[

# [

## Abstract

In 1924 S. Bernstein [4] asked for conditions on a uniformly bounded on Borel function (weight) which imply the denseness of algebraic polynomials in the seminormed space defined as the linear set equipped with the seminorm . In 1998 A. Borichev and M. Sodin [6] completely solved this problem for all those weights for which is dense in but there exists a positive integer such that is not dense in . In the present paper we establish that if is dense in for all then for arbitrary there exists a weight such that is dense in for every and for all .

Polynomial approximation, weighted approximation, -spaces, entire functions

Smoothing of weights …] Smoothing of weights in the Bernstein approximation problem Andrew Bakan]Andrew Bakan Jürgen Prestin]Jürgen Prestin

\subjclass

Primary 41A10, 46E30; Secondary 32A15, 32A60

1

## 1 Introduction

Let be the linear space of all continuous real-valued functions on , the set of all uniformly bounded on Borel functions which have an unbounded support and satisfy as for all . Denote by the set of all algebraic polynomials with real coefficients and by the family of all real-valued infinitely continuously differentiable functions on .

For the seminormed space consists of the linear set of all with and the semi-norm , where .

We recall the definition of the so-called upper Baire function of as (see [15, p. 129]). If is locally bounded from above, then is an upper semi-continuous function and , . It is easy to verify that for arbitrary , and we have

 supx∈(A,B)w(x)|f(x)|=supx∈(A,B)Mw(x)|f(x)|.

This means that the seminormed spaces and coincide identically and, in particular, is dense in iff it is dense in . Thus, it is possible to assume everywhere below that where denotes the family of all those which are upper semi-continuous on , i.e., for all .

Introduce

 Wdens(\BbR):={w∈W∗(\BbR) | P is dense in C0w(\BbR)}. (1.1)

In 1924 S. Bernstein [4] asked for conditions on to be in . This problem is known as Bernstein’s approximation problem. Various results towards a final solution of Bernstein”s approximation problem have been obtained independently by L. Carleson [8](1951), H. Pollard [17](1953), S. N. Mergelyan [14](1958) and L. de Branges [5](1959) (see also the surveys of P. Koosis [11], A. Poltoratski [18] and M. Sodin [19]).

The solution of Bernstein’s problem given by L. de Branges [5] in 1959 was slightly improved in 1996 by M. Sodin and P. Yuditskii [20] and attained the following form.

Let be an entire function, be the set of all its zeros, and , where . We say that is of minimal exponential type if . Denote by the family of all entire functions of minimal exponential type which are real on the real axis (in short: real) and have only real simple zeros.

###### Theorem A (L. de Branges, 1959 [5]).

Let . Then is not dense in if and only if there exists an entire function such that and

 ∑λ∈ΛB 1w(λ)|B′(λ)| < +∞ .

In 1958 S. Mergelyan [14] proved that if algebraic polynomials are dense in but are not dense in for some positive integer , then has countable support and the number of points in the set is as . Motivated by this result, A. Borichev and M. Sodin in 1998 [6] divided Bernstein’s approximation problem into two parts.

###### Definition 1.

Let . It is said that algebraic polynomials are regularly dense in if they are dense in for all .

Algebraic polynomials are called to be singularly dense in if they are dense in but not in for a certain .

Similarly to (1.1), we denote

 Wreg(\BbR) := {w∈W∗(\BbR) | P is % regularly dense in C0w(\BbR)}, Wsing(\BbR) := {w∈W∗(\BbR) | P is % singularly dense in C0w(\BbR)}.

It is obvious that and are two non-intersecting classes of weights and

 Wdens(\BbR)=Wreg(\BbR)⊔Wsing(\BbR) ,

where the symbol denotes the union of two non-intersecting sets.

Thus, the finding of conditions on a given weight to be in or in divides Bernstein’s approximation problem into two independent parts: regular and singular, respectively. A complete solution of the singular part was given by A. Borichev and M. Sodin [6] in 1998.

