“Hardy spaces and holomorphic functions of infinitely many variables”

H a r d y s p a c e s a n d h o l o m o r p h i c f u n c t i o n s o f i n f i n i t e l y m a n y v a r i a b l e s

J o n a t h a n C h i r i n o s

Universidad de La Laguna [email protected]

E k i G o n z á l e z

Universidad del País Vasco/Euskal Herriko Unibersiatea [email protected]

N e r e a G o n z á l e z

Universidade de Santiago de Compostela [email protected]

� A n t o n i L ó p e z - M a r t í n e z

Universitat de València [email protected]

H é c t o r M é n d e z

Universidad de Costa Rica [email protected]

� A l i c i a Q u e r o

Universidad de Granada [email protected]

C o u r s e i n s t r u c t o r

P a b l o S e v i l l a - P e r i s

Universitat Politècnica de València [email protected]

A b s t r a c t : We take a classical result on the interplay between complex and Fourier

analysis in one variable (that the space of bounded holomorphic functions on the unit disc and the Hardy space on the torus are isomorphic), and extend it to functions in infinitely many variables. This will present several difficulties that we sort out.

R e s u m e n : Tomamos un resultado clásico sobre la interacción del análisis complejo

y de Fourier de una variable (que el espacio de funciones holomorfas y acotadas en el disco y el espacio de Hardy en el toro son isomorfos) y lo extendemos a funciones en infinitas variables. Esto presenta varias dificultades que deberemos sortear.

K e y w o r d s : holomorphic function, Fourier analysis, Banach space, isometry.

M S C 2 0 1 0 : 46G20, 30H10, 42A16.

A c k n o w l e d g e m e n t s : Queremos expresar nuestro agradecimiento a la Red de Análisis Funcional, por la oportunidad

que nos dio de participar en la Escuela Taller que se desarrolló en Bilbao en marzo de 2019 y por el apoyo que recibimos.

R e f e r e n c e : CHIRINOS, Jonathan; GONZÁLEZ, Eki; GONZÁLEZ, Nerea; LÓPEZ-MARTÍNEZ, Antoni; MÉNDEZ, Héctor;

QUERO, Alicia, and SEVILLA-PERIS, Pablo. “Hardy spaces and holomorphic functions of infinitely many variables”. In:TEMat monográficos, 1 (2020):Artículos de las VIII y IX Escuela-Taller de Análisis Funcional, pp. 113-129. ISSN: 2660-6003. URL:https://temat.es/monograficos/article/view/vol1-p113.

1 . I n t r o d u c t i o n

Let𝔻= {𝑧∈ℂ∶ |𝑧| <1}be the open unit disc and𝕋= {𝑧∈ℂ∶ |𝑧| =1}its boundary (that we call the torus). We denote by𝐻∞(𝔻)the space of bounded holomorphic functions on the disc and by𝐻∞(𝕋)the Hardy space on the torus (precise definitions are given in section2). A classical result in analysis states that

𝐻∞(𝔻) = 𝐻∞(𝕋);

that is: both spaces are isometrically isomorphic as Banach spaces. Our aim in this note is to present an analogous result for functions of infinitely many variables (or, to be more precise, defined on subsets of infinite dimensional spaces, see theorem43). This was done for the first time by Cole and Gamelin [4]. We follow here a different approach, based in the one given by Defant et al. [5] using results from Rudin [6].

We do it in several steps. First we are going to analyse the proof of the1-dimensional case, so that we can transfer it to functions of several complex variables and, finally to infinite dimensional spaces. In order to achieve this goal we have to face several issues: to find proper analogues to𝔻and𝕋for several and infinitely many variables, find a good definition of holomorphy in this setting, and to find a device that allows to extend the results from the finite to the infinite dimensional case. We assume some knowledge of the basic concepts of complex, harmonic and functional analysis.

2 . T h e

1

As we explained before, we are going to look at the interplay between complex and harmonic analysis, and all the time we will keep one foot in each side. We begin by defining the spaces we will be dealing with. First of all, the space of holomorphic functions on𝔻is denoted by𝐻(𝔻). We consider the following subspace.

D e f i n i t i o n 1. We define the space𝐻_∞(𝔻) = {𝑓 ∶𝔻⟶ℂ∶ 𝑓is bounded and holomorphic}. ◀

T h e o r e m 2. 𝐻_∞(𝔻)with the norm

‖𝑓‖_∞=sup

𝑧∈𝔻

|𝑓(𝑧)|

is a Banach space.

P r o o f. Let{𝑓_𝑛}^∞_𝑛=0be a Cauchy sequence in𝐻_∞(𝔻). The space𝐶_∞(𝔻)of bounded continuous functions

on the disc (that obviously contains𝐻_∞(𝔻)) is Banach (see, e.g., Cerdà’s book [2, Chapter 2]). Then, the sequence converges uniformly on𝔻to some bounded continuous𝑓 ∶𝔻→ℂ. But then{𝑓_𝑛}^∞_𝑛=0converges uniformly on the compact subsets of𝔻to the function𝑓, and a straightforward application of Morera’s theorem (see Stein and Shakarchi’s book [8, Theorem 5.2]) shows that𝑓is holomorphic. ▪

R e m a r k 3. If𝑈is an open subset ofℂ, then a function𝑓 ∶ 𝑈 →ℂis analytic on𝑈if, for every point𝑧₀∈ 𝑈,

there exist𝑟 >0and a sequence{𝑐𝑛}^∞𝑛=0⊂ℂwhich depend on𝑧₀such that 𝑓(𝑧) =

∞

∑

𝑛=0

𝑐_𝑛(𝑧−𝑧₀)^𝑛, for every𝑧∈ (𝑧₀+ 𝑟𝔻) ⊂ 𝑈,

where𝑧₀+ 𝑟𝔻= {𝑧∈ℂ∶ |𝑧−𝑧₀| < 𝑟}. This is,𝑓admits a power series expansion in a neighbourhood of each point𝑧₀ ∈ 𝑈. One of the key results (probably one of the most important ones in complex analysis) is that every holomorphic function is analytic [8, Theorem 4.4]. In our particular case, that is, on the disc 𝔻, it is known that𝑓 ∶𝔻→ℂis holomorphic if and only if there exist coefficients{𝑐𝑛}^∞_𝑛=1⊂ℂsuch that

𝑓(𝑧) =

∞

∑

𝑛=0

𝑐_𝑛𝑧^𝑛,

for every𝑧∈𝔻. This convergence is, moreover, absolute on𝔻and uniform on𝑟𝔻= {𝑧∈ℂ∶ |𝑧| < 𝑟}for

every0< 𝑟 <1. ◀

This is our main object in the side of complex analysis. Let us explore now the side of harmonic analysis.

On𝕋we consider the normalised Lebesgue measure, and the corresponding space𝐿₁(𝕋). This means that, for each𝑓, the integral has to be understood in the following sense:

∫

𝕋

𝑓(𝑤)d𝑤 = 1 2π∫

2π 0

𝑓(e^i𝑡)d𝑡.

If𝑓 ∈ 𝐿₁(𝕋), then𝑤 ∈𝕋↦ 𝑓(𝑤)𝑤^−𝑛is again in𝐿₁(𝕋)for every𝑛 ∈ℤ(because|𝑤^−𝑛| =1). Then, we can define the Fourier coefficients of𝑓in the following way.

