mathlib documentation

analysis.analytic.composition

Composition of analytic functions

in this file we prove that the composition of analytic functions is analytic.

The argument is the following. Assume g z = ∑' qₙ (z, ..., z) and f y = ∑' pₖ (y, ..., y). Then

g (f y) = ∑' qₙ (∑' pₖ (y, ..., y), ..., ∑' pₖ (y, ..., y)) = ∑' qₙ (p_{i₁} (y, ..., y), ..., p_{iₙ} (y, ..., y)).

For each n and i₁, ..., iₙ, define a i₁ + ... + iₙ multilinear function mapping (y₀, ..., y_{i₁ + ... + iₙ - 1}) to qₙ (p_{i₁} (y₀, ..., y_{i₁-1}), p_{i₂} (y_{i₁}, ..., y_{i₁ + i₂ - 1}), ..., p_{iₙ} (....))). Then g ∘ f is obtained by summing all these multilinear functions.

To formalize this, we use compositions of an integer N, i.e., its decompositions into a sum i₁ + ... + iₙ of positive integers. Given such a composition c and two formal multilinear series q and p, let q.comp_along_composition p c be the above multilinear function. Then the N-th coefficient in the power series expansion of g ∘ f is the sum of these terms over all c : composition N.

To complete the proof, we need to show that this power series has a positive radius of convergence. This follows from the fact that composition N has cardinality 2^(N-1) and estimates on the norm of qₙ and pₖ, which give summability. We also need to show that it indeed converges to g ∘ f. For this, we note that the composition of partial sums converges to g ∘ f, and that it corresponds to a part of the whole sum, on a subset that increases to the whole space. By summability of the norms, this implies the overall convergence.

Main results

Implementation details

The main technical difficulty is to write down things. In particular, we need to define precisely q.comp_along_composition p c and to show that it is indeed a continuous multilinear function. This requires a whole interface built on the class composition. Once this is set, the main difficulty is to reorder the sums, writing the composition of the partial sums as a sum over some subset of Σ n, composition n. We need to check that the reordering is a bijection, running over difficulties due to the dependent nature of the types under consideration, that are controlled thanks to the interface for composition.

The associativity of composition on formal multilinear series is a nontrivial result: it does not follow from the associativity of composition of analytic functions, as there is no uniqueness for the formal multilinear series representing a function (and also, it holds even when the radius of convergence of the series is 0). Instead, we give a direct proof, which amounts to reordering double sums in a careful way. The change of variables is a canonical (combinatorial) bijection composition.sigma_equiv_sigma_pi between (Σ (a : composition n), composition a.length) and (Σ (c : composition n), Π (i : fin c.length), composition (c.blocks_fun i)), and is described in more details below in the paragraph on associativity.

Composing formal multilinear series

In this paragraph, we define the composition of formal multilinear series, by summing over all possible compositions of n.

def formal_multilinear_series.apply_composition {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {F : Type u_3} [normed_group F] [normed_space 𝕜 F] (p : formal_multilinear_series 𝕜 E F) {n : } (c : composition n) :
(fin n → E)fin c.length → F

Given a formal multilinear series p, a composition c of n and the index i of a block of c, we may define a function on fin n → E by picking the variables in the i-th block of n, and applying the corresponding coefficient of p to these variables. This function is called p.apply_composition c v i for v : fin n → E and i : fin c.length.

Equations
theorem formal_multilinear_series.apply_composition_ones {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {F : Type u_3} [normed_group F] [normed_space 𝕜 F] (p : formal_multilinear_series 𝕜 E F) (n : ) :
p.apply_composition (composition.ones n) = λ (v : fin n → E) (i : fin (composition.ones n).length), (p 1) (λ (_x : fin 1), v (fin.cast_le _ i))

theorem formal_multilinear_series.apply_composition_update {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {F : Type u_3} [normed_group F] [normed_space 𝕜 F] (p : formal_multilinear_series 𝕜 E F) {n : } (c : composition n) (j : fin n) (v : fin n → E) (z : E) :

Technical lemma stating how p.apply_composition commutes with updating variables. This will be the key point to show that functions constructed from apply_composition retain multilinearity.

def formal_multilinear_series.comp_along_composition_multilinear {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {F : Type u_3} [normed_group F] [normed_space 𝕜 F] {G : Type u_4} [normed_group G] [normed_space 𝕜 G] {n : } :
formal_multilinear_series 𝕜 F Gformal_multilinear_series 𝕜 E Fcomposition nmultilinear_map 𝕜 (λ (i : fin n), E) G

Given two formal multilinear series q and p and a composition c of n, one may form a multilinear map in n variables by applying the right coefficient of p to each block of the composition, and then applying q c.length to the resulting vector. It is called q.comp_along_composition_multilinear p c. This function admits a version as a continuous multilinear map, called q.comp_along_composition p c below.

