NOTE ON THE ZEROS OF POLYNOMIALS

OF LEAST SQUARE APPROXIMATION

Jerzy Szczepański

IMUJ Preprint 1998/02

Abstract. Let the pairs (E, μ) and (F, ν) satisfy Leja’s polynomial condition. A version of Siciak’s theorem on separately analytic functions due to Nguyen and Zeriahi implies that a Bernstein type theorem on the distribution of the zeros of polynomials of least square approximation with respect to the measure μ ⊗ ν on the set E × F holds whenever this theorem holds on the sets E and F for the measures μ and ν, respectively.

0. Introduction

It was probably S.N. Bernstein [4], who first observed that the distribution of the zeros of the polynomials of least square approximation in a weighted L^{2}-norm to a function h defined on the interval I = [−1, 1] is tightly associated with the existence of an analytic extension of h in an open neigbourhood of I. Pleśniak [10] extended Bernstein’s result to the case of any compact subset E ⊂ ℂ and any finite positive measure μ such that the pair (E, μ) satisfies Leja’s polynomial condition. To the best of our knowledge, in the case of E ⊂ ℂ^{N}, N > 1, a characterization of analyticity of a function h defined on E in terms of the zeros of polynomials of least square approximation to h is known only for balls in a norm of ℂ^{N} (see [14], Corollary 5.5).

1980 Mathematics Subject Classification (1985 Revision). Primary 41A10, 30C15.

Key words and phrases. approximation by polynomials, zeros of polynomials, complex equilibrium measure.

Research partially supported by the grant No 2 PO3A 057 08 from KBN (Committee for Scientific Research) .

In this note we introduce a condition (Z) concerning a pair (E, μ), where E is a compact subset of ℂ^{N} and μ is a Borel measure on E. Applying a version of Siciak’s theorem on separately analytic functions due to Nguyen and Zeriahi [9], we prove that a pair (E × F, μ ⊗ ν) satisfies (Z) whenever (E, μ) and (F, ν) do it (Theorem 2.2).

As an aplication of Theorem 2.2 we give a Bernstein type characterization of analyticity of functions in terms of the distribution of zeros of polynomials of least square approximation on a compact set E ⊂ ℂ^{N} with respect to the measure μ such that (E, μ) satisfies (Z) (Proposition 2.9).

1. Definitions and notations

Let ℂ^{N} ∋ z → |z| be a fixed norm in ℂ^{N}. We shall denote by P_{n}(ℂ^{N}) the set of all polynomials of degree at most n, n ∈ ℕ_{0}, and by P(ℂ^{N}) the set of all polynomials.

Following [13], given a set E ⊂ ℂ^{N}, we define the pluricomplex extremal function

V_{E}(z) = sup{u(z), u ∈ L, u_{|E} ≤ 0},

where L denotes the Lelong class of plurisubharmonic functions on ℂ^{N} (psh(ℂ^{N})) satisfying

sup_{z∈ℂ^{N}} u(z) − log^{+} |z| < ∞,

where log^{+} |z| = max{0, log |z|}. We shall denote by V_{E}^{∗}(z) = lim sup_{w→z} V_{E}(w) its upper regularization and for r > 1 the set E_{r} = {z ∈ ℂ^{N} : V_{E}(z) < log r}.

Throughout this note we shall assume that the set E ⊂ ℂ^{N} is compact and L-regular i.e. the function V_{E} is continuouos in ℂ^{N}.

In this case it is known (see [13], Theorem 4.12) that

V_{E}(z) = sup_{n≥1} sup\{\frac{1}{n} log |p(z)| : p ∈ P_{n}(ℂ^{N}), ∥p∥_{E} ≤ 1\}.

Then, it follows that for every b > 1 there exists an open neighbourhood U of E such that for any p ∈ P(ℂ^{N})

(B-W)  |p(z)| ≤ b^{degp}∥p∥_{E}, as z ∈ U,

which is refered as to the Bernstein - Walsh inequality.

For a set E ⊂ ℂ^{N} and an open set D ⊂ ℂ^{N} we also define the relative extremal function associated with the pair (E, D) (see eg. [6]) by

ω(z, E, D) = sup{u(z) : u ∈ psh(ℂ^{N}), u ≤ 1, u_{|E∩D} ≤ 0},

and we denote its upper regularization by ω^{∗}(z, E, D) = lim sup_{w→z} ω(w, E, D).