D e f i n i t i o n 4. Given𝑓 ∈ 𝐿₁(𝕋)and𝑛 ∈ℤ, the𝑛-th Fourier coefficient is defined as

̂𝑓(𝑛) = ∫

𝕋

𝑓(𝑤)𝑤^−𝑛d𝑤. ◀

Let us note that

(1) | ̂𝑓(𝑛)| ≤ ∫

𝕋

|𝑓(𝑤)𝑤^−𝑛|d𝑤 = ‖𝑓‖₁,

and the operator𝐿₁(𝕋) →ℂdefined by𝑓 ↦ ̂𝑓(𝑛)is continuous. We can now define the second space we are going to be dealing with.

D e f i n i t i o n 5. The Hardy space on the circumference is defined as

𝐻∞(𝕋) = {𝑓 ∈ 𝐿∞(𝕋) ∶ ̂𝑓(𝑛) =0 for𝑛 <0}. ◀

T h e o r e m 6. 𝐻∞(𝕋)is a closed subspace of 𝐿∞(𝕋), hence Banach.

P r o o f. The result follows as a straightforward consequence of the fact that the operator𝑓 ↦ 𝑓(𝑛)̂ is

continuous. ▪

Thus, the goal of this section is to prove that

(2) 𝐻∞(𝕋) = 𝐻∞(𝔻)

as Banach spaces. That is: there is an isometric isomorphism between these two spaces.

R e m a r k 7. Let us give the first step towards the proof. Each𝑓 ∈ 𝐻∞(𝕋)defines a family of Fourier

coefficients{ ̂𝑓(𝑛)}^∞𝑛=0, and we may consider the (in principle only formal) power series given by∑^∞_𝑛=0𝑓(𝑛)𝑧̂ ^𝑛. Note that (recall (1) and the fact that‖𝑓‖₁ ≤ ‖𝑓‖∞)

∞

∑

𝑛=0|| ̂𝑓(𝑛)|| |𝑧^𝑛| ≤ ‖𝑓‖∞

∞

∑

𝑛=0

|𝑧|^𝑛< ∞ ⟺ |𝑧| <1.

Then, the function𝑔∶𝔻→ℂgiven by𝑔(𝑧) = ∑^∞_𝑛=0𝑓(𝑛)𝑧̂ ^𝑛is well defined and, by remark3, is holomorphic.

In other words, the operator𝐻_∞(𝕋) → 𝐻(𝔻)given by𝑓 ↦𝑔(𝑧) = ∑^∞_𝑛=0𝑓(𝑛)𝑧̂ ^𝑛is well defined. It is an easy exercise to check that it is linear and injective. The main problem now is to show that in fact it takes values in𝐻_∞(𝔻)(that is, the function𝑔defined in this way is bounded) and is surjective. ◀

Our first concern is to show that the function defined by the power series is indeed bounded on𝔻. To do this, we will reformulate the function in more convenient terms. We bring now our tool for this purpose.

D e f i n i t i o n 8. The Poisson kernel𝑝 ∶𝔻×𝕋→ℂis defined as

(3) 𝑝(𝑧,𝑤) = ∑

𝑛∈ℤ

𝑤^−𝑛𝑟^|𝑛|𝑢^𝑛,

for𝑧∈𝔻and𝑤 ∈𝕋, where𝑧= 𝑟𝑢with𝑢 =𝑧/|𝑧| ∈𝕋and𝑟 = |𝑧|. ◀

R e m a r k 9. Let us note that, since|𝑤| = |𝑢| =1and0≤ 𝑟 <1, we have

∑

𝑛∈ℤ

|𝑤^−𝑛𝑟^|𝑛|𝑢^𝑛| ≤ ∑

𝑛∈ℤ

𝑟^|𝑛|=1+2∑^∞

𝑛=1

𝑟^𝑛< ∞ .

Hence, the series in (3) converges (even absolutely) for each fixed𝑤 ∈𝕋and𝑧∈𝔻and𝑝is well defined.

Moreover, by the Weierstrass M-test (see Rudin’s book [6, Theorem 7.10]), the series converges uniformly

on𝑟𝔻×𝕋for every0< 𝑟 <1. ◀

P r o p o s i t i o n 1 0. The following statements hold:

1 . 𝑝(𝑧,𝑤) =|𝑤|²− |𝑧|²

|𝑤 −𝑧|² >0for all𝑤 ∈𝕋and𝑧∈𝔻;

2 . ∫

𝕋

𝑝(𝑧,𝑤)d𝑤 =1for every fixed𝑧∈𝔻.

P r o o f.

1 . Let𝑤 ∈𝕋and𝑧∈𝔻. Observe that 𝑝(𝑧,𝑤) = ∑

𝑛∈ℤ

𝑤^−𝑛𝑟^|𝑛|𝑢^𝑛=

∞

∑

𝑛=1

(𝑤𝑟 𝑢 )

𝑛

+

∞

∑

𝑛=0

(𝑟𝑢 𝑤)

𝑛

= 𝑤𝑟

𝑢 − 𝑤𝑟 + 𝑤

𝑤 − 𝑟𝑢 = 𝑤𝑢(1− 𝑟²) 𝑢𝑤 − 𝑟𝑢²− 𝑟𝑤²+ 𝑟²𝑤𝑢

= 1− 𝑟²

1− 𝑟_𝑤^ᵆ − 𝑟^𝑤_ᵆ + 𝑟² = 1− 𝑟² 1− 𝑟𝑢𝑤 − 𝑟𝑢𝑤 + 𝑟²

= |𝑤|²− |𝑧|²

|𝑤|²−𝑧𝑤 −𝑧𝑤 + |𝑧|² = |𝑤|²− |𝑧|²

|𝑤 −𝑧|² >0.

2 . If we fix𝑧∈𝔻, the series in (3) converges uniformly on𝕋. Then, we may change the sum and the integral as follows:

∫

𝕋

𝑝(𝑧,𝑤) = ∫

𝕋

∑

𝑛∈ℤ

𝑤^−𝑛𝑟^|𝑛|𝑢^𝑛d𝑤 = ∑

𝑛∈ℤ

𝑟^|𝑛|𝑢^𝑛∫

𝕋

𝑤^−𝑛d𝑤.

A straightforward computation shows that

▪

(4) ∫

𝕋

𝑤^−𝑛d𝑤 = ∫

2𝜋 0

e^−i𝑛πd𝑤

2π = {1, if𝑛 =0, 0, if𝑛 ≠0.

A direct consequence of proposition10.1is that𝑝(𝑧,𝑤)is bounded for each fixed𝑧∈𝔻. Then, for every 𝑓 ∈ 𝐿₁(𝕋), the function given by𝑤 ↦ 𝑝(𝑧,𝑤)𝑓(𝑤)again belongs to𝐿₁(𝕋)and the function𝑃[𝑓] ∶𝔻→ℂ given by

𝑃[𝑓](𝑧) = ∫

𝕋

𝑝(𝑧,𝑤)𝑓(𝑤)d𝑤

is well defined. In fact, since the series defining𝑝is uniformly convergent on𝕋(for fixed𝑧∈𝔻), then

∫

𝕋

𝑝(𝑧,𝑤)𝑓(𝑤)d𝑤 = ∫

𝕋

( ∑

𝑛∈ℤ

𝑤^−𝑛𝑟^|𝑛|𝑢^𝑛) 𝑓(𝑤)d𝑤

= ∑

𝑛∈ℤ

(∫

𝕋

𝑓(𝑤)𝑤^−𝑛d𝑤) 𝑟^|𝑛|𝑢^𝑛

= ∑

𝑛∈ℤ

̂𝑓(𝑛)𝑟^|𝑛|𝑢^𝑛

(note that this series converges because| ̂𝑓(𝑛)| ≤ ‖𝑓‖,0≤ 𝑟 ≤1and|𝑢| =1). In this way, we may define an operator𝑃(that we callPoisson operator) acting on𝐿₁(𝕋)by doing𝑓 ↦ 𝑃[𝑓].

