New upper bound for the B-spline basis condition number, II: a proof of de Boor's 2k-conjecture
Journal of Approximation Theory
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It is known that for P"n, the subspace of C([-1,1]) of all polynomials of degree at most n, the least basis condition number @k"~(P"n) (also called the Banach-Mazur distance between P"n and @?"~^n^+^1) is bounded from below by the projection constant of P"n in C([-1,1]). We show that @k"~(P"n) is in fact the generalized interpolating projection constant of P"n in C([-1,1]), and is consequently bounded from above by the interpolating projection constant of P"n in C([-1,1]). Hence the condition number of the Lagrange basis (say, at the Chebyshev extrema), which coincides with the norm of the corresponding interpolating projection and thus grows like O(lnn), is of optimal order, and for n=2, 1.2201...=