1. Power series with Ostrowski gaps. Let (1.1) f (z) =

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A N N A L E S

U N I V E R S I T A T I S M A R I A E C U R I E - S K Ł O D O W S K A L U B L I N – P O L O N I A

VOL. LXV, NO. 2, 2011 SECTIO A 179–190

JÓZEF SICIAK

Some gap power series in multidimensional setting

Professor Jan Krzyż in memoriam

Abstract. We study extensions of classical theorems on gap power series of a complex variable to the multidimensional case.

1. Power series with Ostrowski gaps. Let (1.1) f (z) =

∞

X

0

Q

_j

(z), where Q

_j

(z) = X

|α|=j

a

_α

z

^α

, α ∈ Z

^N+

, be a power series in C

^N

, i.e. a series of homogeneous polynomials Q

_j

of N complex variables of degree j.

The set D given by the formula D := {a ∈ C

^N

; the sequence (1.1) is convergent in a neighborhood of a} is called a domain of convergence of (1.1).

It is known that

(1.2) D = {z ∈ C

^N

; ψ

^∗

(z) < 1}, where

(1.3) ψ(z) := lim sup

j→∞

j

q

|Q

_j

(z)|,

and ψ

^∗

denotes the upper semicontinuous regularization of ψ.

2000 Mathematics Subject Classification. 30B10, 30B30, 30B40, 32A05, 32A07, 32A10, 32D15.

Key words and phrases. Plurisubharmonic functions, negligible sets in C^N, power series, lacunary power series, multiple power series.

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If ψ

^∗

is finite, then it is plurisubharmonic and absolutely homogeneous (i.e. ψ

^∗

(λz) = |λ|ψ

^∗

(z), λ ∈ C, z ∈ C

^N

). Therefore, the domain of con- vergence D is either empty, or it is a balanced (i.e. λz ∈ D for all λ ∈ C with |λ| ≤ 1 and z ∈ D) domain of holomorphy. Every balanced domain of holomorphy is a domain of convergence of a series (1.1).

For every balanced domain D in C

^N

there is a unique nonnegative func- tion h (so-called Minkowski functional of D) such that h(λz) = |λ|h(z) for all λ ∈ C and z ∈ C

^N

, and D = {z ∈ C

^N

; h(z) < 1}. In particular, if D is a domain of convergence of (1.1), then h(z) ≡ ψ

^∗

(z).

It is known that a balanced domain in C

^N

is a domain of holomorphy if and only if its Minkowski functional h is an absolutely homogeneous plurisubharmonic function.

The number

(1.4) ρ := 1/ lim sup

j→∞

j

q kQ

_j

k

_B

,

where B := {z ∈ C

^N

; kzk ≤ 1}, is called a radius of convergence of series (1.1) (with respect to a given norm k · k).

If N = 1, then ψ(z) =

^|z|_ρ

and D = ρB. If N ≥ 2, then ρB ⊂ D but, in general, D 6= ρB.

Series (1.1) is normally geometrically convergent in D, i.e.

(1.5) lim sup

j→∞

j

q

kQ

_j

k

_K

< 1, lim sup

n→∞

pkf − s

n n

k

_K

< 1,

for all compact sets K ⊂ D, where s

_n

:= Q

_o

+ · · · + Q

_n

is the nth partial sum of (1.1).

Definition 1.1. We say that a function f holomorphic in a neighborhood of a point z

^o

∈ C

^N

possesses at the point z

^o

Ostrowski’s gaps (m

_k

, n

k

], if

1

^o

. m

_k

, n

_k

are natural numbers such that m

_k

< n

_k

< m

_k+1

(k ≥ 1),

n_k

mk

→ ∞ as k → ∞;

2

^o

. lim

_{j→∞, j∈I}

pkQ

^j _j

k

_B

= 0, where B is the unit ball in C

^N

, Q

_j

(z) ≡ Q

^(f,z_j ^o⁾

(z) := X

|α|=j

f

^(α)

(z

^o

)

α! z

^α

= 1 j!

d dλ

j

f (z

^o

+ λz)

_|λ=0

, and I := S

∞

k=1

(m

_k

, n

_k

], (m

_k

, n

_k

] denoting the set of integers j with m

_k

<

j ≤ n

_k

.