R e m a r k 1 1. If𝑓 ∈ 𝐻∞(𝕋), then𝑓(𝑛) =̂ 0for𝑛 <0and, for𝑧= 𝑟𝑢 ∈𝔻, we have

∑

𝑛∈ℤ

̂𝑓(𝑛)𝑟^|𝑛|𝑢^𝑛=

∞

∑

𝑛=0

̂𝑓(𝑛)𝑟^|𝑛|𝑢^𝑛=

∞

∑

𝑛=0

̂𝑓(𝑛)𝑟^𝑛𝑢^𝑛=

∞

∑

𝑛=0

̂𝑓(𝑛)𝑧^𝑛.

Then,𝑃(restricted to𝐻∞(𝕋)) is exactly the operator that we already considered in remark7. We have

|𝑃[𝑓](𝑧)| ≤ ∫

𝕋

|𝑝(𝑧,𝑤)||𝑓(𝑤)|d𝑤 ≤ ‖𝑓‖_∞∫

𝕋

|𝑝(𝑧,𝑤)|d𝑤 = ‖𝑓‖_∞

and, hence,

(5) sup

𝑧∈𝔻

|𝑃[𝑓](𝑧)| ≤ ‖𝑓‖∞.

This shows that𝑃[𝑓]is bounded or, in other words,𝑃 ∶ 𝐻_∞(𝕋) → 𝐻_∞(𝔻)is well defined and continuous. ◀

Roughly speaking, what the operator𝑃does is to “extend” functions on𝕋to𝔻. Let us see how this operator acts on some particularly nice functions.

R e m a r k 1 2. We begin by consideringtrigonometric polynomials. There are functions𝑄 ∶𝕋→ℂthat can

be written as

𝑄(𝑤) =

𝑀

∑

𝑛=𝑁

𝑐_𝑛𝑤^𝑛,

where𝑎𝑛∈ℂ,𝑁 < 𝑀 ∈ℤ. First of all, for each𝑛 ∈ℤ, taking (4) we have

̂𝑄(𝑛) = ∫

𝕋 𝑀

∑

𝑘=𝑁

𝑐_𝑘𝑤^𝑘𝑤^−𝑛d𝑤 =

𝑀

∑

𝑘=𝑁

𝑐_𝑘∫

𝕋

𝑤^𝑘−𝑛d𝑤 = {𝑐𝑛 if𝑁 ≤ 𝑛 ≤ 𝑀, 0 otherwise.

Then,

𝑃[𝑄](𝑧) = ∑

𝑛∈ℤ

̂𝑄(𝑛)𝑟^|𝑛|𝑢^𝑛=

𝑀

∑

𝑛=𝑁

𝑐𝑛𝑟^|𝑛|𝑢^𝑛.

This function is continuous on all𝔻and coincides with𝑄on𝕋. ◀

This, in fact, also happens to every continuous function.

P r o p o s i t i o n 1 3. Let𝑓 ∈ 𝐶(𝕋). Then,𝑃[𝑓]extends to a continuous function on𝔻which is equal to𝑓on𝕋.

P r o o f. By the Stone-Weierstrass theorem [6, Chapter 7], there is a sequence of trigonometric polynomials

{𝑄_𝑛}^∞_𝑛=0converging to𝑓with the supremum norm‖ ⋅ ‖_∞. As we already observed in remark12, each𝑃[𝑄_𝑛] is continuous on𝔻. On the other hand, (5) gives

‖𝑃[𝑄𝑛] − 𝑃[𝑄𝑚]‖∞= ‖𝑃[𝑄𝑛− 𝑄𝑚]‖∞≤ ‖𝑄𝑛− 𝑄𝑚‖∞

for every𝑛and𝑚. This implies that{𝑃[𝑄𝑛]}^∞𝑛=0is a Cauchy sequence in𝐶(𝔻)and, hence, converges uniformly to some continuous function𝐹on𝔻. For each𝑤 ∈𝕋we have

𝐹(𝑤) = lim

𝑛→∞𝑃[𝑄𝑛](𝑤) = lim

𝑛→∞𝑄𝑛(𝑤) = 𝑓(𝑤). ▪

One would expect that, whenever a function is defined on𝔻and is restricted to𝕋, then “extending” it to 𝔻with𝑃would give us the original function. We have that, at least if the function is holomorphic on𝔻 and continuous on𝔻, then this is the case. This shows that, when restricted to this space, the operator𝑃is surjective.

P r o p o s i t i o n 1 4. Let𝑓 ∶𝔻→ℂbe holomorphic on𝔻and continuous on𝔻. Let𝑓|𝕋denote the restriction

of 𝑓to𝕋. Then,𝑃[𝑓|𝕋] = 𝑓on𝔻.

The last tools we need to prove (2) are some basic concepts of functional analysis related with the weak topologies. We just recall here what we are going to need later. For a deeper study, the reader is referred to Brezis’s book [1]. Given a Banach space𝐸, we denote its topological dual by𝐸^∗, and given𝑥 ∈ 𝐸 and𝑥 ∈ 𝐸^∗, we write⟨𝑥,𝑥^∗⟩ ≔ 𝑥^∗(𝑥). For each𝑥^∗ ∈ 𝐸^∗, we consider the function𝜑𝑥^∗∶ 𝐸 →ℂ, defined by 𝜑𝑥^∗(𝑥) = ⟨𝑥,𝑥^∗⟩. Then, the weak topology𝜎(𝐸,𝐸^∗) on𝐸is the finest topology that makes all the maps(𝜑𝑥^∗)𝑥^∗∈𝐸^∗continuous. A sequence{𝑥𝑛}^∞𝑛=0in𝐸converges to𝑥in the weak topology if and only if {⟨𝑥𝑛,𝑥^∗⟩}^∞𝑛=0converges to⟨𝑥,𝑥^∗⟩for all𝑥^∗ ∈ 𝐸^∗. Similarly, for each𝑥 ∈ 𝐸we may consider the function 𝜓_𝑥∶ 𝐸^∗→ℂdefined by𝜑_𝑥(𝑥^∗) = ⟨𝑥,𝑥^∗⟩, and the weak-star (or weak*) topology𝜎(𝐸^∗,𝐸)is defined as the finest topology on𝐸^∗making all the maps(𝜓_𝑥)_𝑥∈𝐸continuous. A sequence{𝑥_𝑛^∗}^∞_𝑛=0in𝐸^∗converges to𝑥^∗ in the weak* topology if and only if{⟨𝑥,𝑥^∗_𝑛⟩}^∞_𝑛=0converges to⟨𝑥,𝑥^∗⟩for all𝑥 ∈ 𝐸.

R e m a r k 1 5. A key fact when dealing with weak topologies is the Banach-Alaoglu theorem [1, Theorem 3.16],

by which the closed ball𝐵_𝐸∗= {𝑓 ∈ 𝐸^∗∶ ‖𝑓‖ ≤1}is compact in the weak* topology.

