metrics.
2. Imbedding theorems for functional classes with the modulus of continuity of a variable increment.
3. Quadrature formulas and the functional recovery operators: applications of theory of divisors, and the method of tensor products of functionals.
4. Computational diameters: informative power of the set of linear functionals and the asymptotic error estimates in optimal recovery of fuzzy information