Polynomial approximation of the reciprocal on a proper circular arc

A proper closed arc

S

of a circle centered at zero has connected complement and stays away from zero. The polynomial Runge theorem therefore gives polynomials converging uniformly to

1/ z

on

S

.

Table of contents
- Explicit polynomial approximation of the reciprocal on the left semicircle Polynomial approximation of the reciprocal on a proper circular arc

Explicit polynomial approximation of the reciprocal on the left semicircle

On

K = {z : ∣ z ∣ = 1, Re z \leq 0}

, the polynomials

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

first in powers of

4/ (z - 4)

, whose modulus is at most

4/ 17

on

K

, and then expand each

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

in powers of

z /4

. The estimate

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

controls the diagonal truncation.

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Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 3 / 2F / iii / Solution

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