Notes on series in several variables

Adam Coffman

January 12, 2002

These notes are elementary derivations of well-known, but sometimes hard to find, facts on series in several variables. By “elementary” I mean “avoiding the theory of complex differentiation and integration,” and the basic ideas of the proofs will be natural generalizations of the first-year calculus treatment of power series in one variable. I will also avoid issues of “uniformity,” even though this is the usual approach to some of the theorems. Some books which state some related facts on multi-indexed series are [D] and [GF].

1 Multi-indexed series

Notation 1.1.

• 𝕎 = {0, 1, 2, 3, 4, …} is the set of whole numbers (so ℕ ⊆ 𝕎 ⊆ ℤ).

• n ∈ ℕ will be a fixed natural number.

• An element α ∈ 𝕎^{n} is a “multi-index.” The “order” of α = (α_{1}, α_{2}, …, α_{n}) is |α| = α_{1} + α_{2} + ⋯ + α_{n}. Sometimes to emphasize the number of terms the order will be written |α|_{n}.

• (𝕂, | |) will be either of the fields ℝ or ℂ, with the usual absolute value and complex conjugation (z ↦ z̄).

• (B, ∥ ∥) will be a Banach space over 𝕂.

Definition 1.2. A “multi-indexed sequence in B” is a function

c : 𝕎^{n} → B : α ↦ c_{α}.

Definition 1.3. If the set

V_{c} = {∑_{α∈Λ} ∥c_{α}∥ : Λ ⊆ 𝕎^{n}, Λ finite}

is a bounded subset of ℝ, we will say “c forms a convergent multi-indexed series.”

It looks like an analogue of “absolutely convergent series,” but since there is no canonical way to order 𝕎^{n} for n > 1, we won’t bother with “conditionally convergent” series, where even when n = 1 the sum depends on the ordering.

Theorem 1.4. If c forms a convergent multi-indexed series, then there exists an element L ∈ B with the following property: for any ϵ_{1} > 0, there is some N_{1} ∈ ℕ such that if N_{2} ≥ N_{1}, then

∥(∑_{k=0}^{N_{2}} ( ∑_{|α|=k} c_{α})) − L∥ < ϵ_{1}.

Further, L is unique and satisfies ∥L∥ ≤ lubV_{c}.

Proof. Let β be the least upper bound of the set V_{c}. Then, given any ϵ_{2} > 0, there’s some finite set Λ ⊆ 𝕎^{n} such that

β − ϵ_{2} < ∑_{α∈Λ} ∥c_{α}∥ ≤ β.

Let N_{3} = max{|α| : α ∈ Λ}. Then,

N_{4} ≥ N_{3} ⟹ β − ϵ_{2} < ∑_{α∈Λ} ∥c_{α}∥ ≤ ∑_{k=0}^{N_{4}} ( ∑_{|α|=k} ∥c_{α}∥) ≤ β,

N_{5} ≥ N_{4} ≥ N_{3} ⟹ ∥(∑_{k=0}^{N_{5}} ( ∑_{|α|=k} c_{α})) − (∑_{k=0}^{N_{4}} ( ∑_{|α|=k} c_{α}))∥

= ∥ ∑_{k=N_{4}+1}^{N_{5}} ( ∑_{|α|=k} c_{α})∥ ≤ ∑_{k=N_{4}+1}^{N_{5}} ( ∑_{|α|=k} ∥c_{α}∥)

= (∑_{k=0}^{N_{5}} ( ∑_{|α|=k} ∥c_{α}∥)) − (∑_{k=0}^{N_{4}} ( ∑_{|α|=k} ∥c_{α}∥))

< β − (β − ϵ_{2}) = ϵ_{2}.

This implies that as a sequence depending on N, ∑_{k=0}^{N} ( ∑_{|α|=k} c_{α}) is a Cauchy

sequence in B, so it converges to some L ∈ B. The uniqueness of L is the usual uniqueness of a limit, and the bound for ∥L∥ is given, for N_{2} ≥ N_{1}, by:

∥L∥ ≤ ∥(∑_{k=0}^{N_{2}} ( ∑_{|α|=k} c_{α})) − L∥ + (∑_{k=0}^{N_{2}} ( ∑_{|α|=k} ∥c_{α}∥)) < ϵ_{1} + β.

Notation 1.5. If c forms a convergent multi-indexed series, and L ∈ B is the element from the previous Theorem, the following abbreviations make sense:

∑_{α∈𝕎^{n}} c_{α} = ∑_{α} c_{α} = ∑ c_{α} = L.

The idea of the Theorem and this Notation is that we can group the multiindexed series by its “homogeneous” parts, to get a well-defined “sum” of the series. The Theorem also relates the multi-indexed series ∑_{α} to a single-indexed

series ∑_{k=0}^{∞}, as defined in first-year calculus. It will usually be convenient to denote

the partial sums:

∑_{k=0}^{N} ( ∑_{|α|=k} c_{α}) = ∑_{|α|≤N} c_{α}.

To approximate the sum L by a finite partial sum, it is obviously not sufficient to consider arbitrary finite index sets Λ, but the following two Theorems generalize Theorem 1.4 by showing that it is sufficient to consider finite sets that contain “enough” of the lower-order terms.

Theorem 1.6. If c forms a convergent multi-indexed series, then there exists a unique element L ∈ B with the following property: for any ϵ > 0, there is some N ∈ ℕ such that if Λ ⊆ 𝕎^{n} is a finite set and {α : |α| ≤ N} ⊆ Λ, then

∥(∑_{α∈Λ} c_{α}) − L∥ < ϵ.

Proof. Let L be as in Theorem 1.4, and let ϵ > 0. Then, corresponding to ϵ_{1} = ϵ/2 > 0, there’s some N_{1} ∈ ℕ such that if N_{2} ≥ N_{1}, then

∥( ∑_{|α|≤N_{2}} c_{α}) − L∥ < ϵ/2.

Also as in Theorem 1.4, corresponding to ϵ_{2} = ϵ/2, there’s some N_{3} so that

N_{4} ≥ N_{3} ⟹ β − ϵ/2 < ∑_{|α|≤N_{4}} ∥c_{α}∥ ≤ β.

Let N = max{N_{1}, N_{3}}, and, for any finite Λ containing {α : |α| ≤ N}, let N_{5} = max{|α| : α ∈ Λ} ≥ N ≥ N_{3}. Then,

∥(∑_{α∈Λ} c_{α}) − L∥ = ∥( ∑_{|α|≤N} c_{α}) − L + ∑ ; \begin{matrix}α ∈ Λ \\ |α| > N \end{matrix} c_{α}∥

≤ ∥( ∑_{|α|≤N} c_{α}) − L∥ + ∑ ; \begin{matrix}α ∈ Λ \\ |α| > N \end{matrix} ∥c_{α}∥

≤ ∥( ∑_{|α|≤N} c_{α}) − L∥ + ∑_{N<|α|≤N_{5}} ∥c_{α}∥
< ϵ/2 + ϵ/2 = ϵ.

For the uniqueness, suppose L_{1} and L_{2} have the claimed property. Then, for any ϵ > 0, there’s some N so that if Λ is finite and {α : |α| ≤ N} ⊆ Λ, then

∥(∑_{α∈Λ} c_{α}) − L_{1}∥ < \frac{ϵ}{2},

and there’s some N′ so that if {α : |α| ≤ N′} ⊆ Λ, then

∥(∑_{α∈Λ} c_{α}) − L_{2}∥ < \frac{ϵ}{2}.

Let N′′ = max{N, N′}, so that if {α : |α| ≤ N′′} ⊆ Λ, then

∥L_{1} − L_{2}∥ = ∥L_{1} − (∑_{α∈Λ} c_{α}) + (∑_{α∈Λ} c_{α}) − L_{2}∥

≤ ∥(∑_{α∈Λ} c_{α}) − L_{1}∥ + ∥(∑_{α∈Λ} c_{α}) − L_{2}∥ < \frac{ϵ}{2} + \frac{ϵ}{2} = ϵ.

Theorem 1.7. If c forms a convergent multi-indexed series with sum L, and σ : 𝕎 → 𝕎^{n} is any bijection, then

∑_{k=0}^{∞} c_{σ(k)} = L.