Let us also recall that(𝐿₁(𝕋))^∗= 𝐿∞(𝕋), and the duality is given by

⟨𝑓,𝑔⟩ = ∫

𝕋

𝑓(𝑤)𝑔(𝑤)d𝑤

for𝑓 ∈ 𝐿₁(𝕋)and𝑔∈ 𝐿_∞(𝕋). An important fact for us is that, since𝐿₁(𝕋)is separable, the closed unit ball of𝐿∞(𝕋)is metrizable in the𝜎(𝐿∞,𝐿₁)-topology [1, Theorem 3.28]. These two facts imply that every bounded sequence in𝐿∞(𝕋)has a subsequence that converges in the𝜎(𝐿∞,𝐿₁)-topology. ◀

We are finally ready to state and prove the main result of this section. Given a function𝑔∈ 𝐻_∞(𝔻), we will denote by𝑐_𝑛(𝑔)the𝑛-th coefficient of its power series expansion centered on0.

T h e o r e m 1 6. The Poisson operator𝑃 ∶ 𝐻_∞(𝕋) → 𝐻_∞(𝔻)defined as𝑓 ↦ 𝑃[𝑓]is an isometric isomorphism

so that𝑐_𝑛(𝑃[𝑓]) = ̂𝑓(𝑛)for all𝑛 ∈ℕ₀.

P r o o f. From remarks7and11we already know that it is well defined, continuous and injective. It is only

left, then, to see that it is onto. Let𝑔∈ 𝐻∞(𝔻), and consider its power series expansion (recall remark3)

(6) 𝑔(𝑧) =

∞

∑

𝑛=0

𝑐_𝑛(𝑔)𝑧^𝑛,

which converges absolutely and uniformly on𝑟𝔻for every0< 𝑟 <1. Now, for each𝑛 ∈ℕwe consider the function𝑓𝑛∶𝕋→ℂgiven by𝑓𝑛(𝑤) =𝑔((1−1/𝑛)𝑤). Note that

‖𝑓𝑛‖∞=sup

𝑤∈𝕋

|𝑓𝑛(𝑤)| =sup

𝑤∈𝕋

|𝑔((1−1/𝑛)𝑤) | ≤sup

𝑧∈𝔻

|𝑔(𝑧)| = ‖𝑔‖_∞.

Then,{𝑓_𝑛}^∞_𝑛=1is a bounded sequence in𝐿_∞(𝕋)that, in view of remark15, has a subsequence{𝑓_𝑛𝑘}^∞_𝑘=1that converges to some𝑓 ∈ 𝐿_∞(𝕋). Notice that‖𝑓‖_∞ ≤ ‖𝑔‖_∞. Our aim now is to see that, in fact,𝑃[𝑓] =𝑔.

First, if𝑛 ∈ℤ, the weak* convergence implies

̂𝑓(𝑛) = ∫

𝕋

𝑓(𝑤)𝑤^−𝑛d𝑤 = ⟨𝑓,𝑤^−𝑛⟩ = lim

𝑘→∞⟨𝑓𝑛_𝑘,𝑤^𝑛⟩ = lim

𝑘→∞

𝑓ˆ𝑛_𝑘(𝑛).

But, since the series in (6) converges uniformly on𝕋, we have (recall again (4))

𝑓ˆ_𝑛𝑘(𝑛) = ∫

𝕋

𝑓_𝑛𝑘(𝑤)𝑤^−𝑛d𝑤 = ∫

𝕋

∞

∑

𝑚=0

𝑐_𝑚(𝑔)(1−_𝑛¹

𝑘)^𝑚𝑤^𝑚𝑤^−𝑛d𝑤

=

∞

∑

𝑚=0

𝑐_𝑚(𝑔)(1−_𝑛¹

𝑘)^𝑚∫

𝕋

𝑤^𝑚𝑤^−𝑛d𝑤 = {(1− _𝑛¹

𝑘)^𝑛𝑐_𝑛(𝑔) if𝑛 ≥0,

0 if0> 𝑛.

Hence,

̂𝑓(𝑛) = {𝑐𝑛(𝑔) if𝑛 ≥0, 0 if0> 𝑛,

thus𝑓 ∈ 𝐻∞(𝕋). Moreover, for𝑧∈𝔻,

𝑃[𝑓](𝑧) =

∞

∑

𝑛=0

̂𝑓(𝑛)𝑧^𝑛=

∞

∑

𝑛=0

𝑐_𝑛(𝑔)𝑧^𝑛=𝑔(𝑧),

and by remark11we have‖𝑔‖∞≤ ‖𝑓‖∞. ▪

We have the result for functions of one variable. Our aim is to extend this to an analogous result in infinitely many variables. As an intermediate step we have to look at it for functions of several (but finitely many) variables. We do this in the following section.

3 . T h e

𝑁

We want to reproduce in this section the program that we presented in section2. As there, we have to keep a foot in the world of holomorphic functions and another foot in the world of Fourier analysis. But before we proceed we have to define the concepts and spaces that we are going to work with.

D e f i n i t i o n 1 7. Let𝑈be open inℂ^𝑁. A function𝑓 ∶ 𝑈 →ℂis holomorphic if for every𝑧∈ 𝑈there exists a

unique∇𝑓(𝑧) ∈ℂ^𝑁so that

limℎ→0

𝑓(𝑧+ ℎ) − 𝑓(𝑧) − ⟨∇𝑓(𝑧),ℎ⟩

‖ℎ‖ =0,

where⟨𝑥,𝑦⟩ = ∑^𝑁_𝑖=1𝑥_𝑖𝑦_𝑖for𝑥 = (𝑥₁,…,𝑥_𝑁),𝑦 = (𝑦₁,…,𝑦_𝑁) ∈ ℂ^𝑁, andℎ →0with the usual topology on

ℂ^𝑁. ◀

R e m a r k 1 8. Given a holomorphic function𝑓 ∶ 𝑈 ⊂ ℂ^𝑁 → ℂ and fixed 𝑁 −1 coordinates, that is,

𝑎₁,…,𝑎_𝑗−1,𝑎_𝑗+1,…,𝑎_𝑁 ∈ ℂ, then the restricted function 𝑔(𝑧) = 𝑓(𝑎₁,…,𝑎_𝑗−1,𝑧,𝑎_𝑗+1,…,𝑎_𝑁)is holomor- phic in its domain of definition and𝑔^′(𝑧) = (∇𝑓(𝑎₁,…,𝑎𝑗−1,𝑧,𝑎𝑗+1,…,𝑎𝑁))

𝑗. Because then limℎ→0

𝑔(𝑧+ ℎ) −𝑔(𝑧) −𝑔^′(𝑧)

|ℎ| =lim

ℎ→0

𝑓(𝑧+ ℎ) − 𝑓(𝑧) − ⟨∇𝑓(𝑧),ℎ⟩

‖ℎ‖ ,

where𝑧= (𝑎₁,…,𝑎𝑗−1,𝑧,𝑎𝑗+1,…,𝑎𝑁)andℎ = (0,…, 0,ℎ, 0,…, 0)withℎin the𝑗-th coordinate, and⟨∇𝑓(𝑧),ℎ⟩

as in the previous definition (not a scalar product). This limit equals0since𝑓is holomorphic. ◀

A key fact of the bounded holomorphic functions space is the following version of theWeierstrass theorem [5, Theorem 2.4].

T h e o r e m 1 9. Let(𝑓𝑛)𝑛be a sequence of holomorphic functions on𝑟𝔻^𝑁that converges uniformly on all

compact subsets of 𝑟𝔻^𝑁to some𝑓 ∶ 𝑟𝔻^𝑁→ℂ. Then,𝑓is holomorphic.