Definition 1.1. (cf. [8]) Let E be a compact, L-regular subset of ℂ^{N} and let μ be a finite Borel measure on E. We say that the pair (E, μ) satisfies the Bernstein - Markov condition (BM) with exponent q ∈ (0, ∞) if for any ϵ > 0 there exists A_{ϵ} > 0 such that

∥p∥_{E} ≤ A_{ϵ}(1 + ϵ)^{deg p}∥p∥_{μ,q}

for any p ∈ P(ℂ^{N}), where ∥p∥_{μ,q} = ( ∫_{E} |p(z)|^{q}dμ(z))^{1/q} and ∥p∥_{E} = sup_{z∈E} |p(z)|.

By Hölder’s inequality, the function

(0, ∞) ∋ q ⟶ (\frac{1}{μ(E)} ∫_{E} |p(z)|^{q}dμ(z))^{1/q}

is increasing. Hence, for any q_{1}, q_{2} with 0 < q_{1} ≤ q_{2} < ∞, if the pair (E, μ) satisfies (BM) with q_{1} then it satisfies (BM) with q_{2} as well.

We also recall the following

Definition 1.2. Let E be a Borel subset of ℂ^{N} and μ be a finite Borel measure on E. We say that the pair (E, μ) satisfies Leja’s polynomial condition (and we write (E, μ) ∈ L^{∗}) if for every family F ⊂ P(ℂ^{N}) satisfying

μ({z ∈ E : sup_{f∈F} |f(z)| = ∞}) = 0

and for every b > 1 there exist M > 0 and a neighbourhood U of E such that

∥p∥_{U} ≤ Mb^{deg p}, for all p ∈ F.

Remark 1.3. (see [8], Théorème 3.1). Let E be a compact, L-regular subset of ℂ^{N}. If the pair (E, μ) satisfies Leja’s polynomial condition then it satisfies the Bernstein - Markov condition with any exponent q, 0 < q < ∞. In particular, let μ_{E} be the complex equilibrium measure of an L-regular, compact set E ⊂ ℂ^{N}, i.e. the Monge-Ampère measure (dd^{c}V_{E})^{N} associated with the pluricomplex extremal function V_{E} (see eg. [6], p. 212). Then, by [8], Théorème 2.1 and Théorème 3.1, the pair (E, μ_{E}) satisfies (BM) with any q ∈ (0, ∞).

For further examples and properties of (E, μ) satisfying (BM) see [8].

In this paper we shall consider the Hilbert space L_{P}^{2}(E, μ), the closure in the norm ∥ · ∥_{μ,2} of the algebra P(ℂ^{N}). We shall denote its norm by ∥p∥_{μ} := ∥p∥_{μ,2}.

Let α : ℕ_{0} ⟶ ℕ_{0}^{N} be a bijection satisfying |α(i)| ≤ |α(i+1)|. We denote by (e_{k}), k ∈ ℕ_{0}, the orthonormal polynomial Schauder basis of L_{P}^{2}(E, μ) associated with the monomials

(z^{α(k)}), k ∈ ℕ_{0}. Thus e_{k}(z) = \frac{t_{k}(z)}{∥t_{k}∥_{μ}}, where t_{k} realizes the infimum

∥t_{k}∥_{μ} = inf{∥z^{α(k)} + ∑_{i=0}^{k−1} c_{i}z^{α(i)}∥_{μ}, c_{i} ∈ ℂ}.

Let h ∈ L_{P}^{2}(E, μ). We denote by h_{n}(z) = ∑_{k=0}^{n} ∑_{|α(i)|=k} χ_{i}e_{i}(z), the orthogonal projection of h onto the space of polynomials of degree at most n, where χ_{i} = ∫_{E} h(z)\overline{e_{i}(z)}dμ(z) is the ith Fourier coefficient of h.

Remark 1.4. The polynomial h_{n} does not depend on the choice of the polynomial basis of L_{P}^{2}(E, μ), since L_{P}^{2}(E, μ) is a Hilbert space and h_{n} is the unique polynomial of best L^{2}-approximation in ∥ · ∥_{μ}-norm of degree at most n. In particular, h_{n} is the orthogonal projection of h onto P_{n}(ℂ^{N}).

