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