Proof. Given any ϵ > 0, let N be the corresponding number from the previous Theorem. Then, σ^{−1}({α : |α| ≤ N}) is a finite subset of 𝕎, with largest element M_{1}. For any M_{2} ≥ M_{1}, let Λ = {σ(1), …, σ(M_{2})}, a finite subset of 𝕎^{n} such that {α : |α| ≤ N} = σ(σ^{−1}({α : |α| ≤ N})) ⊆ σ({1, …, M_{1}}) ⊆ Λ. So,

∥(∑_{k=0}^{M_{2}} c_{σ(k)}) − L∥ = ∥(∑_{α∈Λ} c_{α}) − L∥ < ϵ.

Theorem 1.8 (Easy Comparison). If (B_{1}, ∥ ∥_{1}) and (B_{2}, ∥ ∥_{2}) are Banach spaces, and c_{α} is a multi-indexed sequence in B_{1} that forms a convergent multiindexed series, and b_{α} is a multi-indexed sequence in B_{2} such that ∥b_{α}∥_{2} ≤ ∥c_{α}∥_{1} for all but finitely many α ∈ 𝕎^{n}, then b_{α} also forms a convergent multiindexed series.

Proof. Let U be any upper bound for V_{c}, and let Φ be a fixed finite set such that ∥b_{α}∥_{2} > ∥c_{α}∥_{1} ⟹ α ∈ Φ. Then, the set V_{b} is bounded: for any finite Λ ⊆ 𝕎^{n},

∑_{α∈Λ} ∥b_{α}∥_{2} = ( ∑_{α∈Λ\Φ} ∥b_{α}∥_{2}) + ( ∑_{α∈Λ∩Φ} ∥b_{α}∥_{2})

≤ ( ∑_{α∈Λ\Φ} ∥c_{α}∥_{1}) + ( ∑_{α∈Φ} ∥b_{α}∥_{2}) ≤ U + ( ∑_{α∈Φ} ∥b_{α}∥_{2}) .

Corollary 1.9. Given any set Γ ⊆ 𝕎^{n}, and a multi-indexed sequence in B, c_{α}, define another multi-indexed sequence in B:

d_{α} = \{ \begin{matrix}c_{α} & if α ∈ Γ \\ 0 & if α ∉ Γ \end{matrix} .

If c_{α} forms a convergent multi-indexed series, then so does d_{α}.

Notation 1.10. If c_{α} forms a convergent multi-indexed series, and Γ and d_{α} are as in the previous Corollary, with sum M, denote

∑_{α∈Γ} c_{α} = ∑_{α∈𝕎^{n}} d_{α} = M.

Theorem 1.11 (Comparison with Estimate). Given b_{α}, a multi-indexed sequence in B, and c_{α}, a multi-indexed sequence in ℝ, if ∥b_{α}∥ ≤ c_{α} for all α ∈ 𝕎^{n} and ∑ c_{α} = λ, then b_{α} forms a convergent multi-indexed series, with sum L ∈ B such that ∥L∥ ≤ λ.

Proof. Note that the hypothesis implies c_{α} = |c_{α}|. Let β = lubV_{c}, as in the Proof of Theorem 1.4, so that for any ϵ_{2} > 0, there is some N_{3} such that if N_{4} ≥ N_{3}, then

β − ϵ_{2} < ∑_{|α|≤N_{4}} c_{α} ≤ β

⟹ |( ∑_{|α|≤N_{4}} c_{α}) − β| < ϵ_{2}.

This implies β = λ, by the uniqueness of the sum from Theorem 1.4. For any finite Λ ⊆ 𝕎^{n},

∑_{α∈Λ} ∥b_{α}∥ ≤ ∑_{α∈Λ} c_{α} ≤ λ.

This shows b_{α} forms a convergent multi-indexed series, with lubV_{b} ≤ λ. The inequality ∥L∥ ≤ λ follows from the bound from Theorem 1.4.

Theorem 1.12. If ∑_{α∈𝕎^{n}} c_{α} = L, and σ : 𝕎^{m} → 2^{𝕎^{n}} has the property that

𝕎^{n} = ⋃_{γ∈𝕎^{m}} σ(γ)

is a disjoint union, then

∑_{γ∈𝕎^{m}} ( ∑_{α∈σ(γ)} c_{α}) = L.

Proof. (Step 1, establishing convergence.) For each γ ∈ 𝕎^{m}, denote by d_{α}^{γ} the multi-indexed sequence in B corresponding to Corollary 1.9, applied to c_{α} and σ(γ). Then d_{α}^{γ} forms a convergent multi-indexed series, and as in the above Notation, denote for each γ,

∑_{α∈σ(γ)} c_{α} = ∑_{α∈𝕎^{n}} d_{α}^{γ} = L_{γ}.

Given a finite, non-empty subset Λ ⊆ 𝕎^{m} with #Λ elements, Theorem 1.4 applies to ϵ = \frac{1}{#Λ} > 0, giving N_{1}(γ, Λ) ∈ ℕ so that if N_{2} ≥ N_{1}(γ, Λ), then

∥( ∑_{|α|≤N_{2}} d_{α}^{γ}) − L_{γ} ∥ < \frac{1}{#Λ}.

If N_{2} ≥ N_{1}(Λ) = max{N_{1}(γ, Λ) : γ ∈ Λ}, then

∑_{γ∈Λ} ∥ ∑_{α∈σ(γ)} c_{α}∥ = ∑_{γ∈Λ} ∥L_{γ}∥

= ∑_{γ∈Λ} ∥L_{γ} − ( ∑_{|α|≤N_{2}} d_{α}^{γ}) + ( ∑_{|α|≤N_{2}} d_{α}^{γ})∥

< (∑_{γ∈Λ} \frac{1}{#Λ}) + ∑_{γ∈Λ} ( ∑_{|α|≤N_{2}} ∥d_{α}^{γ}∥)

= 1 + ∑_{finite} ∥c_{α}∥ ≤ 1 + β,

the last step using the disjointness property of σ, and the lub β as in Theorem 1.4.

(Step 2, establishing the value of the limit.) Let ϵ > 0. Denote

∑_{γ∈𝕎^{m}} ( ∑_{α∈σ(γ)} c_{α}) = ∑_{γ∈𝕎^{m}} L_{γ} = L_{σ},

with the goal of showing ∥L−L_{σ}∥ < ϵ. Applying Theorem 1.6 to the hypothesis that c_{α} forms a convergent multi-indexed series with sum L, there’s some N corresponding to ϵ/3 so that if Λ is any finite subset of 𝕎^{n} containing {α : |α| ≤ N}, then

∥∑_{α∈Λ} c_{α} − L∥ < \frac{ϵ}{3}.

By the assumed property of σ, for each α ∈ 𝕎^{n} there is a unique γ ∈ 𝕎^{m} so that α ∈ σ(γ). Let Γ_{1} be a finite subset of 𝕎^{m} so that

{α : |α| ≤ N} ⊆ ⋃_{γ∈Γ_{1}} σ(γ).

Then, for any α such that |α| ≤ N, there’s some γ ∈ Γ_{1} so that α ∈ σ(γ), which, by construction, means c_{α} = d_{α}^{γ}, and for any N_{2} ≥ N, c_{α} will be exactly one of the terms of

∑_{γ∈Γ_{1}} ( ∑_{|α|≤N_{2}} d_{α}^{γ}) .

(The “exactly one” refers to c_{α} as a formal symbol, since of course, some values of the multi-indexed sequence c may repeat, or be equal to 0.) This implies, for any N_{2} ≥ N, and any Γ_{2} ⊆ 𝕎^{m} which is finite and contains Γ_{1},

∥( ∑_{γ∈Γ_{2}} ( ∑_{|α|≤N_{2}} d_{α}^{γ})) − L∥ < \frac{ϵ}{3}.  (1)

Similarly applying Theorem 1.6 to the multi-indexed sequence L_{γ}, which was shown to form a convergent multi-indexed series in Step 1, there is some N′ so that if Γ_{3} ⊆ 𝕎^{m} is a finite set containing {γ : |γ| ≤ N′}, then

∥( ∑_{γ∈Γ_{3}} L_{γ} ) − L_{σ}∥ < \frac{ϵ}{3}.  (2)

In particular, both inequalities (1) and (2) hold for the finite set Γ = Γ_{1} ∪ {γ : |γ| ≤ N′}.