###### Theorem B.

Let . Algebraic polynomials are singularly dense in if and only if is discrete and there exist an entire function and a nonnegative integer such that

 w(x)=∑λ∈ΛEw(λ)χλ(x),  x∈\BbR,    χλ(x):={0,  \textupif  x≠λ,1,  \textupif  x=λ,
 ∑λ∈ΛE1(1+λ2)k w(λ)|E′(λ)|  {<+∞,  \textupif  k=n+1,=+∞,  \textupif  k=n,

and

 ∑λ∈ΛF1w(λ)|F′(λ)|=+∞

for arbitrary transcendental entire functions of minimal exponential type such that and is transcendental.

The regular part of Bernstein’s approximation problem is still open but the following important result holds.

###### Theorem C (M. Sodin, 1996 [19]).

If , then for every .

The following statement about perturbations of zeros of an entire function was proved in [2, Lemma 5, p. 237] (2005).

###### Lemma A.

For an arbitrary entire function with zeros there exists a constant and a sequence of real positive numbers such that for any sequence of real numbers satisfying

 |bn−dn|≤δn, n≥1,

one can find an entire function such that and

 |B′(bn)|≤C⋅|D′(dn)|,n≥1.

If the set of real numbers in Lemma A is bounded from below, then the result of Lemma A can be improved as follows.

###### Lemma 1.

Let and denote the set of its zeros. Assume that

 ∑λ∈ΛB1|B′(λ)|<∞. (1.2)

Then, for arbitrary there exist constants such that for any set of real numbers satisfying

 |λ−dλ|≤ρδe−δ|λ|,  λ∈ΛB, (1.3)

one can find an entire function such that and

 ∣∣B′(λ)∣∣≤Cδ∣∣D′(dλ)∣∣,  λ∈ΛB. (1.4)

Observe that the proof of Lemma 1 in Section 3 gives the explicit expressions for the constants and in (1.3) and in (1.4). Lemma 1 is instrumental for the proof of the next statement.

###### Lemma 2.

Let and . Then,

 wε(x):=sup|t|≤e−ε|x|(w(x+t)+e−ε|x+t|)∈Wreg(\BbR). (1.5)
###### Proof.

In view of [10, Example 1, p. 8], the function

 βε(x):=w(x)+e−ε|x| (1.6)

is upper semi-continuous on and an application of [10, Theorem 1.2, p. 4] to the supremum in (1.5) yields for each the existence of such that

 wε(x)=βε(x+θε(x)e−ε|x|),   x∈\BbR. (1.7)

To prove , let , , , and let us choose an infinite sequence such that . Since for every we have , (1.7) and (1.5) yield , . Thus, if is infinite, then . Otherwise, it suffices to consider the case in which and therefore , by virtue of (1.7) and the upper semi-continuity of . This completes the proof of (see [16, Theorem 2, p.150]).

Assume that . Then, for some we have and by Theorem A there exists an entire function such that

 ∑λ∈ΛF1(1+λ2m)wε(λ)|F′(λ)|<∞. (1.8)

It follows from that and therefore (1.2) holds for .

By Theorem C,

 βε∈Wreg(\BbR). (1.9)

From (1.8) and (1.7) we obtain

 ∑λ∈ΛF1(1+λ2m)βε(λ+θε(λ)e−ε|λ|)|F′(λ)|<∞. (1.10)

Applying Lemma 1 for , we find such that , , and then we find an entire function with zeros , where

 dλ=λ,   λ∈ΛF∩[−Tε,Tε],   dλ=λ+θ(λ)e−ε|λ|,   λ∈ΛF∖[−Tε,Tε].