Observe that f

_o

(z) := P

j∈I

Q

j

(z − z

^o

) is an entire function such that

the function g := f − f

_o

possesses Ostrowski’s gaps (m

_k

, n

_k

] at z

^o

with

Q

^(g,z_j ^o⁾

= 0 for m

k

< j ≤ n

k

, k ≥ 1. Hence, a holomorphic function f

possesses Ostrowski’s gaps (m

_k

, n

k

] at a point z

^o

if and only if there exists

an entire function f

_o

such that Q

^{(f −f}_j ^o^,z^o⁾

= 0 for m

_k

< j ≤ n

_k

, k ≥ 1.

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Moreover, the maximal domain of existence G = G

_f

of f is identical with the maximal domain of existence of f − f

_o

.

Definition 1.2. We say that a function f holomorphic in a neighborhood of a point z

^o

possesses Ostrowski’s gaps relative to a sequence of positive integers {n

_k

}, if {n

_k

} is increasing and there exists a sequence of positive real numbers {q

_k

} such that q

_k

→ 0 as k → ∞ and lim

_{j→∞,j∈I}

pkQ

^j _j

k

_B

= 0, where I := S

∞

k=1

(bq

_k

n

_k

c, n

_k

].

A function f possesses Ostrowski’s gaps according to Definition 1.1 if and only if f possesses Ostrowski’s gaps according to Definition 1.2.

Indeed, if the conditions of Definition 1.1 are satisfied, then it is sufficient to put q

_k

:= m

_k

/n

k

.

If the conditions of Definition 1.2 are satisfied, consider two cases. If m := lim inf

k→∞

q

k

n

k

is finite, then the function f is entire, so that f has Ostrowski’s gaps (m

_k

, n

_k

] according to Definition 1 for any sequence m

_k

, n

_k

satisfying 1

^o

.

If lim inf

_k→∞

q

_k

n

_k

= ∞, then f possesses Ostrowski’s gaps (bq

_k_l

n

_k_l

c, n

_k_l

] for a suitable chosen increasing subsequence k

_l

of positive integers.

We say that a compact subset K of C

^N

is polynomially convex if K is identical with its polynomially convex hull K := {a ∈ C ˆ

^N

; |P (a)| ≤ kP k

K

for every polynomial P of N complex variables}. We say that an open set Ω in C

^N

is polynomially convex, if for every compact subset K of Ω the polynomially convex hull ˆ K of K is contained in Ω.

The following theorem is known (see [7]). It is a multidimensional version of the classical Ostrowski’s Theorem (see Theorem 3.1.1 in [1]).

Theorem 1. If a holomorphic function f possesses Ostrowski’s gaps (m

_k

, n

_k

] at a point z

^o

∈ C

^N

, then the maximal domain of existence G = G

_f

of f is one-sheeted and polynomially convex. Moreover, for every compact subset K of G we have

(1.6) lim sup

k→∞

kf − s

_n_k

k

_K^1/n^k

< 1, where

s

n

(z) ≡ s

^(f,z_n ^o⁾

(z) =

n

X

j=0

Q

^(f,z_j ^o⁾

(z − z

^o

)

is the nth partial sum of the Taylor series development of f around z

^o

. Corollary 1.1. If

f (z

^o

+ z) =

∞

X

k=1

Q

^(f,z_m ^o⁾

k

(z),

where m

_k

/m

_k+1

→ 0 as k → ∞, then Q

^(f,z_j ^o⁾

= 0 for j / ∈ {m

_k

} so that f

has Ostrowski’s gaps (m

_k

, n

_k

] with n

_k

:= m

_k+1

− 1. Therefore, the maximal

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domain of existence G

_f

of f is identical with the domain of convergence D

_f

of the Taylor series development of f around z

^o

, i.e.

G

_f

= D

_f

:= z ∈ C

^N

: ψ

^∗

(z − z

^o

) < 1 , where ψ(z) := lim sup

_k→∞ ^mk

q

|Q

^(f,z_m_k^o⁾

(z)|.