As in Step 1, there is some N_{1}(Γ) = max{N_{1}(γ, Γ) : γ ∈ Γ} corresponding to the above Γ and \frac{ϵ}{3·#Γ} > 0, so that if N_{2} ≥ N_{1}(Γ), then

∑_{γ∈Γ} ∥L_{γ} − ∑_{|α|≤N_{2}} d_{α}^{γ}∥ < \frac{ϵ}{3}.  (3)

Let N_{1} = max{N, N_{1}(Γ)}, so that for any N_{2} ≥ N_{1}, inequalities (1), (2), and (3) all hold, and:

∥L − L_{σ}∥ = ∥L − (∑_{γ∈Γ} L_{γ} ) + (∑_{γ∈Γ} L_{γ} ) − L_{σ}∥

≤ ∥∑_{γ∈Γ} (L_{γ} − ∑_{|α|≤N_{2}} d_{α}^{γ})∥

+ ∥(∑_{γ∈Γ} ( ∑_{|α|≤N_{2}} d_{α}^{γ})) − L∥

+ ∥(∑_{γ∈Γ} L_{γ} ) − L_{σ}∥ < \frac{ϵ}{3} + \frac{ϵ}{3} + \frac{ϵ}{3}.

Theorem 1.7 could be considered a special case. The converse statement, that if the double sum converges, then the multi-indexed sum also converges: ∑_{α∈𝕎^{n}} c_{α} = L, is clearly false. However, under a stronger “absolute convergence”

assumption, the following result holds.

Theorem 1.13. Given a multi-indexed sequence c_{α} in B, and a map σ as in Theorem 1.12, if

∑_{γ∈𝕎^{m}} ( ∑_{α∈σ(γ)} ∥c_{α}∥)

forms a convergent multi-indexed series, with sum λ ∈ ℝ, then

∑_{α∈𝕎^{n}} c_{α}

and

∑_{γ∈𝕎^{m}} ( ∑_{α∈σ(γ)} c_{α})

both form convergent multi-indexed series, with the same sum L ∈ B, and ∥L∥ ≤ λ.

Proof. Let d_{α}^{γ} be the multi-indexed sequence in B as in Notation 1.10, corresponding to the c_{α} terms with indices in the set σ(γ). The hypothesis means that

∑_{α∈𝕎^{n}} ||d_{α}^{γ}|| = ∑_{α∈σ(γ)} ∥c_{α}∥

converges, with a sum λ_{γ}, which as in the Proof of Theorem 1.11, is the lub of finite sums of terms ∥c_{α}∥, α ∈ σ(γ). Theorem 1.11 then applies to show that

∑_{α∈𝕎^{n}} d_{α}^{γ} = ∑_{α∈σ(γ)} c_{α}

is convergent, with sum L_{γ} ∈ B, and ∥L_{γ}∥ ≤ λ_{γ}. The hypothesis also means that ∑_{γ∈𝕎^{m}} λ_{γ} = λ, which by Theorem 1.11 again, implies that ∑_{γ∈𝕎^{m}} L_{γ} is a

convergent series, with sum L ∈ B such that ∥L∥ ≤ λ.

To show that ∑_{α∈𝕎^{n}} c_{α} is convergent, let Λ be a finite subset of 𝕎^{n}. Then,

there is some finite set Γ so that Λ = ⋃_{γ∈Γ}(Λ ∩ σ(γ)), and

∑_{α∈Λ} ∥c_{α}∥ = ∑_{γ∈Γ} ( ∑_{α∈Λ∩σ(γ)} ∥c_{α}∥) ≤ ∑_{γ∈Γ} λ_{γ} ≤ λ.

By Theorem 1.4, ∑_{α∈𝕎^{n}} c_{α} has sum L′ ∈ B; to show L′ = L, suppose ϵ > 0.

By Theorem 1.6, corresponding to ϵ/3 > 0, there is some N ∈ ℕ such that if Λ is a finite subset of 𝕎^{n} and {α : |α| ≤ N} ⊆ Λ, then

∥(∑_{α∈Λ} c_{α}) − L′∥ < \frac{ϵ}{3}.

Also by Theorem 1.4, there is some N_{3} ∈ ℕ such that if N_{4} ≥ N_{3}, then

∥( ∑_{|γ|≤N_{4}} L_{γ} ) − L∥ < \frac{ϵ}{3}.

We can further pick N_{4} large enough so that {α : |α| ≤ N} ⊆ ⋃_{|γ|≤N_{4}} σ(γ). Let C be the number of such indices:

C = #{γ ∈ 𝕎^{m} : |γ| ≤ N_{4}}.

For each γ, there is, corresponding to \frac{ϵ}{3C} > 0, some N_{5}(γ) such that if N_{6}(γ) ≥ N_{5}(γ), then

∥( ∑_{|α|≤N_{6}(γ)} d_{α}^{γ}) − L_{γ} ∥ < \frac{ϵ}{3C} .

If we choose each N_{6}(γ) larger than N, then

{α : |α| ≤ N} ⊆ ⋃_{|γ|≤N_{4}}{α ∈ σ(γ) : |α| ≤ N_{6}(γ)},

and

∥L − L′∥ ≤ ∥L − ∑_{|γ|≤N_{4}} L_{γ} ∥

+ ∑_{|γ|≤N_{4}} ∥( ∑_{|α|≤N_{6}(γ)} d_{α}^{γ}) − L_{γ} ∥

+ ∥( ∑_{|γ|≤N_{4}} ( ∑_{|α|≤N_{6}(γ)} d_{α}^{γ})) − L′∥

< \frac{ϵ}{3} + C · \frac{ϵ}{3C} + \frac{ϵ}{3}.

2 The geometric series

Lemma 2.1. Given k ∈ 𝕎, the number of multi-indices α ∈ 𝕎^{n} such that

|α| = k is ( \begin{matrix}k + n − 1 \\ n − 1 \end{matrix} ).

Proof. We will first find the number of multi-indices α ∈ ℕ^{n} such that |α| = K ≥ n. The sum α_{1} + … + α_{n} = K can be visualized as K dots in a row, separated into blocks of size α_{i} by n − 1 dividers, for example, 6 = 2 + 3 + 1 is represented:

· · | ⋯ |·

Each divider fits between two of the dots, and between any two adjacent dots is at most one divider (since α_{i} > 0). The number of ways to assign n − 1 dividers

to the K − 1 spaces between the K dots is ( \begin{matrix}K − 1 \\ n − 1 \end{matrix} ).

The function (α_{1}, …, α_{n}) ↦ (α_{1} + 1, …, α_{n} + 1) is obviously a bijection 𝕎^{n} → ℕ^{n}, which, for any k ≥ 0, restricts to a bijection from the set of multiindices of order k in 𝕎^{n} to the set of multi-indices of order k+n in ℕ^{n}. Applying the previous paragraph’s formula to K = k+n gives the claim of the Lemma.

Theorem 2.2 (Geometric series: convergence). Given v ∈ B and r = (r_{1}, r_{2}, …, r_{n}) ∈ 𝕂^{n} such that |r_{i}| < 1 for i = 1, …, n, the multi-indexed sequence in B:

v · r^{α} = v · r_{1}^{α_{1}} · r_{2}^{α_{2}} · . . . · r_{n}^{α_{n}}

forms a convergent multi-indexed series. Its sum is

∑_{α} v · r^{α} = v · ∏_{i=1}^{n} \frac{1}{(1 − r_{i})}.

Proof. (Step 1, establishing convergence.) Let ρ = max{|r_{1}|, …, |r_{n}|}, and given any finite Λ ⊆ 𝕎^{n}, let N = max{|α| : α ∈ Λ}.