Equations
theorem formal_multilinear_series.comp_along_composition_multilinear_bound {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {F : Type u_3} [normed_group F] [normed_space 𝕜 F] {G : Type u_4} [normed_group G] [normed_space 𝕜 G] {n : } (q : formal_multilinear_series 𝕜 F G) (p : formal_multilinear_series 𝕜 E F) (c : composition n) (v : fin n → E) :

The norm of q.comp_along_composition_multilinear p c is controlled by the product of the norms of the relevant bits of q and p.

def formal_multilinear_series.comp_along_composition {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {F : Type u_3} [normed_group F] [normed_space 𝕜 F] {G : Type u_4} [normed_group G] [normed_space 𝕜 G] {n : } :

Given two formal multilinear series q and p and a composition c of n, one may form a continuous multilinear map in n variables by applying the right coefficient of p to each block of the composition, and then applying q c.length to the resulting vector. It is called q.comp_along_composition p c. It is constructed from the analogous multilinear function q.comp_along_composition_multilinear p c, together with a norm control to get the continuity.

Equations
@[simp]
theorem formal_multilinear_series.comp_along_composition_apply {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {F : Type u_3} [normed_group F] [normed_space 𝕜 F] {G : Type u_4} [normed_group G] [normed_space 𝕜 G] {n : } (q : formal_multilinear_series 𝕜 F G) (p : formal_multilinear_series 𝕜 E F) (c : composition n) (v : fin n → E) :

theorem formal_multilinear_series.comp_along_composition_norm {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {F : Type u_3} [normed_group F] [normed_space 𝕜 F] {G : Type u_4} [normed_group G] [normed_space 𝕜 G] {n : } (q : formal_multilinear_series 𝕜 F G) (p : formal_multilinear_series 𝕜 E F) (c : composition n) :

The norm of q.comp_along_composition p c is controlled by the product of the norms of the relevant bits of q and p.

theorem formal_multilinear_series.comp_along_composition_nnnorm {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {F : Type u_3} [normed_group F] [normed_space 𝕜 F] {G : Type u_4} [normed_group G] [normed_space 𝕜 G] {n : } (q : formal_multilinear_series 𝕜 F G) (p : formal_multilinear_series 𝕜 E F) (c : composition n) :
nnnorm (q.comp_along_composition p c) (nnnorm (q c.length)) * ∏ (i : fin c.length), nnnorm (p (c.blocks_fun i))

def formal_multilinear_series.comp {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {F : Type u_3} [normed_group F] [normed_space 𝕜 F] {G : Type u_4} [normed_group G] [normed_space 𝕜 G] :

Formal composition of two formal multilinear series. The n-th coefficient in the composition is defined to be the sum of q.comp_along_composition p c over all compositions of n. In other words, this term (as a multilinear function applied to v_0, ..., v_{n-1}) is ∑'_{k} ∑'_{i₁ + ... + iₖ = n} pₖ (q_{i_1} (...), ..., q_{i_k} (...)), where one puts all variables v_0, ..., v_{n-1} in increasing order in the dots.

Equations
theorem formal_multilinear_series.comp_coeff_zero {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {F : Type u_3} [normed_group F] [normed_space 𝕜 F] {G : Type u_4} [normed_group G] [normed_space 𝕜 G] (q : formal_multilinear_series 𝕜 F G) (p : formal_multilinear_series 𝕜 E F) (v : fin 0 → E) (v' : fin 0 → F) :
(q.comp p 0) v = (q 0) v'

The 0-th coefficient of q.comp p is q 0. Since these maps are multilinear maps in zero variables, but on different spaces, we can not state this directly, so we state it when applied to arbitrary vectors (which have to be the zero vector).