The following result gives an N -dimensional version of W. Luh’s Theo- rem 1 in [4]. In particular, it says that if a function f holomorphic in a domain G in C

^N

possesses Ostrowski’s gaps at some point z

^o

∈ G, then f possesses the same property at every other point a of the maximal domain of existence of f .

Theorem 2. Let f possess Ostrowski’s gaps (m

_k

, n

_k

] at a point z

^o

∈ C

^N

. Then

1

^o

. f possesses Ostrowski’s gaps

m

_k_l

, l

_n

kl

l

mi

at every point a ∈ G

_f

, where the sequence of natural numbers {k

_l

} (independent of a) is chosen in such a way that n

_k_l

≥ m

_k_l

l

²

and l

_n

kl

l

m

< m

_k_l+1

for l ≥ 1;

2

^o

. If Q

^(f,z_j ^o⁾

= 0 for m

_k

< j ≤ n

_k

, k ≥ 1

¹

, then the sequence n

s

^(f,zmk^o⁾

− s

^(f,a)mk

o

converges to zero normally with order n

_k

on C

^N

, i.e.

lim sup

k→∞

s

^(f,z_m ^o⁾

k

− s

^(f,a)_m

k

1/nk

K

< 1 for every compact set K ⊂ C

^N

.

By 2

^o

and Theorem 1 we get the following:

Corollary 1.2. If f possesses ordinary Ostrowski’s gaps (m

_k

, n

_k

] at a fixed point z

^o

∈ G, then

lim sup

k→∞

nk

r

f − s

^(f,a)m_k

K

< 1 for every point a ∈ G

_f

and every compact subset K of G

_f

.

Proof of Theorem 2. 1

^o

. Without loss of generality we may assume that z

^o

= 0 and

Q

^(f,z_j ^o⁾

= 0, m

k

< j ≤ n

k

, k ≥ 1.

Given a fixed point a ∈ G

_f

, we have Q

^(f,a)_j

(z) = 1

2πi Z

|λ|=r

f (a + λz) − s

_n_k

(a + λz)

λ

^j+1

dλ,

1In such a case we say that f possesses ordinary Ostrowski’s gaps at z^o

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kzk ≤ 1, j > m

_k

, k ≥ 1, where s

_n_k

= s

^(f,znk ^o⁾

(Observe that s

_n_k

is a polyno- mial of degree at most m

_k

), and 0 < r < min(dist(a, ∂G

_f

), dist(z

^o

, ∂G

_f

)).

By Theorem 1 there exist M > 1 and 0 < θ < 1 such that (1.7) kf − s

_n_k

k

_B(a,r)

≤ M θ

ⁿ^k

, k ≥ 1.

Therefore, by Cauchy inequalities, (1.8)

Q

^(f,a)_j

B

≤ M

r

^j

θ

ⁿ^k

, j > m

_k

, k ≥ 1.

Let {k

_l

} be an increasing sequence of natural numbers such that m

_k_l+1

> l n

_k_l

l m

, n

_k_l

m

_k_l

≥ l

²

, l ≥ 1.

By (1.8) we get Q

^(f,a)_j

1/j B

≤ M

r θ

ⁿ^kl^/j

≤ M

l θ

^l

, m

_k_l

< j ≤ l n

_k_l

l

m

, l ≥ 1.

The choice of the sequence {k

_l

} does not depend on a ∈ G

_f

. Therefore, f possesses Ostrowski’s gaps

m

_k_l

, l

_n

kl

l

mi

at every point a of G

_f

(according to Definition 1.1). The proof of the case 1

^o

is ended.

2

^o

. Observe that for kz − ak ≤

¹₂

r we have

f (z) − s

^(f,a)_m_k

(z) =

∞

X

m_k+1

Q

^(f,a)_j

(z − a) ≤

∞

X

m_k+1

Q

^(f,a)_j

B

r 2

j

, which by (1.8) gives

(1.9)

f (z) − s

^(f,a)_m

k

(z) ≤

∞

X

pk+1

2

^−j

M θ

ⁿ^k

≤ M θ

ⁿ^k

, k ≥ 1, kz − ak ≤ r 2 . By (1.7) and (1.9) we get

(1.10)

s

^(f,z_m_k^o⁾

− s

^(f,a)_m

k

B(a,¹₂r)

≤ 2M θ

ⁿ^k

, k ≥ 1.