∑_{α∈Λ} ∥v · r^{α}∥ = ∑_{α∈Λ} ∥v∥ · |r_{1}|^{α_{1}} · |r_{2}|^{α_{2}} · . . . · |r_{n}|^{α_{n}}

≤ ∥v∥ ∑_{k=0}^{N} ( ∑_{|α|=k} |r_{1}|^{α_{1}} · |r_{2}|^{α_{2}} · . . . · |r_{n}|^{α_{n}} )

≤ ∥v∥ ∑_{k=0}^{N} ( \begin{matrix}k + n − 1 \\ n − 1 \end{matrix} ) ρ^{k},

using the previous Lemma. The above finite sum is a partial sum of a singleindexed series, which converges by the Ratio test ([C]):

lim_{k→∞} |\frac{( \begin{matrix}k + 1 + n − 1 \\ n − 1 \end{matrix} ) ρ^{k+1}}{( \begin{matrix}k + n − 1 \\ n − 1 \end{matrix} ) ρ^{k}} | = lim_{k→∞} \frac{k + n}{k + 1}ρ = ρ < 1.

(Step 2, approximating the geometric series.) The following claim will be proved by induction on n. For any N ∈ 𝕎, there is some multi-indexed sequence in 𝕂, δ_{α}^{N,n}, such that |δ_{α}^{N,n}| ≤ 2^{n−1} and

(∏_{i=1}^{n}(1 − r_{i})) ∑_{k=0}^{N} ( ∑_{|α|_{n}=k} r^{α}) = 1 − ∑_{k=N+1}^{N+n} ( ∑_{|α|_{n}=k} δ_{α}^{N,n}r^{α}) .

For n = 1, let δ_{(α_{1})}^{N,1} = 1 if α_{1} = N + 1, or 0 otherwise. This works, by the usual calculation:

LHS = (∏_{i=1}^{1}(1 − r_{i})) ∑_{k=0}^{N} ( ∑_{|α|_{1}=k} r^{α}) = (1 − r_{1}) ∑_{k=0}^{N} r_{1}^{k} = 1 − r_{1}^{N+1},

RHS = 1 − ∑_{k=N+1}^{N+1} ( ∑_{|α|_{1}=k} δ_{α}^{N,1}r^{α}) = 1 − δ_{(N+1)}^{N,1}r_{1}^{N+1}.

Suppose, inductively, that the claim holds for some n ∈ ℕ. Then, it also holds for n + 1, applied to the vector (r_{1}, r_{2}, …, r_{n}, r_{n+1}), although we will continue to use the symbol r for an n-tuple: (r_{1}, r_{2}, …, r_{n}). Starting with the LHS,

(∏_{i=1}^{n+1}(1 − r_{i})) ∑_{k=0}^{N} ( ∑_{|α|_{n+1}=k}(r_{1}, r_{2}, …, r_{n}, r_{n+1})^{α})

= (1 − r_{n+1}) (∏_{i=1}^{n}(1 − r_{i})) ∑_{j=0}^{N} ( ∑_{k=0}^{N−j} ( ∑_{|α|_{n}=k} r^{α})) r_{n+1}^{j}

= (1 − r_{n+1}) ∑_{j=0}^{N} (1 − ∑_{k=N−j+1}^{N−j+n} ( ∑_{|α|_{n}=k} δ_{α}^{N−j,n}r^{α})) r_{n+1}^{j}

= (∑_{j=0}^{N} (1 − ∑_{k=N−j+1}^{N−j+n} ( ∑_{|α|_{n}=k} δ_{α}^{N−j,n}r^{α})) r_{n+1}^{j})

− (∑_{j=0}^{N} (1 − ∑_{k=N−j+1}^{N−j+n} ( ∑_{|α|_{n}=k} δ_{α}^{N−j,n}r^{α})) r_{n+1}^{j+1})

= (1 − ∑_{k=N+1}^{N+n} ( ∑_{|α|_{n}=k} δ_{α}^{N,n}r^{α}))

+ (∑_{j=1}^{N} (1 − ∑_{k=N−j+1}^{N−j+n} ( ∑_{|α|_{n}=k} δ_{α}^{N−j,n}r^{α})) r_{n+1}^{j})

− ( ∑_{j=1}^{N+1} (1 − ∑_{k=N−(j−1)+1}^{N−(j−1)+n} ( ∑_{|α|_{n}=k} δ_{α}^{N−(j−1),n}r^{α})) r_{n+1}^{j})

= 1 − ( ∑_{k=N+1}^{N+n} ( ∑_{|α|_{n}=k} δ_{α}^{N,n}r^{α}))

+(∑_{j=1}^{N}(( ∑_{k=N−j+2}^{N−j+1+n}( ∑_{|α|_{n}=k}δ_{α}^{N−j+1,n}r^{α}))−( ∑_{k=N−j+1}^{N−j+n}( ∑_{|α|_{n}=k}δ_{α}^{N−j,n}r^{α})))r_{n+1}^{j})

− (1 − ∑_{k=1}^{n} ( ∑_{|α|_{n}=k} δ_{α}^{0,n}r^{α})) r_{n+1}^{N+1}

= 1 − ∑_{k=N+1}^{N+n+1} ( ∑_{|α|_{n+1}=k} δ_{α}^{N,n+1}(r_{1}, r_{2}, …, r_{n}, r_{n+1})^{α}) = RHS,

where δ_{α}^{N,n+1} is either 0, ±1, a number from a δ^{∗,n} multi-indexed sequence, or the difference of two of these numbers.

(Step 3, establishing the value of the limit.) If v = 0, the sum claimed in the Theorem is obvious. If v ≠ 0, and ϵ > 0, then, by the Cauchy property of the convergent series from Step 1, there’s some N_{1} ∈ ℕ so that for all N ≥ N_{1},

∑_{k=N+1}^{N+n} ( \begin{matrix}k + n − 1 \\ n − 1 \end{matrix} ) ρ^{k} < \frac{∏_{i=1}^{n} |1 − r_{i}|}{2^{n−1}∥v∥} · ϵ.

By the equality from Step 2,

|(∏_{i=1}^{n}(1 − r_{i})) (∑_{k=1}^{N} ( ∑_{|α|=k} r^{α})) − 1|

= | ∑_{k=N+1}^{N+n} ( ∑_{|α|=k} δ_{α}^{N,n}r^{α})|

≤ ∑_{k=N+1}^{N+n} ( ∑_{|α|=k} |δ_{α}^{N,n}r^{α}|)

≤ ∑_{k=N+1}^{N+n} 2^{n−1} ( \begin{matrix}k + n − 1 \\ n − 1 \end{matrix} ) ρ^{k} < \frac{∏_{i=1}^{n} |1 − r_{i}|}{∥v∥} · ϵ,

and this is enough to find the limit from Theorem 1.4:

∥(∑_{k=1}^{N} ( ∑_{|α|=k} v · r^{α})) − v · ∏_{i=1}^{n} \frac{1}{(1 − r_{i})}∥ < ϵ.

Theorem 2.3 (Geometric series: divergence). For v, r, as in the previous Theorem, but with v ≠ 0 and |r_{i}| ≥ 1 for some i = 1, …, n, v · r^{α} does not form a convergent multi-indexed series.

Proof. Finite sets of the form

Λ = {(0, 0, …, 0, k, 0, …, 0) : N_{1} ≤ k ≤ N_{2}} ⊆ 𝕎^{n},

with α_{j} = 0 for j ≠ i, give sums of the form

∑_{α∈Λ} ∥v · r^{α}∥ = ∑_{k=N_{1}}^{N_{2}} ∥v∥ · |r_{i}|^{k} ≥ ∥v∥(N_{2} − N_{1} + 1),

which are unbounded. (Here, as always, we are using the convention that r_{j}^{0} = 1 for any r_{j} ∈ 𝕂.)

3 Power series

Notation 3.1. For r = (r_{1}, …, r_{n}) ∈ ℝ^{n}, and a = (a_{1}, …, a_{n}) ∈ 𝕂^{n}, define the “polydisc with center a and polyradius r,” Δ(a, r) ⊆ 𝕂^{n}, by

Δ(a, r) = {(x_{1}, …, x_{n}) ∈ 𝕂^{n} : |x_{i} − a_{i}| < r_{i}, i = 1, …, n}.

Note that if some r_{i} ≤ 0, then Δ(a, r) = Ø.

