Lemma 14, p. 16 — the Chebyshev series (55) truncated at j₀ = √(b log(4b/ε)), b = κ² log(κ/ε), is 2ε-close to 1/x on D_κ

Let $\kappa \ge 1$ and $\varepsilon > 0$. Put $b = \kappa^2 \log(\kappa/\varepsilon)$ and $j_0 = \sqrt{b \log(4b/\varepsilon)}$, and let

where $\mathcal T_m$ is the $m$-th Chebyshev polynomial of the first kind. Then $g$ is $2\varepsilon$-close to $1/x$ on $D_\kappa = [-1,-1/\kappa] \cup [1/\kappa,1]$:

This is the decomposition of $1/x$ into odd Chebyshev polynomials of degree $O(\kappa\log(\kappa/\varepsilon))$ on which the Chebyshev-based quantum linear systems algorithm (Theorem 4 of the paper) is built: applied to a Hermitian matrix with spectrum in $D_\kappa$, the same combination of Chebyshev polynomials of the matrix approximates its inverse.

Formalization Note $\kappa \ge 1$ is a condition number (ratio of the largest to the smallest singular value, p. 3); it makes $D_\kappa$ nonempty and keeps $0$ out of it. The page's real $b$ is rounded up, $b = \lceil \kappa^2\log(\kappa/\varepsilon)\rceil$ (it is an exponent, and this keeps the hypothesis of Lemma 17), and the sum runs over $j = 0,\dots,\lfloor j_0\rfloor$. No upper bound on $\varepsilon$ is assumed: if $\varepsilon \ge \kappa$ then $b = 0$, $g \equiv 0$ and $|1/x| \le \kappa \le \varepsilon$ on $D_\kappa$, so the statement is still true there.