The Scottish Book by R. Daniel Mauldin

Author:R. Daniel Mauldin
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

One can obtain estimates on the left-hand side of (4) by estimating the right-hand side of (4) and minimizing. (A reasonable first choice is x ij  = k i ∕2 − j.) To prove (4), observe that if p(t) is a polynomial of degree ≤ k i , then the coefficient of in the Lagrange interpolation formula for p is and apply this to each variable of W.

For example, we show that the improved estimate

(5)

holds in Problem 74, where

Indeed, choose x i0 = 2, x i1 = 0, x i2 = −2 for i = 1, …, ℓ, x i0 = 1, x i1 = −1 for i = ℓ + 1, …, n − ℓ, and x i0 = 0 for i = n − ℓ + 1, …, n. Then by (4),

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