Definition 3.2. For c_{α}, a multi-indexed sequence in B, a = (a_{1}, …, a_{n}) ∈ 𝕂^{n}, and x = (x_{1}, …, x_{n}) ∈ 𝕂^{n}, denote a multi-indexed sequence in B:

c_{α}(x − a)^{α} = c_{α} · (x_{1} − a_{1})^{α_{1}} · (x_{2} − a_{2})^{α_{2}} · . . . · (x_{n} − a_{n})^{α_{n}}.

If it forms a convergent multi-indexed series, call its sum, ∑_{α∈𝕎^{n}} c_{α}(x − a)^{α}, a

“convergent (B-valued) power series.” Given c_{α}, and a, call the set

{x : ∑_{α∈𝕎^{n}} c_{α}(x − a)^{α} is a convergent power series} ⊆ 𝕂^{n}

the “set of convergence of the power series with coefficients c_{α} and center a.” Such a set always contains a. Its (possibly empty) interior is the “domain of convergence.” If S is any subset of the set of convergence, we will say “the power series ∑ c_{α}(x − a)^{α} converges for x ∈ S.”

Theorem 3.3. If c_{α} is a multi-indexed sequence in B, and a, y ∈ 𝕂^{n}, and {c_{α}(y_{1} −a_{1})^{α_{1}} ·. . .·(y_{n} −a_{n})^{α_{n}} : α ∈ 𝕎^{n}} is a bounded set in B, then ∑ c_{α}(x− a)^{α}, ∑ ∥c_{α}∥(x − a)^{α}, and ∑ ∥c_{α}(x − a)^{α}∥ all converge for x ∈ Δ(a, (|y_{1} − a_{1}|, …, |y_{n} − a_{n}|)).

Proof. By definition of “bounded,” there’s some M ∈ ℝ so that for all α,

∥c_{α}(y_{1} − a_{1})^{α_{1}} · . . . · (y_{n} − a_{n})^{α_{n}}∥ = ∥c_{α}∥ · |y_{1} − a_{1}|^{α_{1}} · . . . · |y_{n} − a_{n}|^{α_{n}} ≤ M.

If x ∈ Δ(a, (|y_{1} − a_{1}|, …, |y_{n} − a_{n}|)), then

∥c_{α}(x − a)^{α}∥ = |∥c_{α}∥(x − a)^{α}|
= ∥c_{α}∥ · |x_{1} − a_{1}|^{α_{1}} · . . . · |x_{n} − a_{n}|^{α_{n}}
≤ M · |\frac{x_{1} − a_{1}}{y_{1} − a_{1}} |^{α_{1}} · . . . · |\frac{x_{n} − a_{n}}{y_{n} − a_{n}} |^{α_{n}} ,

so ∑ c_{α}(x−a)^{α}, ∑ ∥c_{α}∥(x−a)^{α}, and ∑ ∥c_{α}(x−a)^{α}∥ converge by comparison to the geometric series.

Corollary 3.4. Given c_{α}, a, and y, if ∑ c_{α}(y − a)^{α} is a convergent power series, then the polydisc Δ(a, (|y_{1} − a_{1}|, …, |y_{n} − a_{n}|)) is a subset of the set of convergence of the power series with coefficients c_{α} and center a. The same polydisc is also a subset of the set of convergence of the power series with coefficients ∥c_{α}∥ and center a. There exists a constant M such that for all x ∈ Δ(a, (|y_{1} − a_{1}|, …, |y_{n} − a_{n}|)), the sum ∑ c_{α}(x − a)^{α} satisfies

∥∑ c_{α}(x − a)^{α}∥ ≤ ∑ ∥c_{α}(x − a)^{α}∥ ≤ M ∏_{i=1}^{n} \frac{1}{1 − \frac{|x_{i}−a_{i}|}{|y_{i}−a_{i}|}} .

Similarly,

|∑ ∥c_{α}∥(x − a)^{α}| ≤ ∑ ∥c_{α}(x − a)^{α}∥ ≤ M ∏_{i=1}^{n} \frac{1}{1 − \frac{|x_{i}−a_{i}|}{|y_{i}−a_{i}|}} .

Proof. The boundedness of the terms follows immediately from the definition of convergent series. The estimates follow from Theorems 1.11 and 2.2.

Notation 3.5. For a multi-index α = (α_{1}, …, α_{n}) ∈ 𝕎^{n}, we’ll use a “prime” to denote α′ = (α_{1}, …, α_{n−1}), and then denote α = (α′, α_{n}). Similarly for vectors y = (y_{1}, …, y_{n}) ∈ 𝕂^{n}, let y′ = (y_{1}, …, y_{n−1}) and y = (y′, y_{n}).

Theorem 3.6. Given n ≥ 2, a multi-indexed sequence c in B, a sequence b : 𝕎 → 𝕂, and y ∈ 𝕂^{n}, if

∑_{α′∈𝕎^{n−1}} ∥c_{(α′,α_{n})}(y′)^{α′} ∥

forms a convergent multi-indexed series for each α_{n} ∈ 𝕎, and

{( ∑_{α′∈𝕎^{n−1}} ∥c_{(α′,α_{n})}(y′)^{α′} ∥) · b_{α_{n}} · y_{n}^{α_{n}} : α_{n} ∈ 𝕎}

is a bounded subset of 𝕂, then, for all x ∈ Δ(0, (|y_{1}|, …, |y_{n}|)),

∑_{α_{n}∈𝕎} ( ∑_{α′∈𝕎^{n−1}} c_{(α′,α_{n})}(x′)^{α′}) · b_{α_{n}} · x_{n}^{α_{n}}

and

∑_{α∈𝕎^{n}} c_{α} · b_{α_{n}} · x^{α}

are both convergent, with the same sum.

Proof.

x ∈ Δ(0, (|y_{1}|, …, |y_{n}|)) ⟹ ∥c_{(α′,α_{n})}(x′)^{α′} ∥ ≤ ∥c_{(α′,α_{n})}(y′)^{α′} ∥ ,

so ∑_{α′∈𝕎^{n−1}} c_{(α′,α_{n})}(x′)^{α′} and ∑_{α′∈𝕎^{n−1}} ∥c_{(α′,α_{n})}(x′)^{α′} ∥ converge by comparison

(Theorem 1.11), and

∑_{α′∈𝕎^{n−1}} ∥c_{(α′,α_{n})}(x′)^{α′} ∥ ≤ ∑_{α′∈𝕎^{n−1}} ∥c_{(α′,α_{n})}(y′)^{α′} ∥ ⟹

|( ∑_{α′∈𝕎^{n−1}} ∥c_{(α′,α_{n})}(x′)^{α′} ∥)b_{α_{n}}y_{n}^{α_{n}}| ≤ |( ∑_{α′∈𝕎^{n−1}} ∥c_{(α′,α_{n})}(y′)^{α′} ∥)b_{α_{n}}y_{n}^{α_{n}}| .

By hypothesis, the RHS is bounded by M ≥ 0, so

|( ∑_{α′∈𝕎^{n−1}} ∥c_{(α′,α_{n})}(x′)^{α′} ∥) b_{α_{n}}x_{n}^{α_{n}} | ≤ M |\frac{x_{n}}{y_{n}} |^{α_{n}}

(assuming y_{n} ≠ 0, since otherwise the Theorem is trivial). The convergence of the first claimed sum from the Theorem follows from comparison with the single-variable geometric series.

The convergence of

( ∑_{α′∈𝕎^{n−1}} ∥c_{(α′,α_{n})}(x′)^{α′} ∥) · |b_{α_{n}}x_{n}^{α_{n}}| = ∑_{α′∈𝕎^{n−1}} ∥c_{α}b_{α_{n}}x^{α}∥

for each α_{n}, and the convergence of

∑_{α_{n}∈𝕎} ( ∑_{α′∈𝕎^{n−1}} ∥c_{α}b_{α_{n}}x^{α}∥)

are enough, by Theorem 1.13, to establish the convergence of ∑_{α} c_{α}b_{α_{n}}x^{α}, and

the claimed equality.

