ZEROS OF POLYNOMIALS

APPROXIMATING ANALYTIC FUNCTIONS

Jerzy Szczepański

IMUJ Preprint 1997/23

Abstract. We give some sufficient conditions for the analyticity of an extension of a continuous function f outside a compact set E ⊂ ℂ^{N} in terms of the zeros of polynomials of best uniform or least square approximation to f on the set E.

1. Introduction

The constructive theory of functions discusses the relationship between the regularity of a given function f and the rate of convergence to zero of the sequence

dist(f, A_{n}) = inf{d(f, p) : p ∈ A_{n}},

where (A_{n}) is most often a sequence of finite dimensional subspaces of a complete metric space (X, d), which is increasing with respect to inclusion, A_{k} ⊂ A_{k+1}, and whose union ⋃_{n∈ℕ_{0}} A_{k} is dense in X.

In the case of polynomial approximation of functions defined on a compact set E ⊂ ℂ, an answer to the problem is given by the Bernstein-Walsh theorem ([11]), which may be to some extent enriched by the information about the distribution of the zeros of polynomials of best or near best approximation. This was first observed in one-dimensional case by Bernstein [1], who stated the following Theorem 1.1. Let f be a complex-valued continuous function defined on the real interval I = [−1, 1] ⊂ {z ∈ ℂ : ℜz = 0}. Then f admits an analytic continuation onto interior of the ellipse E_{ρ} = {z ∈ ℂ : |z − 1| + |z + 1| ≤ 2ρ} if and only if all limit points of the zeros of polynomials which are nearest to f in a weighted L^{2}-norm on I lay outside E_{ρ}.

1991 Mathematics Subject Classification. Primary 41A10, 30C15.

Key words and phrases. zeros of polynomials, best uniform approximation, Fourier expansion.

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

This result was a subject of numerous attempts to generalize it. Pleśniak [6] extended Bernstein’s theorem to the case of polynomial approximation in a L^{2}(E, μ)- norm for an arbitrary compact set E ⊂ ℂ and a finite positive measure μ satisfying a Leja polynomial condition. Borwein [4] proved Bernstein’s theorem for uniform polynomial approximation on the interval I = [−1, 1]. Later, Blatt and Saff [2] and, independently, Wójcik [12] generalized Borwein’s result in the case of polynomial approximation of an analytic function on a compact and regular (with respect to the Dirichlet problem) subset E ⊂ ℂ.

However, all these results concerned the approximation by holomorphic polynomials in one complex variable.

In this paper we give some sufficient conditions for the existence of the holomorphic extension of a function f continuous on a compact set E ⊂ ℂ^{N} in terms of the zeros of polynomials p_{n} of best uniform approximation (Theorem 3.4) or the zeros of polynomials of least square approximation (Theorem 5.4) on the set E and two constants ρ(E) and ρ_{∗}(E) which depend on the geometry of the set E.

2. Definitions and preliminary remarks

Let ℂ^{N} ∋ z → |z| be a fixed norm in ℂ^{N}. We shall denote by

p̂_{n}(z) = ∑_{|α|=n} a_{α}z^{α} ∈ H_{n}(ℂ^{N})

the homogeneous part of degree n of the polynomial

p_{n}(z) = ∑_{|α|≤n} a_{α}z^{α} ∈ P_{n}(ℂ^{N}),

where P_{n}(ℂ^{N}) = { ∑_{|α|≤n} a_{α}z^{α}, a_{α} ∈ ℂ, α ∈ ℕ_{0}^{N}, |α| = α_{1} + ⋯ + α_{N} ≤ n} is the

set of all polynomials of degree at most n, n ∈ ℕ_{0}, and

H_{k}(ℂ^{N}) = { ∑_{|α|=k} a_{α}z^{α}, a_{α} ∈ ℂ, α ∈ ℕ_{0}^{N}, |α| = α_{1} + ⋯ + α_{N} = k}

is the set of all homogeneous polynomials of degree k, k ≥ 0. Here ℕ_{0} denotes the set of all nonnegative integers and z^{α} = z_{1}^{α_{1}} … z_{N}^{α_{N}}.

Moreover, by P = P(ℂ^{N}) = ⋃_{n≥0} P_{n}(ℂ^{N}) and H = H(ℂ^{N}) = ⋃_{k≥0} H_{k}(ℂ^{N}) we shall denote the set of all polynomials and homogeneous polynomials, respectively.

For a polynomial p ∈ P(ℂ^{N}) we denote by {p = 0} = {z ∈ ℂ^{N}, p(z) = 0} the set of its zeros.

