Description of main results (1983-2001).

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

where is the set of polynomials with zero mean. It is proved that

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

is found for

and for

In particularly, it is proved that

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

for any fixed Here is the best -approximation of a function by spherical polynomials of degree at most is the th modulus of continuity of based on translation

is the measure of the unit Euclidean sphere is the first positive zero of the Gegenbauer's cosine polynomial .

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

for or and for Here is the best approximation of a function by a cosine polynomial of degree at most in -metric and is the th modulus of continuity based on the operator of generalized translation in is the first positive zero of Jacobi cosine polynomial

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

between the best approximation of a function by the space of entire functions of exponential type , on the left hand, and its th modulus of continuity on the other hand. We proved that is continuous with respect to

2002-01-24