Poisson's summation formula #

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We prove Poisson's summation formula ∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n, where 𝓕 f is the Fourier transform of f, under the following hypotheses:

f is a continuous function ℝ → ℂ.
The sum ∑ (n : ℤ), 𝓕 f n is convergent.
For all compacts K ⊂ ℝ, the sum ∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ K } is convergent. See real.tsum_eq_tsum_fourier_integral for this formulation.

These hypotheses are potentially a little awkward to apply, so we also provide the less general but easier-to-use result real.tsum_eq_tsum_fourier_integral_of_rpow_decay, in which we assume f and 𝓕 f both decay as |x| ^ (-b) for some b > 1, and the even more specific result schwartz_map.tsum_eq_tsum_fourier_integral, where we assume that both f and 𝓕 f are Schwartz functions.

TODO #

At the moment schwartz_map.tsum_eq_tsum_fourier_integral requires separate proofs that both f and 𝓕 f are Schwartz functions. In fact, 𝓕 f is automatically Schwartz if f is; and once we have this lemma in the library, we should adjust the hypotheses here accordingly.

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theorem real.fourier_coeff_tsum_comp_add {f : C(ℝ , ℂ )} (hf : ∀ (K : topological_space.compacts ℝ), summable (λ (n : ℤ), ‖continuous_map.restrict ↑K (f.comp (continuous_map.add_right ↑n))‖)) (m : ℤ) :

fourier_coeff _.lift m = real.fourier_integral ⇑f ↑m

The key lemma for Poisson summation: the m-th Fourier coefficient of the periodic function ∑' n : ℤ, f (x + n) is the value at m of the Fourier transform of f.

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theorem real.tsum_eq_tsum_fourier_integral {f : C(ℝ , ℂ )} (h_norm : ∀ (K : topological_space.compacts ℝ), summable (λ (n : ℤ), ‖continuous_map.restrict ↑K (f.comp (continuous_map.add_right ↑n))‖)) (h_sum : summable (λ (n : ℤ), real.fourier_integral ⇑f ↑n)) :

∑' (n : ℤ), ⇑f ↑n = ∑' (n : ℤ), real.fourier_integral ⇑f ↑n

Poisson's summation formula, most general form.

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theorem is_O_norm_Icc_restrict_at_top {E : Type u_1} [normed_add_comm_group E] {f : C(ℝ , E)} {b : ℝ} (hb : 0 < b) (hf : ⇑f =O[filter.at_top ] λ (x : ℝ), |x| ^ -b) (R S : ℝ) :

(λ (x : ℝ), ‖continuous_map.restrict (set.Icc (x + R) (x + S)) f‖) =O[filter.at_top ] λ (x : ℝ), |x| ^ -b

If f is O(x ^ (-b)) at infinity, then so is the function λ x, ‖f.restrict (Icc (x + R) (x + S))‖ for any fixed R and S.

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theorem is_O_norm_Icc_restrict_at_bot {E : Type u_1} [normed_add_comm_group E] {f : C(ℝ , E)} {b : ℝ} (hb : 0 < b) (hf : ⇑f =O[filter.at_bot ] λ (x : ℝ), |x| ^ -b) (R S : ℝ) :

(λ (x : ℝ), ‖continuous_map.restrict (set.Icc (x + R) (x + S)) f‖) =O[filter.at_bot ] λ (x : ℝ), |x| ^ -b

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theorem is_O_norm_restrict_cocompact {E : Type u_1} [normed_add_comm_group E] (f : C(ℝ , E)) {b : ℝ} (hb : 0 < b) (hf : ⇑f =O[filter.cocompact ℝ] λ (x : ℝ), |x| ^ -b) (K : topological_space.compacts ℝ) :

(λ (x : ℝ), ‖continuous_map.restrict ↑K (f.comp (continuous_map.add_right x))‖) =O[filter.cocompact ℝ] λ (x : ℝ), |x| ^ -b

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theorem real.tsum_eq_tsum_fourier_integral_of_rpow_decay_of_summable {f : ℝ → ℂ} (hc : continuous f) {b : ℝ} (hb : 1 < b) (hf : f =O[filter.cocompact ℝ] λ (x : ℝ), |x| ^ -b) (hFf : summable (λ (n : ℤ), real.fourier_integral f ↑n)) :

∑' (n : ℤ), f ↑n = ∑' (n : ℤ), real.fourier_integral f ↑n

Poisson's summation formula, assuming that f decays as |x| ^ (-b) for some 1 < b and its Fourier transform is summable.

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theorem real.tsum_eq_tsum_fourier_integral_of_rpow_decay {f : ℝ → ℂ} (hc : continuous f) {b : ℝ} (hb : 1 < b) (hf : f =O[filter.cocompact ℝ] λ (x : ℝ), |x| ^ -b) (hFf : real.fourier_integral f =O[filter.cocompact ℝ] λ (x : ℝ), |x| ^ -b) :

∑' (n : ℤ), f ↑n = ∑' (n : ℤ), real.fourier_integral f ↑n

Poisson's summation formula, assuming that both f and its Fourier transform decay as |x| ^ (-b) for some 1 < b. (This is the one-dimensional case of Corollary VII.2.6 of Stein and Weiss, Introduction to Fourier analysis on Euclidean spaces.)

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theorem schwartz_map.tsum_eq_tsum_fourier_integral (f g : schwartz_map ℝ ℂ) (hfg : real.fourier_integral ⇑f = ⇑g) :

∑' (n : ℤ), ⇑f ↑n = ∑' (n : ℤ), ⇑g ↑n

Poisson's summation formula for Schwartz functions.