Definition 2.1. (see e.g. [8]) For a compact subset E ⊂ ℂ^{N} we define the 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} satisfying

sup_{z∈ℂ^{N}} u(z) − log^{+} |z| < ∞,

where log^{+} |z| = max{0, log |z|}.

Note that the definition of the Lelong class does not depend on the choice of a norm ℂ^{N} ∋ z → |z|.

Remark 2.2. If ℂ^{N} ∋ z → |z| ∈ ℂ^{N} is a norm in ℂ^{N} and B = {z ∈ ℂ^{N}, |z − a| ≤ r\} then V_{B}(z) = log^{+} |\frac{z−a}{r} |.

Throughout the paper we shall assume that the compact E ⊂ ℂ^{N} is regular i.e. the function V_{E} is continuouos in ℂ^{N}.

For R > 1 and δ > 0 we shall consider the open sets

E_{R} = {z ∈ ℂ^{N}, V_{E}(z) < log R}

and

E_{(δ)} = {z ∈ ℂ^{N}, dist(z, E) < δ},

where dist(z, E) = inf_{w∈E} |z − w|.

For a continuous function f : ℂ^{N} ⊃ E → ℂ we denote its supremum norm on E by

∥f∥_{E} = sup_{z∈E} |f(z)|

and the error of the polynomial approximation of f on E by

E_{n}(f, E) = dist_{E}(f, P_{n}(ℂ^{N})) = inf{∥f − p∥_{E} : p ∈ P_{n}(ℂ^{N})}.

Let W (E) denote the closure of the set P(ℂ^{N}) in the space of continuous functions C(E) with the norm f → ∥f∥_{E}.

Any polynomial p_{n} ∈ P_{n}(ℂ^{N}) satisfying E_{n}(f, E) = ∥f − p_{n}∥_{E} is called a polynomial of best uniform approximation of f on the set E. Notice that p_{n} need not be unique nor deg p_{n} = n.

Let A = [a_{ij}] ∈ ℂ^{N×N} be a complex matrix. Let us consider the following sets of affine nondegenerate maps in ℂ^{N}

\begin{aligned}G_{0} & = \{z → z + a, a ∈ ℂ^{N}\}, \\ G_{1} & = \{z → Az + a, | det A| = 1, a ∈ ℂ^{N}\}, \\ G_{2} & = \{z → Az + a, det A ≠ 0, a ∈ ℂ^{N}\}. \end{aligned}

Remark 2.3. For any E ⊂ ℂ^{N} and g ∈ G_{2} we have

V_{g(E)} = V_{E} ∘ g^{−1} and g(E_{R}) = (g(E))_{R}.

For i = 0, 1, 2, let us define the following classes of polynomials

\begin{aligned}H_{k} ∘ G_{i} & = \{h ∘ g_{i}, h ∈ H_{k}(ℂ^{N}), g_{i} ∈ G_{i}\}, \\ P_{k} ∘ G_{i} & = \{p ∘ g_{i}, p ∈ P_{k}(ℂ^{N}), g_{i} ∈ G_{i}\}. \end{aligned}

Remark 2.4. Notice that

H_{k} ⊂ H_{k} ∘ G_{0} ⊂ H_{k} ∘ G_{1} ⊂ H_{k} ∘ G_{2} ⊂ H_{k} ∘ G_{0} ⊂ P_{k}

and

P_{k} ⊂ P_{k} ∘ G_{0} ⊂ P_{k} ∘ G_{1} ⊂ P_{k} ∘ G_{2} ⊂ P_{k} ∘ G_{0} ⊂ P_{k}

and, in particular, deg(h ∘ g) = k for any h ∈ H_{k}, h ≠ 0, and g ∈ G_{2}.

Indeed, for any h ∈ H(ℂ^{N}) and g_{2}(z) = Az + a, the polynomial

h_{0} = h ∘ A : ℂ^{N} ∋ z → h(Az) ∈ ℂ

is homogeneous and g_{0}(z) = z + A^{−1}a is a translation, which means that h ∘ g ≡ h_{0} ∘ g_{0} ∈ H ∘ G_{0}.

We shall denote by ∂^{∗}E the closure of the set

⋃_{\begin{matrix}h∈H∘G_{0} \\ h≠const \end{matrix}}\{z ∈ E : |h(z)| = ∥h∥_{E}\}.

By the maximum principle we have ∂^{∗}E ⊂ ∂E, where ∂E denotes the topological boundary of E.