Observe that for z ∈ C

^N

s

^(f,z_n ^o⁾

(z) ≤

n

X

j=0

Q

^(f,z_j ^o⁾

B

kz − z

^o

k

^j

≤

n

X

0

kf k

_B(zo,r)

r

^j

kz − z

^o

k

^j

≤ (n + 1)kf k

_B(zo,r)

1 + kzk + kz

_o

k r

n

.

Put M := kf k

_B(zo,r)∪B(a,r)

and c := max{kz

^o

k, kak}. Then for z ∈ C

^N

u

k

(z) := 1

n

k

log

s

^(f,z_m_k^o⁾

(z) − s

^(f,a)_m_k

(z)

≤ 1 n

k

log[2M (m

k

+ 1)] + m

_k

n

k

log

1 + kzk + kck r

.

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It follows that the sequence of plurisubharmonic functions {u

_k

} is locally uniformly upper bounded in C

^N

, and

u(z) := lim sup

k→∞

u

_k

(z) ≤ 0, z ∈ C

ⁿ

. Therefore, the plurisubhamonic function u

^∗

= const.

By (1.10) u

_k

(z) ≤

_n¹

k

log 2M + log θ for z ∈ B(a, r), k ≥ 1. Hence u

^∗

≤ log θ in C

^N

which ends the proof of 2

^o

. 2. E. Fabry’s Theorem. Now we shall present a multidimensional version of E. Fabry’s Theorem (Theorem 2.2.1 in [1]). Let f be a function of N complex variables holomorphic in a neighborhood of 0 with a gap Taylor series development

(2.1) f (z) =

∞

X

k=1

Q

m_k

(z), m

_k

< m

_k+1

.

Put ψ(z) := lim sup

_k→∞ ^mk

p|Q

_m_k

(z)|, h(z) := ψ

^∗

(z). It is known that D := {z ∈ C

^N

; h(z) < 1} = {a ∈ C

^N

; series (2.1) is convergent in a neighborhood of a} is a domain of convergence of (2.1).

Theorem 3. If lim

_k→∞_m^k

k

= 0, then the domain of convergence D of the series (2.1) is identical with the maximal domain of existence G

_f

of f .

Proof. Without loss of generality we may assume that D 6= C

^N

.

Due to Fabry we know that Theorem 3 is true for N = 1. It is also well known (by Bedford–Taylor Theorem on negligible sets) that the set E := {z ∈ C

^N

; ψ(z) < ψ

^∗

(z)} is pluripolar. Therefore, in particular, the set E is of 2N -dimensional Lebesgue measure zero.

Suppose Theorem 3 is not true for some N > 1. Then there is a function g holomorphic in a ball B(z

_o

, R) with z

_o

∈ D, R > r := dist(z

_o

, ∂D) such that g(z) = f (z) for z ∈ B(z

_o

, r).

Let b

_o

be a fixed point of ∂D such that kb

_o

− z

_o

k = r.

Since the ball B(z

_o

, r) is non-thin at the point b

_o

,we have lim sup

z→bo,z∈B(zo,r)

ψ

^∗

(z) = ψ

^∗

(b

_o

).

Therefore, there is a sequence {z

_k⁰

} ⊂ B(z

_o

, r) such that z

_k⁰

→ b

_o

, and ψ

^∗

(z

_k⁰

) → ψ

^∗

(b

_o

) as k → ∞. It follows that ψ

^∗

(b

_o

) ≤ 1. Since b

_o

∈ ∂D, we have ψ

^∗

(b

o

) ≥ 1. Therefore, ψ

^∗

(b

o

) = 1.

We know that the 2N -dimensional Lebesgue measure v

_2N

(E) = 0. There- fore, by the sub-mean-value property, for every k ≥ 1 there is a point z

_k

∈ B(z

_k⁰

,

¹_k

) ∩ B(z

o

, r) \ E such that ψ(z

_k

) = ψ

^∗

(z

_k

), |ψ

^∗

(z

_k⁰

) − ψ(z

_k

)| <

_k¹

. It is clear that the sequence {z

_k

} satisfies the following properties:

z

_k

∈ B(z

_o

, r), z

_k

→ b

_o

, ψ(z

_k

) = ψ

^∗

(z

_k

), ψ(z

_k

) → ψ

^∗

(b

_o

).