Following exactly the same proof as in the one-variable case (theorem2) we have that it is a Banach space.

T h e o r e m 2 0. 𝐻_∞(𝔻^𝑁) = {𝑓 ∶𝔻^𝑁 → ℂ ∶ 𝑓is bounded and holomorphic}is a Banach space with the

norm‖𝑓‖_∞=sup_𝑧∈𝔻𝑁|𝑓(𝑧)|.

In the one variable case we had that every holormorphic function is also analytic. This is also true for finitely many variables, as we shall see in theorem23. But before we get into that we have to make clear what it means that a series converges in this context.

R e m a r k 2 1. If(𝑐_𝑖)_𝑖∈𝐼is a family of scalars with𝐼being a countable family of indexes, we say that(𝑐_𝑖)_𝑖∈𝐼is

summable if there exists𝑠 ∈ℂ(which we call the sum of(𝑐_𝑖)_𝑖∈𝐼and write𝑠 = ∑_𝑖∈𝐼𝑐_𝑖) such that for all𝜖 >0 there exists a finite set𝐹₀⊆ 𝐼such that|𝑠 − ∑_𝑖∈𝐹𝑐_𝑖| < 𝜖for all finite sets𝐹₀⊆ 𝐹 ⊆ 𝐼. This is equivalent to the following three statements:

1 . (Cauchy’s criterion) for all𝜖 >0there exists a finite set𝐹₀ ⊆ 𝐼such that| ∑_𝑖∈𝐹𝑐𝑖| < 𝜖for all finite sets 𝐹 ⊆ 𝐼\𝐹₀;

2 . (absolute summability)(|𝑐𝑖|)𝑖∈𝐼is summable, which equivalently means thatsup_{𝐹⊆𝐼finite}∑_𝑖∈𝐹|𝑐𝑖| < ∞;

in this case∑_𝑖∈𝐼|𝑐𝑖| =sup_{𝐹⊆𝐼finite}∑_𝑖∈𝐹|𝑐𝑖|;

3 . (unconditional summability) for every bijection 𝜎 ∶ℕ → 𝐼, ∑^∞_𝑛=1𝑐_𝜍(𝑛) converges; in this case

∑_𝑖∈𝐼𝑐_𝑖= ∑^∞_𝑛=1𝑐_𝜍(𝑛). ◀

Let us fix some notation. Amulti-indexis an𝑁-tuple𝛼 = (𝛼₁,…,𝛼𝑁) ∈ℕ^𝑁₀. Given such a multi-index and some𝑧= (𝑧₁,…,𝑧𝑁) ∈ℂ^𝑁, we denote

𝑧^𝛼=𝑧^𝛼₁¹⋯𝑧^𝛼_𝑁^𝑁. Amonomialis any mapping of the form𝑧↦𝑧^𝛼.

E x a m p l e 2 2. The first example of power series in𝑁variables is

∑

𝛼∈ℕ^𝑁₀

𝑧^𝛼,

that converges if and only if|𝑧𝑗| <1for every𝑗 =1,…,𝑁and, in this case,

∑

𝛼∈ℕ^𝑁₀

𝑧^𝛼=

𝑁

∏

𝑗=1

1−1𝑧_𝑗.

This follows immediately from the fact that every finite subset ofℕ^𝑁₀ is contained in{0, 1,…,𝑀}^𝑁for some 𝑀, that

∑

𝛼∈{0,1,…,𝑀}^𝑁

|𝑧|^𝛼= (

𝑀

∑

𝑘=0

|𝑧₁|^𝑘) ⋯ (

𝑀

∑

𝑘=0

|𝑧_𝑁|^𝑘)

and the formula of the geometric series. ◀

T h e o r e m 2 3. Let𝑓 ∶𝔻^𝑁→ℂ. The following two statements are equivalent:

1 . 𝑓is holomorphic;

2 . there exist coefficients(𝑐_𝛼(𝑓))

𝛼∈ℕ^𝑁₀ ⊂ℂso that 𝑓(𝑧) = ∑

𝛼∈ℕ^𝑁₀

𝑐_𝛼(𝑓)𝑧^𝛼,

for every 𝑧∈𝔻^𝑁.

Moreover, in this case, the convergence is absolute and uniform on each compact set of 𝔻^𝑁, and the coefficients are unique and can be computed as

𝑐_𝛼(𝑓) = 1 (2πi)^𝑁∫

|𝜁₁|=𝜌₁

… ∫

|𝜁_𝑁|=𝜌_𝑁

𝑓(𝜁₁,…,𝜁𝑁)

𝜁₁^𝛼1+1… 𝜁_𝑁^𝛼𝑁+1 d𝜁_𝑁⋯d𝜁₁ for any0< 𝜌𝑖<1and 𝛼 = (𝛼₁,…,𝛼𝑁) ∈ℕ^𝑁₀.

P r o o f. Suppose first that𝑓is holomorphic. Let𝑧= (𝑧₁,…,𝑧_𝑁) ∈𝔻^𝑁and choose = (𝜌₁,…,𝜌𝑁)such that

|𝑧_𝑗| < 𝜌𝑗<1for all1≤ 𝑗 ≤ 𝑁. Then, applying𝑁times the Cauchy integral formula for one one variable, we obtain

𝑓(𝑧) = 1 (2πi)^𝑁∫

|𝜁₁|=𝜌₁

… ∫

|𝜁_𝑁|=𝜌_𝑁

𝑓(𝜁₁,…,𝜁_𝑁)

(𝜁₁−𝑧₁)… (𝜁𝑁−𝑧𝑁)d𝜁_𝑁⋯d𝜁₁. But1/(𝜁_𝑗−𝑧_𝑗) = ∑^∞_𝑘

𝑗=0𝑧_𝑗^𝑘𝑗/𝜁_𝑗^𝑘𝑗+1, so this can be rewritten as 𝑓(𝑧) = 1

(2πi)^𝑁∫

|𝜁₁|=𝜌₁

… ∫

|𝜁_𝑁|=𝜌_𝑁

(

∞

∑

𝑘₁=0

𝑧^𝑘₁¹ 𝜁₁^𝑘1+1)… (

∞

∑

𝑘_𝑁=0

𝑧^𝑘_𝑁^𝑁

𝜁_𝑁^𝑘𝑁+1) 𝑓(𝜁₁,…,𝜁𝑁)d𝜁_𝑁⋯d𝜁₁.