We denote by B(a, r) = {z ∈ ℂ^{N}, |z − a| < r} the open ball of radii r centered at a in a fixed norm of ℂ^{N}. We put B(a, 0) = ∅.

Definition 2.5. For a compact set E ⊂ ℂ^{N}, we define two constants ρ(E) and ρ_{∗}(E) and two sets E^{ρ} and E^{ρ_{∗}} as follows:

ρ(E) = sup_{a∈E} sup{r ≥ 0, B(a, r) ∩ ∂E = ∅}

E^{ρ} = {a ∈ E : B(a, ρ(E)) ∩ ∂E = ∅}.

Similarly, we define

ρ_{∗}(E) = sup_{a∈E^{ρ}} sup{r ≥ 0 : B(a, r) ∩ ∂^{∗}E = ∅},

E^{ρ_{∗}} = {a ∈ E^{ρ} : B(a, ρ_{∗}(E)) ∩ ∂^{∗}E = ∅}.

Let us note that the sets E^{ρ} and E^{ρ_{∗}} are nonempty and compact whenever the set E ⊂ ℂ^{N} is nonempty and compact. Moreover, if ρ(E) = 0 (ρ_{∗}(E) = 0, respectively) then E^{ρ} = E (E^{ρ_{∗}} = E^{ρ}, respectively).

Since ∂^{∗}E ⊂ ∂E we have E \ ∂^{∗}E ⊃ E \ ∂E, which implies the inequality ρ_{∗}(a) ≥ ρ(a) for all a ∈ E^{ρ} and \frac{1}{2}diam(E) ≥ ρ_{∗}(E) ≥ ρ(E).

In general ∂^{∗}E is much smaller than ∂E. For the interval E = [−1, 1] ⊂ ℂ we have ∂_{∗}E = {−1, 1} whereas ∂E = E. In this case we have also E^{ρ} = E and ρ(E) = 0 whereas E^{ρ_{∗}} = {0} and ρ_{∗}(E) = 1.

Remark 2.6. Notice that the following two inequalities hold (i) dist(a, ℂ^{N} \ \overline{E_{(δ)}}) ≥ δ + ρ(E) for any a ∈ E^{ρ}; (ii) dist(a, ∂^{∗}E) ≤ diamE − ρ_{∗}(E) for any a ∈ E^{ρ_{∗}}.

It is evident that the definitions of E^{ρ}, E^{ρ_{∗}}, ρ(E) and ρ_{∗}(E) may depend on the choice of a norm in ℂ^{N} if N > 1. If E is a closed ball (in the norm of ℂ^{N} which has been fixed before defing E^{ρ}, E^{ρ_{∗}}, ρ(E) and ρ_{∗}(E)) i.e. E = {z ∈ ℂ^{N} : |z − a| ≤ r} then E^{ρ} = E^{ρ_{∗}} = {a} and ρ(E) = ρ_{∗}(E) = r.

3. Zeros of polynomials of best uniform approximation

Let E be a compact regular subset of ℂ^{N} and let (p_{n})_{n∈ℕ_{0}}, p_{n} ∈ P_{n}(ℂ^{N}), be any sequence of polynomials of best uniform approximation to a nonconstant function f ∈ W (E).

In the multidimensional case, Siciak ([7] Theorem 10.3) generalized the Bernstein - Walsh theorem as follows:

Theorem 3.1. A necessary and sufficient condition that a function f be holomorphic in E_{R}, R > 1, is that

lim sup_{n→∞} ∥f − p_{n}∥_{E}^{1/n} ≤ \frac{1}{R}.

We recall also a one-dimensional Wójcik’s result ([12], Theorem 3) concerning the zeros of polynomials of best uniform approximation of a function f.

Theorem 3.2. Let E be a compact and regular subset of ℂ^{N} and let f ∈ W (E). Let R > 1. Then the following conditions are equivalent:

(i) there exists an extension F of f which is holomorphic in E_{R};

(ii) for any r ∈ (1, R), there exists A_{r} ∈ ℂ such that

p_{n}(z) − A_{r} ≠ 0 for all n ∈ ℕ_{0} and z ∈ \overline{E_{r}};

(iii) lim sup_{n→∞} |a_{n}|^{\frac{1}{n}} ≤ \frac{1}{Rd(E)},

where a_{n} is the nth coefficient of the polynomial p_{n}(z) = a_{n}z^{n} + ⋯ + a_{0} and d(E) is the Tchebyshev constant of the set E, i.e. d(E) = lim_{n→∞} d_{n}(E)^{1/n} where d_{n}(E) = inf{∥z^{n} + q∥_{E} : q ∈ P_{n−1}(ℂ^{N})}.