Hence, in view of (1.4) and (1.10) we have

 ∞ > ∑λ∈ΛF∖[−Tε,Tε]1(1+λ2m)βε(λ+θε(λ)e−ε|λ|)|F′(λ)| ≥ 1Cε/2∑λ∈ΛF∖[−Tε,Tε]1+d2mλ1+λ2m 1(1+d2mλ)βε(dλ)|D′(dλ)| ≥ 122mCε/2∑λ∈ΛF∖[−Tε,Tε]1(1+d2mλ)βε(dλ)|D′(dλ)| ,

from which it follows that

 ∑λ∈ΛD(1+λ2m)−1βε(λ)−1 ∣∣D′(λ)∣∣−1<∞ .

By Theorem A this means that and therefore . This contradicts (1.9) and finishes the proof of Lemma 2. ∎

We are now ready to prove our main result.

###### Theorem 1.

For arbitrary and there exists such that and for all .

###### Proof.

Since the statement of the theorem for implies its validity for all , we can assume without loss of generality that .

Let be defined as in (1.5), as in (1.6) and

 Ωρ(x):=sup|s|≤ρe−ε|x|βε(x+s),   x∈\BbR,  ρ∈(0,1].

Since

 w(x)≤w(x)+e−ε|x|=βε(x)≤Ωρ(x)≤wε(x),  x∈\BbR, (1.11)

by Lemma 2,

 Ωρ∈Wreg(\BbR),  ρ∈(0,1].

For arbitrary satisfying

 0<ω(x)≤e−ε|x|/4,   x∈\BbR, (1.12)

let us introduce

 (1.13)

where , and . For example, we may take . Obviously,

 ∫ω(x)−ω(x)Kω(x,t)dt=1,   x∈\BbR,

and therefore the weight

 Wε(x):=∫ω(x)−ω(x)Kω(x,t)Ω1/2(x+t)dt=∫x+ω(x)x−ω(x)Kω(x,t−x)Ω1/2(t)dt (1.14)

belongs to .

Let be arbitrary and let satisfy . Then, by (1.12) we have and the inequalities and imply . Thus, for every ,

 (3ρ−1)e−ε|x|/4≤ρe−ε|x+t|+t≤(16ρ+3)e−ε|x|/12,

and therefore

 Ω(3ρ−1)/4(x)≤Ωρ(x+t)≤Ω(16ρ+3)/12(x),  ρ∈(1/3,1),  |t|≤e−ε|x|/4,  x∈\BbR,

from which we infer for that

 βε(x)≤Ω1/8(x)≤Ω1/2(x+t)≤Ω11/12(x)≤Ω1(x)=wε(x),|t|≤ω(x),x∈\BbR. (1.15)

In view of (1.11) this means that the weight satisfies

 w(x)+e−ε|x|≤Wε(x)≤wε(x),  x∈\BbR. (1.16)

It follows from the right-hand side inequality of (1.16) that and therefore the left-hand side inequality of (1.16) completes the proof. ∎

Since the weight defined in (1.14) depends on an arbitrary function satisfying (1.12), we prove in the next corollary that the special choice yields a good upper estimate for . Here,

 ϕε(x):=e−ε4κ∫1−1e−11−t2e−ε|x+t|dt,  x∈\BbR,  ε>0 , (1.17) κ:=∫1−1e−11−t2dt=K1(1/2)−K0(1/2)√e∈(1.2e ,1.21e), (1.18)

, are modified Bessel functions (see [9, (13), p.5]) and (1.18) is proved in Section 4.

###### Corollary 1.

Let , and be defined as in (1.5). Then there exists a weight such that and for all .

Theorem 1 allows to assume without loss of generality that each weight in the regular part of Bernstein’s approximation problem is continuous and positive on the whole real axis. It also allows to apply for this part of the problem the sufficient conditions for the denseness of algebraic polynomials in obtained earlier under this assumption (see [17, p.869], [14, p.80]). On the other hand, Lemma 2 makes it possible to replace any weight by the greater step function

 ˆw(x) = ∑n∈\BbZw\footnotesize{n}χ% \footnotesize{[σnlog(1+|n|),σnlog(1+|n+1|))}(x),   x∈\BbR, wn := sup\footnotesize{x∈[σnlog(1+|n|),σnlog(1+|n+1|)]}w(x),  σn:=sign(n),  n∈\BbZ,

such that algebraic polynomials are regularly dense in if and only if they are regularly dense in . Here, is equal to if , if and if .

Notice also that Theorem 1 can be efficiently applied to a representation of the so-called -regular measures for . Recall (see [6, p.250]) that a non-negative Borel measure on is called -regular if all its moments , , are finite and algebraic polynomials are dense in for every . Here, for arbitrary non-negative Borel measures , on and , we write or if for arbitrary Borel subset of . According to [3, Lemma 4, p.203], if is -regular, then there exists a finite non-negative Borel measure on and such that (the converse is evident). Taking for this the weight from Theorem 1, we obtain where is also a non-negative finite Borel measure on as follows from and for all . Thus, the following assertion holds.

###### Corollary 2.

Let and a measure is -regular. Then, for every , there exists a finite non-negative Borel measure on and a weight such that for all and .

## 2 Auxiliary Results

###### Lemma 3.

Let the real numbers , , and satisfy

 b∈(a−Δ2,a+Δ2),   0∉(a−Δ,a+Δ)  and   x∉(a−2Δ,a+2Δ). (2.1)

Then,

 ∣∣∣(1−xa)(1−xb)−1∣∣∣≤(1+Δ)2.
###### Proof.

The conditions (2.1) imply , and therefore . Thus, and , i.e. . Finally,

 ∣∣∣1−x/a1−x/b∣∣∣ =|b||a||x−a||x−b|=|b||a||b−a+(x−b)||x−b| ≤|b||a||b−a|+|x−b||x−b|≤(1+Δ)⋅(1+|b−a||x−b|)≤(1+Δ)2,

which completes the proof. ∎

###### Lemma 4.

Let , and be an entire function satisfying

 |f(z)|≤Cεe\footnotesize{ε|z|},   z∈\BbC . (2.2)

Then,

 |f′(z)|,   ∣∣∣f(z)z−λ∣∣∣≤Cεe\footnotesize{ε|z|},  λ∈Λf,  z∈\BbC .
###### Proof.

Cauchy’s formula [21, (3), p. 81]

 f′(z)=12πi∫|z−ζ|=1/εf(ζ)dζ(ζ−z)2

and (2.2) for any yield

 ∣∣f′(z)∣∣ ≤εCεe\footnotesize{% ε(|z|+1/ε)}=εeCεe\footnotesize{ε|z|}≤Cεe\footnotesize{ε|z|}.

For arbitrary and satisfying it follows from (2.2) that

 ∣∣∣f(z)z−λ∣∣∣≤2εCεe\footnotesize{ε|z|}≤Cεe\footnotesize{ε|z|},

which by the maximum modulus principle [21, p. 165] yields

 ∣∣∣f(z)z−λ∣∣∣ ≤max|ζ−λ|=1/(2ε)∣∣∣f(ζ)ζ−λ∣∣∣=2εmax|ζ−λ|=1/(2ε)|f(ζ)| ≤2εCεmax|ζ−λ|=1/(2ε)e\footnotesize{ε|ζ|}≤2εCεe\footnotesize{ε(|z|+1/ε)}≤2εeCεe\footnotesize{ε|z|}≤Cεe\footnotesize{ε|z|},

provided that . This finishes the proof of Lemma 4. ∎

###### Lemma 5.

Let , and be an entire function from the class satisfying

 (a)  |B(z)|≤Cεe\footnotesize{ε|z|},  z∈\BbC,    (b)  ΘB:=∑λ∈ΛB1|B′(λ)|<∞ . (2.3)

Then, for arbitrary the inequality

 ∣∣∣B(x)x−λ∣∣∣≥∣