In this work we study an extremal problem for functions continuous
on an interval which are representable by series with respect to
orthogonal polynomials with restrictions imposed on the values of
the functions and the expansion coefficients, which is encountered
in the investigations of Ph.Delsarte of the bounds of packings in
certain metric spaces. Delsarte idea was developed and successfully
used in the works of Ph.Delsarte, J.M.Goethalts and
J.J.Seidel, G.A.Kabatyanskii and V.I.Levenstein,
A.M.Odlyzko and N.J.A.Sloane, V.I.Levenstein, and
V.M.Sidel'nikov, in particular, for investigation of contact
number of the Euclidean space
which is equal
to the maximal number of balls of unit radius with nonintersecting
interiors which simultaneously touch the unit ball of the space.
The exposition of these results and rich bibliography on this
subjects can be found in the monograph of J.H.Conway and
N.J.A.Sloane, Sphere packings, lattices and groups, Springer
Verlag, New York, 1988. At present, the exact value of
is
known only for
, namely,
,
. For arbitrary values of
the
lower and upper estimates of the constant
are known; for
instance, in the four-dimensional case we have
.
Suppose that
and that
,
is a system of ultraspherical polynomials which
are orthogonal on the interval
with respect to weight
function
and which are normalized by the
condition
We denote by
a set consisting of all continuous functions
on
which
are non-positive on
and
represented on
by series
with non-negative
coefficients
and
On this set of
functions we consider the quantity
The following is done in this article. A dual problem is written
out and the corresponding duality theorem is given for a problem
which is somewhat more general than The exact solution of
problem
is given for
. It has turned out that the
solution is a polynomial which is close to the polynomial written
out earlier in the work of A.M.Odlyzko and N.J.A.Sloane, New
bounds on the number of unit spheres that can touch a unit sphere
in
dimensions, Journal of Combinatorial Theory, Series A, 26
(2) (1979), 210-214. Our result signifies that inequality
cannot give, for the number
, an upper
estimate that would be better than the upper estimate
obtained by A.M.Odlyzko and N.J.A.Sloane.
Bibliogr. 21 ref.