In the case of several complex variables, some counterparts of the implications (i) ⟹ (iii) and (iii) ⟹ (i) from the above theorem were studied by the author [10] and recently by Bloom [3].

Remark 3.3. The implication (i) ⟹ (ii) is valid for any dimension N ≥ 1.

Indeed, if f admits a continuation F that is holomorphic in E_{R} then, by Theorem 3.1, for any r_{1} ∈ (r, R) there exists a sequence of polynomials q_{n}, n ∈ ℕ_{0}, which converges to F uniformly on \overline{E_{r_{1}}} and

lim sup_{n→∞} ∥F − q_{n}∥_{E}^{1/n} ≤ \frac{1}{r_{1}} .

This implies that there exist M > 0 and n_{0} ∈ ℕ_{0} such that

∥F − q_{n}∥_{E} ≤ Mr_{1}^{−n} for all n ≥ n_{0}.

Since ∥f − p_{n}∥_{E} ≤ ∥f − q_{n}∥_{E} and F_{|E} = f, by the Bernstein-Walsh inequality, we have

∥p_{n} − q_{n}∥\overline{E_{r}} ≤ r^{n}∥p_{n} − q_{n}∥_{E} ≤ 2M(\frac{r}{r_{1}} )^{n},

which implies that

∥p_{n}∥\overline{E_{r}} ≤ 2M(\frac{r}{r_{1}} )^{n} + ∥q_{n}∥\overline{E_{r}}.

Thus,

A_{r} = 1 + sup_{n≥0} ∥p_{n}∥\overline{E_{r}} < ∞

and we obtain that p_{n}(z) − A_{r} ≠ 0 for all z ∈ \overline{E_{r}} and n ∈ ℕ_{0}.

Theorem 3.4. Assume that there exist a norm in ℂ^{N}, a positive number δ > 0, a complex number A and a positive integer n_{0} such that

E_{(δ)} ∩ ⋃_{n≥n_{0}}{p_{n} = A} = ∅.

If δ > diamE − ρ^{∗}(E) − ρ(E) then f admits an extension that is holomorphic in

E_{R} with R = \frac{δ + ρ(E)}{diamE − ρ^{∗}(E)}.

If E is the closed ball of radii r centred at a point a then we have ρ(E) = ρ_{∗}(E) = r. Since E_{1+ \frac{δ}{r}} = E_{(δ)} we get by Remark 3.3 and Theorem 3.4 the following

Corollary 3.5. Let E be a closed ball in a norm of ℂ^{N} and let f ∈ W (E). Let R > 1. Then the following two conditions are equivalent

1) f admits an extension that is holomorphic in E_{R};

2) for any r ∈ (1, R) there exists A_{r} ∈ ℂ such that p_{n}(z) − A_{r} ≠ 0 for all n ∈ ℕ and all z ∈ \overline{E_{r}}.

4. Auxiliary lemmas

Following an idea of Bloom [3], for a homogeneous polynomial h ∈ H_{n}(ℂ^{N}) we denote by T ch_{E}h a Tchebyshev polynomial associated with h minimizing the norm ∥h + q∥_{E}, q ∈ P_{n−1}(ℂ^{N}) i.e.

∥T ch_{E}h∥_{E} = inf{∥h + q∥_{E}, q ∈ P_{n−1}(ℂ^{N})}.

We recall a version of the Bernstein-Walsh-Siciak theorem (see Theorem 3.1) due to Bloom [3]:

Theorem 4.1. Let E be a compact and regular subset of ℂ^{N} and let f ∈ W (E). Then the following conditions are equivalent:

1) f admits an extension that is holomorphic in E_{R};

2) lim sup_{n→∞} ∥f − p_{n}∥_{E}^{1/n} ≤ \frac{1}{R};

3) lim sup_{n→∞} ∥T ch_{E}p̂_{n}∥_{E}^{1/n} ≤ \frac{1}{R}, where (p_{n}), p_{n} ∈ P_{n}(ℂ^{N}), is a sequence of polynomials of best uniform approximation to f on E. (Let us recall that p̂_{n} = 0 if deg p_{n} < n.)

We owe the idea of the proof of the next lemma to M. Simkani [9], who first proved it in the case where E = {(z, w) ∈ ℂ^{2}, |z| ≤ 1, |w| ≤ 1} and g(z, w) = (z, w) and announced his result during his stay in Cracow in 1991.

