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abuse of notation achieved Algebra for Tchebycheffian alternating signs applies Assume Bernstein polynomials best approximation best TSF block of equal bound characterized by property coefficients continuous function convex cone convex hull counting zeros depends continuously determinant Doctor of Philosophy equal y values extremal polynomial following lemma fundamental theorem Hence implicit function theorem inductive hypothesis Johnson k^xl k+i+1 Karlin and Studden Karlin and Ziegler knots l+t)n Legendre polynomials Let f lim f(t linearly independent n+1 extreme rays n+2k n+k+1 n+k+2 nonnegative nonzero number of zeros obtained Problem 6.1 proceeds by induction property 9 reduced TSP Remarks 2.6 representation result satisfying Schoenberg Section sequence strict sign changes strictly positive Studden 9 sublinear functional Tchebycheffian monosplines Tchebycheffian spline function Theorem 3.2 theorem of Algebra tn+2k total positivity u-polynomial unique T-monospline W(uQ wQ(t xi^l Xj+i+l zero of multiplicity zeros on a,b