Notation 3.7. For any α ∈ 𝕎^{n}, there exists a multi-indexed sequence in ℝ,

𝕎^{n} → ℝ : β ↦ ( \begin{matrix}α \\ β \end{matrix} ) ,

with these properties:

• ( \begin{matrix}α \\ β \end{matrix} ) ≥ 0,

• If for some i, β_{i} > α_{i}, then ( \begin{matrix}α \\ β \end{matrix} ) = 0; otherwise, if β_{i} ≤ α_{i} for all

i = 1, …, n, denote this property of β by “β ≤ α.”

• For any x, y ∈ 𝕂^{n}, (x + y)^{α} = ∑_{β≤α} ( \begin{matrix}α \\ β \end{matrix} ) x^{β}y^{α−β}.

We won’t need any exact values for ( \begin{matrix}α \\ β \end{matrix} ) until Section 5. It will sometimes

be convenient to write

∑_{β≤α} ( \begin{matrix}α \\ β \end{matrix} ) x^{β}y^{α−β} = ∑_{β∈𝕎^{n}} ( \begin{matrix}α \\ β \end{matrix} ) x^{β}y^{α−β},

with the understanding that all terms where “β ≤ α” is false are zero, even though negative exponents formally appear.

Theorem 3.8. Suppose Δ(0, r) is a subset of the set of convergence of a power series with coefficients c_{α} and center 0, and a ∈ Δ(0, r). Then, there is a multiindexed sequence in B, c′_{α}, so that for all x ∈ Δ(a, (r_{1} − |a_{1}|, …, r_{n} − |a_{n}|)), ∑ c′_{α}(x − a)^{α} is a convergent power series, and

∑ c′_{α}(x − a)^{α} = ∑ c_{α}x^{α}.

Proof. (Step 1, establishing convergence of a multi-indexed series.) Given any x ∈ Δ(a, (r_{1} − |a_{1}|, …, r_{n} − |a_{n}|)),

|x_{i}| ≤ |x_{i} − a_{i}| + |a_{i}| < (r_{i} − |a_{i}|) + |a_{i}| = r_{i}

implies both x and (|x_{1} − a_{1}| + |a_{1}|, …, |x_{n} − a_{n}| + |a_{n}|) are elements of Δ(0, r), so Δ(a, (r_{1} − |a_{1}|, …, r_{n} − |a_{n}|)) ⊆ Δ(0, r), the RHS of the claimed equation is a convergent power series, and ∑ c_{α}(|x_{1} − a_{1}| + |a_{1}|, …, |x_{n} − a_{n}| + |a_{n}|)^{α} is also a convergent power series. By definition, there is some upper bound U(x) for the partial sums:

∑_{finite} ∥c_{α} · (|x_{1} − a_{1}| + |a_{1}|)^{α_{1}} · . . . · (|x_{n} − a_{n}| + |a_{n}|)^{α_{n}}∥ ≤ U(x).

For α, β ∈ 𝕎^{n}, let (α, β) denote the element (α_{1}, …, α_{n}, β_{1}, …, β_{n}) ∈ 𝕎^{2n}. Define a multi-indexed sequence

𝕎^{2n} → B : (α, β) ↦ c_{α} · ( \begin{matrix}α \\ β \end{matrix} ) (x − a)^{β}a^{α−β}.

It forms a convergent multi-indexed series: let Λ be a finite subset of 𝕎^{2n}, and N = max{|α| : (α, β) ∈ Λ}. Then

∑_{(α,β)∈Λ} ∥c_{α} ( \begin{matrix}α \\ β \end{matrix} ) (x − a)^{β}a^{α−β}∥

≤ ∑_{|α|≤N}( ∑_{β≤α} ∥c_{α}∥ ( \begin{matrix}α \\ β \end{matrix} ) |x_{1} − a_{1}|^{β_{1}} ⋯ |x_{n} − a_{n}|^{β_{n}}|a_{1}|^{α_{1}−β_{1}} ⋯ |a_{n}|^{α_{n}−β_{n}})

= ∑_{|α|≤N} ∥c_{α}∥ · (|x_{1} − a_{1}| + |a_{1}|)^{α_{1}} · . . . · (|x_{n} − a_{n}| + |a_{n}|)^{α_{n}} ≤ U(x).

(Step 2., establishing the claimed equality.) Define, as in Theorem 1.12, a map

σ_{1} : 𝕎^{n} → 2^{𝕎^{2n}} : α ↦ {(α, β) : β ∈ 𝕎^{n}}.

It, and the multi-indexed series from Step 1, satisfy the hypotheses of that Theorem, so

∑_{(α,β)∈𝕎^{2n}} c_{α} · ( \begin{matrix}α \\ β \end{matrix} ) (x − a)^{β}a^{α−β}

= ∑_{α∈𝕎^{n}} ( ∑_{(α,β)∈σ_{1}(α)} c_{α} · ( \begin{matrix}α \\ β \end{matrix} ) (x − a)^{β}a^{α−β})

= ∑_{α∈𝕎^{n}} (c_{α} · ( ∑_{β∈𝕎^{n}} ( \begin{matrix}α \\ β \end{matrix} ) (x − a)^{β}a^{α−β}))

= ∑_{α∈𝕎^{n}} c_{α}x^{α}.

The Theorem also applies to another map

σ_{2} : 𝕎^{n} → 2^{𝕎^{2n}} : β ↦ {(α, β) : α ∈ 𝕎^{n}},

to give

∑_{(α,β)∈𝕎^{2n}} c_{α} · ( \begin{matrix}α \\ β \end{matrix} ) (x − a)^{β}a^{α−β}

= ∑_{β∈𝕎^{n}} ( ∑_{(α,β)∈σ_{2}(β)} c_{α} · ( \begin{matrix}α \\ β \end{matrix} ) (x − a)^{β}a^{α−β})

= ∑_{β∈𝕎^{n}} (( ∑_{α∈𝕎^{n}} c_{α} · ( \begin{matrix}α \\ β \end{matrix} ) a^{α−β}) (x − a)^{β}) .

Technically, the last expression follows from the previous one only for the terms where (x − a)^{β} ≠ 0. Since Δ(a, (r_{1} − |a_{1}|, …, r_{n} − |a_{n}|)) is non-empty, it has some element x so that (x − a)^{β} ≠ 0 for all β, and we can use this to establish the convergence of

∑_{α∈𝕎^{n}} c_{α} · ( \begin{matrix}α \\ β \end{matrix} ) a^{α−β},

which defines c′_{β} not depending on x.

4 Geometry of the ball

Definition 4.1. A “positive semidefinite Hermitian form” on 𝕂^{n} is a function g : 𝕂^{n} × 𝕂^{n} → 𝕂 such that:

• (homogeneity) For all x, y ∈ 𝕂^{n}, λ ∈ 𝕂, g(λ · x, y) = λg(x, y).

• (additivity) For all x, y, z ∈ 𝕂^{n}, g(x + y, z) = g(x, z) + g(y, z).

• (Hermitian symmetry) For all x, y ∈ 𝕂^{n}, g(x, y) = \overline{g(y, x)}. (so, for any x ∈ 𝕂^{n}, g(x, x) ∈ ℝ.)

• (positivity) For all x ∈ 𝕂^{n}, g(x, x) ≥ 0.

Lemma 4.2 (CBS). Given a positive semidefinite Hermitian form g, for any x, y ∈ 𝕂^{n},

|g(x, y)|^{2} ≤ g(x, x)g(y, y).

Proof. For any λ, μ ∈ 𝕂,

0 ≤ g(λ · x + μ · y, λ · x + μ · y)
= λλ̄g(x, x) + μλ̄g(y, x) + λμ̄g(x, y) + μμ̄g(y, y).

In particular, for λ = g(y, y) and μ = −g(x, y),

0 ≤ λλ̄g(x, x) + μλ̄(−μ̄) + λμ̄(−μ) + μμ̄λ
= λ̄(g(x, x)g(y, y) − |g(x, y)|^{2}),

and if g(y, y) ≠ 0, this proves the claim. Similarly, for λ = −g(y, x) and μ = g(x, x),

0 ≤ λλ̄μ + μλ̄(−λ) + λμ̄(−λ̄) + μμ̄g(y, y)
= μ̄(g(x, x)g(y, y) − |g(y, x)|^{2}),

and if g(x, x) ≠ 0, this proves the claim. Finally, if g(x, x) = g(y, y) = 0, let λ = 1 and μ = −g(x, y), so

0 ≤ 0 − g(x, y)g(y, x) − g(y, x)g(x, y) + 0
= −2|g(x, y)|^{2},

proving g(x, y) = 0, and the claim.