Lemma 4.2. If {p_{n} = 0} ∩ E = ∅ and deg p_{n} = n > 1 then for any g ∈ G_{2} and any point z_{g}^{∗} ∈ ∂^{∗}E satisfying ∥(p_{n} ∘ g)ˆ ∘ g^{−1}∥_{E} = |(p_{n} ∘ g)ˆ (g^{−1}(z_{g}^{∗}))| we have

∥(p_{n} ∘ g)ˆ ∘ g^{−1}∥_{E} = |p_{n}(g(0))|\frac{|z_{g}^{∗} − g(0)|^{n}}{∏_{l=1}^{n} |z_{l} − g(0)|}

where {z_{l}, l = 1, 2, …, n} = {p_{n} = 0} ∩ {(1 − ζ)g(0) + ζz_{g}^{∗}, ζ ∈ ℂ}.

Proof. By Remark 2.4, for any g ∈ G_{2} we have deg(p_{n} ∘ g)ˆ ∘ g^{−1} = n. The regularity of E implies that E is determining for polynomials. In particular, E is determining for the family H ∘ G_{0}, which means that ∥(p_{n} ∘ g)ˆ ∘ g^{−1}∥_{E} > 0 and z_{g}^{∗} ≠ g(0). If it were z_{g}^{∗} = g(0) then (p_{n} ∘ g)ˆ g^{−1}(g(0)) = (p_{n} ∘ g)ˆ (0) = 0, contradiction.

Let us define h_{g} : ℂ ∋ ζ → (1 − ζ)g(0) + ζz_{g}^{∗} ∈ ℂ^{N}. Since g(z) = Az + g(0) we have g^{−1}(w) = A^{−1}w − A^{−1}g(0) = A^{−1}(w − g(0)), which gives

(g^{−1} ∘ h_{g})(ζ) = A^{−1}(ζ(z_{g}^{∗} − g(0))) = ζA^{−1}(z_{g}^{∗} − g(0)) = ζg^{−1}(z_{g}^{∗}).

Let (p_{n} ∘ g)(w) = ∑_{|α|≤n} a_{α}w^{α}, in particular, (p_{n} ∘ g)ˆ (w) = ∑_{|α|=n} a_{α}w^{α}. Next, let us compute

\begin{aligned}(p_{n} ∘ h_{g})(ζ) & = (p_{n} ∘ g) ∘ (g^{−1} ∘ h_{g})(ζ) = ∑_{|α|≤n} a_{α}(ζg^{−1}(z_{g}^{∗}))^{α} \\ & = ∑_{k=0}^{n}( ∑_{|α|=k} a_{α}(g^{−1}(z_{g}^{∗}))^{α})ζ^{k} = ( ∑_{|α|=n} a_{α}(g^{−1}(z_{g}^{∗}))^{α}) ∏_{l=1}^{n}(ζ − ζ_{l}), \end{aligned}

with

| ∑_{|α|=n} a_{α}(g^{−1}(z_{g}^{∗}))^{α}| = ∥(p_{n} ∘ g)ˆ ∘ g^{−1}∥_{E} > 0.

This implies that

|(p_{n} ∘ h_{g})(0)| = | ∑_{|α|=n} a_{α}(g^{−1}(z_{g}^{∗}))^{α}| ∏_{l=1}^{n} |ζ_{l}|

where

{ζ_{l}, l = 1, 2, …, n} = {ζ ∈ ℂ : (p_{n} ∘ h_{g})(ζ) = 0}

(the zeros of the polynomial p_{n} ∘ h_{g} are to be counted with their multiplicities). Putting z_{l} = (1 − ζ_{l})g(0) + ζ_{l}z_{g}^{∗} = ζ_{l}(z_{g}^{∗} − g(0)) + g(0) we have

|z_{l} − g(0)| = |ζ_{l}| · |z_{g}^{∗} − g(0)|.

Finally, we obtain

|p_{n}(g(0))| = ∥(p_{n} ∘ g)ˆ ∘ g^{−1}∥_{E} \frac{∏_{l=1}^{n} |z_{l} − g(0)|}{|z_{g}^{∗} − g(0)|^{n}}

where

{z_{l}, l = 1, 2, …, n} = {p_{n} = 0} ∩ {(1 − ζ)g(0) + ζz_{g}^{∗}, ζ ∈ ℂ}.
□

Let p_{n} ∈ P_{n}(ℂ^{N}). Since for any g ∈ G_{2} and any constant A ∈ ℂ the homogeneous parts of the polynomials ((p_{n} − A) ∘ g)ˆ ∘ g^{−1} and T ch_{E}p̂_{n} coincide we get the following estimate of the norm ∥T ch_{E}p̂_{n}∥_{E}.