Since these series converge uniformly on compact subsets, they commute with the integration and, therefore,

𝑓(𝑧) =

∞

∑

𝑘₁=0

…

∞

∑

𝑘_𝑁=0

( 1 (2πi)^𝑁∫

|𝜁₁|=𝜌₁

… ∫

|𝜁_𝑁|=𝜌_𝑁

𝑓(𝜁₁,…,𝜁𝑁)

𝜁₁^𝑘1+1… 𝜁_𝑁^𝑘𝑁+1 d𝜁_𝑁⋯d𝜁₁)𝑧^𝑘₁¹…𝑧^𝑘_𝑁^𝑁. So for each𝛼 = (𝑘₁,…,𝑘𝑁)we can define

(7) 𝑐_𝛼(𝑓) = 1

(2πi)^𝑁∫

|𝜁₁|=𝜌₁

… ∫

|𝜁_𝑁|=𝜌_𝑁

𝑓(𝜁₁,…,𝜁_𝑁)

𝜁₁^𝑘1+1… 𝜁_𝑁^𝑘𝑁+1 d𝜁_𝑁⋯d𝜁₁

and notice that it does not depend on the choice of . With this, we get that𝑓is analytic as 𝑓(𝑧) = ∑

𝛼∈ℕ^𝑁₀

𝑐𝛼(𝑓)𝑧^𝛼 for all𝑧∈𝔻^𝑁

and the convergence is absolute (recall remark21). Let us see that the series converges uniformly in every𝐾, compact subset of𝔻^𝑁. Since𝔻^𝑁is open, there exists𝜖 >0such that(1+ 𝜖)𝐾 ⊂𝔻. Now, given 𝑧 = (𝑧₁,…,𝑧_𝑁) ∈ 𝐾, define𝑠(𝑧) = ((1+ 𝜖)|𝑧₁|,…,(1+ 𝜖)|𝑧_𝑁|) ∈ [0, 1)^𝑁, so𝑧 ∈ 𝑠(𝑧)𝔻^𝑁and, therefore, {𝑠(𝑧)𝔻^𝑁∶𝑧∈ 𝐾}is an open cover of𝐾. Hence, there exist𝑧₁,…,𝑧_𝑛∈ 𝐾such that𝐾 ⊂ ∪_𝑗=^𝑛₁𝑠(𝑧_𝑗)𝔻^𝑁. Then it is enough to check the uniform convergence on subsets of the form𝑠𝔻^𝑁with𝑠 ∈ (0, 1)^𝑁. To see this we choose = (𝜌₁,…,𝜌𝑁)such that𝑠𝑗< 𝜌𝑗<1for all𝑗and use (7) to get

|𝑐𝛼(𝑓)|𝑠^𝛼≤ 𝑠^𝛼 𝜌^𝛼‖𝑓‖∞. Since∑_𝛼∈ℕ𝑁

0 𝑠^𝛼/𝜌^𝛼converges (recall example22), the Weierstrass M-test [6, Theorem 7.10] implies that

∑_𝛼∈ℕ𝑁

0 𝑐𝛼(𝑓)𝑧^𝛼converges uniformly in𝑠𝔻^𝑁. Let us assume now that𝑓(𝑧) = ∑_𝛼∈ℕ𝑁

0 𝑏𝛼𝑧^𝛼(pointwise) for every𝑧∈𝔻^𝑁. Pick some0< 𝑟 <1and note that|𝑏𝛼𝑧^𝛼| < |𝑏𝛼𝑟^𝛼1⋯ 𝑟^𝛼𝑁|for every𝑧∈ 𝑟𝔻^𝑁. But the series

∑_𝛼|𝑏𝛼𝑟^𝛼1⋯ 𝑟^𝛼𝑁|converges (by assumption) and, by the Weierstrass M-test, the series∑_𝛼𝑏𝛼𝑧^𝛼converges uniformly on𝑠𝔻^𝑁for every0< 𝑠 < 𝑟. In particular, the polynomials given by

∑

𝛼∈{0,1,…,𝑘}^𝑁

𝑏𝛼𝑧^𝛼

converge (as𝑘 → ∞) uniformly to𝑓on𝑠𝔻^𝑁 for every0 < 𝑠 < 𝑟. By theorem19we have that𝑓 is holomorphic and bounded on𝑠𝔻^𝑁and, since𝑠is arbitrary,𝑓is holomorphic in𝔻^𝑁. In particular,

𝑐_𝛼(𝑓) = 1 (2πi)^𝑁∫

|𝜁₁|=𝜌₁

… ∫

|𝜁_𝑁|=𝜌_𝑁

𝑓(𝜁₁,…,𝜁_𝑁)

𝜁₁^𝛼1+1… 𝜁_𝑁^𝛼𝑁+1d𝜁_𝑁⋯d𝜁₁ =

= ∑

𝛽∈ℕ^𝑁₀

𝑏_𝛽 1 (2πi)^𝑁∫

|𝜁₁|=𝜌₁

… ∫

|𝜁_𝑁|=𝜌_𝑁

𝜁^𝛽

𝜁₁^𝛼1+1… 𝜁_𝑁^𝛼𝑁+1 d𝜁_𝑁⋯d𝜁₁= 𝑏_𝛼. ▪

Now we move to the Fourier side. Analogously to the one dimensional case, we have to define the Fourier coefficients and then the Hardy space which we are going to work with.

D e f i n i t i o n 2 4. Given𝑓 ∈ 𝐿₁(𝕋^𝑁), for each𝛼 ∈ℤ^𝑁the𝛼-th Fourier coefficient is defined as

̂𝑓(𝛼) = ∫

𝕋^𝑁

𝑓(𝑤)𝑤^−𝛼d𝑤. ◀

Just as in (1), we have| ̂𝑓(𝛼)| ≤ ‖𝑓‖₁and the operator𝐿₁(𝕋^𝑁) →ℂgiven by𝑓 ↦ ̂𝑓(𝛼)is continuous. This immediately gives the following.

T h e o r e m 2 5. The Hardy space

𝐻_∞(𝕋^𝑁) = {𝑓 ∈ 𝐿_∞(𝕋^𝑁) ∶ ̂𝑓(𝛼) =0for𝛼 ∉ℕ^𝑁₀} is a closed subspace of 𝐿∞(𝕋^𝑁)and, therefore, it is a Banach space.

The goal now is to show that there exists an isometric isomorphism between these two spaces, the space of holomorphic functions and the Hardy space. For that purpose we define the analogous tool to the one we used in the one dimensional case:

D e f i n i t i o n 2 6. The𝑁-dimensional Poisson kernel𝑝_𝑁∶𝔻^𝑁×𝕋^𝑁→ℂis defined as

𝑝_𝑁(𝑧,𝑤) =

𝑁

∏

𝑗=1

𝑝(𝑧_𝑗,𝑤_𝑗)

for𝑧∈𝔻^𝑁and𝑤 ∈𝕋^𝑁, where𝑢 ∈𝕋^𝑁is given by𝑢𝑗=𝑧_𝑗/|𝑧_𝑗|and𝑟 = (𝑟₁,…,𝑟𝑁)by𝑟𝑗= |𝑧_𝑗|, and we write

𝑟^|𝛼|= 𝑟₁^|𝛼1|⋯ 𝑟_𝑁^|𝛼𝑁|and𝑧= 𝑟𝑢. ◀

The absolute convergence of the series defining𝑝(𝑧,𝑤)in (4) gives 𝑝_𝑁(𝑧,𝑤) = ∑

𝛼∈ℤ^𝑁

𝑤^−𝛼𝑟^|𝛼|𝑢^𝛼,

for every𝑧∈𝔻^𝑁and𝑤 ∈𝕋^𝑁. Also, the series converges uniformly on𝑟𝔻^𝑁for every0< 𝑟 <1.

P r o p o s i t i o n 2 7. The following statements hold:

1 . 𝑝_𝑁(𝑧,𝑤) >0for every 𝑤 ∈𝕋^𝑁and 𝑧∈𝔻^𝑁;

2 . ∫

𝕋^𝑁

𝑝_𝑁(𝑧,𝑤)𝑑𝑤 =1for every fixed𝑧∈𝔻^𝑁.