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Put b

_k

= z

_k

/ψ(z

_k

) (k ≥ 1). Then ψ(b

_k

) = ψ

^∗

(b

_k

) = 1, in particular, b

k

∈ ∂D for k ≥ 1, and b

_k

→ b

_o

as k → ∞.

Fix k so large that b := b

_k

∈ B(z

_o

, R). Put G

r

:= {λ ∈ C; λb ∈ B(z

o

, r)}, G

R

:= {λ ∈ C; λb ∈ B(z

o

, R)}.

One can easily check that the sets G

_r

, G

_R

are open, convex, nonempty (because λ

_o

b ∈ G

r

for λ

_o

:= ψ(z

_k

), and G

r

⊂ G

_R

). Moreover, G

_r

⊂ ∆ :=

{|λ| < 1}, and 1 ∈ G

_R

.

The function f (λb) (resp., g(λb)) is holomorphic in ∆ (resp., in G

_R

), and f (λb) = g(λb) for λ ∈ G

r

. Therefore, f (λb) = g(λb) on ∆ ∩ G

_R

. It follows that g(λb) is an analytic continuation of f (λb) across λ = 1, contrary to the Fabry Theorem for N = 1. We have got a contradiction showing that

Theorem 3 is true.

Remark. The present proof of Theorem 3 – with no assumption on the continuity of the function ψ

^∗

– is a joint result of the author and Professor Azimbay Sadullaev.

3. Fatou–Hurwitz–Polya Theorem. First we shall state Fatou–Hurwitz –Polya Theorem for a series of homogeneous polynomials of N complex vari- ables.

Theorem 4. Let f be a function holomorphic in a neighborhood of 0 ∈ C

^N

. Let

(3.0) f (z) =

∞

X

0

Q

_j

(z), Q

_j

(z) = X

|α|=j

f

^(α)

(0) α! z

^α

,

be its Taylor series development around 0. Then there exists a sequence

= {

_j

} with

_j

∈ {−1, 1} (resp.,

_j

∈ {0, 1}) such that the function f

(z) :=

∞

X

j=0

_j

Q

_j

(z), z ∈ D,

has no analytic continuation across any boundary point of the domain of convergence D := {ψ

^∗

(z) < 1} of series (3.0), where

ψ(z) := lim sup

j→∞

j

q

|Q

_j

(z)|.

For N = 1 this theorem (with

_j

∈ {−1, 1}) is due to Fatou–Hurwitz–

Polya (Theorem 4.2.8 in [1]).

Now, we shall present an N -dimensional version of the Fatou–Hurwitz–

Polya theorem for N -tuple power series

(3.1) f (z) = X

|α|≥0

c

α

z

^α

,

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where c

_α

z

^α

is a monomial of N complex variables z = (z

₁

, . . . , z

_N

) of degree

|α| := α

₁

+ · · · + α

N

. The set D := {a ∈ C

^N

; the series (3.1) is absolutely convergent in a neighborhood of a} is called a domain of convergence of the multiple power series (3.1).

It is known that D = {z ∈ C

^N

; h(z) < 1} is a complete N -circular (hence, in particular, D is balanced) domain whose Minkowski’s functional h ≡ h

_D

is given by the formula h(z) = M

^∗

(z), where

(3.2)

M (z) := lim sup

|α|→∞

|α|

p|c

α

z

^α

|

= lim sup

k→∞

max

n

|α|

p|c

_α

z

^α

|; |α| = k o

, z ∈ C

^N

.

Moreover, h(z

₁

, . . . , z

_N

) = h(|z

1

|, . . . , |z

_N

|) for all z ∈ C

^N

, and h is contin- uous (see [2], Lemma 1.7.1 (b)).

Theorem 5. If the domain of convergence D of (3.1) is not empty, then there exists a multiple sequence = {

_α

} with

_α

∈ {−1, 1} (resp., with

α

∈ {0, 1}) such that the function f

(z) := X

|α|≥0

_α

c

_α

z

^α

, z ∈ D,

has no analytic continuation across any boundary point of D.