@[simp]
theorem formal_multilinear_series.comp_coeff_zero' {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {F : Type u_3} [normed_group F] [normed_space 𝕜 F] {G : Type u_4} [normed_group G] [normed_space 𝕜 G] (q : formal_multilinear_series 𝕜 F G) (p : formal_multilinear_series 𝕜 E F) (v : fin 0 → E) :
(q.comp p 0) v = (q 0) (λ (i : fin 0), 0)

theorem formal_multilinear_series.comp_coeff_zero'' {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {F : Type u_3} [normed_group F] [normed_space 𝕜 F] (q : formal_multilinear_series 𝕜 E F) (p : formal_multilinear_series 𝕜 E E) :
q.comp p 0 = q 0

The 0-th coefficient of q.comp p is q 0. When p goes from E to E, this can be expressed as a direct equality

The identity formal power series

We will now define the identity power series, and show that it is a neutral element for left and right composition.

def formal_multilinear_series.id (𝕜 : Type u_1) [nondiscrete_normed_field 𝕜] (E : Type u_2) [normed_group E] [normed_space 𝕜 E] :

The identity formal multilinear series, with all coefficients equal to 0 except for n = 1 where it is (the continuous multilinear version of) the identity.

Equations
@[simp]
theorem formal_multilinear_series.id_apply_one (𝕜 : Type u_1) [nondiscrete_normed_field 𝕜] (E : Type u_2) [normed_group E] [normed_space 𝕜 E] (v : fin 1 → E) :

The first coefficient of id 𝕜 E is the identity.

theorem formal_multilinear_series.id_apply_one' (𝕜 : Type u_1) [nondiscrete_normed_field 𝕜] (E : Type u_2) [normed_group E] [normed_space 𝕜 E] {n : } (h : n = 1) (v : fin n → E) :
(formal_multilinear_series.id 𝕜 E n) v = v 0, _⟩

The nth coefficient of id 𝕜 E is the identity when n = 1. We state this in a dependent way, as it will often appear in this form.

@[simp]
theorem formal_multilinear_series.id_apply_ne_one (𝕜 : Type u_1) [nondiscrete_normed_field 𝕜] (E : Type u_2) [normed_group E] [normed_space 𝕜 E] {n : } :

For n ≠ 1, the n-th coefficient of id 𝕜 E is zero, by definition.

@[simp]
theorem formal_multilinear_series.comp_id {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {F : Type u_3} [normed_group F] [normed_space 𝕜 F] (p : formal_multilinear_series 𝕜 E F) :

theorem formal_multilinear_series.id_comp {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {F : Type u_3} [normed_group F] [normed_space 𝕜 F] (p : formal_multilinear_series 𝕜 E F) :
p 0 = 0(formal_multilinear_series.id 𝕜 F).comp p = p

Summability properties of the composition of formal power series

theorem formal_multilinear_series.comp_summable_nnreal {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {F : Type u_3} [normed_group F] [normed_space 𝕜 F] {G : Type u_4} [normed_group G] [normed_space 𝕜 G] (q : formal_multilinear_series 𝕜 F G) (p : formal_multilinear_series 𝕜 E F) :
0 < q.radius0 < p.radius(∃ (r : ℝ≥0), 0 < r summable (λ (i : Σ (n : ), composition n), (nnnorm (q.comp_along_composition p i.snd)) * r ^ i.fst))

If two formal multilinear series have positive radius of convergence, then the terms appearing in the definition of their composition are also summable (when multiplied by a suitable positive geometric term).

theorem formal_multilinear_series.le_comp_radius_of_summable {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {F : Type u_3} [normed_group F] [normed_space 𝕜 F] {G : Type u_4} [normed_group G] [normed_space 𝕜 G] (q : formal_multilinear_series 𝕜 F G) (p : formal_multilinear_series 𝕜 E F) (r : ℝ≥0) :
summable (λ (i : Σ (n : ), composition n), (nnnorm (q.comp_along_composition p i.snd)) * r ^ i.fst)r (q.comp p).radius

Bounding below the radius of the composition of two formal multilinear series assuming summability over all compositions.

Composing analytic functions

Now, we will prove that the composition of the partial sums of q and p up to order N is given by a sum over some large subset of Σ n, composition n of q.comp_along_composition p, to deduce that the series for q.comp p indeed converges to g ∘ f when q is a power series for g and p is a power series for f.

This proof is a big reindexing argument of a sum. Since it is a bit involved, we define first the source of the change of variables (comp_partial_source), its target (comp_partial_target) and the change of variables itself (comp_change_of_variables) before giving the main statement in comp_partial_sum.