P r o o f. Both statements follow from the one-dimensional case (proposition10) and Fubini’s theorem. ▪

D e f i n i t i o n 2 8. Given𝑓 ∈ 𝐿₁(𝕋^𝑁), we define the𝑁-dimensional Poisson operator for𝑓by the function

𝑃_𝑁[𝑓] ∶𝔻^𝑁→ℂgiven by

𝑃_𝑁[𝑓](𝑧) = ∫

𝕋^𝑁

𝑝_𝑁(𝑧,𝑤)𝑓(𝑤)d𝑤 = ∑

𝛼∈ℤ^𝑁

̂𝑓(𝛼)𝑟^|𝛼|𝑢^𝛼. ◀

𝑃𝑁[𝑓]is well defined for every𝑓 ∈ 𝐿₁(𝕋^𝑁)since| ̂𝑓(𝛼)| ≤ ‖𝑓‖₁,𝑟 ∈ [0, 1]^𝑁and𝑢 ∈𝕋^𝑁.

T h e o r e m 2 9. For each𝑁 ∈ℕ, the𝑁-dimensional Poisson operator

𝑃_𝑁∶ 𝐻_∞(𝕋^𝑁) → 𝐻_∞(𝔻^𝑁) is an isometric isomorphism such that𝑐𝛼(𝑃𝑁[𝑓]) = ̂𝑓(𝛼)for all𝛼 ∈ℕ^𝑁₀.

P r o o f. On the one hand, observe that, for𝑓 ∈ 𝐻∞(𝕋^𝑁),

(8) 𝑃𝑁[𝑓](𝑧) = ∑

𝛼∈ℕ^𝑁₀

̂𝑓(𝛼)𝑧^𝛼

for every𝑧∈𝔻^𝑁and (recall theorem23)𝑃𝑁[𝑓]is holomorphic on𝔻^𝑁. Moreover, using the properties of the𝑁-dimensional Poisson kernel (proposition27), we have that

‖𝑃𝑁[𝑓]‖∞≤ sup

𝑧∈𝔻^𝑁

∫

𝕋^𝑁

|𝑝𝑁(𝑧,𝑤)𝑓(𝑤)|d𝑤 ≤ ‖𝑓‖_∞ sup

𝑧∈𝔻^𝑁

∫

𝕋^𝑁

𝑝𝑁(𝑧,𝑤)d𝑤 = ‖𝑓‖_∞.

Therefore,𝑃_𝑁[𝑓] ∈ 𝐻_∞(𝔻^𝑁)and the operator𝑃_𝑁is well defined and continuous such that𝑓(𝛼) = 𝑐̂ _𝛼(𝑃_𝑁[𝑓]) for every𝑓 ∈ 𝐻_∞(𝕋^𝑁)and𝛼 ∈ℕ^𝑁₀. The uniqueness of the coefficients shows that the operator defined is injective. It only remains to see that𝑃_𝑁is surjective. We take some𝑔∈ 𝐻_∞(𝔻^𝑁)and, for each𝑛 ∈ℕ, consider the function defined by𝑓_𝑛(𝑢) = 𝑔((1−1/𝑛)𝑢)for every𝑢 ∈ 𝕋^𝑁, which is in𝐻_∞(𝕋^𝑁)and has Fourier coefficients𝑓_𝑛̂(𝛼) = 𝑐_𝛼(𝑔)(1−1/𝑛)^|𝛼|. Indeed, since the monomial series expansion of𝑔converges uniformly on𝑟𝕋^𝑁for every0< 𝑟 <1, we have that

̂𝑓_𝑛(𝛼) = ∫

𝕋^𝑁

𝑓_𝑛(𝑤)𝑤^−𝛼d𝑤 = {𝑐_𝛼(𝑔)(1−1/𝑛)^|𝛼| for𝛼 ∈ℕ^𝑁₀,

0 otherwise.

Clearly,‖𝑓𝑛‖∞≤ ‖𝑔‖∞for every𝑛 ∈ ℕ. With exactly the same argument as in remark15we can find a subsequence{𝑓𝑛_𝑘}^∞_𝑘=1which𝜎(𝐿∞,𝐿₁)-converges to some𝑓 ∈ 𝐵𝐿_∞(𝕋^𝑁)(0,‖𝑔‖∞). As a consequence of the weak convergence,

̂𝑓(𝛼) = ∫

𝕋^𝑁

𝑓(𝑤)𝑤^−𝛼d𝑤 = lim

𝑘→∞∫𝕋^𝑁𝑓_𝑛𝑘(𝑤)𝑤^−𝛼d𝑤 = {𝑐𝛼(𝑔) for𝛼 ∈ℕ^𝑁₀, 0 otherwise.

This implies𝑓 ∈ 𝐻_∞(𝕋^𝑁). Moreover, by (8) we get𝑃_𝑁[𝑓](𝑧) =𝑔, and then‖𝑓‖_∞= ‖𝑔‖_∞. ▪

4 . T h e i n f i n i t e - d i m e n s i o n a l c a s e

We jump now from finitely to infinitely many variables. To do so, we will restrict the problem to finite variables, we will apply the finite-dimensional theorem and, using some powerful tools, we will go back to infinitely many variables. We have, then, to face two problems:

• to define a proper setting for our problem in the infinite dimensional setting,

• to find tools that allow us to jump from the finite to the infinite dimensional case.

We begin by tackling the first issue: to translate our problem to the setting of infinite dimensions. We start by defining the main components of our result. Firstly, we need to find a proper substitute for𝔻^𝑁. Then, we need to understand the concept of holomophic function in infinite dimensions. Finally, we define the Fourier coefficients for infinitely many variables.

As substitute for𝔻^𝑁we could think of the unit ball of the Banach space ℓ_∞ (the space of bounded sequences with the supremum norm‖ ⋅ ‖). However, this candidate presents some problems, since the space𝑐₀₀(of sequences with only finitely many non-zero elements) is not dense in(ℓ_∞,‖ ⋅ ‖). So, we may choose a “smaller” space: this is going to be the Banach space𝑐₀= {{𝑧𝑛}^∞_𝑛=1 ⊂ℂ∶lim𝑛→∞𝑧𝑛=0}, and we will consider its unit ball𝐵𝑐₀ as analogous to𝔻^𝑁. Recall that the dual space of𝑐₀is the space (𝑐₀)^∗ = ℓ₁= {{𝑧𝑛}^∞𝑛=1⊂ℂ∶ ∑^∞_𝑛=1|𝑧𝑛| < ∞}.

The analogue to𝕋^𝑁will be the infinite dimensional torus𝕋^∞ = {{𝑤𝑛}^∞𝑛=1 ∶ 𝑤𝑛 ∈ 𝕋for each𝑛 ∈ ℕ}, which is a compact space by Tychonoff’s theorem. Also, since𝕋^∞is a grup with the product coordinate to coordinate, we are able to work with the Haar measure (see Cohn’s book [3, Chapter 9]).

The definition of holomorphic functions on𝐵_𝑐0is a particular case of the Fréchet differentiability, which is valid for any normed space and any open subset.

D e f i n i t i o n 3 0. Let𝑈be an open subset of a normed space𝑋. A function𝑓 ∶ 𝑈 → ℂ is said to be

holomorphic if it is Fréchet differentiable at every𝑥 ∈ 𝑈, that is, if there exists a continuous linear functional𝑥^∗∈ 𝑋^∗such that

limℎ→0

𝑓(𝑥 + ℎ) − 𝑓(𝑥) − 𝑥^∗(ℎ)

‖ℎ‖ =0.

