On the best conditioned bases of quadratic polynomials

It is known that for Pn, the subspace of C([−1, 1]) of all polynomials of degree at most n, the least basis condition number κ∞(Pn) (also called the Banach-Mazur distance between Pn and `n+1 ∞ ) is bounded from below by the projection constant of Pn in C([−1, 1]). We show that κ∞(Pn) is in fact the generalized interpolating projection constant of Pn in C([−1, 1]), and is consequently bounded from above by the interpolating projection constant of Pn 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 . . . ≤ κ∞(P2) ≤ 1.25. We prove that there is a basis u of P2 such that κ∞(u) ≈ 1.24839. This result means that no Lagrange basis of P2 is best conditioned. It also seems likely that the previous value is actually the least basis condition number of P2, which therefore would not equal the projection constant of P2 in C([−1, 1]). As for trigonometric polynomials of degree at most 1, we present numerical evidence that the Lagrange bases at equidistant points are best conditioned.