Lemma 4.3 (Δ ≠). Given a positive semidefinite Hermitian form g, the function

𝕂^{n} → ℝ : x ↦ ∥x∥_{g} = +\sqrt{g(x, x)}

satisfies, for all x, y ∈ 𝕂^{n},

∥x + y∥_{g} ≤ ∥x∥_{g} + ∥y∥_{g}.

Proof.

∥x + y∥_{g}^{2} = g(x + y, x + y)
= |g(x, x) + g(y, x) + g(x, y) + g(y, y)|
≤ g(x, x) + g(y, y) + 2|g(x, y)|
≤ g(x, x) + g(y, y) + 2\sqrt{g(x, x)g(y, y)}
= (∥x∥_{g} + ∥y∥_{g})^{2},

using the previous Lemma.

Definition 4.4. For i = 1, …, n, denote the “reflections in the coordinate hyperplanes”

R_{i} : (x_{1}, …, x_{i}, …, x_{n}) ↦ (x_{1}, …, −x_{i}, …, x_{n}).

A positive semidefinite Hermitian form g is in “standard position” if all of the reflections satisfy the “isometry” equation: for all x, y ∈ 𝕂^{n},

g(R_{i}(x), R_{i}(y)) = g(x, y).

Lemma 4.5. If g is in standard position, then it is of the form

g(x, y) = ∑_{i=1}^{n} g_{i}x_{i}ȳ_{i},

for nonnegative real constants g_{1}, …, g_{n}.

Proof. First, any Hermitian form can be expressed in terms of a matrix, with respect to the usual basis of row vectors {e^{i} = (0, …, 0, 1, 0, …, 0)}. For x = ∑ x_{i}e^{i} and y = ∑ y_{j}e^{i}, the linearity properties give

g(x, y) = ∑_{i=1}^{n} x_{i} (∑_{j=1}^{n} ȳ_{j}g(e^{i}, e^{j})) = xG\overline{y}^{T} .

The “standard position” hypothesis, applied to the basis vectors, gives, for j ≠ i,

g(e^{i}, e^{j}) = g(R_{i}(e^{i}), R_{i}(e^{j})) = g(−e^{i}, e^{j}) = −g(e^{i}, e^{j}),

so G is a diagonal matrix, with diagonal entries g_{i} = g(e^{i}, e^{i}) ≥ 0.

Notation 4.6. For a positive semidefinite Hermitian form g, denote the “ball with center a ∈ 𝕂^{n} and radius R ∈ ℝ” by

B_{g}(a, R) = {(x_{1}, …, x_{n}) : ∥(x_{1} − a_{1}, …, x_{n} − a_{n})∥_{g} < R} ⊆ 𝕂^{n}.

Geometrically, this shape will be the interior of an ellipsoid (if g is positive definite), or of an ellipsoidal cylinder (if degenerate), or all of 𝕂^{n} (if g = 0).

Lemma 4.7. If g is in standard position, then any ball B_{g}(a, R) is a union of polydiscs with center a.

Proof. Given x ∈ B_{g}(a, R), pick any constant ρ such that ∥x − a∥_{g}^{2} < ρ^{2} < R^{2}.

Then, pick any δ_{1}, …, δ_{n} > 0 so that ∑_{i=1}^{n} g_{i}δ_{i}^{2} < R^{2} − ρ^{2}. Define r by

r_{i} = \{ \begin{matrix}\frac{|x_{i}−a_{i}|}{∥x−a∥_{g}} · ρ & if x_{i} − a_{i} ≠ 0 \\ δ_{i} & if x_{i} − a_{i} = 0. \end{matrix}

Then x ∈ Δ(a, r), and a + r ∈ B_{g}(a, R):

∑_{i=1}^{n} g_{i}|a_{i} + r_{i} − a_{i}|^{2} = ∑_{i=1}^{n} g_{i}r_{i}^{2}

≤ ∑_{i=1}^{n} g_{i}δ_{i}^{2} + ∑_{i=1}^{n} g_{i} (\frac{|x_{i} − a_{i}|}{∥x − a∥_{g}} · ρ)^{2}

≤ ∑_{i=1}^{n} g_{i}δ_{i}^{2} + ρ^{2} < R^{2}.

For any element y ∈ Δ(a, r),

∥y − a∥_{g}^{2} = ∑_{i=1}^{n} g_{i}|y_{i} − a_{i}|^{2} ≤ ∑_{i=1}^{n} g_{i}r_{i}^{2} < R^{2}.

So, for any x ∈ B_{g}(a, R), there is a polydisc such that x ∈ Δ(a, r) ⊆ B_{g}(a, R).

Theorem 4.8. Given c, a multi-indexed sequence in B, a complex Banach space, and a vector a ∈ ℝ^{n}, if g is in standard position and ∑ c_{α}(x − a)^{α} converges for all x in a real ball,

{x ∈ ℝ^{n} : ∑_{i=1}^{n} g_{i}(x_{i} − a_{i})^{2} < R^{2}} = B_{g}(a, R) ∩ ℝ^{n},

then ∑ c_{α}(z − a)^{α} and ∑ ∥c_{α}∥(z − a)^{α} converge on the complex ball with the same radius,

B_{g}(a, R) = {z ∈ ℂ^{n} : ∑_{i=1}^{n} g_{i}|z_{i} − a_{i}|^{2} < R^{2}}.

Proof. Given any complex vector z ∈ B_{g}(a, R), the real vector (|z_{1} − a_{1}| + a_{1}, …, |z_{n} − a_{n}| + a_{n}) is an element of B_{g}(a, R) ∩ ℝ^{n}. From the Proof of the previous Lemma, there is some r such that a+r ∈ B_{g}(a, R)∩ℝ^{n} and (|z_{1} −a_{1}|+ a_{1}, …, |z_{n} − a_{n}| + a_{n}) ∈ Δ(a, r). It follows that z is in the complex polydisc Δ(a, r). By hypothesis, ∑ c_{α}(a + r − a)^{α} is convergent, and by Corollary 3.4, ∑ c_{α}(z − a)^{α} and ∑ ∥c_{α}∥(z − a)^{α} are also convergent.

Theorem 4.9. If g is in standard position and ∑ c_{α}x^{α} converges on B_{g}(0, R), and a ∈ B_{g}(0, R), then there is some multi-indexed sequence c′_{α} so that for all x ∈ B_{g}(a, R−∥a∥_{g}), ∑ c′_{α}(x−a)^{α} is a convergent power series, with sum equal to ∑ c_{α}x^{α}.

Proof. By Lemma 4.3, B_{g}(a, R − ∥a∥_{g}) ⊆ B_{g}(0, R). Given x ∈ B_{g}(a, R − ∥a∥_{g}), there is, by the construction of the previous Lemma, some r ∈ ℝ^{n} such that ∥r∥_{g} < R − ∥a∥_{g} and x ∈ Δ(a, r). The claim is that

Δ(a, r) ⊆ Δ(0, (|a_{1}| + r_{1}, …, |a_{n}| + r_{n})) ⊆ B_{g}(0, R).

For the first subset, suppose y ∈ Δ(a, r). Then

|y_{i}| ≤ |y_{i} − a_{i}| + |a_{i}| < r_{i} + |a_{i}|.

For the second subset, suppose y ∈ Δ(0, (|a_{1}| + r_{1}, …, |a_{n}| + r_{n})). Then, using the “standard position” hypothesis, and Lemmas 4.5 and 4.2 (CBS),

∥y∥_{g}^{2} = ∑_{i=1}^{n} g_{i}|y_{i}|^{2}

< ∑_{i=1}^{n} g_{i}(|a_{i}| + r_{i})^{2}
= ∥a∥_{g}^{2} + ∥r∥_{g}^{2} + 2g((|a_{1}|, …, |a_{n}|), r)
≤ (∥a∥_{g} + ∥r∥_{g})^{2} < R^{2}.

