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Bounds for the truncated logarithm: $∣ L_{N} (h) z_{i} / h ∣ \leq 8 M / 3$ and $∣ X_{j} ∣ \leq (8 M / 3) (C_{0} M j / ρ)^{j - 1}$

Statement

Let $d \geq 1$ , let $(U, U_{ρ}, \tilde{A}, \tilde{B}, M, K, δ, Ψ_{h})$ be analytic splitting data, and use the notation of The truncated binomial family $T_{h}^{s, N}$ , the truncated logarithm $L_{N} (h)$ , its coefficient fields $X_{j}$ and the truncations $X^{[N]}$ and The difference operator $Δ_{h} = Ψ_{h}^{*} - i d$ loses a controlled domain margin and is bounded by $C_{d} M ∣ h ∣ / b$ : $b = ρ / (8 N)$ , $r_{N} = ρ / (C_{0} M N)$ with $C_{0} = \max {16, 128 d^{2}}$ , the domains $D_{k}$ , the difference operator $Δ_{h}$ , the truncated logarithm $L_{N} (h)$ and the coefficient fields $X_{j}$ , $1 \leq j \leq N$ . Then for every integer $N \geq 2$ , every $∣ h ∣ \leq r_{N}$ and every coordinate index $i$ :

$∥ Δ_{h}^{k} z_{i} ∥_{D_{k}} \leq 2 M ∣ h ∣ q^{k - 1}$ for $1 \leq k \leq N$ , where $q = C_{d} M ∣ h ∣ / b \leq 1 / 4$ .
The function $h \mapsto \frac{(L_{N} (h) z_{i}) (Z)}{h}$ , initially defined for $0 < ∣ h ∣ \leq r_{N}$ , extends holomorphically to the disc $∣ h ∣ \leq r_{N}$ , and $∣ \frac{(L_{N} (h) z_{i}) (Z)}{h} ∣ \leq \frac{2 M}{1 - q} \leq \frac{8 M}{3} (∣ h ∣ \leq r_{N}, Z \in D_{N}) .$
For $1 \leq j \leq N$ and every $Z \in D_{N}$ , $\max_{i} ∣ (X_{j})_{i} (Z) ∣ \leq \frac{8 M}{3} r_{N}^{- (j - 1)},$ and the sharper estimate $\max_{i} ∣ (X_{j})_{i} (Z) ∣ \leq \frac{8 M}{3} {(\frac{C_{0} M j}{ρ})}^{j - 1},$ obtained for $j \geq 2$ by computing the $j$ -th field with truncation order $j$ , and for $j = 1$ from $X_{1} = X_{\tilde{A}} + X_{\tilde{B}}$ ; the two bounds agree in the form $r_{N}^{- (j - 1)} = (C_{0} M N / ρ)^{j - 1}$ . The corresponding Euclidean field bounds are these component bounds multiplied by $\sqrt{2 d}$ .
The field $X_{j}$ is independent of the truncation order: if $N, N^{'} \geq j$ then the $j$ -th coefficient fields of $L_{N}$ and $L_{N^{'}}$ agree on $D (ρ / 4)$ .

Facts & Assumptions

Given: The analytic splitting data, $N \geq 2$ , $b = ρ / (8 N)$ , $r_{N} = ρ / (C_{0} M N)$ , the fields $X_{j}$ and functions $L_{N} (h) z_{i}$ of The truncated binomial family $T_{h}^{s, N}$ , the truncated logarithm $L_{N} (h)$ , its coefficient fields $X_{j}$ and the truncations $X^{[N]}$ .

[L1]

Difference-operator bounds (The difference operator $Δ_{h} = Ψ_{h}^{*} - i d$ loses a controlled domain margin and is bounded by $C_{d} M ∣ h ∣ / b$ ): $∥ Ψ_{h} (Z) - Z ∥ \leq 2 M ∣ h ∣ \leq b$ on $D_{k}$ , and for $f$ holomorphic on $D_{k - 1}$ with $∣ f ∣ \leq F$ , $∣ Δ_{h} f ∣ \leq q F$ on $D_{k}$ with $q = C_{d} M ∣ h ∣ / b \leq 1 / 4$ ; hence $∣ Δ_{h}^{k} f ∣ \leq q^{k} F$ on $D_{k}$ .

