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 Incoming links: Quadratic Bernstein coefficient bound

Use the order of the three Bernstein basis functions given in this part. Endpoint evaluation gives

a_{1} = p (1), a_{3} = p (0),

so both endpoint coefficients have absolute value at most one. At the midpoint,

p (1/2) = \frac{1}{4} a_{1} + \frac{1}{2} a_{2} + \frac{1}{4} a_{3}, a_{2} = 2 p (1/2) - \frac{1}{2} [p (0) + p (1)] .

∣ a_{1} ∣ \leq 1, ∣ a_{2} ∣ \leq 3, ∣ a_{3} ∣ \leq 1.

All three bounds are sharp. Indeed

p (x) = 8 x^{2} - 8 x + 1 = T_{2} (2 x - 1)

(1, - 3, 1)

in this basis. The reversed ordering of endpoint Bernstein functions does not affect these symmetric endpoint bounds.