 Explicit polynomial approximation of the reciprocal on the left semicircle

ID: explicit-polynomial-approximation-of-the-reciprocal-on-the-left-semicircle

K = {z : ∣ z ∣ = 1, Re z \leq 0}

p_{n} (z) = - \frac{1}{4} \sum_{j = 0}^{n} \sum_{k = 0}^{n^{2}} (k j + k) (\frac{z}{4})^{k}

converge uniformly to

1/ z

. Expand

1/ z

4/ (z - 4)

, whose modulus is at most

4/ 17

K

, and then expand each

(z - 4)^{- j - 1}

in powers of

z /4

. The estimate

(k j + k) \leq 2^{j + k}

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