GrowsPolynomially — Mathlib

English

If f is polynomially growing, then composing with the smoothingFn keeps polynomial growth for the derivative-related terms, i.e., a congruence of growth under smoothing.

Русский

Если f растет POLY, то под действием сглаживающей функции производные сохраняют полиномиальный рост.

LaTeX

$$GrowsPolynomially f → GrowsPolynomially (fun x => deriv(...))$$

Lean4

/-- The growth condition that the function `g` must satisfy for the Akra-Bazzi theorem to apply.
It roughly states that `c₁ g(n) ≤ g(u) ≤ c₂ g(n)`, for `u` between `b * n` and `n`, for any
constant `b ∈ (0, 1)`. -/
def GrowsPolynomially (f : ℝ → ℝ) : Prop :=
  ∀ b ∈ Set.Ioo 0 1, ∃ c₁ > 0, ∃ c₂ > 0, ∀ᶠ x in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * (f x)) (c₂ * f x)

Похожие утверждения