Definition 1.5. Let E be a compact, L-regular subset of ℂ^{N} and let μ be a finite Borel measure on E. We say that the pair (E, μ) satisfies condition (Z) if there exists a continuous strictly increasing function ζ : (1, +∞) ∋ t → ζ(t) ∈ (1, +∞) such that for every continuous function h ∈ L_{P}^{2}(E, μ) the hypothesis that

(a1) there exists R > 1 such that for every r ∈ (1, R) there exists a complex number A such that h_{n}(z) ≠ A, for all n ∈ ℕ and all z ∈ \overline{E_{r}}

ensures that

(a2) h admits an analytic extension onto E_{ζ(R)}.

Remark 1.6. (cf. [14], Remark 3.3) If the pair (E, μ) satisfies (BM) with q = 2 then the converse implication (a2) ⟹ (a1) holds with ζ(t) = t whenever E is a compact, L-regular subset of ℂ^{N}. This follows from the Bernstein - Walsh inequality (B − W) and the Bernstein - Walsh - Siciak characterization of analytic functions ([12], Theorem 10.3).

Remark 1.7. (see [10], Theorem 4.1) Let E ⊂ ℂ be a compact, L-regular set and μ be a finite Borel measure on E. Then (E, μ) satisfies (Z) with ζ(t) = t whenever (E, μ) satisfies (BM) with q = 2.

To show this it suffices to use the following

Lemma 1.8. (cf. [10], Lemma 1.4) Let R > 1 and let p(z) = a(z − z_{1}) … (z − z_{n}) be a polynomial. If z_{i} ∈ ℂ \ E_{R}, for all i = 1, 2, …, n, then

|a| ≤ \frac{∥p∥_{E}}{(Rd(E))^{n}} ,

d(E) being the transfinite diameter of the set E. □

For, let h_{n}, R, r and A be as in hypothesis (a1) of condition (Z). Applying the above lemma to the polynomial p(z) := h_{n}(z) − A = ∑_{k=0}^{n} χ_{k}z^{k} − A we have, for all n ∈ ℕ_{0},

\frac{|χ_{n}|}{∥t_{n}∥_{μ}} ≤ \frac{∥h_{n}∥_{E} + |A|}{(rd(E))^{n}}.

Fix ϵ > 0. By (BM) there exists A_{ϵ} such that ∥h_{n}∥_{E} ≤ A_{ϵ}(1+ϵ)^{n}∥h_{n}∥_{μ}, for all n ∈ ℕ_{0}. Since lim_{n→∞} ∥t_{n}∥_{μ}^{1/n} = d(E) and since by the Bessel inequality ∥h_{n}∥_{μ} ≤ ∥h∥_{μ}, we get

lim sup_{n→∞} |χ_{n}|^{1/n} < (1 + ϵ)/r. This implies that lim sup_{n→∞} |χ_{n}|^{1/n} ≤ 1/r, since ϵ > 0 was chosen arbitrary. Thus, by the Bernstein - Walsh inequality (B-W) and condition (BM), the Fourier series ∑_{k=0}^{∞} χ_{k}e_{k} converges uniformly on compact subsets of E_{r} with r ∈ (1, R). Hence, h admits an analytic extension onto E_{R}. □

Let us note that in [10] Pleśniak originally assumed that the pair (E, μ) satisfies Leja’s polynomial condition. By Remark 1.3, (E, μ) satisfies then (BM).

It is known that if E ⊂ ℂ^{N} (N ≥ 1) is a ball in a norm in ℂ^{N} and μ is a finite Borel measure, then the pair (E, μ) satisfies (Z) (with ζ(t) = t) if it satisfies (BM) with q = 2. (cf. [14], Corollary 5.5).

2. Main result

Let E ⊂ ℂ^{N} and F ⊂ ℂ^{M} be compact, L-regular sets and let μ and ν be finite Borel measures on E and F, respectively. We fix two bijections α : ℕ_{0} → ℕ_{0}^{N} and β : ℕ_{0} → ℕ_{0}^{M} such that |α(i)| ≤ |α(i + 1)| and |β(j)| ≤ |β(j + 1)|, for all i, j ∈ ℕ_{0}. By the Hilbert-Schmidt orthogonalization process we can find two orthonormal polynomial Schauder bases {e_{i}(z)} and {f_{j}(w)} for L_{P}^{2}(E, μ) and L_{P}^{2}(F, ν), respectively, associated with the monomials {z^{α(i)}}, i ∈ ℕ_{0}, and {w^{β(j)}}, j ∈ ℕ_{0}.