The Theorem follows from the claimed inclusion: since ∑ c_{α}x^{α} converges on Δ(0, (|a_{1}|+r_{1}, …, |a_{n}|+r_{n})), there exist coefficients c′_{α}, defining a power series ∑ c′_{α}(x−a)^{α} which converges to ∑ c_{α}x^{α} on Δ(a, r), by Theorem 3.8. From the Proof of that Theorem, these coefficients c′_{α} do not depend on x or the choice of r, so B_{g}(a, R − ∥a∥_{g}) is a subset of the set of convergence of ∑ c′_{α}(x − a)^{α}.

5 Functions defined by power series

Theorem 5.1. If ∑ c_{α}x^{α} converges on some polydisc Δ(0, r), then the function

f : Δ(0, r) → B : x ↦ f(x) = ∑ c_{α}x^{α}

is continuous at a for all a ∈ Δ(0, r).

Proof. “Continuity at the point a” means that for any ϵ > 0, there are positive numbers δ_{i}, i = 1, …, n, so that if x ∈ Δ(a, (δ_{1}, …, δ_{n})), then ∥f(x)−f(a)∥ < ϵ.

(Step 1, showing continuity at 0.) Fix some w ∈ Δ(0, r), such that w_{i} > 0 for i = 1, …, n. Theorem 1.12 applies to the series ∑ c_{α}w^{α} and the map

σ : 𝕎^{1} → 2^{𝕎^{n}} : \{ \begin{matrix}(0) & ↦ & \{0\} \\ (1) & ↦ & \{α : α_{1} > 0\} \\ (i) & ↦ & \{α : α_{1} = … = α_{i−1} = 0, α_{i} > 0\} & if 2 ≤ i ≤ n \\ (j) & ↦ & Ø & if j > n \end{matrix}

to give

∑_{α∈𝕎^{n}} c_{α}w^{α} = c_{0} + ∑_{i=1}^{n} ( ∑_{α∈σ(i)} c_{α}w^{α})

= c_{0} + ∑_{i=1}^{n} w_{i} ( ∑_{α∈σ(i)} c_{α}w_{i}^{α_{i}−1}w_{i+1}^{α_{i+1}} ⋯ w_{n}^{α_{n}} ) .

For each i = 1, …, n, Corollary 3.4 applies to the convergent power series

∑_{α∈σ(i)} c_{α}w_{i}^{α_{i}−1}w_{i+1}^{α_{i+1}} ⋯ w_{n}^{α_{n}},

so there’s some M_{i} > 0 so that for all x ∈ Δ(0, w),

∥ ∑_{α∈σ(i)} c_{α}x_{i}^{α_{i}−1}x_{i+1}^{α_{i+1}} ⋯ x_{n}^{α_{n}} ∥ ≤ M_{i} ∏_{i=1}^{n} \frac{1}{1 − \frac{|x_{i}|}{w_{i}}} .

Multiplying both sides by |x_{i}| gives

∥ ∑_{α∈σ(i)} c_{α}x^{α}∥ ≤ |x_{i}|M_{i} ∏_{i=1}^{n} \frac{1}{1 − \frac{|x_{i}|}{w_{i}}} .

So, given ϵ > 0, let δ_{i} = min\{\frac{ϵ}{n2^{n}M_{i}} , \frac{w_{1}}{2} , …, \frac{w_{n}}{2} \}. Then,

|x_{i}| < δ_{i} ⟹ 1 − \frac{|x_{i}|}{w_{i}} > \frac{1}{2} ⟹ ∏_{i=1}^{n} \frac{1}{1 − \frac{|x_{i}|}{w_{i}}} < 2^{n},

and

∥f(x) − f(0)∥ = ∥f(x) − c_{0}∥ = ∥∑_{i=1}^{n} ( ∑_{α∈σ(i)} c_{α}x^{α})∥

≤ ∑_{i=1}^{n} (|x_{i}|M_{i} ∏_{i=1}^{n} \frac{1}{1 − \frac{|x_{i}|}{w_{i}}} ) < ϵ.

(Step 2, showing continuity everywhere else.) By Theorem 3.8, for any point a ∈ Δ(0, r), there are coefficients c′_{α}, and a polydisc with center a, so that for x in that polydisc, ∑ c′_{α}(x − a)^{α} converges, with sum f(x). By the construction from the Proof of that Theorem, and the fact that the multinomial coefficient

( \begin{matrix}α \\ 0 \end{matrix} ) has value 1 for all α,

c′_{0} = ∑_{α∈𝕎^{n}} c_{α} · ( \begin{matrix}α \\ 0 \end{matrix} ) a^{α} = f(a).

So, Step 1 applies to show

lim_{x→a} f(x) = lim_{x−a→0} ∑ c′_{α}(x − a)^{α} = c′_{0} = f(a).

The following Theorem is for single-indexed series, with coefficients c : 𝕎 → B, but Step 2 uses the methods of multi-indexed series (Theorem 3.8).

Theorem 5.2. If ∑_{k=0}^{∞} c_{k}z^{k} converges on some disc {z : |z| < r} ⊆ 𝕂^{1}, then the

(B-valued) function f(z) = ∑_{k=0}^{∞} c_{k}z^{k} is differentiable at a for all a in the disc,

with f′(a) = ∑_{k=1}^{∞} c_{k} · ka^{k−1}.

Proof. “Differentiability at the point a” means that there’s an element f′(a) ∈ B so that for any ϵ > 0, there is a δ > 0 so that if 0 < |z − a| < δ, then ∥\frac{f(z)−f(a)}{z−a} − f′(a)∥ < ϵ.

(Step 1, showing differentiability at 0.) Fix w ∈ 𝕂 with 0 < |w| < r, so

\frac{f(w) − f(0)}{w − 0} − c_{1} = \frac{c_{0} + c_{1}w + (∑_{k=2}^{∞} c_{k}w^{k}) − c_{0}}{w} − c_{1} = w ∑_{k=2}^{∞} c_{k}w^{k−1}.

Just as in the Proof of the previous Theorem, Corollary 3.4 applies to the

convergent power series ∑_{k=2}^{∞} c_{k}w^{k−1}, giving some M so that if |z| < |w|, then

∥\frac{f(z) − f(0)}{z − 0} − c_{1}∥ ≤ |z|M \frac{1}{1 − \frac{|z|}{|w|}} ,

and this can be made less than any ϵ > 0 by choosing δ = min\{\frac{ϵ}{2M} , \frac{|w|}{2}\}.

(Step 2, showing differentiability everywhere else.) By Theorem 3.8, for any point a such that |a| < r, there are coefficients c′_{k}, and a disc with center a,

so that for z in that disc, ∑_{k=0}^{∞} c′_{k}(z − a)^{k} converges, with sum f(z). By the

construction from the Proof of that Theorem, and the fact that the binomial

coefficient ( \begin{matrix}(k) \\ (1) \end{matrix} ) = ( \begin{matrix}k \\ 1 \end{matrix} ) has value k for all k ≥ 1 (and in particular, value

0 for k = 0),

c′_{1} = ∑_{k=0}^{∞} c_{k} · ( \begin{matrix}k \\ 1 \end{matrix} ) a^{k−1} = ∑_{k=1}^{∞} c_{k} · ka^{k−1}.

So, Step 1 applies to show

lim_{z→a} \frac{f(z) − f(a)}{z − a} = lim_{z−a→0} \frac{(∑_{k=0}^{∞} c′_{k}(z − a)^{k}) − c′_{0}}{z − a} = c′_{1} = f′(a).

[C] gives a proof that ∑_{k=0}^{∞} c_{k}z^{k} and ∑_{k=1}^{∞} c_{k} · kz^{k−1} have the same radius of

convergence.

References

[C] A. Coffman, Notes on sequences and series in the calculus of one variable,

     unpublished course notes. http://users.pfw.edu/CoffmanA/

[D] J. Dieudonné, Foundations of Modern Analysis (Vol. I of Treatise on Analysis), Pure and Applied Mathematics 10, Academic Press, New York, 1969.

[GF] H. Grauert and K. Fritzsche, Several Complex Variables, GTM 38, Springer-Verlag, New York, 1976.