[L2]

Cauchy coefficient estimates in one variable (Cauchy's inequalities bound the Taylor coefficients by the circle supremum): if $G$ is holomorphic on $D (0, R)$ and $∣ G ∣ \leq A$ on $∣ h ∣ = r < R$ , then the $j$ -th Taylor coefficient obeys $∣ c_{j} ∣ \leq A / r^{j}$ .

[L3]

Holomorphy in $h$ : each $h \mapsto (Δ_{h}^{k} f) (Z)$ is holomorphic on $∣ h ∣ \leq r_{k}$ (The difference operator $Δ_{h} = Ψ_{h}^{*} - i d$ loses a controlled domain margin and is bounded by $C_{d} M ∣ h ∣ / b$ , Hamiltonian flows and the exact splitting map preserve the complexified canonical form and the Poisson bracket, The flow of a bounded holomorphic field on a polydisc: existence in complex time, displacement bound, holomorphy, and the composition law, Calculus of holomorphic maps: chain rule, Wirtinger derivatives, coefficient extraction in one variable, and locally uniform limits clause 1), so $h \mapsto (L_{N} (h) z_{i}) (Z)$ is holomorphic on $∣ h ∣ \leq r_{N}$ (The truncated binomial family $T_{h}^{s, N}$ , the truncated logarithm $L_{N} (h)$ , its coefficient fields $X_{j}$ and the truncations $X^{[N]}$ ).

[L4]

Removable singularities and coefficient extraction: a function holomorphic and bounded on a punctured disc extends holomorphically to the centre, and the coefficients coincide with the Cauchy coefficients of Calculus of holomorphic maps: chain rule, Wirtinger derivatives, coefficient extraction in one variable, and locally uniform limits clause 3; the truncated logarithm is a finite linear combination of the $Δ_{h}^{k} z_{i}$ , whose lowest $h$ -degree in the $k$ -th term is $k$ , so $(L_{N} (h) z_{i}) (Z) = O (h)$ at $h = 0$ (Iterated Cauchy formula, power-series expansion and Cauchy estimates for functions continuous and separately holomorphic on a polydisc, Calculus of holomorphic maps: chain rule, Wirtinger derivatives, coefficient extraction in one variable, and locally uniform limits).

[L5]

Taylor coefficients of $L_{N}$ versus $L_{N^{'}}$ : $L_{N} - L_{N^{'}} = \sum_{k = \min + 1}^{\max} \frac{(- 1)^{k + 1}}{k} Δ_{h}^{k}$ is a sum of terms of $h$ -degree $\geq \min (N, N^{'}) + 1$ (The truncated binomial family $T_{h}^{s, N}$ , the truncated logarithm $L_{N} (h)$ , its coefficient fields $X_{j}$ and the truncations $X^{[N]}$ ).

Proof

technique · direct; combine the displacement bound with the single-step operator bound, sum the geometric series, and apply Cauchy's coefficient estimate in $h$

1.1givenL1algebra

Iterate bound. For $Z \in D_{1}$ , $∣ Δ_{h} z_{i} (Z) ∣ = ∣ z_{i} (Ψ_{h} (Z)) - z_{i} (Z) ∣ \leq ∥ Ψ_{h} (Z) - Z ∥ \leq 2 M ∣ h ∣$ by [L1]; applying the operator bound [L1] a further $k - 1$ times gives $∣ Δ_{h}^{k} z_{i} (Z) ∣ \leq q^{k - 1} \cdot 2 M ∣ h ∣$ on $D_{k}$ . This is clause 1.

2.1givenstep 1.1L3L4algebra

Bound for the truncated logarithm. On $D_{N}$ , $∣ \frac{L_{N} (h) z_{i} (Z)}{h} ∣ \leq \sum_{k = 1}^{N} \frac{∣ Δ_{h}^{k} z_{i} (Z) ∣}{k ∣ h ∣} \leq 2 M \sum_{k = 1}^{N} \frac{q^{k - 1}}{k} \leq \frac{2 M}{1 - q} \leq \frac{8 M}{3},$ using $q \leq 1 / 4$ . The quantity $(L_{N} (h) z_{i}) (Z)$ is $O (h)$ at $h = 0$ by [L4], and $h \mapsto (L_{N} (h) z_{i}) (Z) / h$ is holomorphic for $0 < ∣ h ∣ \leq r_{N}$ by [L3]; the bound just proved makes $h = 0$ a removable singularity, so the extension is holomorphic on $∣ h ∣ \leq r_{N}$ . This is clause 2.