Remark 2.1. {e_{i}(z)f_{j}(w)}, i, j = 0, 1, 2, …, is an orthonormal polynomial Schauder basis for L_{P}^{2}(E × F, μ ⊗ ν).

Indeed, if h ∈ L_{P}^{2}(E × F, μ ⊗ ν) then h = lim_{n→∞} h_{n} in L_{P}^{2}(E × F, μ ⊗ ν), where h_{n} ∈ P_{n}(ℂ^{N+M}) is the polynomial of degree at most n of the L^{2}− approximation to h in the ∥ · ∥_{μ⊗ν} norm (see Remark 1.4). For a fixed z ∈ ℂ^{N}, h_{n}(z, ·) is a polynomial of degree at most n in variable w ∈ ℂ^{M}. Since {f_{j}} is a Schauder basis of L_{P}^{2}(F, ν), we have h_{n}(z, ·) = ∑_{|β(l)|=0}^{n} χ_{l}(z)f_{l}(·), where χ_{l}(z) is a polynomial of degree at most n − |β(l)| in variable z. Since {e_{i}} is a Schauder basis of L_{P}^{2}(E, μ) there are complex numbers χ_{kl},

where k ∈ ℕ_{0}, |α(k)| ≤ n − |β(l)|, such that χ_{l}(z) = ∑_{|α(k)|=0}^{n−|β(l)|} χ_{kl}e_{k}(z). Hence h_{n}(z, w) =

∑_{|α(k)|+|β(l)|≤n} χ_{kl}e_{k}(z)f_{l}(w). Since ∫_{E×F} h_{n}\overline{e_{k}f_{l}}dμ ⊗ ν = χ_{kl} and h − h_{n} is orthogonal to

P_{n}(ℂ^{N+M}) we have ∫_{E×F} (h − h_{n})\overline{e_{k}f_{l}}dμ ⊗ ν = 0 for k, l such that |α(k)| + |β(l)| ≤ n. This implies that

χ_{kl} = ∫_{E×F} (h_{n} + (h − h_{n}))\overline{e_{k}f_{l}}dμ ⊗ ν = ∫_{E×F} h\overline{e_{k}f_{l}}dμ ⊗ ν.

Thus, h = ∑_{n=0}^{∞} ∑_{\begin{matrix}k,l \\ |α(k)|+|β(l)|=n \end{matrix}} χ_{kl}e_{k}f_{l}, where χ_{kl} = ∫_{E×F} h\overline{e_{k}f_{l}}dμ ⊗ ν. This means that

{e_{i}(z)f_{j}(w)}, i, j = 0, 1, 2, ... is a polynomial Schauder basis for L_{P}^{2}(E × F, μ ⊗ ν). By the Fubini theorem, ∫_{E×F} e_{i}f_{j}\overline{e_{i′}f_{j′}}dμ ⊗ ν = δ_{ii′}δ_{jj′}. Thus, the basis is orthonormal.

Theorem 2.2. Let E ⊂ ℂ^{N} and F ⊂ ℂ^{M} be compact sets. Assume that the pairs (E, μ) and (F, ν) satisfy Leja’s polynomial condition. If the pairs (E, μ) and (F, ν) satisfy (Z), so does the pair (E × F, μ ⊗ ν).

We shall need the following result of Nguyen and Zeriahi ([9], Théorème 1.1)

Theorem 2.3. Let E_{1} ⊂ ℂ^{N_{1}} and E_{2} ⊂ ℂ^{N_{2}} be nonpluripolar sets such that at least one of them is K-analytic. Let D_{1} ⊂ ℂ^{N_{1}} and D_{2} ⊂ ℂ^{N_{2}} be bounded open pseudoconvex sets such that E_{1} ⊂ D_{1} and E_{2} ⊂ D_{2}. Let f be a complex-valued function defined on X := D_{1} × E_{2} ∪ E_{1} × D_{2} that is separately analytic on X i.e. f(·, z_{2}) is analytic on D_{1} for all z_{2} ∈ E_{2} and f(z_{1}, ·) is analytic on D_{2} for all z_{2} ∈ E_{2}. Then there exists a function f̃ that is analytic on X̃ := {(z_{1}, z_{2}) ∈ D_{1} × D_{2} : ω^{∗}(z_{1}, E_{1}, D_{1}) + ω^{∗}(z_{2}, E_{2}, D_{2}) < 1} and coincides with f on X ∩ X̃.