Lemma 4.3. Let p_{n} be any polynomial of degree not greater than n. Let A be a complex number such that p_{n}(z) − A ≠ 0 for all z ∈ E_{(δ)}. Then the following estimate holds

∥T ch_{E}p̂_{n}∥_{E} ≤ (∥p_{n}∥_{E} + |A|)\frac{diamE − ρ_{∗}(E)}{δ + ρ(E)}.

Proof. Applying Remark 2.6 and Lemma 4.2 to the polynomial z → p_{n}(z) − A and g ∈ G_{2} such that g(0) ∈ E^{ρ_{∗}} we get the assertion of the lemma in the case of deg p_{n} = n. If deg p_{n} < n then p̂_{n} = 0 and T ch_{E}p̂_{n} = 0, thus the estimate given by the lemma is valid as well. □

Proof of Theorem 3.4. Let (p_{n})_{n∈ℕ} be any sequence of polynomials of best uniform approximation to the function f ∈ W (E). If for δ > diamE −ρ^{∗}(E)−ρ(E) there exists a complex number A and a positive integer n_{0} such that p_{n}(z) ≠ A for all n ≥ n_{0} and all z ∈ \overline{E_{(δ)}} then, by Lemma 4.3 and the fact that the norms ∥p_{n}∥_{E} are bounded, we get

lim sup_{n→∞} ∥T ch_{E}p̂_{n}∥_{E}^{1/n} ≤ \frac{diamE − ρ_{∗}(E)}{δ + ρ(E)} < 1.

Hence, by Theorem 4.1, f admits an extension that is holomorphic in E_{R} with R = \frac{δ+ρ(E)}{diamE−ρ_{∗}(E)}.

5. An application to least square approximation

Let E be a compact and 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 if for any ϵ > 0 there exists a constant A_{ϵ} such that

∥p∥_{E} ≤ A_{ϵ}(1 + ϵ)^{deg p}( ∫_{E} |p(z)|^{2}dμ(z))^{1/2}, for any p ∈ P(ℂ^{N}),

(see e.g. [5]).

We denote by L_{P}^{2}(E, μ) the closure of the algebra P(ℂ^{N}) in the norm f →

∥f∥_{μ} = ( ∫_{E} |f(z)|^{2}dμ(z))^{1/2}.

Following an idea of Bloom [3] we introduce the Tchebyshev polynomial T ch_{μ}h_{n} associated with the polynomial h_{n} ∈ H_{n}(ℂ^{N}) and the measure μ as follows

∥T ch_{μ}h_{n}∥_{μ} = inf{∥h − q∥_{μ}, q ∈ P_{n−1}(ℂ^{N})}.

The polynomial T ch_{μ}h_{n} is unique since h_{n} − T ch_{μ}h_{n} is the nearest polynomial to h_{n} in the subspace P_{n−1}(ℂ^{N}) of the Hilbert space L_{P}^{2}(E, μ).

Let α : ℕ_{0} → ℕ_{0}^{N} be any bijection satisfying |α(i)| ≤ |α(i + 1)| for any i ∈ ℕ_{0}, where |α| = α_{1} + ⋯ + α_{N} denotes the length of α = (α_{1}, …, α_{N}).

We choose any sequence of polynomials (e_{i}), i ∈ ℕ_{0} such that

(b1) deg e_{i} = |α(i)|;

(b2) ∫_{E} e_{i}\overline{e_{j}}dμ = 0 if i ≠ j and ∫_{E} |e_{i}|^{2}dμ = 1;

(b3) the homogeneous parts of degree n of the polynomials e_{i} with |α(i)| = n form a basis of H_{n}(ℂ^{N}).

We may take e_{i}(z) := t_{i}(z)/∥t_{i}∥_{μ} where t_{i} realizes the infimum

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

Remark 5.1. Let (e_{i}) be a sequence of polynomials satisfying conditions b1, b2 and b3 with respect to a measure μ and let h ∈ H_{n}(ℂ^{N}). If h(z) = ∑_{|α(i)|=n} a_{i}ê_{i}(z) then T ch_{μ}h_{n}(z) = ∑_{|α(i)|=n} a_{i}e_{i}(z). Moreover, the norm

∥T ch_{μ}h_{n}(z)∥_{μ} = ( ∑_{|α(i)|=n} |a_{i}|^{2})^{1/2}

does not depend on the choice of the sequence (e_{i}).

