mathlib documentation

analysis.asymptotics.superpolynomial_decay

Super-Polynomial Function Decay #

This file defines a predicate asymptotics.superpolynomial_decay f for a function satisfying one of following equivalent definitions (The definition is in terms of the first condition):

These conditions are all equivalent to conditions in terms of polynomials, replacing x ^ c with p(x) or p(x)⁻¹ as appropriate, since asymptotically p(x) behaves like X ^ p.nat_degree. These further equivalences are not proven in mathlib but would be good future projects.

The definition of superpolynomial decay for f : α → β is relative to a parameter k : α → β. Super-polynomial decay then means f x decays faster than (k x) ^ c for all integers c. Equivalently f x decays faster than p.eval (k x) for all polynomials p : β[X]. The definition is also relative to a filter l : filter α where the decay rate is compared.

When the map k is given by n ↦ ↑n : ℕ → ℝ this defines negligible functions: https://en.wikipedia.org/wiki/Negligible_function

When the map k is given by (r₁,...,rₙ) ↦ r₁*...*rₙ : ℝⁿ → ℝ this is equivalent to the definition of rapidly decreasing functions given here: https://ncatlab.org/nlab/show/rapidly+decreasing+function

Main Theorems #

def asymptotics.superpolynomial_decay {α : Type u_1} {β : Type u_2} [topological_space β] [comm_semiring β] (l : filter α) (k f : α β) :
Prop

f has superpolynomial decay in parameter k along filter l if k ^ n * f tends to zero at l for all naturals n

Equations
theorem asymptotics.superpolynomial_decay.congr' {α : Type u_1} {β : Type u_2} {l : filter α} {k f g : α β} [topological_space β] [comm_semiring β] (hf : asymptotics.superpolynomial_decay l k f) (hfg : f =ᶠ[l] g) :
theorem asymptotics.superpolynomial_decay.congr {α : Type u_1} {β : Type u_2} {l : filter α} {k f g : α β} [topological_space β] [comm_semiring β] (hf : asymptotics.superpolynomial_decay l k f) (hfg : (x : α), f x = g x) :
@[simp]
theorem asymptotics.superpolynomial_decay_zero {α : Type u_1} {β : Type u_2} [topological_space β] [comm_semiring β] (l : filter α) (k : α β) :
theorem asymptotics.superpolynomial_decay.mul_const {α : Type u_1} {β : Type u_2} {l : filter α} {k f : α β} [topological_space β] [comm_semiring β] [has_continuous_mul β] (hf : asymptotics.superpolynomial_decay l k f) (c : β) :
asymptotics.superpolynomial_decay l k (λ (n : α), f n * c)
theorem asymptotics.superpolynomial_decay.const_mul {α : Type u_1} {β : Type u_2} {l : filter α} {k f : α β} [topological_space β] [comm_semiring β] [has_continuous_mul β] (hf : asymptotics.superpolynomial_decay l k f) (c : β) :
asymptotics.superpolynomial_decay l k (λ (n : α), c * f n)
theorem asymptotics.superpolynomial_decay_iff_abs_tendsto_zero {α : Type u_1} {β : Type u_2} (l : filter α) (k f : α β) [topological_space β] [linear_ordered_comm_ring β] [order_topology β] :
asymptotics.superpolynomial_decay l k f (n : ), filter.tendsto (λ (a : α), |k a ^ n * f a|) l (nhds 0)
theorem asymptotics.superpolynomial_decay_mul_const_iff {α : Type u_1} {β : Type u_2} (l : filter α) (k f : α β) [topological_space β] [field β] [has_continuous_mul β] {c : β} (hc0 : c 0) :
theorem asymptotics.superpolynomial_decay_const_mul_iff {α : Type u_1} {β : Type u_2} (l : filter α) (k f : α β) [topological_space β] [field β] [has_continuous_mul β] {c : β} (hc0 : c 0) :
theorem asymptotics.superpolynomial_decay_iff_zpow_tendsto_zero {α : Type u_1} {β : Type u_2} {l : filter α} {k : α β} (f : α β) [topological_space β] [linear_ordered_field β] [order_topology β] (hk : filter.tendsto k l filter.at_top) :
asymptotics.superpolynomial_decay l k f (z : ), filter.tendsto (λ (a : α), k a ^ z * f a) l (nhds 0)
theorem asymptotics.superpolynomial_decay.param_zpow_mul {α : Type u_1} {β : Type u_2} {l : filter α} {k f : α β} [topological_space β] [linear_ordered_field β] [order_topology β] (hk : filter.tendsto k l filter.at_top) (hf : asymptotics.superpolynomial_decay l k f) (z : ) :
asymptotics.superpolynomial_decay l k (λ (a : α), k a ^ z * f a)
theorem asymptotics.superpolynomial_decay.mul_param_zpow {α : Type u_1} {β : Type u_2} {l : filter α} {k f : α β} [topological_space β] [linear_ordered_field β] [order_topology β] (hk : filter.tendsto k l filter.at_top) (hf : asymptotics.superpolynomial_decay l k f) (z : ) :
asymptotics.superpolynomial_decay l k (λ (a : α), f a * k a ^ z)
theorem asymptotics.superpolynomial_decay_iff_norm_tendsto_zero {α : Type u_1} {β : Type u_2} (l : filter α) (k f : α β) [normed_linear_ordered_field β] :
asymptotics.superpolynomial_decay l k f (n : ), filter.tendsto (λ (a : α), k a ^ n * f a) l (nhds 0)
theorem asymptotics.superpolynomial_decay_iff_is_O {α : Type u_1} {β : Type u_2} {l : filter α} {k : α β} (f : α β) [normed_linear_ordered_field β] [order_topology β] (hk : filter.tendsto k l filter.at_top) :
asymptotics.superpolynomial_decay l k f (z : ), f =O[l] λ (a : α), k a ^ z
theorem asymptotics.superpolynomial_decay_iff_is_o {α : Type u_1} {β : Type u_2} {l : filter α} {k : α β} (f : α β) [normed_linear_ordered_field β] [order_topology β] (hk : filter.tendsto k l filter.at_top) :
asymptotics.superpolynomial_decay l k f (z : ), f =o[l] λ (a : α), k a ^ z