Proof of Theorem 2.2. We assume that the pairs (E, μ) and (F, ν) satisfy (Z) with continuous increasing functions ζ_{E} and ζ_{F}, respectively. We shall prove that the pair

(E × F, μ ⊗ ν) satisfies (Z) with ζ(t) := exp ((1/ log ζ_{E}(t) + 1/ log ζ_{F} (t))^{−1}). It is seen

that ζ : (1, +∞) → (1, +∞) is a strictly increasing, continuous function whenever ζ_{E}, ζ_{F} : (1, +∞) → (1, +∞) are strictly increasing, continuous functions.

Fix a continuous function h ∈ L_{P}^{2}(E × F, μ ⊗ ν) and let

h_{n}(z, w) = ∑_{k=0}^{n} ∑_{|α(i)|+|β(j)|=k} χ_{ij}e_{i}(z)f_{j}(w),

where χ_{ij} = ∫_{E×F} h\overline{e_{i}f_{j}}dμ ⊗ ν. Assume that (a1) holds. Let

R = sup\{ρ > 1 : ℂ \ ⋃_{n∈ℕ_{0}} h_{n}(\overline{(E × F)_{ρ}}) ≠ ∅\}.

By [13], Proposition 5.9, we have V_{E×F} (z, w) = max{V_{E}(z), V_{F} (w)}. For every ρ > 1, the set (E × F)_{ρ} = {(z, w) ∈ ℂ^{N+M} : V_{E×F} (z, w) < log ρ} is open since E and F are L-regular. Since (a1) of (Z) holds, we get R > 1.

Let r ∈ (1, R). Chose A_{r} ∈ ℂ \ ⋃_{n∈ℕ_{0}} h_{n}(\overline{(E × F)_{r}}) ≠ ∅. Then h_{n}(z, w) − A_{r} ≠ 0 for

any (z, w) ∈ \overline{(E × F)_{r}} and any n ∈ ℕ_{0}.

Define

E′ = {z ∈ E : h_{n}(z, ·) converges to h(z, ·) in L_{P}^{2}(F, ν)},
F′ = {w ∈ F : h_{n}(·, w) converges to h(·, w) in L_{P}^{2}(E, μ)}.

If we denote by E_{n,k} = \{z ∈ E : ∫_{F} |h_{n}(z, w) − h(z, w)|^{2}dν(w) ≤ \frac{1}{k} \} and by F_{n,k} = \{w ∈ F : ∫_{F} |h_{n}(z, w) − h(z, w)|^{2}dμ(z) ≤ \frac{1}{k} \}, we see that

E′ = ⋂_{k∈ℕ_{0}} ⋃_{n_{0}∈ℕ_{0}} ⋂_{\begin{matrix}n∈ℕ_{0} \\ n≥n_{0} \end{matrix}} E_{n,k} and F′ = ⋂_{k∈ℕ_{0}} ⋃_{n_{0}∈ℕ_{0}} ⋂_{\begin{matrix}n∈ℕ_{0} \\ n≥n_{0} \end{matrix}} F_{n,k}.

Since h is continuous and the sets E and F are compact, the sets E_{n,k} and F_{n,k} are compact, thus E′ and F′ are Borel sets. In particular, E′ and F′ are K-analytic. (For a definition of K-analytic (or Souslin) sets see eg.[5]).

Since h_{n} converges to h in L_{P}^{2}(E × F, μ ⊗ ν), we have μ(E \ E′) = ν(F \ F′) = 0.

Fix z′ ∈ E′. Then \overline{\{z′\} × F_{r}} ⊂ \overline{(E × F)_{r}} and h_{n}(z′, w) − A_{r} ≠ 0 for any w ∈ F_{r} and any n ∈ ℕ_{0}.

Since the pair (F, ν) satisfies (Z) with a continuous increasing function ζ_{F}, the function h(z′, ·) admits an analytic extension onto F_{ζ_{F} (r)}. Repeating this argument for all r ∈ (1, R) we conclude that h(z′, ·) admits an analytic extension onto F_{ζ_{F} (R)}, since F is determining for analytic functions and ζ_{F} is continuous, increasing function.

