Comparing sums and integrals

Summary

It is often the case that error terms in analysis can be computed by comparing an infinite sum to the improper integral of an antitone function. This file will eventually enable that.

At the moment it contains several lemmas in this direction, for antitone or monotone functions (or products of antitone and monotone functions), formulated for sums on range i or Ico a b.

TODO: Add more lemmas to the API to directly address limiting issues

Main Results

• AntitoneOn.integral_le_sum: The integral of an antitone function is at most the sum of its values at integer steps aligning with the left-hand side of the interval
• AntitoneOn.sum_le_integral: The sum of an antitone function along integer steps aligning with the right-hand side of the interval is at most the integral of the function along that interval
• MonotoneOn.integral_le_sum: The integral of a monotone function is at most the sum of its values at integer steps aligning with the right-hand side of the interval
• MonotoneOn.sum_le_integral: The sum of a monotone function along integer steps aligning with the left-hand side of the interval is at most the integral of the function along that interval
• sum_mul_Ico_le_integral_of_monotone_antitone: the sum of f i * g i on an interval is bounded by the integral of f x * g (x - 1) if f is monotone and g is antitone.
• integral_le_sum_mul_Ico_of_antitone_monotone: the sum of f i * g i on an interval is bounded below by the integral of f x * g (x - 1) if f is antitone and g is monotone.

Tags

analysis, comparison, asymptotics

theorem sum_Ico_le_integral_of_le {a b : ℕ} {f g : ℝ → ℝ} (hab : a ≤ b) (h : ∀ i ∈ Set.Ico a b, ∀ x ∈ Set.Ico ↑i ↑(i + 1), f ↑i ≤ g x) (hg : MeasureTheory.IntegrableOn g (Set.Ico ↑a ↑b) MeasureTheory.volume) : ∑ i ∈ Finset.Ico a b, f ↑i ≤ ∫ (x : ℝ) in ↑a..↑b, g x

theorem integral_le_sum_Ico_of_le {a b : ℕ} {f g : ℝ → ℝ} (hab : a ≤ b) (h : ∀ i ∈ Set.Ico a b, ∀ x ∈ Set.Ico ↑i ↑(i + 1), g x ≤ f ↑i) (hg : MeasureTheory.IntegrableOn g (Set.Ico ↑a ↑b) MeasureTheory.volume) : ∫ (x : ℝ) in ↑a..↑b, g x ≤ ∑ i ∈ Finset.Ico a b, f ↑i

theorem AntitoneOn.integral_le_sum {x₀ : ℝ} {a : ℕ} {f : ℝ → ℝ} (hf : AntitoneOn f (Set.Icc x₀ (x₀ + ↑a))) : ∫ (x : ℝ) in x₀..x₀ + ↑a, f x ≤ ∑ i ∈ Finset.range a, f (x₀ + ↑i)

theorem AntitoneOn.integral_le_sum_Ico {a b : ℕ} {f : ℝ → ℝ} (hab : a ≤ b) (hf : AntitoneOn f (Set.Icc ↑a ↑b)) : ∫ (x : ℝ) in ↑a..↑b, f x ≤ ∑ x ∈ Finset.Ico a b, f ↑x

theorem AntitoneOn.sum_le_integral {x₀ : ℝ} {a : ℕ} {f : ℝ → ℝ} (hf : AntitoneOn f (Set.Icc x₀ (x₀ + ↑a))) : ∑ i ∈ Finset.range a, f (x₀ + ↑(i + 1)) ≤ ∫ (x : ℝ) in x₀..x₀ + ↑a, f x

theorem AntitoneOn.sum_le_integral_Ico {a b : ℕ} {f : ℝ → ℝ} (hab : a ≤ b) (hf : AntitoneOn f (Set.Icc ↑a ↑b)) : ∑ i ∈ Finset.Ico a b, f ↑(i + 1) ≤ ∫ (x : ℝ) in ↑a..↑b, f x

theorem MonotoneOn.sum_le_integral {x₀ : ℝ} {a : ℕ} {f : ℝ → ℝ} (hf : MonotoneOn f (Set.Icc x₀ (x₀ + ↑a))) : ∑ i ∈ Finset.range a, f (x₀ + ↑i) ≤ ∫ (x : ℝ) in x₀..x₀ + ↑a, f x

theorem MonotoneOn.sum_le_integral_Ico {a b : ℕ} {f : ℝ → ℝ} (hab : a ≤ b) (hf : MonotoneOn f (Set.Icc ↑a ↑b)) : ∑ x ∈ Finset.Ico a b, f ↑x ≤ ∫ (x : ℝ) in ↑a..↑b, f x

theorem MonotoneOn.integral_le_sum {x₀ : ℝ} {a : ℕ} {f : ℝ → ℝ} (hf : MonotoneOn f (Set.Icc x₀ (x₀ + ↑a))) : ∫ (x : ℝ) in x₀..x₀ + ↑a, f x ≤ ∑ i ∈ Finset.range a, f (x₀ + ↑(i + 1))

theorem MonotoneOn.integral_le_sum_Ico {a b : ℕ} {f : ℝ → ℝ} (hab : a ≤ b) (hf : MonotoneOn f (Set.Icc ↑a ↑b)) : ∫ (x : ℝ) in ↑a..↑b, f x ≤ ∑ i ∈ Finset.Ico a b, f ↑(i + 1)

theorem sum_mul_Ico_le_integral_of_monotone_antitone {a b : ℕ} {f g : ℝ → ℝ} (hab : a ≤ b) (hf : MonotoneOn f (Set.Icc ↑a ↑b)) (hg : AntitoneOn g (Set.Icc (↑a - 1) (↑b - 1))) (fpos : 0 ≤ f ↑a) (gpos : 0 ≤ g (↑b - 1)) : ∑ i ∈ Finset.Ico a b, f ↑i * g ↑i ≤ ∫ (x : ℝ) in ↑a..↑b, f x * g (x - 1)

theorem integral_le_sum_mul_Ico_of_antitone_monotone {a b : ℕ} {f g : ℝ → ℝ} (hab : a ≤ b) (hf : AntitoneOn f (Set.Icc ↑a ↑b)) (hg : MonotoneOn g (Set.Icc (↑a - 1) (↑b - 1))) (fpos : 0 ≤ f ↑b) (gpos : 0 ≤ g (↑a - 1)) : ∫ (x : ℝ) in ↑a..↑b, f x * g (x - 1) ≤ ∑ i ∈ Finset.Ico a b, f ↑i * g ↑i