We shall see that Theorems 4 and 5 are direct consequences of the fol- lowing Lemma 3.2.

Let X := {0, 1}

^N

(resp. {−1, 1}

^N

) be the space of all sequences x = (x

₁

, x

₂

, . . . ) where x

_j

= 0, or x

_j

= 1 (resp. x

_j

= −1, or x

_j

= 1) for j = 1, 2, . . . . Endow X in the topology determined by the metric

ρ(x, y) :=

∞

X

j=1

1 2

^j

|x − y|

_j

1 + |x − y|

j

, where

|x − y|

_j

:= max{|x

_k

− y

_k

|; k = 1, . . . , j}.

One can easily check that X is a complete metric space, and therefore, it has Baire property.

Moreover, in the topology a sequence {x(n)} of elements of X converges to an element x ∈ X if and only if for every k

_o

∈ N there exists n

o

∈ N such that x

_k

(n) = x

_k

for k = 1, . . . , k

_o

, n ≥ n

_o

.

Remark 3.1. Let {f

_k

} be a sequence of holomorphic functions in an open subset Ω of C

ⁿ

. Then the following three conditions are equivalent:

(1) the series P

∞

1

|f

_k

(z)| converges at each point z ∈ Ω, and its sum ϕ(z) := P

∞

1

|f

_k

(z)| is locally bounded on Ω;

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(2) the series P

∞

1

f

_k

converges locally normally in Ω, i.e. for every point a of Ω there exists a neighborhood U of a such that the series P

∞

1

kf

_k

k

_U

is convergent;

(3) the series P

∞

1

|f

_k

| converges locally uniformly in Ω.

Proof. It is clear that (2) ⇒ (3) ⇒ (1).

Suppose now (1) is true, and let E(a, r) := {z ∈ C

ⁿ

; |z

j

− a

_j

| < r (j = 1, . . . , n)} be a polydisk whose closure is contained in Ω. Then there is a positive constant M such that P

∞

1

|f

_k

(z)| ≤ M for all z ∈ E(a, r). By the Cauchy integral formula

|f

_k

(z)| ≤ µ

k

:= 1 πr

n

Z

2π 0

. . . Z

2π

0

|f

_k

(a

1

+ re

^it¹

, . . . , a

n

+ re

^itⁿ

)|dt

1

. . . dt

n

, for all z ∈ E(a,

^r₂

) and k ≥ 1.

By Lebesgue monotonous convergence theorem the series P

∞

1

µ

k

is con- vergent, and so is the series P

∞

1

kf

_k

k

_U

with U := E(a,

^r₂

). We shall see that our extensions of the classical Fatou–Hurwitz–Polya Theorem (Theorem 4.2.8 in [1]) are a direct consequence of the following Lemma 3.2 (slight modification of Lemma 5, p. 97 in [5]).

Lemma 3.2. Let X denote any of the two metric spaces {0, 1}

^N

or {−1, 1}

^N

. Let {f

_k

} be a sequence of holomorphic functions in an open neighborhood Ω of the closure of a ball B = B(w, r) such that the series P

∞

1

|f

_k

(z)|

converges at every point z ∈ B. Let a be a boundary point of B.

Then, either the series P

∞

1

f

k

is normally convergent on a neighborhood of a, or there exists a subset R of X of the first category such that for every x ∈ X \ R the function f

x

(z) := P

k

x

k

f

k

(z), z ∈ B, has a singular point at a (in other words, f

_x

cannot be analytically continued to any neighborhood of a).

Proof. Given a natural number m, let R

_m

denote the set of all x ∈ X such that there exists a holomorphic function ˜ f

_x

on E

_m

(where E

_m

is the polydisk E

_m

:= E a,

_m¹

with center a and radius

_m¹

) such that | ˜ f

_x

(z)| ≤ m on the polydisk, and ˜ f

x

(z) = f

x

(z) for all z ∈ B ∩ E

m

. By definition, we put R

_m

= ∅, if m < 1/ dist(a, ∂Ω).