Since the pair (E, μ) satisfies (Z) with a continuous inreasing function ζ_{E}, the same argument shows that h(·, w′) admits an analytic extension onto E_{ζ_{E}(R)}. Thus function h is the restriction of a function separately analytic on X = E′ × F_{ζ_{F} (R)} ∪ E_{ζ_{E}(R)} × F′. Since (E, μ) and (F, ν) satisfy L^{∗} and μ(E \ E′) = ν(F \ F′) = 0, we conclude that the sets E′ and F′ are nonpluripolar (see [11]). The sets E_{ζ_{E}(R)} and F_{ζ_{F} (R)} are polynomialy convex. Thus, by Levi’s theorem they are pseudoconvex.

By Theorem 2.3, there exists a function h̃ that is analytic in

X̃ = {(z, w) ∈ E_{R} × F_{R} : ω^{∗}(z, E′, E_{ζ_{E}(R)}) + ω^{∗}(w, F′, F_{ζ_{F} (R)}) < 1}

and coincides with h on X.

By [6], Proposition 5.3.3, we have

ω^{∗}(z, E′, E_{ζ_{E}(R)}) = \frac{V_{E′}^{∗}(z)}{log ζ_{E}(R)}, for z ∈ E_{ζ_{E}(R)},

ω^{∗}(z, F′, F_{ζ_{F} (R)}) = \frac{V_{F′}^{∗}(z)}{log ζ_{F} (R)}, for z ∈ F_{ζ_{F} (R)}.

Since the pairs (E, μ) and (F, ν) satisfy L^{∗} and μ(E\E′) = ν(F \F′) = 0, by Levenberg’s theorem ([7], Theorem 3.1, see also [11]), we get V_{E′}^{∗} = V_{E}^{∗} and V_{F′}^{∗} = V_{F}^{∗}. This implies that

X̃ = \{(z, w) ∈ ℂ^{N+M} : \frac{V_{E}(z)}{log ζ_{E}(R)} + \frac{V_{F} (w)}{log ζ_{F} (R)} < 1\}.

Let (z, w) ∈ (E × F)_{ζ(R)}. Then V_{E}(z) < log ζ(R) and V_{F} (w) < log ζ(R) and we get

\frac{V_{E}(z)}{log ζ_{E}(R)} + \frac{V_{F} (w)}{log ζ_{F} (R)} < log ζ(R)(\frac{1}{log ζ_{E}(R)} + \frac{1}{log ζ_{F} (R)}) = 1,

i.e. (E × F)_{ζ(R)} ⊂ X̃ and h admits an analytic extension onto (E × F)_{ζ(R)}.

This means that the pair (E × F, μ ⊗ ν) satisfies (Z). □

Remark 2.4. In the proof of Theorem 2.2 we can put ζ_{1}(t) := min\{\sqrt{ζ_{E}(t)}, \sqrt{ζ_{F} (t))}\}

instead of ζ(t) := exp ((1/ log ζ_{E}(t) + 1/ log ζ_{F} (t))^{−1}). There arises a natural question

of whether the pair (E, μ) satisfies (Z) with ζ_{1}(t) = t if it satisfies (Z) with a continuous increasing function ζ.

If B := E×F is a ball in a norm of ℂ^{N+M}, using a different technique (see [14], Corollary 5.5) one may prove that if (B, μ) satisfies (BM) then it satisfies (Z) with ζ(t) = t.

Let μ_{E} be the complex equilibrium measure of the compact set E ⊂ ℂ^{N}. It is known that μ_{E} is supported on the boundary of the polynomial hull of E and posseses the following product property.

Proposition 2.5. (cf. [3], Proposition 2.2) Let E_{1}, ..., E_{N} be compact, L-regular subsets of ℂ. Then μ_{E_{1}×···×E_{N}} = μ_{E_{1}} ⊗ ⋯ ⊗ μ_{E_{N}} . □

Question 2.6. Does the above property hold for compact sets E_{i} ⊂ ℂ^{N_{i}} for any dimension N_{i} ≥ 1?

Recently Baran proved that the answer to the question is affirmative whenever the sets E_{i} ⊂ ℝ^{N_{i}} ⊂ ℂ^{N_{i}}, i = 1, 2, ..., are balls in a norm of R^{N_{i}} (see [2], Theorem 3.12, see also [1], where several explicite formulae for μ_{E} are given).

