**1983-1984**. An arbitrary space arising from a
-periodic Chebyshev system
and invariant
with respect to a translation of
is
considered. Let

The important example is the space of trigonometric polynomials of degree The problem appropriate to trigonometric case was put by L.V.Taikov in the beginning 60th years of XX century.

**1985**. Let be an infinite-dimesional linear normed
space and
its unit ball. We
prove that for any fixed number
there
exists a continuous mapping of the ball into
itself such that
for all

**1986-1988**. Let
be the best trigonometric
-approximation of order of a -periodic
function and let
be the
modulus of continuity of in The smallest
constant
in the Jackson inequality

The analogous results are obtained for mean square approximations of periodic functions by trigonometric polynomials on a uniform net.

**1996-1998**. It is proved exact Jackson-Stechkin
inequality

Let be a space of real quadratically integrable functions on the segment with the Jacobi weight function or The following exact Jackson-Stechkin inequality has been proved

The analogous exact inequalities are obtained for mean square approximations of functions of several variables given on the Euclidean and projective spaces. These results expand the known sharp results of N.I.Chernykh, V.A.Yudin, V.V.Arestov and V.Yu.Popov.

**2000-2001**. It is considered the Jackson-Stechkin
inequality between the best mean square
approximation of an arbitrary -periodic
function by finite-dimensional subspace
and modulus of continuity (smoothness)
of generated by a finite difference operator with
coefficients (weights) depending continuously on the step
of the operator. The universal (independent of ) lower
bound for the exact constant in the described inequality
is obtained. This bound is sharp in a number of cases.

The exact results in this direction for the case of the classical modulus of continuity of the first order in spaces and were established by N.P.Korneichuk (1962-1984), N.I.Chernykh (1967-1992), V.I.Berdyshev (1967-1985) and other mathematicians.

The mentioned above finite difference operator and corresponding modulus of continuity are connected with divided and finite differences, generalized moduli of continuity (smoothness) which were investigated and used by T.Popoviciu (1959), J.Boman and H.S.Shapiro (1971), G.Mühlbach (1973), A.Sharma and J.Tzimbalario (1977), Ch.A.Micchelli (1979), V.T.Shevaldin (1981), R.B.Barrar and H.L.Loeb (1984), Z.Wronicz (1984), Gh.Toader and S.Toader (1985) and other mathematicians.

**Below we describe the results received jointly with
V.V.Arestov.**

**1997-2000**. Let
be an
uniformly bounded sequence of continues real functions on
compactum
, and denote set
of all non-negative summable sequences
such that
If
, let us consider the minimization
problem for the functional
on .
Delsarte's method on upper bounds for packings in some
metric spaces lead to the problem for special functional
system . We wrote dual problem for mentioned
problem and proved corresponding duality theorem
including existence of solutions of the both problems.
Additional properties of the solutions was indicated
in the case when
and is the
system
of Jacobi polynomials where
. For
, it was found exact
solutions of these problems which are connected with the
kissing number problem in .

Possibilities of the Delsarte scheme for upper bounds of minimal distances in the sets which consist of 24 and 25 points on the unit sphere of the Euclidean space were shown. As consequence, it is obtained that among arbitrary points located on unit sphere in there exist two points with the angular distance between them strictly less than .

**2001-2002**. Denote by
the
least constant in the Jackson inequality

2002-01-24