It is clear that the set R := S

∞

1

R

_m

≡ {x ∈ X ; f

_x

has an analytic continuation across a}.

The lemma will be proved if we show that the following two claims are true.

Claim 1. The set R

m

is closed in the space X .

Claim 2. If the interior of R

m

is not empty, then the series P

∞ 1

f

k

is

normally convergent on a neighborhood of a.

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Indeed, if the series f

_x

:= P

∞

1

x

_k

f

_k

converges normally on no neighbor- hood U of a, then for every m ≥ 1 the set R

_m

is closed and has empty interior. Hence, the set R := S

∞

1

R

_m

≡ {x ∈ X ; f

_x

has an analytic con- tinuation ˜ f

x

across a} is of the first category, and for every x ∈ X \ R the function f

_x

has a singular point at a, i.e. f

_x

has no analytic continuation across a. We say that a function ˜ f

x

holomorphic on a polydisk E with center a is an analytic continuation of f

x

across a, if ˜ f

x

(z) = f

x

(z) on B ∩ E.

Proof of Claim 1. Let {x(j)} be a sequence of elements of R

_m

convergent to x ∈ X . Let {h

_j

} ≡ { ˜ f

_x(j)

} be a sequence of holomorphic functions on E

m

such that |h

_j

(z)| ≤ m on E

m

and h

_j

(z) = f

_x(j)

(z) on the intersection B ∩ E

_m

for j ≥ 1 . Observe that for every k

_o

there exists j

_o

such that

|f

_x(j)

(z) − f

x

(z)| ≤ P

k>ko

2|f

_k

(z)| for all z ∈ B ∩ E

m

and for all j > j

_o

. It follows that the sequence {h

_j

} is convergent at each point of B ∩ E

_m

. By Vitali’s theorem the sequence {h

_j

} is locally uniformly convergent on E

_m

to a holomorphic function h bounded by m and identical with f

_x

on E

_m

∩ B, which shows that x ∈ R

_m

.

Proof of Claim 2. If R

_m

has a nonempty interior, then there exist x(0) = (x

1

(0), x

2

(0), . . . ) ∈ R

m

and a natural number k

_o

such that

(*) x ∈ X , x

_j

= x

_j

(0) (j = 1, . . . , k

_o

) =⇒ x ∈ R

_m

. Put

M := sup (

_k₀

X

k=1

|f

_k

(z)|; z ∈ E

_m

)

, u

_k

:= <f

_k

, v

_k

:= =f

_k

.

By implication (2) ⇒ (3) of Remark 3.1 it is sufficient to show that (**)

∞

X

k=1

|f

_k

(z)| ≤ M + 4m, z ∈ E

_m

.

Let A be a finite subset of N \ [1, k

0

]. Given a fixed point z of E

_m

, put A

1

:= {k ∈ A; u

_k

(z) ≥ 0}, A

2

:= {k ∈ A; u

_k

(z) < 0}.

It is clear that A = A

₁

∪ A

₂

, A

₁

∩ A

₂

= ∅. Consider two cases.

Case 1: X = {0, 1}

^N

. Let x(j) = (x

₁

(j), x

₂

(j), . . . ) (j = 1, 2) be two points of the interior of R

_m

defined by the formulas:

x

_k

(j) = x

_k

(0), k = 1, . . . , k

0

, j = 1, 2;

x

_k

(j) = x

_k

(0), k > k

₀

, k / ∈ A, j = 1, 2;

x

_k

(1) = 1, x

_k

(2) = 0, k ∈ A

₁

;

x

_k

(1) = 0, x

_k

(2) = 1, k ∈ A

₂

.

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Then X

k∈A

|u

_k

(z)| ≤

X

k∈A

(x

_k

(1) − x

_k

(2))f

_k

(z)

= | ˜ f

_x(1)

(z) − ˜ f

_x(2)

(z)| ≤ 2m.

By the arbitrary property of A and z one gets

∞

X

k=k0+1

|u

_k

(z)| ≤ 2m, z ∈ E

m

. The same argument gives

∞

X

k=k0+1

|v

_k

(z)| ≤ 2m, z ∈ E

m

. Hence

∞

X

k=1

|f

_k

(z)| =





k0

X

k=1

+

∞

X

k=k0+1



 |f

_k

(z)| ≤ M + 4m, z ∈ E

_m

.