Hence by Remark 1.3, Remark 1.7, Proposition 2.5 and by Theorem 2.2 we get

Example 2.7. Let E_{1}, ..., E_{N} be compact, L-regular subsets of ℂ. Then the pair

(E_{1} × ⋯ × E_{N}, μ_{E_{1}×···×E_{N}})

satisfies condition (Z).

In [14], Theorem 5.3, Corollary 5.5, we have proved that if E ⊂ ℂ^{N} is a ball in a norm of ℂ^{N} and the pair (E, μ) satisfies (BM) then it satisfies (Z) (with ζ(t) = t). This gives another

Example 2.8. Let E_{i} ⊂ ℂ^{N_{i}} be a compact set and let μ_{i} be a finite Borel measure such that the pair (E_{i}, μ_{i}) satisfy L^{∗}, i = 1, …, k. If for every i = 1, …, k we have N_{i} = 1 or E_{i} is a ball in a norm of ℂ^{N_{i}} then the pair (E_{1} × ⋯ × E_{k}, μ_{1} ⊗ ⋯ ⊗ μ_{k}) satisfies condition (Z).

The above example includes such pairs (E, μ) satisfying (Z) that E is neither a ball in ℂ^{N} nor a Cartesian product of one-dimensional compact, L-regular sets, e.g. E := E_{1} ×E_{2} ⊂ ℂ^{3}, where E_{1} := {(u, v) ∈ ℂ^{2} : |u|^{2} +|v|^{2} ≤ 1}, E_{2} = {z ∈ ℂ : |ℜz| ≤ 1, ℑz = 0} with the measure μ = μ_{E_{1}} ⊗ μ_{E_{2}}. (If the answer to Question 2.6 were affirmative then μ = μ_{E}).

As an application of Theorem 2.2 we present the following Bernstein type characterization of analyticity of functions defined on a compact set E ⊂ ℂ^{N} that we proved in [14], Corollary 5.5, in the particular case of E ⊂ ℂ^{N} being a ball in a norm of ℂ^{N} (cf. [10], Theorem 4.1, for N=1).

Proposition 2.9. Let E be a compact, L-regular subset of ℂ^{N} and let μ be a finite Borel measure such that the pair (E, μ) satisfies (BM). Let h ∈ L_{P}^{2}(E, μ) be a continuous function and let R > 1. Consider the following conditions

1) lim sup_{i→∞} |χ_{i}|^{1/|α(i)|} < \frac{1}{R};

2) function h admits an analytic extension onto E_{R};

3) for every r ∈ (1, R) there exists A_{r} ∈ ℂ such that h_{n}(z) − A ≠ 0 for any n ∈ ℕ_{0} and any z ∈ \overline{E_{r}};

4) function h admits an analytic extension onto an open neighbourhood of E;

5) lim sup_{i→∞} |χ_{i}|^{1/|α(i)|} < 1.

If the pair (E, μ) satisfies (Z) then (3) ⟹ (4)⟺(5).

If the pair (E, μ) satisfies (Z) with ζ(t) = t then (1)⟺(2)⟺(3).

Here h_{n}(z) = ∑_{k=0}^{n} ∑_{|α(i)|=k} χ_{i}e_{i}(z), χ_{i} = ∫_{E} f(z)\overline{e_{i}(z)}dμ(z) and (e_{i}), i ∈ ℕ_{0}, is the orthonormal polynomial Schauder basis for L_{P}^{2}(E, μ) associated with the monomials {z^{α(i)}}, i ∈ ℕ_{0}, where α : ℕ_{0} → ℕ_{0} is a bijection satisfying |α(n)| ≤ |α(n + 1)| for all n ∈ ℕ_{0}. □

Proof. The implications (2) ⟹ (1) and (4) ⟹ (5) easily follow from the Bernstein-Walsh-Siciak characterization of analytic functions ((BWS), [12], Theorem 10.3). The implications (1) ⟹ (2) and (5) ⟹ (4) follow from (BWS) and from condition (BM). The implication (2) ⟹ (3) easily follows from the Bernstein - Walsh inequality (B-W) and from (BWS) (see eg. [14], Remark 3.3).

Since the pair (E, μ) satisfies (Z) by Theorem 2.2, we get the implication (3) ⟹ (4) and (3) ⟹ (2).

□

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Institute of Mathematics, Reymonta 4, 30 059 Kraków, Poland e-mail: szczepan@im.uj.edu.pl