Rational Approximation of Holomorphic Semigroups Revisited

Belangrijkste concepten

Using the H-calculus for A(ψ, m)-stable rational functions provides sharp estimates near infinity.

Samenvatting

The content delves into the rational approximation of holomorphic semigroups using a functional calculus approach. It revisits established theories and aims to improve norm estimates for rational approximations. The paper introduces the H-calculus as a unified approach to norm estimates for rational approximations, enhancing existing results. Stability and convergence issues are addressed, emphasizing optimal approximation rates. Various theorems and propositions are presented to support the core message.

Statistieken

Stability does not hold for θ = π/2 in Example 3.4.
In Example 3.5, (3.3) does not hold for θ = 0 and small t > 0.
Theorem 3.6 provides an estimate ∥PKn,r∥Hθ,0 ≤ C(1 + log(K1/K0)).
Theorem 3.7 establishes ∥PKn,r∥Hθ,0 ≤ C independent of Kn if |r(∞)| < 1.
Lemma 3.9 gives bounds on |r(z)| near infinity: e^(-b2/|z|^m) ≤ |r(z)| ≤ e^(-b1/|z|^m).

Citaten

"Stability does not hold for θ = π/2 in Example 3.4."
"In Example 3.5, (3.3) does not hold for θ = 0 and small t > 0."
"Theorem 3.6 provides an estimate ∥PKn,r∥Hθ,0 ≤ C(1 + log(K1/K0))."
"Theorem 3.7 establishes ∥PKn,r∥Hθ,0 ≤ C independent of Kn if |r(∞)| < 1."
"Lemma 3.9 gives bounds on |r(z)| near infinity: e^(-b2/|z|^m) ≤ |r(z)| ≤ e^(-b1/|z|^m)."

Belangrijkste Inzichten Gedestilleerd Uit

Rational approximation of holomorphic semigroups revisited

by Charles Batt... om arxiv.org 03-26-2024

https://arxiv.org/pdf/2403.15894.pdf

Diepere vragen

How do the stability issues at θ = π/2 impact the overall applicability of the findings

The stability issues at θ = π/2 have significant implications for the overall applicability of the findings. When stability does not hold at θ = π/2, it restricts the range of angles where certain rational approximations can be effectively applied. This limitation impacts the generalizability and robustness of the results obtained in the study. It indicates that there are specific conditions or constraints under which certain rational functions may not provide stable approximations, highlighting a critical boundary in the analysis.

What implications arise from (3.3) not holding for θ = 0 in Example 3.5

The fact that (3.3) does not hold for θ = 0 in Example 3.5 has important implications regarding the behavior and characteristics of rational functions within different sectors. This observation suggests that there are variations in how these functions approximate exponential values based on their properties and structure, particularly when approaching specific angles such as θ = 0. The non-applicability of (3.3) at this angle signifies a unique behavior or pattern exhibited by certain rational approximations, indicating potential limitations or challenges in achieving stable estimates under those circumstances.

How do the optimal approximation rates discussed in the content contribute to advancements in mathematical analysis

The optimal approximation rates discussed in the content represent a significant advancement in mathematical analysis by providing sharper estimates and more precise bounds for rational approximations to holomorphic semigroups. These optimal rates offer improved accuracy and efficiency in modeling complex systems governed by differential equations or operators, enhancing our ability to predict behaviors and outcomes with higher precision. By establishing optimality through rigorous mathematical proofs, these results contribute to refining numerical methods, advancing computational techniques, and deepening our understanding of functional calculus applications within various mathematical contexts.