3.1givenstep 2.1L2algebra

Coefficient bound. Fix $Z \in D_{N}$ and let $G (h) : = (L_{N} (h) z_{i}) (Z) / h$ denote the holomorphic extension of clause 2. It is bounded by $8 M / 3$ throughout the open disc $∣ h ∣ < r_{N}$ . For any radius $0 < r < r_{N}$ , the Taylor coefficient of order $j - 1$ , namely $(X_{j})_{i} (Z)$ (The truncated binomial family $T_{h}^{s, N}$ , the truncated logarithm $L_{N} (h)$ , its coefficient fields $X_{j}$ and the truncations $X^{[N]}$ ), obeys $∣ (X_{j})_{i} (Z) ∣ \leq (8 M / 3) r^{- (j - 1)}$ by [L2], whose circle hypothesis is strict. Letting $r ↑ r_{N}$ gives $∣ (X_{j})_{i} (Z) ∣ \leq (8 M / 3) r_{N}^{- (j - 1)}$ . Taking the maximum over $i$ gives the first estimate. The finite difference $L_{N} - L_{N^{'}}$ starts at degree $\min (N, N^{'}) + 1$ by [L5], so coefficients through that degree minus one coincide. For $j \geq 2$ we may therefore use truncation order $j$ , giving the sharper estimate. For $j = 1$ , differentiating the three subflows at zero gives $X_{1} = X_{\tilde{A}} + X_{\tilde{B}}$ , with each coordinate bounded by $2 M \leq 8 M / 3$ .

4.1givenL5algebra

Independence of the truncation order. If $N, N^{'} \geq j$ , then by [L5] the difference $L_{N} (h) z_{i} - L_{N^{'}} (h) z_{i}$ has no terms of $h$ -degree $\leq \min (N, N^{'})$ , hence none of degree $\leq j$ ; dividing by $h$ and extracting the coefficient of $h^{j - 1}$ gives the same value for both truncation orders, on the common domain $D (ρ / 4)$ . This is clause 4, and it justifies the substitution made in step 3.1.

5.1step 1.1step 2.1step 3.1step 4.1∎

Conclusion. Clauses 1, 2 and 4 are steps 1.1, 2.1 and 4.1; clause 3 is step 3.1. All constants depend only on $d$ , $M$ and $ρ$ .

Remarks

Note the displayed first estimate of clause 3 contains a redundant form; the usable statement is $\max_{i} ∣ (X_{j})_{i} (Z) ∣ \leq (8 M / 3) (C_{0} M j / ρ)^{j - 1}$ , which exhibits the factorial-type growth of the modified-field coefficients and is the input to the truncation choice of Optimal finite truncation gives an exponential local splitting defect. The bound is uniform on $D_{N}$ and depends on $M$ , $ρ$ and $d$ only.
No convergent infinite series is used: $L_{N}$ is a finite sum and the coefficients are extracted with Cauchy's formula in the single complex variable $h$ .

Bounds for the truncated logarithm: $∣ L_{N} (h) z_{i} / h ∣ \leq 8 M / 3$ and $∣ X_{j} ∣ \leq (8 M / 3) (C_{0} M j / ρ)^{j - 1}$

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources

Bounds for the truncated logarithm: ∣LN(h)zi/h∣≤8M/3∣LN​(h)zi​/h∣≤8M/3 and ∣Xj∣≤(8M/3)(C0Mj/ρ)j−1∣Xj​∣≤(8M/3)(C0​Mj/ρ)j−1

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources

Bounds for the truncated logarithm: $∣ L_{N} (h) z_{i} / h ∣ \leq 8 M / 3$ and $∣ X_{j} ∣ \leq (8 M / 3) (C_{0} M j / ρ)^{j - 1}$