Source set in the change of variables to compute the composition of partial sums of formal power series. See also comp_partial_sum.

Equations

Change of variables appearing to compute the composition of partial sums of formal power series

Equations

Target set in the change of variables to compute the composition of partial sums of formal power series, here given a a set.

Equations

Target set in the change of variables to compute the composition of partial sums of formal power series, here given a a finset. See also comp_partial_sum.

Equations

The auxiliary set corresponding to the composition of partial sums asymptotically contains all possible compositions.

theorem formal_multilinear_series.comp_partial_sum {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {F : Type u_3} [normed_group F] [normed_space 𝕜 F] {G : Type u_4} [normed_group G] [normed_space 𝕜 G] (q : formal_multilinear_series 𝕜 F G) (p : formal_multilinear_series 𝕜 E F) (N : ) (z : E) :
q.partial_sum N (∑ (i : ) in finset.Ico 1 N, (p i) (λ (j : fin i), z)) = ∑ (i : Σ (n : ), composition n) in formal_multilinear_series.comp_partial_sum_target N, (q.comp_along_composition_multilinear p i.snd) (λ (j : fin i.fst), z)

Composing the partial sums of two multilinear series coincides with the sum over all compositions in comp_partial_sum_target N. This is precisely the motivation for the definition of comp_partial_sum_target N.

theorem has_fpower_series_at.comp {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {F : Type u_3} [normed_group F] [normed_space 𝕜 F] {G : Type u_4} [normed_group G] [normed_space 𝕜 G] {g : F → G} {f : E → F} {q : formal_multilinear_series 𝕜 F G} {p : formal_multilinear_series 𝕜 E F} {x : E} :

If two functions g and f have power series q and p respectively at f x and x, then g ∘ f admits the power series q.comp p at x.

theorem analytic_at.comp {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {F : Type u_3} [normed_group F] [normed_space 𝕜 F] {G : Type u_4} [normed_group G] [normed_space 𝕜 G] {g : F → G} {f : E → F} {x : E} :
analytic_at 𝕜 g (f x)analytic_at 𝕜 f xanalytic_at 𝕜 (g f) x

If two functions g and f are analytic respectively at f x and x, then g ∘ f is analytic at x.

Associativity of the composition of formal multilinear series

In this paragraph, we us prove the associativity of the composition of formal power series. By definition,

(r.comp q).comp p n v
= _{i₁ + ... + iₖ = n} (r.comp q) (p_{i₁} (v₀, ..., v_{i₁ -1}), p_{i₂} (...), ..., p_{iₖ}(...))
= _{a : composition n} (r.comp q) a.length (apply_composition p a v)

decomposing r.comp q in the same way, we get

(r.comp q).comp p n v
= _{a : composition n} _{b : composition a.length}
  r b.length (apply_composition q b (apply_composition p a v))

On the other hand,

r.comp (q.comp p) n v = _{c : composition n} r c.length (apply_composition (q.comp p) c v)

Here, apply_composition (q.comp p) c v is a vector of length c.length, whose i-th term is given by (q.comp p) (c.blocks_fun i) (v_l, v_{l+1}, ..., v_{m-1}) where {l, ..., m-1} is the i-th block in the composition c, of length c.blocks_fun i by definition. To compute this term, we expand it as ∑_{dᵢ : composition (c.blocks_fun i)} q dᵢ.length (apply_composition p dᵢ v'), where v' = (v_l, v_{l+1}, ..., v_{m-1}). Therefore, we get

r.comp (q.comp p) n v =
_{c : composition n} _{d₀ : composition (c.blocks_fun 0),
  ..., d_{c.length - 1} : composition (c.blocks_fun (c.length - 1))}
  r c.length (λ i, q dᵢ.length (apply_composition p d v'))

To show that these terms coincide, we need to explain how to reindex the sums to put them in bijection (and then the terms we are summing will correspond to each other). Suppose we have a composition a of n, and a composition b of a.length. Then b indicates how to group together some blocks of a, giving altogether b.length blocks of blocks. These blocks of blocks can be called d₀, ..., d_{a.length - 1}, and one obtains a composition c of n by saying that each dᵢ is one single block. Conversely, if one starts from c and the dᵢs, one can concatenate the dᵢs to obtain a composition a of n, and register the lengths of the dᵢs in a composition b of a.length.

