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Representation theorems for Tchebychcff polynomials with homogeneous boundary conditions are proved, and a number of extremal problems are solved. Representation theorems for non-negative polynomials were considered by Lukhcs in the early decades of this century. Luk~ics proved (see Szegti [13, p. 4-]) that every non-trivial polynomial pn(t) = E~=0 ak, non-negative on a finite interval [a, b], admits a representation which is essentially a sum of squares of polynomials. In 1953, Karlin and Shapley 1-8, p. 35-] established for p,(t) as above, the existence of a unique representation of the form ~-I(t-t2j-1) 2+fl(t-a)(b-t) H (t-t2j) 2, for n = 2m (1) f m-1 pn(t) =-.I = t j = 1- f

Year: 1973

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