Case 2: X = {−1, 1}

^N

. Now we define two elements x(1), x(2) of the interior of R

_m

by the formulas:

x

_k

(j) = x

_k

(0), k = 1, . . . , k

₀

, j = 1, 2;

x

k

(j) = x

k

(0), k > k

0

, k / ∈ A, j = 1, 2;

x

_k

(1) = 1, x

_k

(2) = −1, k ∈ A

1

; x

_k

(1) = −1, x

_k

(2) = 1, k ∈ A

₂

. Then

2 X

k∈A

|u

_k

(z)| ≤

X

k∈A

(x

_k

(1) − x

2

(2)) f

_k

(z)

= | ˜ f

_x(1)

(z) − ˜ f

_x_k₍₂₎

(z)| ≤ 2m.

Hence, by the analogous argument as in the proof of the case 1, we get

∞

X

k=1

|f

_k

(z)| ≤ M + 4m, z ∈ E

_m

,

which ends the proof of the case 2.

Corollary 3.3. Let {f

k

} be a sequence of holomorphic functions on an open set Ω ⊂ C

^N

. Let D denote the set of all points a in Ω such that the series P

∞

1

f

k

is absolutely convergent at every point of a neighborhood of a. Assume that the sum ϕ(z) := P

∞

1

|f

_k

(z)| is locally bounded in D, and D ⊂ Ω. Let X be any of the two metric spaces {0, 1} ¯

^N

or {−1, 1}

^N

.

Then there exists a subset R of X of the first category such that for every point x ∈ X \ R the holomorphic function f

_x

(z) := P

∞

k=1

x

_k

f

_k

(z), z ∈ D,

cannot be continued analytically across any boundary point of D.

(12)

Proof. Let {w

_j

} be the sequence of all rational points of D (or any count- able dense subset of D). Let a

_j

be a point of ∂D such that kw

_j

− a

_j

k = dist(w

j

, ∂D). By Lemma 3.2 for every j there exists a subset R

j

of X of the first category such that for every x ∈ X \ R

_j

the function f

_x

has a singular point at a

_j

. The set R := S R

_j

is again of the first category such that for every x ∈ X \ R the function f

_x

has analytic extension across no boundary

point of D.

Proof of Theorems 4 and 5. It is sufficient to apply Lemma 3.2 with Ω = C

^N

, with f

_k

= Q

k

and f

_k

= c

_α(k)

z

^α(k)

(k ∈ Z

+

), respectively, where α : Z

+

3 k 7→ α(k) ∈ Z

^N+

is a one-to-one mapping, and with D replaced by the domain of convergence D of the corresponding power series. Remark 3.4. The author would like to draw reader’s attention to the fact that, unfortunately, the proofs of Theorems 4 and 5 published in [6] contain flaws.

References

[1] Bieberbach, L., Analytische Fortsetzung, Springer-Verlag, Berlin 1955.

[2] Jarnicki, M., Jakóbczak, P., Wstęp do teorii funkcji holomorficznych wielu zmiennych zespolonych, Wydawnictwo Uniwersytetu Jagiellońskiego, Kraków, 2002.

[3] Klimek, M., Pluripotential Theory, Oxford University Press, New York, 1991.

[4] Luh, W., Universal approximation properties of convergent power series on open sets, Analysis 6 (1986), 191–207.

[5] Saint Raymond, J., Transformations s´epar´ement analytiques, Ann. Inst. Fourier (Grenoble) 40 (1990), 79–101.

[6] Siciak, J., Generalizations of a theorem of Fatou, Zeszyty Naukowe UJ 11 (1966), 81–84.

[7] Siciak, J., Sets in C^N with vanishing global extremal function and polynomial approx- imation, Ann. Fac. Sci. Toulouse 20 (2011), no. S2, 189–209.

Józef Siciak

Institute of Mathematics Jagiellonian University Łojasiewicza 6

30-348 Kraków Poland

e-mail: Jozef.Siciak@im.uj.edu.pl Received May 16, 2011