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 four lemmas in this direction: AntitoneOn.integral_le_sum, AntitoneOn.sum_le_integral and versions for monotone functions, which can all be paired with a Filter.Tendsto to estimate some errors.

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

Tags

analysis, comparison, asymptotics

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 ≤ (Finset.range a).sum fun (i : ℕ) => f (x₀ + ↑i)

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

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

theorem AntitoneOn.sum_le_integral_Ico {a : ℕ} {b : ℕ} {f : ℝ → ℝ} (hab : a ≤ b) (hf : AntitoneOn f (Set.Icc ↑a ↑b)) : ((Finset.Ico a b).sum fun (i : ℕ) => 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))) : ((Finset.range a).sum fun (i : ℕ) => f (x₀ + ↑i)) ≤ ∫ (x : ℝ) in x₀..x₀ + ↑a, f x

theorem MonotoneOn.sum_le_integral_Ico {a : ℕ} {b : ℕ} {f : ℝ → ℝ} (hab : a ≤ b) (hf : MonotoneOn f (Set.Icc ↑a ↑b)) : ((Finset.Ico a b).sum fun (x : ℕ) => 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 ≤ (Finset.range a).sum fun (i : ℕ) => f (x₀ + ↑(i + 1))

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