Let (p_{n}) be a sequence of any polynomials such that deg p_{n} ≤ n. Assume that the pairs (E, μ) and (E, ν) satisfy the Bernstein-Markow condition. We shall consider the Tchebyshev polynomials T ch_{E}p̂_{n}, T ch_{μ}p̂_{n} and T ch_{ν} p̂_{n} associated with the homogeneous parts of degree n of the polynomial p_{n}. (Note that p̂_{n} = 0 if deg p_{n} < n.)

Lemma 5.2. Let (p_{n})_{n∈ℕ} be a sequence of any polynomials such that deg p_{n} ≤ n. Then the following equalities hold

lim sup_{n→∞} ∥T ch_{E}p̂_{n}∥_{E}^{1/n} = lim sup_{n→∞} ∥T ch_{E}p̂_{n}∥_{μ}^{1/n} = lim sup_{n→∞} ∥T ch_{μ}p̂_{n}∥_{μ}^{1/n}

= lim sup_{n→∞} ∥T ch_{μ}p̂_{n}∥_{E}^{1/n} = lim sup_{n→∞} ∥T ch_{μ}p̂_{n}∥_{ν}^{1/n}

= lim sup_{n→∞} ∥T ch_{ν} p̂_{n}∥_{μ}^{1/n}.

Proof. Since the pairs (E, μ) and (E, ν) satisfy the Bernstein-Markov condition, by the definition of the Tchebyshev polynomials we have that for any ϵ > 0 there are A_{ϵ} and B_{ϵ} such that for any polynomial p_{n} ∈ P_{n}(ℂ^{N}) the following inequalities hold

\begin{aligned}∥T ch_{E}p̂_{n}∥_{E} & ≤ ∥T ch_{μ}p̂_{n}∥_{E} ≤ A_{ϵ}(1 + ϵ)^{n}∥T ch_{μ}p̂_{n}∥_{μ} ≤ A_{ϵ}(1 + ϵ)^{n}∥T ch_{E}p̂_{n}∥_{μ} \\ & ≤ A_{ϵ}(1 + ϵ)^{n}μ(E)^{1/2}∥T ch_{E}p̂_{n}∥_{E} \end{aligned}

and, similarly,

∥T ch_{E}p̂_{n}∥_{E} ≤ ∥T ch_{ν} p̂_{n}∥_{E} ≤ A_{ϵ}(1 + ϵ)^{n}∥T ch_{ν} p̂_{n}∥_{μ}
≤ A_{ϵ}(1 + ϵ)^{n}μ(E)^{1/2}∥T ch_{ν} p̂_{n}∥_{E}
≤ A_{ϵ}B_{ϵ}(1 + ϵ)^{2n}μ(E)^{1/2}∥T ch_{ν} p̂_{n}∥_{ν}
≤ A_{ϵ}B_{ϵ}(1 + ϵ)^{2n}μ(E)^{1/2}∥T ch_{E}p̂_{n}∥_{ν}
≤ A_{ϵ}B_{ϵ}(1 + ϵ)^{2n}μ(E)^{1/2}ν(E)^{1/2}∥T ch_{E}p̂_{n}∥_{E}.

Raising the above inequalities to the power \frac{1}{n} and taking the lim sup we get the assertion of the lemma since ϵ > 0 has been chosen arbitrarily. □

Let (e_{i}) be a sequence of polynomials satisfying conditions (b1-b3) and let f ∈ L_{P}^{2}(E, μ). We denote by ϕ_{i} = ∫_{E} f(z)\overline{e_{i}(z)}dμ(z) the ith Fourier coefficient and by f_{n}(z) = ∑_{k=0}^{n} ∑_{|α(i)|=k} ϕ_{i}e_{i}(z) the nth partial sum of the Fourier expansion of f.

We give the following counterpart of Theorem 4.1.

Theorem 5.3. Let E be a compact and regular subset of ℂ^{N} and let (E, μ) satisfy the Bernstein-Markov condition. Let f ∈ L_{P}^{2}(E, μ) and let R > 1. The following conditions are equivalent:

(1) f admits an extension that is holomorphic in E_{R};

(2) lim sup_{i→∞} |ϕ_{i}|^{1/|α(i)|} ≤ \frac{1}{R};

(3) lim sup_{n→∞} ∥T ch_{E}f̂_{n}∥_{E} ≤ \frac{1}{R}.

Proof. In order to prove the implication (1) ⟹ (2) we use an idea of Zeriahi ([13], Theorem 3.10). Since for i with |α(i)| = n the polynomial ϕ_{i}e_{i}(z) is orthogonal to the subspace P_{n−1}(ℂ^{N}) we get for all i ∈ ℕ with |α(i)| = n the estimate

|ϕ_{i}| = | ∫_{E} f\overline{e_{i}}dμ| ≤ inf\{| ∫_{E}(f − p)\overline{e_{i}}dμ|, p ∈ P_{n−1}(ℂ^{N})\}.

Thus,

|ϕ_{i}| ≤ inf{∥f − p∥_{E}, p ∈ P_{n−1}(ℂ^{N})}.

Since f admits an extension that is holomorphic in E_{R}, appling Theorem 3.1 we obtain lim sup_{i→∞} |ϕ_{i}|^{1/|α(i)|} ≤ \frac{1}{R}.

The implication (2) ⟹ (1) holds since the Fourier series

∑_{k=0}^{∞} ∑_{|α(i)|=k} ϕ_{i}e_{i}(z)

converges uniformly on compact subsets of E_{R}.

Let r ∈ (1, R). We choose ϵ > 0 and ρ ∈ (1, R) such that r(1+ϵ) < ρ < R. Since the number of i such that |α(i)| = n is equal to \binom{N−1+n}{N−1} and lim_{n→∞} \binom{N−1+n}{N−1}^{1/n} = 1, by (2) we find n_{0} ∈ ℕ such that for all n ≥ n_{0} we have ∑_{|α(i)|=n} |ϕ_{i}| ≤ 1/ρ. Thus, by the Bernstein-Walsh inequality and since μ satisfies the Bernstein-Markov condition we get for any n ≥ n_{0} the estimate

\begin{aligned}∑_{k=n}^{∞} ∥ ∑_{|α(i)|=k} ϕ_{i}e_{i}∥\overline{E_{r}} & ≤ ∑_{k=n}^{∞} r^{k} ∑_{|α(i)|=k} ∥ϕ_{i}e_{i}∥_{E} ≤ ∑_{k=n}^{∞} r^{k}A_{ϵ}(1 + ϵ)^{k} ∑_{|α(i)|=k} |ϕ_{i}| \\ & ≤ ∑_{k=n}^{∞} A_{ϵ}(\frac{r(1 + ϵ)}{ρ})^{k}, \end{aligned}

A_{ϵ} being a constant depending only on ϵ.

Since \frac{r(1+ϵ)}{ρ} < 1, the Fourier series ∑_{k=0}^{∞} ∑_{|α(i)|=k} ϕ_{i}e_{i}(z) converges uniformly on \overline{E_{r}}.

The equivalence of conditions (2) and (3) follows from Remark 5.1, Lemma 5.2 and the identity

lim sup_{i→∞} |ϕ_{i}|^{1/|α(i)|} = lim sup_{n→∞} ( ∑_{|α(i)|=n} |ϕ_{i}|^{2})^{1/2n} = lim sup_{n→∞} ∥T ch_{μ}f̂_{n}∥_{μ}^{1/n}.

□

As a corollary from Theorem 5.3 and Lemma 4.3 we get the following

Theorem 5.4. Let E be a compact and regular subset of ℂ^{N} and let the pair (E, μ) satisfy the Bernstein-Markov condition. Let f ∈ L_{P}^{2}(E, μ). Assume that there exist a norm in ℂ^{N}, a positive number δ > 0, a complex number A and a positive integer n_{0} such that

E_{(δ)} ∩ ⋃_{n≥n_{0}}{f_{n} = A} = ∅.

If δ > diamE − ρ^{∗}(E) − ρ(E) then f admits an extension that is holomorphic in

E_{R} with R = \frac{δ + ρ(E)}{diamE − ρ^{∗}(E)}.

□

Proceeding similarly as in the proof of Remark 3.3, we also derive a counterpart of Corollary 3.5.

Corollary 5.5. Let E be a ball in a norm in ℂ^{N} and let (E, μ) satisfy the Bernstein - Markov condition. Let f ∈ L_{P}^{2}(E, μ) and R > 1. Then the necessary and sufficient condition that the function f be holomorphic in E_{R} is that for any r ∈ (1, R) there exists a complex number A_{r} such that f_{n}(z) ≠ A_{r} for all n ∈ ℕ and all z ∈ \overline{E_{r}}.

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