Lobachevskii Journal of Mathematics

http://ljm.ksu.ru
ISSN 1818-9962

Vol. 25, 2007, 197–216 \begin{matrix}c \\ ◯ \end{matrix} M. S. Martirosyan and S.V. Samarchyan

M. S. Martirosyan and S.V. Samarchyan

Q-BOUNDED SYSTEMS: COMMON APPROACH TO FISHER-MICCHELLI’S AND BERNSTEIN-WALSH’S TYPE PROBLEMS

(submitted by F. G. Avkhadiev)

Abstract. We have developed a new common method to investigate geometrically fast approximation problems. Fisher-Micchelli’s, Bernstein-Walsh’s and Batirov-Varga’s well known results are obtained as applications.

Introduction: Fisher-Micchelli’s & Bernstein-Walsh’s type problems

Let K be a compact subset of the open unit disk D, H^{∞}(D) be the set of bounded analytic functions on D and C(K) be the set of continuous functions on K.

Then each function f ∈ H^{∞}(D) is approximable by finite linear combinations of the system of powers {z^{k}}_{0}^{∞} uniformly on K at a rate of geometrical progression and as approximants one can take Taylor polynomials, i.e.

lim sup_{n→∞} \sqrt[n]{∥f − P_{n}f∥_{C(K)}} < 1,

where P_{n}(f)(z) := ∑_{k=0}^{n} \frac{f^{(k)}(0)}{k!}z^{k}.

tems.

2000 Mathematical Subject Classification. 41A, 30E10, 30E15.

Key words and phrases. fast approximation, n-widths, biorthogonal sys-

If f ∈ H(ℂ) then one can obtain faster approximation

lim_{n→∞} \sqrt[n]{∥f − P_{n}f∥_{C(K)}} = 0.

From now on approximation which is faster than geometrical progression will be called fast approximation.

In case when logarithmic capacity of K is positive, γ(K) > 0, Bernstein and Walsh [BW] obtained the following result.

BW type result (Bernstein-Walsh): The class of functions f : K → ℂ permitting the fast approximation by finite linear combinations of the system {z^{k}}_{0}^{∞} in C(K), coincides with H(ℂ).

We have seen that to make the fast approximation of the class H^{∞}(D) we have to miniaturize it up to the class of entire functions. The natural question arises if we can fast approximate whole class H^{∞}(D) using some other system instead of {z^{k}}_{0}^{∞}. Due to Fisher and Micchelli [FM] the answer is negative.

FM type result (Fisher-Micchelli) There is no system of functions

e_{k}^{(n)} : K → ℂ, k = 1, 2, …, n; n = 1, 2, …

such that every function f ∈ H^{∞}(D) admits the fast approximation by polynomials

∑_{k=1}^{n} a_{k}^{(n)}e_{k}^{(n)}

in C(K).

Reasoning from these results, let us consider the following problems named Fisher-Micchelli’s and Bernstein-Walsh’s type problem, respectively.

Problems:

FM)For a given space H, find a system e_{k}^{(n)} such that each element

of H admits the fast approximation by linear combinations ∑_{k=1}^{n} a_{k}^{(n)}e_{k}^{(n)}.

BW) For a given system e_{k}^{(n)}, find the class of elements permitting the

fast approximation by linear combinations ∑_{k=1}^{n} a_{k}^{(n)}e_{k}^{(n)}.

We have developed a common method to investigate both problems based on the notion of q-bounded systems.

Q-BOUNDED SYSTEMS 199

1. q-bounded systems

Definition 1.1 Let X be a Banach space, (e_{k}^{(n)})_{k≤n} ⊂ X, n = 1, 2, … be a triangle matrix, and L_{n} be the linear span of the finite system {e_{k}^{(n)}}_{k=1}^{n}. For q > 0 the matrix (e_{k}^{(n)})_{k≤n} is called

i. q-lower bounded in X, if for each sequence P_{n} = ∑_{k=1}^{n} a_{k}^{(n)}e_{k}^{(n)} ∈ L_{n}

one has

lim sup_{n→∞} \sqrt[n]{∥P_{n}∥_{X}} ≥ q lim sup_{n→∞} \sqrt[n]{max_{1≤k≤n} |a_{k}^{(n)}|}.

ii. q-upper bounded in X, if

lim sup_{n→∞} \sqrt[n]{max_{1≤k≤n} ∥e_{k}^{(n)}∥_{X}} ≤ q.

Definition 1.2 The matrix (e_{k}^{(n)})_{k≤n} is called 0-lower bounded (∞- upper bounded), if it is q-lower (upper) bounded for some q ∈ (0, ∞).

Definition 1.3 The system {e_{k}}_{k=1}^{∞} ⊂ X is called q-lower (upper) bounded, if the matrix (e_{k}^{(n)})_{k≤n} is q-lower (upper) bounded for e_{k}^{(n)} := e_{k}, k ≤ n.

Checking of q-upper boundedness is usually easy. As regards checking of q-lower boundedness, it seems difficult. The following lemma shows that checking of q-lower boundedness can be reduced to checking of qupper boundedness of biorthogonal system.

Lemma 1.1 (Checking lower boundedness, [F]) If the finite systems {φ_{k}^{(n)}}_{k=1}^{n} and {e_{k}^{(n)}}_{k=1}^{n} are biorthogonal for all n ∈ N and the

matrix (φ_{k}^{(n)})_{k≤n} is q-upper bounded in a normed space Y then the

matrix (e_{k}^{(n)})_{k≤n} is \frac{1}{q}-lower bounded in the dual space Y^{∗}.

2. Basic lemma

There is a close relation between Kolmogorov n-widths and q-bounded systems.

Definition 2.1 Let K be a subset of a normed linear space X. The quantity

d_{n}(K, X) = inf_{X_{n}⊂X} sup_{x∈K} inf_{y∈X_{n}} ∥y − x∥_{X} ,

where the leftmost infimum is taken over all subspaces X_{n} ⊂ X of dimension n, is called the Kolmogorov n-width of K in X.

Lemma 2.1(q-bounded systems and Kolmogorov n-widths)

Let H, X be Banach spaces and H ⊂ X. If there exists a matrix (φ_{k}^{(n)})_{k≤n} ⊂ H which is q_{1}-lower bounded in X and q_{2}-upper bounded in H, then

lim sup_{n→∞} \sqrt[n]{d_{n}(B_{H}, X)} ≥ \frac{q_{1}}{q_{2}} ,

for the unit ball B_{H} = {x ∈ H : ∥x∥_{H} ≤ 1} of H.

Lemma 2.1 could be established by Tikhomirov’s well known result. Lemma (Tikhomirov, [T]) If B_{n+1} is a unit ball of some n + 1-

dimensional subspace of a Banach space X, then d_{n}(B_{n+1}, X) = 1.

Instead, we shall give here an elementary proof in the sense that it does not depend on Borsuk’s theorem.

Proof. Assuming the converse, there is a positive number δ and the n-dimensional (n ≥ n_{0}) subspaces Y_{n} ⊂ X such that

sup_{x∈B_{H}} inf_{y∈Y_{n}} \sqrt[n]{∥y − x∥_{X}} < \frac{q_{1}}{q_{2}} − δ (n ≥ n_{0}).

Let {e_{1}^{(n)}, …, e_{n}^{(n)}} be a basis of Y_{n}. Denoting

r_{n}(x) = inf_{a_{k}^{(n)}} \sqrt[n]{∥x − ∑_{k=1}^{n} a_{k}^{(n)}e_{k}^{(n)}∥_{X}},

we have

sup_{x∈B_{H}} r_{n}(x) < \frac{q_{1}}{q_{2}} − δ (n ≥ n_{0}).  (1)

For each x ∈ B_{H} take coefficients a_{k}^{(n)}(x) such that

lim sup_{n→∞} \sqrt[n]{∥x − ∑_{k=1}^{n} a_{k}^{(n)}(x)e_{k}^{(n)}∥_{X}} = lim sup_{n→∞} r_{n}(x).  (2)

Since (φ_{k}^{(n)})_{k≤n} is q_{2}-upper bounded in H, it follows that

ψ_{k}^{(n)} := \frac{φ_{k}^{(n)}}{(q_{2} + ε)^{n}} ∈ B_{H}, k = 1, 2, …, n,  (3)

for each positive ε beginning from some N(ε).

The linear dependence of the system

{⟨a_{1}^{(n)}(ψ_{i}^{(n+1)}), a_{2}^{(n)}(ψ_{i}^{(n+1)}), …, a_{n}^{(n)}(ψ_{i}^{(n+1)})⟩}_{i=1}^{n+1} in ℂ^{n}

implies the existence of coefficients c_{i}^{(n+1)}, i = 1, …, n + 1 such that

∑_{i=1}^{n+1} c_{i}^{(n+1)}a_{k}^{(n)}(ψ_{i}^{(n+1)}) = 0, k = 1, 2, …, n and

max_{1≤i≤n+1} |c_{i}^{(n+1)}| = 1.  (4)

Denote P_{n+1} = ∑_{i=1}^{n+1} c_{i}^{(n+1)}ψ_{i}^{(n+1)}. Combining (1) – (4), we obtain

\frac{q_{1}}{q_{2} + ε} ≤ lim sup_{n→∞} \sqrt[n+1]{∥P_{n+1}∥_{X}}

= lim sup_{n→∞} \sqrt[n+1]{∥∑_{i=1}^{n+1} c_{i}^{(n+1)} [ψ_{i}^{(n+1)} − ∑_{k=1}^{n} a_{k}^{(n)}(ψ_{i}^{(n+1)})e_{k}^{(n)}]∥_{X}}

≤ lim sup_{n→∞} \sqrt[n+1]{∑_{i=1}^{n+1} |c_{i}^{(n+1)}|}

× lim sup_{n→∞} \sqrt[n+1]{max_{1≤i≤n+1} ∥ψ_{i}^{(n+1)} − ∑_{k=1}^{n} a_{k}^{(n)}(ψ_{i}^{(n+1)})e_{k}^{(n)}∥_{X}}

= lim sup_{n→∞} max_{1≤i≤n+1} r_{n+1}(ψ_{i}^{(n+1)}) ≤ lim sup_{n→∞} sup_{x∈B_{H}} r_{n+1}(x) ≤ \frac{q_{1}}{q_{2}} − δ.

This yields that \frac{q_{1}}{q_{2}} − δ ≥ \frac{q_{1}}{q_{2}} and the contradiction proves the lemma.

The following basic lemma shows that checking of q-boundedness of even one system leads to solution of both problems at once.

Lemma 2.2 (Basic lemma)

FM) Suppose H ⊂ X are Banach spaces and ∥x∥_{X} ≤ C ∥x∥_{H} , x ∈ H. If exists a matrix (φ_{k}^{(n)})_{k≤n} ⊂ H, which is q_{1}-lower bounded in X

and q_{2}-upper bounded in H, then for every matrix (e_{k}^{(n)})_{k≤n} ⊂ X there is an element x ∈ H such that

lim sup_{n→∞} \sqrt[n]{∥x − ∑_{k=1}^{n} a_{k}^{(n)}e_{k}^{(n)}∥_{X}} ≥ \frac{q_{1}}{q_{2}} ,

for all numerical matrices (a_{k}^{(n)})_{k≤n}.

BW) Let {e_{k}}_{k=1}^{∞} be a 0-lower and ∞-upper bounded system in a Banach space X. For x ∈ X there are polynomials P_{n} = ∑_{k=1}^{n} a_{k}^{(n)}e_{k}

satisfying \sqrt[n]{∥x − P_{n}∥_{X}} →_{n→∞} 0 if and only if x = ∑_{k=1}^{∞} x_{k}e_{k}, \sqrt[k]{|x_{k}|} →_{k→∞} 0.

Proof. One can find BW) proof in [F] under even more general conditions.

FM) Assuming the converse, it is possible to take (e_{k}^{(n)})_{k≤n} ⊂ X such

that lim sup_{n→∞} r_{n}(x) < \frac{q_{1}}{q_{2}} (∀x ∈ H) for r_{n}(x) := inf_{a_{k}^{(n)}} \sqrt[n]{∥x − ∑_{k=1}^{n} a_{k}^{(n)}e_{k}^{(n)}∥_{X}}.

Then let H = ⋃_{i=1}^{∞} H_{i}, where H_{i} := \{x ∈ H : lim sup_{n→∞} r_{n}(x) < \frac{q_{1}}{q_{2}} − \frac{1}{i}\}.

Since H is a Banach space, one can take some H_{i_{0}} being a set of the second category there.

Denote q_{0} = \frac{q_{1}}{q_{2}} − \frac{1}{i_{0}} and

E_{k} = {x ∈ H : r_{n}(x) ≤ q_{0} for n ≥ k} .

As ∥∗∥_{H} is stronger than ∥∗∥_{X}, the sets E_{k} are closed in H. Since

H_{i_{0}} ⊂ ⋃_{k=1}^{∞} E_{k}, then some E_{k_{0}} contains a ball in H, that is the estimates

r_{n}(x_{0} + μx) ≤ q_{0} (n ≥ k_{0})

hold for some positive number μ, for some x_{0} ∈ X and for each x chosen from the unit ball B_{H} = {x ∈ H : ∥x∥_{H} ≤ 1}. As

\sqrt[n]{μ} · r_{n}(x) = r_{n}(μx)

≤ \sqrt[n]{inf_{a_{k}^{(n)}} ∥x_{0} + μx − ∑_{k=1}^{n} a_{k}^{(n)}e_{k}^{(n)}∥_{X} + inf_{a_{k}^{(n)}} ∥x_{0} − ∑_{k=1}^{n} a_{k}^{(n)}e_{k}^{(n)}∥_{X}}

≤ \sqrt[n]{2}q_{0} (∀x ∈ B_{H}),

then

lim sup_{n→∞} sup_{x∈B_{H}} r_{n}(x) ≤ q_{0}.

Therefore

lim sup_{n→∞} \sqrt[n]{d_{n}(B_{H}, X)} ≤ q_{0}

that contradicts lemma 2.1.

To each pair (X, φ_{k}^{(n)}), where (φ_{k}^{(n)})_{k≤n} is some matrix of elements

from a Banach space X, let’s put in correspondence the set ℑ_{X} (φ_{k}^{(n)}), consisting of elements x ∈ X that

∑_{j=1}^{n} ∑_{i=1}^{j} a_{ij}^{(n)}φ_{i}^{(j)} →_{n→∞}^{X} x, sup_{n} ∑_{j=1}^{n} ∑_{i=1}^{j} |a_{ij}^{(n)}| < ∞.

Corollary. Let (φ_{k}^{(n)})_{k≤n} be q-lower bounded matrix in the Banach

space X and ∥φ_{k}^{(n)}∥_{X} ≤ C, k ≤ n, n = 1, 2, . . .. Then for each matrix

(e_{k}^{(n)})_{k≤n} ⊂ X there is an element x ∈ ℑ_{X} (φ_{k}^{(n)}) such that

lim sup_{n→∞} \sqrt[n]{∥x − ∑_{k=1}^{n} a_{k}^{(n)}e_{k}^{(n)}∥_{X}} ≥ q,

for all numerical matrices (a_{k}^{(n)})_{k≤n}.

Proof. Let M_{n} be the Banach space of all n × n triangular numerical matrices a = (a_{ij}), 1 ≤ i ≤ j ≤ n, equipped with the norm |a|_{n} =

∑_{j=1}^{n} ∑_{i=1}^{j} |a_{ij}|. For a = (a_{ij}) ∈ M_{n} denote

aφ = ∑_{j=1}^{n} ∑_{i=1}^{j} a_{ij}φ_{i}^{(j)} ∈ X.

Consider the set M of sequences A = {a^{(n)}}_{n=1}^{∞} , a^{(n)} ∈ M_{n} that converges in X,

Aφ = lim_{n→∞} a^{(n)}φ,

and sup_{n} |a^{(n)}|_{n} < ∞. Then M is a Banach space, with the norm

∥A∥_{M} = sup_{n} |a^{(n)}|_{n} .

Therefore the set ℑ_{X} (φ_{k}^{(n)}) that coincide with {Aφ : A ∈ M} turns to a Banach space, equipped with the norm

∥x∥_{ℑ_{X}} = inf_{A:Aφ=x} ∥A∥_{M} .

Indeed, ℑ_{X} (φ_{k}^{(n)}) is isometrically isomorphic to the factor space M/M_{0}, which is a Banach space as M_{0} = {A : Aφ = 0} is a closed subspace of M.

To complete the proof it is enough to note that

∥φ_{k}^{(n)}∥_{ℑ_{X}} ≤ 1, ∥x∥_{X} ≤ C ∥x∥_{ℑ_{X}} , x ∈ ℑ_{X} (φ_{k}^{(n)})

and use basic lemma FM).

3. Application 1

Let K be a compact subset of the open unit disk D with positive logarithmic capacity γ(K), C(K) be the set of continuous functions on K and H^{∞}(D) be the set of bounded analytic functions on the unit disk D.

Take H = H^{∞}(D), X = C(K) then H ⊂ X and ∥∗∥_{X} ≤ ∥∗∥_{H}.

It is obvious that the system φ_{k}^{(n)}(z) = φ_{k}(z) = z^{k} is 1-upper bounded in H. On the other hand, one can easily establish \frac{γ(K)}{4}-lower boundedness of {z^{k}}_{0}^{∞} in X using the following

Proposition ([F]). For each polynomial P_{n}(z) = ∑_{k=0}^{n} a_{k}z^{n−k}, a_{0} ≠ 0

there is a polynomial Q_{n}(z) = ∑_{k=0}^{n} b_{k}z^{n−k}, b_{0} = 1 satisfying

|P_{n}(z)| ≥ \frac{max_{0≤k≤n} |a_{k}|}{4^{n}} |Q_{n}(z)| , z ∈ \overline{D}.

Now applying basic lemma we obtain Fisher-Micchelli’s and Bernstein-Walsh’s results.

(FM) Let (e_{k}^{(n)})_{k≤n} be a matrix of continuous functions on K. Then there is a function f ∈ H^{∞}(D) such that

lim sup_{n→∞} \sqrt[n]{∥f − ∑_{k=1}^{n} a_{k}^{(n)}e_{k}^{(n)}∥_{C(K)}} ≥ \frac{γ(K)}{4},

for all numerical matrices (a_{k}^{(n)})_{k≤n}.

(BW) The class of functions f : K → ℂ, permitting the fast approximation by finite linear combinations of the system {z^{k}}_{0}^{∞} in C(K), coincide with H (ℂ).

As you could see we have obtained FM and BW results by checking q-boundedness of just one system {z^{k}}_{k=0}^{∞}.

4. Application 2

It was mentioned above that to make the fast approximation of H^{∞} (D) we have to miniaturize it up to the class of entire functions. Now let’s consider another miniaturization.

Suppose 0 ≤ n_{1} < n_{2} < … < n_{k} < … are integers with density τ, i.e.

lim_{k→∞} \frac{k}{n_{k}} = τ.

Denote

H_{{n_{k}}}^{∞}(D) = {f ∈ H^{∞}(D) : f(z) = ∑_{k=1}^{∞} a_{k}z^{k}, a_{k} = 0, k ∉ {n_{k}}} ,

∥f∥_{H_{{n_{k}}}^{∞}} = sup_{z∈D} |f(z)| .

Let K be a compact subset of the unit disk D = {z ∈ ℂ : |z| < 1} with positive logarithmic capacity γ (K).

Theorem 4.1

FM 1) If τ = 0 then lim_{n→∞} \sqrt[n]{d_{n} (B, C(K))} = 0, for the unit ball B of H_{{n_{k}}}^{∞}(D).

FM 2) If τ > 0 then for each matrix (e_{k}^{(n)})_{k≤n} of continuous functions on K there is a function f ∈ H_{{n_{k}}}^{∞}(D) satisfying

lim sup_{n→∞} \sqrt[n]{∥f − ∑_{k=1}^{n} a_{k}^{(n)}e_{k}^{(n)}∥_{C(K)}} ≥ (\frac{γ(K)}{4})^{\frac{1}{τ}} ,

for all numerical matrices (a_{k}^{(n)})_{k≤n}.

BW) If τ > 0 then the class of functions f : K → ℂ, permitting the fast approximation by the finite linear combinations of the system {z^{n_{k}}}_{k=1}^{∞} in C(K), coincide with H_{{n_{k}}}^{∞}(D) ∩ H(ℂ).

Proof. To prove FM1) one can take the partial sums of corresponding lacunary series as approximants. Proofs of FM2) and BW) can be established by the basic lemma.

Indeed, take H = H_{{n_{k}}}^{∞}(D), X = C(K) and φ_{k}(z) = z^{n_{k}} ∈ H_{{n_{k}}}^{∞}(D). It can be easily checked that {z^{n_{k}}}_{k=1}^{∞} is 1-upper bounded in H and

(\frac{γ(K)}{4})^{\frac{1}{τ}} -lower bounded in X.

5. Application 3

Consider the system of exponents

{e^{−λ_{k}x}}_{k=1}^{∞} ,  (5)

where λ_{k} are disjoint numbers, which satisfy

Reλ_{k} ≥ a > 0, |λ_{k}| ≤ M < ∞, k = 1, 2, ….  (6)

To every function f ∈ L^{2}(0, ∞) we put in correspondence its approximation error by finite linear combinations of first n elements of (5),

i.e. E_{n}(f) = inf_{a_{k}^{(n)}} ∥f(x) − ∑_{k=1}^{n} a_{k}^{(n)}e^{−λ_{k}x}∥_{L^{2}(0,∞)}. There is an analogue of

Bernstein theorem for exponents.

Theorem (Musoyan, [M1]) Let f ∈ L^{2}(0, ∞). Then the estimate lim sup_{n→∞} \sqrt[n]{E_{n}(f)} < 1 holds if and only if there exists an entire function of exponential type with indicator diagram located in the open left half-plane coinciding with f on (0, ∞) almost everywhere.

In [Z] it has been shown that in a sense such rate of approximation can’t be improved.

Theorem ([Z]) Let D be a set of positive measure μ(D) and D ⊂ {z : |z| ≤ M_{1}, Rez < 0}. If (6) takes place then

lim sup_{n→∞} \sqrt[n]{E_{n}(f_{λ})} ≥ \frac{1}{4e\sqrt{π}} \begin{matrix}√ & \overline{μ(D)} \\ max & \{M, M_{1}\} \end{matrix},

where f_{λ}(z) = e^{−λz} for some λ chosen from −\overline{D} = \{z : −\overline{z} ∈ D\}.

For entire function f of exponential type introduce its Borel transform β_{f}(z) = ∑_{n=0}^{∞} f^{(n)}(0)z^{−n−1}. Let K be a compact subset of the open

right half-plane with positive logarithmic capacity γ(K) and −K = {z : −z ∈ K}. For 1 ≤ p ≤ ∞ we denote by L_{K}^{p} the class of functions f ∈ L^{p}(0, ∞) admitting the extension up to entire function of exponential type with β_{f} holomorphic on the complement of −K.

For f ∈ L^{p}(0, ∞) and any matrix (e_{k}^{(n)})_{k≤n} ⊂ L^{p}(0, ∞) consider

E_{n}^{(p)}(f) := inf_{a_{k}^{(n)}} ∥f − ∑_{k=1}^{n} a_{k}^{(n)}e_{k}^{(n)}∥_{L^{p}(0,∞)} .

Theorem 5.1 (FM) For each matrix (e_{k}^{(n)})_{k≤n} ⊂ L^{p}(0, ∞), 1 ≤ p ≤ ∞, there is a function f ∈ L_{K}^{p} satisfying

lim sup_{n→∞} \sqrt[n]{E_{n}^{(p)}(f)} ≥ \frac{γ(K)}{d(K)},

where d(K) = inf {2M : K ⊂ {z : |z| ≤ M}}.

Proof. There exists a matrix of numbers (λ_{kn})_{k≤n} ⊂ K such that the matrix of exponents (e^{−λ_{kn}x})_{k≤n} is \frac{γ(K)}{d(K)}-lower bounded in L^{p}(0, ∞). To prove this, we introduce Fekete’s n-th transfinite diameter and Chebyshev n-th constant of K ([L], p. 606) that are τ_{n} = max_{z_{1},...,z_{n}∈K} ∏_{j≠k} |z_{k} − z_{j}|^{1/n(n−1)} and c_{n} = min_{z_{1},...,z_{n}∈ℂ} max_{z∈K} ∏_{k=1}^{n} |z_{k} − z|^{1/n} respectively. The theorem of Fekete

– Szego states that both τ_{n} and c_{n} tend to γ(K) as n → ∞.

We take (λ_{kn})_{k≤n} in such a way that

∏_{j≠k} |λ_{kn} − λ_{jn}|^{1/n(n−1)} = τ_{n}, n = 1, 2, ….  (7)

Consider the finite system of exponents

{e^{−λ_{1n}x}, …, e^{−λ_{nn}x}}  (8)

and Blashke product B_{n}(λ) = ∏_{k=1}^{n} \frac{λ−λ_{kn}}{λ+\overline{λ_{kn}}} for \{λ_{kn}\}_{k=1}^{n}. The biorthogonal

system generated by (8) is the system of functions [M2]

φ_{k}^{(n)}(x) = \frac{1}{\overline{B′_{n}(λ_{kn})}} ∑_{m=1}^{n} \frac{e^{−λ_{mn}x}}{B′_{n}(λ_{mn})(λ_{mn} + \overline{λ_{kn}})}, k = 1, …, n; x > 0, (9)

that is, ∫_{o}^{∞}e^{−λ_{pn}x}\overline{φ_{qn}(x)}dx = δ_{pq} (δ_{pq} is Kronecker’s delta) and the linear

spans of (8) and (9) coincide. According to lemma 1.1, we just need \frac{d(K)}{γ(K)}-

upper boundedness of (φ_{k}^{(n)})_{k≤n} in all spaces L^{p}(0, ∞), 1 ≤ p ≤ ∞.

The function φ_{k}^{(n)}(x) can be written in the integral form

φ_{k}^{(n)}(x) = \frac{1}{\overline{B′_{n}(λ_{kn})}} \frac{1}{2πi} ∫_{Γ_{r,R}} \frac{e^{−ζx}dζ}{B_{n}(ζ)(ζ + \overline{λ_{kn}})},  (10)

where Γ_{r,R} is the contour consisting of the segment [r + iR, r − iR] and semicircle ζ = r + Re^{iφ}, −\frac{π}{2} ≤ φ ≤ \frac{π}{2} running in positive direction. Besides, interior of Γ_{r,R} contains the compact set K. For any positive number ε one can fix r such small and R such large that |\frac{ζ−λ}{ζ+\overline{λ}} | > 1 − ε when (ζ, λ) ∈ Γ_{r,R} × K. Consequently, (10) implies

∥φ_{k}^{(n)}∥_{L^{p}(0,∞)} ≤ \frac{C}{|B′_{n}(λ_{kn})|} \frac{1}{(1 − ε)^{n}} , k = 1, 2, …, n; n = 1, 2, …,

where the constant C does not depend on k and n. As ε was arbitrary, the last estimate leads to

lim sup_{n→∞} \sqrt[n]{max_{1≤m≤n} ∥φ_{m}^{(n)}∥_{L^{p}(0,∞)}} ≤ \frac{1}{lim inf_{n→∞} \sqrt[n]{min_{1≤m≤n} |B′_{n}(λ_{mn})|}} ≤

≤ \frac{d(K)}{lim inf_{n→∞} \sqrt[n]{min_{1≤m≤n} ∏_{k=1,k≠m}^{n} |λ_{kn} − λ_{mn}|}} ≤ \frac{d(K)}{γ(K)}.

Indeed, according to (7) we get

\sqrt[n]{∏_{k=1,k≠m}^{n} |λ_{kn} − λ_{mn}|} = \sqrt[n]{max_{z∈K} ∏_{k=1,k≠m}^{n} |λ_{kn} − z|} ≥ c_{n−1}^{1− \frac{1}{n}}

for all m = 1, 2, …, n, therefore

lim inf_{n→∞} \sqrt[n]{min_{1≤m≤n} ∏_{k=1,k≠m}^{n} |λ_{kn} − λ_{mn}|} ≥ lim_{n→∞} c_{n−1}^{1− \frac{1}{n}} = γ(K).

Now let’s prove ℑ_{L^{p}(0,∞)} (e^{−λ_{kn}x}) ⊂ L_{K}^{p}.

For f ∈ ℑ_{L^{p}(0,∞)} (e^{−λ_{kn}x}) consider the exponential polynomials f_{n}(z) =

∑_{j=1}^{n} ∑_{i=1}^{j} a_{ij}^{(n)}e^{−λ_{ij}z} (z ∈ ℂ) such that

sup_{n} ∑_{j=1}^{n} ∑_{i=1}^{j} |a_{ij}^{(n)}| < ∞  and  ∥f_{n} − f∥_{L^{p}(0,∞)} →_{n→∞} 0.

The chosen sequence of polynomials is a normal family of entire functions because of its uniformly boundedness inside of ℂ. Similarly, the sequence of Borel transforms

β_{f_{n}} (z) = ∑_{j=1}^{n} ∑_{i=1}^{j} \frac{a_{ij}^{(n)}}{z + λ_{ij}}

is a normal family on the complement of −K. So f_{n}(z) →_{n→∞} f̃(z) uniformly on each compact subset of the complex plane and β_{f_{n}} (z) →_{n→∞} β (z) uniformly on each compact subset of the complement of −K.

As entire function f̃ is of exponential type and f̃ = f almost everywhere on (0, ∞), it remains to prove βf̃(z) = β(z), z ∈ ℂ\ (−K). Indeed, if Rez ≥ δ > 0 then

β_{f_{n}}(z) − βf̃(z) = ∫_{0}^{∞}[f_{n}(t) − f(t)] e^{−zt}dt.

Therefore

|β_{f_{n}}(z) − βf̃(z)| = O(1) ∥f_{n} − f∥_{L^{p}(0,∞)} (n → ∞).

Thus βf̃(z) = β(z) (Rez > 0). Consequently, βf̃ and β coincide on the complement of −K. Now theorem 5.1 follows from the corollary of basic lemma.

Remark. It is known [J] that the sequence {λ_{k}}_{1}^{∞} , Reλ_{k} > 0 of disjoint numbers satisfies Carleson’s separability condition [C]

inf_{1≤j≤n} ∏_{k=1,k≠j}^{n} |\frac{λ_{k} − λ_{j}}{λ_{k} + \overline{λ_{j}}} | ≥ δ > 0, n = 1, 2, …  (11)

if and only if the system {e^{−λ_{k}x}}_{k=1}^{∞} is basis in its closed linear span in the space L^{2}(0, ∞). If (6) holds then {e^{−λ_{k}x}}_{k=1}^{∞} isn’t minimal, so (11) doesn’t hold. However, a bounded sequence of powers can be taken geometrically separable, i.e.

lim inf_{n→∞} \sqrt[n]{inf_{1≤j≤n} ∏_{k=1,k≠j}^{n} |\frac{λ_{k} − λ_{j}}{λ_{k} + \overline{λ_{j}}} |} ≥ δ > 0.  (12)

Moreover,

Lemma 5.1 Let the sequence {λ_{k}}_{1}^{∞} satisfies the condition (6). Then (12) takes place if and only if the system {e^{−λ_{k}x}}_{k=1}^{∞} is δ-lower bounded in L^{2}(0, ∞).

Proof. Suppose

B_{n}(λ) = ∏_{k=1}^{n} \frac{λ − λ_{k}}{λ + \overline{λ_{k}}}

and

φ_{k}^{(n)}(x) = \frac{1}{\overline{B′_{n}(λ_{k})}} ∑_{m=1}^{n} \frac{e^{−λ_{m}x}}{B′_{n}(λ_{m})(λ_{m} + \overline{λ_{k}})}.

The inequality

lim sup_{n→∞} \sqrt[n]{max_{1≤j≤n} ∥φ_{j}^{(n)}∥_{L^{p}(0,∞)}} ≤ \frac{1}{lim inf_{n→∞} \sqrt[n]{min_{1≤j≤n} |B′_{n}(λ_{j})|}} =

= (lim inf_{n→∞} \sqrt[n]{inf_{1≤j≤n} ∏_{k=1,k≠j}^{n} |\frac{λ_{k} − λ_{j}}{λ_{k} + \overline{λ_{j}}} |})^{−1}  (13)

holds for all 1 ≤ p ≤ ∞. Taking into account lemma 1.1, (13) completes the proof of necessity.

To prove sufficiency we use Fourier transforms of functions φ_{k}^{(n)} ∈ L^{2}(0, ∞)

φ̂_{k}^{(n)}(τ) = \frac{1}{\overline{B′_{n}(λ_{k})}} \frac{1}{2πi} ∫_{0}^{∞}e^{−iτx} ∫_{Γ} \frac{e^{−ζx}dζ}{B_{n}(ζ)(ζ + \overline{λ_{k}})}dx, τ ∈ (−∞, ∞),

where Γ is a contour that lies in the open right half – plane and contains all points λ_{k}, k = 1, 2, … inside. Applying Fubini’s and residue theorems, one can obtain

φ̂_{k}^{(n)}(τ) = \frac{1}{\overline{B′_{n}(λ_{k})}} \frac{1}{B_{n}(−iτ)(iτ − \overline{λ_{k}})}.

Hence

∥φ_{k}^{(n)}∥_{L^{2}(0,∞)} = \sqrt{2π} ∥φ̂_{k}^{(n)}∥_{L^{2}(−∞,∞)}

= \begin{matrix}√ & \overline{2π} \\ | & B′_{n}(λ_{k}) & | \end{matrix} \{∫_{−∞}^{∞} \frac{dτ}{|iτ + λ_{k}|^{2}} \}^{1/2} = \frac{π\sqrt{2}}{|B′_{n}(λ_{k})| \sqrt{Reλ_{k}}} .

On the other hand, since φ_{k}^{(n)} was generated by {e^{−λ_{k}x}}_{k=1}^{n}, it is 1/δupper bounded in the space L^{2}(0, ∞) [F]. Therefore

lim inf_{n→∞} \sqrt[n]{min_{1≤j≤n} |B′_{n}(λ_{j})|} ≥ δ.

The lemma is proved.

Combining (13), lemma 1.1 and basic lemma, we get the following BW type theorem.

Theorem 5.2 (BW) Let the sequence of disjoint numbers {λ_{k}}_{1}^{∞} satisfying (6) be geometrically separable. Then the class of functions f ∈ L^{p}(0, ∞), 1 ≤ p ≤ ∞, permitting the fast approximation by finite linear combinations of the system {e^{−λ_{k}x}}_{k=1}^{∞}, coincide with the set of series

∑_{k=1}^{∞} a_{k}e^{−λ_{k}x}, \sqrt[k]{|a_{k}|} →_{k→∞} 0,

where the convergence is in the sense of L^{p}(0, ∞) topology.

6. Application 4

Let H^{p}(G^{+}), 1 < p < ∞, be the Hardy space of functions f analytic on the upper half - plane G^{+} = {z : Imz > 0}, with the norm

∥f∥_{H^{p}(G^{+})} = sup_{y>0} {∫_{−∞}^{∞} |f(x + iy)|^{p} dx}^{1/p} < ∞.

For f ∈ H^{p}(G^{+}) and any matrix (e_{k}^{(n)})_{k≤n} ⊂ H^{p}(G^{+}) consider the approximation error

E_{n}^{(p)}(G^{+})(f) := inf_{a_{k}^{(n)}} ∥f − ∑_{k=1}^{n} a_{k}^{(n)}e_{k}^{(n)}∥_{H^{p}(G^{+})} .

The theorem of Paley and Wiener states: the class H^{2}(G^{+}) coincides with the set of functions representable in the form

f(z) = \frac{1}{\sqrt{2π}} ∫_{0}^{∞} f̂(t)e^{izt}dt, z ∈ G^{+},

where f̂ ∈ L^{2}(0, ∞). If f̂(t) = e^{−λt}, Reλ > 0 then the corresponding function f ∈ H^{2}(G^{+}) is

f(z) = \frac{i}{2π} \frac{1}{z − \overline{μ}}, μ = i\overline{λ} ∈ G^{+}.

Reasoning from this, let’s consider the system of rational functions

e_{k}(z) = \frac{1}{z − \overline{λ_{k}}} , k = 1, 2, …,

where λ_{k} are disjoint complex numbers chosen from some compact set K ⊂ G^{+}. Denote \overline{K} = \{z : \overline{z} ∈ K\}. Using Musoyan’s theorem, one

can establish lim sup_{n→∞} \sqrt[n]{E_{n}^{(2)}(G^{+})(f)} < 1 for f ∈ H^{2}(G^{+}) ∩ Hol(ℂ\\overline{K}),

f(∞) = 0.

Then, let γ(K) > 0 and

H_{K}^{p}(G^{+}) = \{f : f ∈ H^{p}(G^{+}) ∩ Hol(ℂ\\overline{K}) , f(∞) = 0\} .

Theorem 6.1 (FM) For each matrix (e_{k}^{(n)})_{k≤n} ⊂ H^{p} (G^{+}) there is a function f ∈ H_{K}^{p}(G^{+}) satisfying

lim sup_{n→∞} \sqrt[n]{E_{n}^{(p)} (G^{+}) (f)} ≥ \frac{γ (K)}{d (K)},

where d(K) = inf {2M : K ⊂ {z : |z| ≤ M}}.

Proof. As in theorem 5.1, there exists a matrix of numbers (λ_{kn})_{k≤n} ⊂ K such that (\frac{1}{z−\overline{λ_{kn}}} )_{k≤n} is \frac{γ(K)}{d(K)}-lower bounded in H^{p}(G^{+}) (see [M3] for integral representation of generated biorthogonal system and [F]). The embedding ℑ_{H^{p}(G^{+})} (\frac{1}{z−\overline{λ_{kn}}} ) ⊂ H_{K}^{p}(G^{+}) holds as well. To complete the proof it remains to apply the corollary of basic lemma.

Now consider the Hardy space H^{p}(D), 1 < p < ∞, of analytic functions on the unit disk D = {z : |z| < 1}, with the norm

∥f∥_{H^{p}(D)} = sup_{0<r<1} \{\frac{1}{2π} ∫_{−π}^{π} |f(re^{iθ})|^{p} dθ\}^{1/p} < ∞.

For f ∈ H^{p}(D) and any matrix (e_{k}^{(n)})_{k≤n} ⊂ H^{p}(D) denote

E_{n}^{(p)}(D)(f) := inf_{a_{k}^{(n)}} ∥f − ∑_{k=1}^{n} a_{k}^{(n)}e_{k}^{(n)}∥_{H^{p}(D)} .

Let K be a compact subset of the unit disk with positive logarithmic capacity γ(K) and 1/K = {z : 1/z ∈ K}. If e_{k}^{(n)}(z) = e_{k}(z) = z^{k} then

lim sup_{n→∞} \sqrt[n]{E_{n}^{(p)} (D) (f)} < 1,

for every function f, holomorphic on the complement of 1/K.

Theorem 6.2 (FM) For each matrix (e_{k}^{(n)})_{k≤n} ⊂ H^{p} (D) there is a function f ∈ H^{p} (D), holomorphic on the complement of 1/K , satisfying

lim sup_{n→∞} \sqrt[n]{E_{n}^{(p)} (D) (f)} ≥ \frac{γ (K)}{2}.

Proof. First of all, using technique of the theorem 5.1 once more, we find a matrix (λ_{kn})_{k≤n} ⊂ K such that the matrix of rational functions (\frac{1}{1−\overline{λ_{kn}}z})_{k≤n} is \frac{γ(K)}{2}-lower bounded in H^{p}(D) (see [M] for representation of generated biorthogonal system and [F]). Secondly, the checking of embedding ℑ_{H^{p}(D)} (\frac{1}{1−\overline{λ_{kn}}z}) ⊂ H(ℂ\1/K) is trivial. Finally, we use the corollary of basic lemma.

Corresponding BW results are presented below.

Theorem 6.3 (BW) Let the sequence of disjoint complex numbers {λ_{k}}_{k=1}^{∞} be chosen from a compact subset of G^{+} and be geometrically separable, i.e.

lim inf_{n→∞} \sqrt[n]{inf_{1≤j≤n} ∏_{k=1,k≠j}^{n} |\frac{λ_{k} − λ_{j}}{λ_{k} − \overline{λ_{j}}} |} ≥ δ > 0.

Then the class of functions f ∈ H^{p} (G^{+}), permitting the fast approximation by finite linear combinations of the system \{\frac{1}{z−\overline{λ_{k}}} \}_{k=1}^{∞}, coincides with the set of series

∑_{k=1}^{∞} \frac{a_{k}}{z − \overline{λ_{k}}} , \sqrt[k]{|a_{k}|} →_{k→∞} 0,

where the convergence is in the sense of H^{p} (G^{+}) topology.

Theorem 6.4 (BW) Let the sequence of disjoint complex numbers {λ_{k}}_{k=1}^{∞} be chosen from a compact subset of the unit disk D and be geometrically separable, i.e.

lim inf_{n→∞} \sqrt[n]{inf_{1≤j≤n} ∏_{k=1,k≠j}^{n} |\frac{λ_{k} − λ_{j}}{1 − \overline{λ_{j}}λ_{k}} |} ≥ δ > 0.

Then the class of functions f ∈ H^{p} (D), permitting the fast approximation by finite linear combinations of the system \{\frac{1}{1−\overline{λ_{k}}z}\}_{k=1}^{∞}, coincides with the set of series

∑_{k=1}^{∞} \frac{a_{k}}{1 − \overline{λ_{k}}z} , \sqrt[k]{|a_{k}|} →_{k→∞} 0,

where the convergence is in the sense of H^{p} (D) topology.

7. Application 5

Let φ̂ be the Fourier transform of φ ∈ L^{2} (ℝ). Further, assume that |φ̂ (ξ)| ≥ m > 0, ξ ∈ (a, a + 2σ) and {λ_{k}}_{k=1}^{∞} ⊂ ℝ satisfies strong separability condition

\sqrt[n]{inf_{1≤j≤n} ∏_{k=1,k≠j}^{n} sin \frac{σ}{n} |λ_{k} − λ_{j}|} ≥ δ > 0, n ≥ n_{0}.  (14)

Then the following statement takes place

Theorem 7.1 (BW) f ∈ L^{2} (ℝ) is fast approximable by the system of translates {φ (x − λ_{k})}_{k=1}^{∞}, i.e.

\sqrt[n]{inf_{a_{k}^{(n)}} ∥f (x) − ∑_{k=1}^{n} a_{k}^{(n)}φ (x − λ_{k})∥_{L^{2}(ℝ)}} →_{n→∞} 0

if and only if

f (x) =^{L^{2}(ℝ)} ∑_{k=1}^{∞} c_{k}φ (x − λ_{k}), \sqrt[k]{|c_{k}|} → 0.

Proof. It is enough to show that under conditions of the theorem the system {φ (x − λ_{k})}_{k=1}^{∞} is 0-lower bounded and ∞-upper bounded in L^{2} (ℝ). As regards ∞-upper boundedness it is obvious. To prove 0-lower boundedness, at first we denote P_{n} (x) = ∑_{k=1}^{n} a_{k}^{(n)}φ (x − λ_{k}). Then note

that

∥P_{n} (x)∥_{L^{2}(ℝ)} = ∥P̂_{n} (ξ)∥_{L^{2}(ℝ)} = ∥φ̂ (ξ) ∑_{k=1}^{n} a_{k}^{(n)}e^{iλ_{k}ξ}∥_{L^{2}(ℝ)} .

So it remains to prove that {e^{iλ_{k}ξ}}_{k=1}^{∞} is 0-lower bounded in L^{2} (a, a + 2σ), that is

lim sup_{n→∞} \sqrt[n]{∥∑_{k=1}^{n} a_{k}^{(n)}e^{iλ_{k}ξ}∥_{L^{2}(a,a+2σ)}} ≥ q lim sup_{n→∞} \sqrt[n]{max_{1≤k≤n} |a_{k}^{(n)}|}  (15)

for some positive number q.

As |e^{iλ_{k}(a+σ)}| = 1, k = 1, 2, … one can replace (15) by

lim sup_{n→∞} \sqrt[n]{∥∑_{k=1}^{n} a_{k}^{(n)}e^{iλ_{k}ξ}∥_{L^{2}(−σ,σ)}} ≥ q lim sup_{n→∞} \sqrt[n]{max_{1≤k≤n} |a_{k}^{(n)}|}.

To establish this inequality we construct biorthogonal matrix (φ_{k}^{(n)})_{k≤n}

and use lemma 1.1. Let φ̂_{k}^{(n)} (λ) = \frac{sin \frac{σ(λ−λ_{k})}{n}}{\frac{σ(λ−λ_{k})}{n}} ∏_{i=1,i≠k}^{n} \begin{matrix}sin & \frac{σ(λ−λ_{i})}{n} \\ sin & \frac{σ(λ_{k}−λ_{i})}{n} \end{matrix}. Then

φ̂_{k}^{(n)} (λ) ∈ P W_{σ}, where P W_{σ} is Paley-Wiener class of entire functions of exponential type ≤ σ which belong to L^{2} (ℝ). By the theorem of Paley and Wiener one has φ_{k}^{(n)} (ξ) ∈ L^{2} (−σ, σ). Now taking into account

φ̂_{k}^{(n)} (λ_{i}) = \{ \begin{matrix}1, i = k \\ 0, i ≠ k \end{matrix}

we obtain that the systems \{\frac{1}{\sqrt{2π}}φ_{k}^{(n)}\}_{k=1}^{n} and \{e^{iλ_{k}ξ}\}_{k=1}^{n} are biorthogonal for all natural numbers n. On the other hand, (14) implies

∥φ_{k}^{(n)}∥ = ∥φ̂_{k}^{(n)}∥ ≤ \frac{Cn}{δ^{n}},

for some positive constant C.

Finally, we apply lemma 1.1 and establish the theorem.

8. Application 6

It is well known (see [B] and [V]) that there is a close relation between the order of entire function and the rate of its approximation by polynomials.

Theorem (Batirov-Varga) Let K ⊂ ℂ be a compact set of positive logarithmic capacity. Then for each entire function f one has

lim sup_{n→∞} \frac{n ln n}{− ln E_{n} (f, K)} = ρ,

where ρ is the order of function f, E_{n} (f, K) is the error of approximation of function f by algebraic polynomials in the uniform norm on K.

One can easily obtain an analogy of basic lemma’s BW result in this direction.

Theorem 8.1 (BW) Let {e_{k}}_{k=1}^{∞} be a 0-lower and ∞-upper bounded system in a Banach space X, E_{n} (x) be the error of approximation of

x ∈ X by polynomials ∑_{k=1}^{n} a_{k}^{(n)}e_{k} in X and, finally, ρ be some non-negative

number. Then

lim sup_{n→∞} \frac{n ln n}{− ln E_{n} (x)} ≤ ρ

if and only if

x = ∑_{k=1}^{∞} x_{k}e_{k}, lim sup_{n→∞} \frac{n ln n}{− ln |x_{k}|} ≤ ρ.

Taking X = C (K) and e_{k} = z^{k} we immediately establish Batirov-Varga’s result.

References

[BW] Bernstein S. N., On best approximation of continuous functions by polynomials of a given degree. Soobshcheniya of Kharkov Society of Mathematics (2), 13, 1920.

[W] Walsh I. L., Uber den grad der Approximation einer analytishen funktion, Minchner Bericte, 1926, 223-229.

[FM] S. D. Fisher and C. A. Micchelli, The n-width of sets of analytic functions, Duke Math. J. 47, 1980, 789-801.

[F] Samarchyan S. V., Fast approximation in Hardy and Lebesgue spaces, Journal of Contemporary Mathematical Analysis, vol. 39, no. 6, 2004, 69-81.

[T] Tikhomirov V. M., Diameters of sets in function spaces and the theory of best approximations, Uspekhi Mat. Nauk, 15, 1960, 81-120; English transl. in Russian Math. Surveys, 15, 1960, 75-111.

[M1] Musoyan V. Kh., On rate of approximation of entire functions by complete system of Dirichlet, Mat. Zametki, vol. 36, no. 6, 1984, 857-863.

216 M. S. MARTIROSYAN AND S.V. SAMARCHYAN

[Z] Martirosyan M. S. and Samarchyan S. V., On speed of approximation of entire functions by a system of exponents with bounded sequence of powers, submitted for publication.

[L] Lorentz G. G., Golitschek M. V., Makovoz Y., Constructive Approximation: Advanced Problems, Springer Verlag, Berlin, 1996.

[M2] Musoyan V. Kh., Extremal properties of Dirichlet polynomials, Izv. AN Arm. SSR, Mat, vol. 18, no. 4, 1983, 253-270.

[J] Jrbashyan M., Basicity of some biorthogonal systems and the solution of a multiple interpolation problem in H^{p} classes in the half plane, Izv. Akad. Nauk, SSSR Ser. Mat. Vol. 42, 1978, 1323-1384.

[C] Carleson L., Interpolation by bounded analytic functions and the corona problem, Ann. Math., 76, 1962, 547-559.

[M3] Musoyan V. Kh., Summation of biorthogonal expansions by incomplete systems of exponents and rational functions, Izv. AN Arm. SSR, Mat., vol. 21, no. 2, 1986, 163-186.

[M] Martirosyan M. S. and Musoyan V. Kh., Summation of biorthogonal expansions by incomplete system of rational functions in spaces H^{p}(1 < p < ∞) on the disk, Journal of Contemporary Math. Anal., vol. 32, no. 5, 1997, 27-38.

[B] Batirov A. V., On the best approximation of analytic function by polynomials, DAN SSSR, vol. 76, 1951, 173-175.

[V] Varga R. S., On an extension of a result of S. N. Bernstein, JAT 1, 1968, 176-179.

E-mail address: approximation@gmail.com

Received November 23, 2006