In that case we denote the unique𝑥^∗byd𝑓(𝑥), and call it the differential of𝑓at𝑥. ◀

R e m a r k 3 1. The restriction of every holomorphic function to finite dimensional subspaces is again holo-

morphic. More precisely, given a holomorphic function𝑓 ∶ 𝑈 →ℂand𝑀an𝑁dimensional subspace of𝑋 with basis𝑒₁,…,𝑒𝑁, then we just take the inclusion𝑖𝑀∶ 𝑀 → 𝑋,𝑒𝑖↦ 𝑖𝑀(𝑒𝑖) = 𝑏𝑖and consider, for each 𝑧₀∈ 𝑈 ∩ 𝑀, the vector∇(𝑓 ∘ 𝑖_𝑀)(𝑧₀) = ([d𝑓(𝑧₀)](𝑖_𝑀(𝑒_𝑘)))^𝑁

𝑘=1= ([d𝑓(𝑧₀)](𝑏_𝑘))^𝑁

𝑘=1, which is the differential of𝑓|𝑈∩𝑀. Indeed,

limℎ→0

𝑓(𝑧₀+ ℎ) − 𝑓(𝑧₀) − ⟨∇(𝑓 ∘ 𝑖𝑀)(𝑧₀),ℎ⟩

‖ℎ‖ =lim

ℎ→0

𝑓(𝑧₀+ ℎ) − 𝑓(𝑧₀) − ∑_𝑘[d𝑓(𝑧₀)](𝑏𝑘)ℎ𝑘

‖ℎ‖ =0.

In particular, if𝑓 ∶ 𝐵𝑐₀ →ℂis a holomorphic function and𝑁 ∈ℕ, the restriction𝑓𝑁∶𝔻^𝑁→ℂdefined by 𝑓𝑁(𝑧₁,…,𝑧_𝑁) = 𝑓(𝑧₁,…,𝑧_𝑁, 0, 0,…)is holomorphic. ◀

R e m a r k 3 2. Given a holomorphic function𝑓 ∶𝔻^𝑁→ℂ, we may see it as a function on𝐵_𝑐0(let us denote it

by𝑓) just by adding zeros̃ 𝑓(𝑧) = 𝑓(𝑧̃ ₁,…,𝑧_𝑁, 0,…), which is holomorphic. ◀

T h e o r e m 3 3. The space𝐻∞(𝐵𝑐₀) = {𝑓 ∶ 𝐵𝑐₀ →ℂ∶ 𝑓is holomorphic and bounded}with the norm‖𝑓‖∞= sup_𝑧∈𝐵

𝑐0|𝑓(𝑧)|is a Banach space.

This fundamental fact is a consequence of the following simplified Weierstrass type theorem, a proof of which can be found in the book of Defant et al. [5, Theorem 2.13].

T h e o r e m 3 4. Let 𝑋 be a normed space, and (𝑓_𝑛) a bounded sequence in𝐻_∞(𝐵_𝑋) that converges to

𝑓 ∶ 𝐵_𝑋→ℂuniformly on each compact subset of 𝐵_𝑋(i.e., with respect to the compact-open topology).

Then,𝑓 ∈ 𝐻_∞(𝐵_𝑋)and‖𝑓‖_∞≤sup_𝑛‖𝑓_𝑛‖_∞.

Before introducing Taylor coefficients, let us fix some notation that will be used frequently throughout this section. We will write

ℕ^(ℕ)₀ = ⋃

𝑁∈ℕ

ℕ^𝑁₀ and ℤ^(ℕ)= ⋃

𝑁∈ℤ

ℤ^𝑁.

When convenient, we will also identify(𝛼₁,𝛼₂,…,𝛼_𝑁)with(𝛼₁,𝛼₂,…,𝛼_𝑁, 0, 0,…). Given𝑓 ∈ 𝐻_∞(𝐵_𝑐0)and 𝑁, let𝑓𝑁∶𝔻^𝑁 →ℂbe its restriction, that is,𝑓𝑁(𝑧₁,…,𝑧_𝑁) = 𝑓(𝑧₁,…,𝑧_𝑁, 0, 0,…). This is a holomorphic function on𝔻^𝑁(see remark31) and, by theorem23, can be expanded as a power series

𝑓_𝑁(𝑧) = ∑

𝛼∈ℕ^𝑁₀

𝑐_𝛼(𝑓_𝑁)𝑧^𝛼,

for every𝑧 ∈𝔻^𝑁. This, in principle, provides us a set of coefficients{𝑐_𝛼(𝑓_𝑁)}_𝛼∈ℕ^𝑁

0 for each𝑁. But, as a matter of fact, when we increase the dimension we only add “new” coefficients for the “new” dimensions.

Let us be more precise. If𝑀 ≥ 𝑁and𝛼 ∈ ℕ^𝑁₀ (that we identify with(𝛼₁,…,𝛼𝑁, 0,…, 0) ∈ℕ^𝑀₀ and call again𝛼), then

𝑐𝛼(𝑓𝑁) = 𝑐𝛼(𝑓𝑀).

Indeed, by theorem23we have

𝑐_(𝛼,0)(𝑓_𝑁+1) = 1 (2πi)^𝑁+1 ∫

𝜌₁𝕋

⋯ ∫

𝜌_𝑁+1𝕋

𝑓(𝜁₁,…,𝜁𝑁+1, 0,…) 𝜁₁^𝛼1+1… 𝜁_𝑁^𝛼𝑁+1𝜁𝑁+1

d𝜁_𝑁+1⋯d𝜁₁,

and using Cauchy’s integral formula,

𝑐(𝛼,0)(𝑓𝑁+1) = 2πi (2πi)^𝑁+1 ∫

𝜌₁𝕋

⋯ ∫

𝜌_𝑁+1𝕋

𝑓(𝜁₁,…,𝜁_𝑁, 0,…)

𝜁₁^𝛼1+1… 𝜁_𝑁^𝛼𝑁+1 d𝜁𝑁⋯d𝜁₁

= 1

(2πi)^𝑁∫

𝜌₁𝕋

⋯ ∫

𝜌_𝑁𝕋

𝑓(𝜁₁,…,𝜁_𝑁, 0,…)

𝜁₁^𝛼1+1… 𝜁_𝑁^𝛼𝑁+1 d𝜁_𝑁⋯d𝜁₁= 𝑐_𝛼(𝑓_𝑁).

Then, each function𝑓 ∈ 𝐻∞(𝐵𝑐₀)defines a unique family of coefficients{𝑐𝛼(𝑓)}_𝛼ℕ(ℕ)

0 , that we callTaylor coefficients. In other words, each function𝑓defines aformalpower series

𝑓 ∼ ∑

𝛼∈ℕ^(ℕ)₀

𝑐_𝛼(𝑓)𝑧^𝛼.

The problem now is that𝑓(𝑧)may not coincide with∑ 𝑐_𝛼(𝑓)𝑧^𝛼. Toeplitz [9] gave an example of a function 𝑓 ∈ 𝐻_∞(𝐵_𝑐0)and a point𝑧∈ 𝐵_𝑐0for which∑ 𝑐_𝛼(𝑓)𝑧^𝛼does not converge. In other words, when we deal with functions of infinitely many variables,holomorphicandanalyticare no longer equivalent. This is a problem for us, since the proof of the isometry between the spaces, both in the case of one and several variables (recall theorems 16and29) depends heavily on the fact that a holomorphic function has a representation as a power series.

We now move on to the Fourier part, as we did in the previous sections. Given𝑓 ∈ 𝐿₁(𝕋^∞)and𝛼 ∈ℤ^(ℕ), we define the𝛼-th Fourier coefficient as

̂𝑓(𝛼) = ∫

𝕋^∞

𝑓(𝑤)𝑤^−𝛼d𝑤