Theorems · Theorem · functional analysis

Function.HasTemperateGrowth.isBigO_uniform

∀ {E : Type u_5} {F : Type u_6} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [inst_2 : NormedAddCommGroup F]
  [inst_3 : NormedSpace ℝ F] {f : E → F},
  Function.HasTemperateGrowth f → ∀ (N : ℕ), ∃ k, ∀ n ≤ N, iteratedFDeriv ℝ n f =O[⊤] fun x => (1 + ‖x‖) ^ k

If f has temperate growth, then for any N : ℕ one can find k such that all iterated derivatives of f of order ≤ N are O((1 + ‖x‖) ^ k). Note that the O here is with respect to the ⊤ filter, meaning that the bound holds everywhere.

Defined in

Mathlib.Analysis.Distribution.TemperateGrowth

Cited by

2 results in Mathlib

Foundations

Depth 178 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace