Published in: Trudy Mat. Inst. im. V.A. Steklova 219 (1997), pp. 44-73;

English translation: Proc. of the Steclov Inst. of Math. 219 (1997), pp. 36-65.

English translation: Proc. of the Steclov Inst. of Math. 219 (1997), pp. 36-65.

A B S T R A C T

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

In the papers 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 it was proved that arbitrary function gives the estimation therefore the inequality holds. In such a way at the expense of a choice of concrete functions , V.I.Levenstein, A.M.Odlyzko and N.J.A.Sloane have received good estimations from above of contact numbers in particular, have calculated .

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.