An example might be enlightening. Suppose a = [2, 2, 3, 4, 2]. It is a composition of length 5 of 13. The content of the blocks may be represented as 0011222333344. Now take b = [2, 3] as a composition of a.length = 5. It says that the first 2 blocks of a should be merged, and the last 3 blocks of a should be merged, giving a new composition of 13 made of two blocks of length 4 and 9, i.e., c = [4, 9]. But one can also remember that the new first block was initially made of two blocks of size 2, so d₀ = [2, 2], and the new second block was initially made of three blocks of size 3, 4 and 2, so d₁ = [3, 4, 2].

This equivalence is called composition.sigma_equiv_sigma_pi n below.

We start with preliminary results on compositions, of a very specialized nature, then define the equivalence composition.sigma_equiv_sigma_pi n, and we deduce finally the associativity of composition of formal multilinear series in formal_multilinear_series.comp_assoc.

Rewriting equality in the dependent type Σ (a : composition n), composition a.length) in non-dependent terms with lists, requiring that the blocks coincide.

theorem composition.sigma_pi_composition_eq_iff {n : } (u v : Σ (c : composition n), Π (i : fin c.length), composition (c.blocks_fun i)) :
u = v list.of_fn (λ (i : fin u.fst.length), (u.snd i).blocks) = list.of_fn (λ (i : fin v.fst.length), (v.snd i).blocks)

Rewriting equality in the dependent type Σ (c : composition n), Π (i : fin c.length), composition (c.blocks_fun i) in non-dependent terms with lists, requiring that the lists of blocks coincide.

When a is a composition of n and b is a composition of a.length, a.gather b is the composition of n obtained by gathering all the blocks of a corresponding to a block of b. For instance, if a = [6, 5, 3, 5, 2] and b = [2, 3], one should gather together the first two blocks of a and its last three blocks, giving a.gather b = [11, 10].

Equations

An auxiliary function used in the definition of sigma_equiv_sigma_pi below, associating to two compositions a of n and b of a.length, and an index i bounded by the length of a.gather b, the subcomposition of a made of those blocks belonging to the i-th block of a.gather b.

Equations
theorem composition.size_up_to_size_up_to_add {n : } (a : composition n) (b : composition a.length) {i j : } (hi : i < b.length) :
j < b.blocks_fun i, hi⟩a.size_up_to (b.size_up_to i + j) = (a.gather b).size_up_to i + (a.sigma_composition_aux b i, _⟩).size_up_to j

Auxiliary lemma to prove that the composition of formal multilinear series is associative.

Consider a composition a of n and a composition b of a.length. Grouping together some blocks of a according to b as in a.gather b, one can compute the total size of the blocks of a up to an index size_up_to b i + j (where the j corresponds to a set of blocks of a that do not fill a whole block of a.gather b). The first part corresponds to a sum of blocks in a.gather b, and the second one to a sum of blocks in the next block of sigma_composition_aux a b. This is the content of this lemma.

Natural equivalence between (Σ (a : composition n), composition a.length) and (Σ (c : composition n), Π (i : fin c.length), composition (c.blocks_fun i)), that shows up as a change of variables in the proof that composition of formal multilinear series is associative.

Consider a composition a of n and a composition b of a.length. Then b indicates how to group together some blocks of a, giving altogether b.length blocks of blocks. These blocks of blocks can be called d₀, ..., d_{a.length - 1}, and one obtains a composition c of n by saying that each dᵢ is one single block. The map ⟨a, b⟩ → ⟨c, (d₀, ..., d_{a.length - 1})⟩ is the direct map in the equiv.

Conversely, if one starts from c and the dᵢs, one can join the dᵢs to obtain a composition a of n, and register the lengths of the dᵢs in a composition b of a.length. This is the inverse map of the equiv.

Equations
theorem formal_multilinear_series.comp_assoc {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {F : Type u_3} [normed_group F] [normed_space 𝕜 F] {G : Type u_4} [normed_group G] [normed_space 𝕜 G] {H : Type u_5} [normed_group H] [normed_space 𝕜 H] (r : formal_multilinear_series 𝕜 G H) (q : formal_multilinear_series 𝕜 F G) (p : formal_multilinear_series 𝕜 E F) :
(r.comp q).comp p = r.comp (q.comp p)