Theorems · Theorem · functional analysis

Function.hasTemperateGrowth_iff_isBigO

∀ {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 ↔ ContDiff ℝ (↑⊤) f ∧ ∀ (n : ℕ), ∃ k, iteratedFDeriv ℝ n f =O[⊤] fun x => (1 + ‖x‖) ^ k

A function has temperate growth if and only if it is smooth and its n-th iterated derivative is O((1 + ‖x‖) ^ k) for some k : ℕ (depending on n). Note that the O here is with respect to the ⊤ filter, meaning that the bound holds everywhere. TODO: when E is finite dimensional, this is equivalent to the derivatives being O(‖x‖ ^ k) as ‖x‖ → ∞.

Defined in

Mathlib.Analysis.Distribution.TemperateGrowth

Cited by

3 results in Mathlib

Foundations

Depth 176 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace