Strong Converse Inequalities for Bernstein Polynomials with Explicit Asymptotic Constants
José A. Adell Daniel Cárdenas-Morales


Hardware Acceleration of LLMs: A comprehensive survey and comparison
    Nikoletta Koilia
Department of Electrical 

and Electronics Engineering 

University of West Attica

Athens, Greece 

eee19387106@uniwa.gr
Christoforos Kachris
Department of Electrical 

and Electronics Engineering 

University of West Attica

Athens, Greece 

kachris@uniwa.gr

    
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§ ABSTRACT 
We obtain strong converse inequalities for the Bernstein polynomials with explicit asymptotic constants. We give different estimation procedures in the central and non-central regions of [0,1]. The main ingredients in our approach are the following: representation of the derivatives of the Bernstein polynomials in terms of the Krawtchouk polynomials, estimates of different inverse moments of various random variables, sharp estimates of both absolute central moments of Bernstein polynomials and the total variation distance between binomial and Poisson distributions, and iterates of the Bernstein polynomials, together with their probabilistic representations.






Mathematics Subject Classification 41A10 · 41A25 · 41A27 · 41A36 · 60E05






§ INTRODUCTION AND STATEMENT OF THE MAIN RESULTS


The Bernstein polynomials represent the most paradigmatic example of positive linear operators. Recall that for a function f:[0,1]→ℝ and a natural number n∈ℕ, the nth Bernstein polynomial of f is defined as

    B_nf(x)=∑_k=0^n
    f(k/n)nkx^k(1-x)^n-k,   x∈ [0,1].

In the nineties of the last century, Ditzian and Ivanov <cit.> and Totik <cit.> characterized the rates of uniform convergence of sequences of positive linear operators L_nf towards the function f, as n tends to infinity, in terms of the so called second order Ditzian-Totik modulus of smoothness of f (cf. Ditzian and Totik <cit.>). In the case of Bernstein polynomials, it turns out that

    K_1ω_2^φ(f;1/√(n))≤B_nf-f≤ K_2ω_2^φ(f;1/√(n)),   f∈ C[0,1], n∈ℕ,

for some absolute positive constants K_1 and K_2. We recall here  some necessary definitions and notations. As usual, we denote by C[0,1] the space of all real continuous functions defined on [0,1] endowed with the supremum norm ·, and by C^m[0,1] the subspace of all m-times continuously differentiable functions. Besides, f^(m) and f^m represent, respectively, the mth derivative of f, and f raised to the power of m. The second order central difference of f is given by

    Δ_h^2f(x)=f(x+h)-2f(x)+f(x-h),   h≥ 0,   x± h∈ [0,1],

and the Ditzian-Totik modulus of smoothness of f with weight function

    φ (x)=√(x(1-x)),   0≤ x≤ 1,

is defined as

    ω_2^φ(f;δ )=sup{|Δ_hφ (x)^2f(x) |  : 0≤ h≤δ, x± hφ (x)∈ [0,1] },  δ≥ 0.



The second inequality in (<ref>) is called a direct inequality. Different authors completed this inequality by showing specific values for the constant K_2. In this regard, Gavrea et al. <cit.> and Bustamante <cit.> provided K_2=3, whereas Păltănea <cit.> obtained K_2=2.5, which is the best result up to date and up to our knowledge (see also <cit.> for an asymptotic result). It was also observed that if additional smoothness conditions on f are added, then a direct inequality  may be valid for values of K_2 smaller than 2.5. In this respect, the authors recently proved <cit.> that for any f∈ C^2[0,1] and n∈ℕ, the following inequality holds true

    |B_nf-f-1/2ω_2^φ(f;1/√(n))|≤1/4n(ω_1(f^(2);1/3√(n))+1/4ω_2^φ(f^(2);1/√(n))),

where ω_1 (f^(2);· ) stands for the usual first modulus of continuity of f^(2). This inequality completes asymptotic results previously shown by Bustamante and Quesada <cit.> and Păltănea <cit.>.

The first inequality in (<ref>) is called strong converse inequality, according to the terminology given in Ditzian and Ivanov <cit.>, and it turns out to be the best possible one among different types of converse inequalities. It is a consequence of the works by Totik <cit.>, Knopp and Zhou <cit.>, and Sangüesa <cit.> for general sequences of positive linear operators.

Direct and converse inequalities in the L_p-norm, 1≤ p≤∞, have been widely considered in the literature (see, for instance, Totik <cit.>, Ditzian and Ivanov <cit.>, Chen and Ditzian <cit.>, Della Vecchia <cit.>, Guo and Qi <cit.>, Finta <cit.>, Gadjev <cit.>, and Bustamante <cit.>, among many others). The main tool to prove converse inequalities is the use of K-
functionals, shown to be equivalent to the corresponding Ditzian-Totik modulus of smoothness. In some papers (cf. Knopp and Zhou <cit.> and Sangüesa <cit.>), the authors use an arbitrary number of iterates of the operators under consideration as the main tool. In any case, the proofs of converse inequalities are very involved and do not provide explicit constants in general.

This is the motivation of the present paper, whose main result reads as follows.


Let 𝔉 denote the set of all non-affine functions f∈ C[0,1]. Then,

    5≤ _n→∞sup_f∈𝔉ω_2^φ(f;1/√(n))/B_nf-f< 75.



An analogous result to Theorem <ref> referring to a non-centered gamma-type operator was given in <cit.>. The upper constant there was 105, instead of 75. On the other hand, as seen in (<ref>), the estimate of the constant K_2 is improved when we restrict our attention to functions in C^2[0,1]. A similar property holds concerning strong converse inequalities, as the following result shows. In this regard, denote by ω_2(f;· ) the usual second modulus of continuity of f. Let 𝔊⊆𝔉 be the subset of those functions f such that

    lim _δ→ 0ω_2(f;δ)/δ=0.




We have

    _n→∞sup_f∈𝔊ω_2^φ(f;1/√(n))/B_nf-f≤ 4+ √(2)(√(2)+1)/1-0.99/√(3)log 4=15.0477…

Moreover, 𝔉∩ C^1[0,1]⊆𝔊.




To prove the aforementioned results, we follow an approach inspired by the work of Sangüesa <cit.>, using different estimation procedures in the central and non-central regions of [0,1]. In the central region, the starting point is a Taylor's formula of third order for the third iterate of the Bernstein polynomials. As specific tools in this region, we use sharp estimates for the first absolute central moments of the Bernstein polynomials, as well as for the total variation distance between binomial and Poisson distributions. In the non-central region, we start from a formula that involves the second derivatives of the iterates of the Bernstein polynomials, providing at the same time probabilistic representations of such iterates.

Two common tools applied in both regions should be emphasized. On the one hand, several expressions of the derivatives of the Bernstein polynomials, particularly that in terms of the orthogonal polynomials with respect to the binomial distribution, namely, the Krawtchouk polynomials. On the other hand, accurate estimates of various inverse moments involving different random variables.

The main auxiliary results to give estimates in the central (resp. non-central) region are Theorems <ref>, <ref>, and <ref> (resp. Theorems <ref> and <ref>). On the other hand, in some stages of the proofs of Theorems <ref> and <ref>, we use numerical computations performed with the software Mathematica 10.2 in order to obtain explicit constants. We finally mention that the problem of giving non-asymptotic estimates for K_1 remains open.







§ GENERAL AUXILIARY RESULTS

Let ℕ_0=ℕ∪{0}. From now on, whenever we write f,  x, and n, we respectively assume that f∈ C[0,1], x∈ (0,1), and n∈ℕ is large enough.

Let (β_m)_m≥ 1 be a sequence of random variables such that β_m has the beta density

    ρ _m(θ )=m(1-θ)^m-1,   0≤θ≤ 1.

These random variables allow us to write the remainder of Taylor's formula in closed form. In fact, if g∈ C^m[0,1], m∈ℕ, we have

    g(y)=∑_j=0^m-1g^(j)(x)/j! (y-x)^j+(y-x)^m/m!𝔼g^(m)(x+(y-x)β_m),   0≤ y ≤ 1,

where 𝔼 stands for mathematical expectation.




We have

    ω_2^φ(f;1/√(n))≤ 4B_nf-f+log 4/nφ ^2(B_nf)^(2).



Proof.Suppose that [x-φ (x)h,x+φ (x)h]⊆ [0,1], with 0≤ h≤ 1/√(n). This implies that

    r(x):=φ (x)h/x∈ [0,1],     s(x):=φ (x)h/1-x∈ [0,1].

We claim that

    |Δ ^2_hφ (x)g(x)|≤ h^2φ ^2 g^(2)log 4,   g∈ C^2[0,1].

Actually, we have from (<ref>)

    |Δ ^2_hφ (x)g(x)|    =h^2 φ ^2(x)/2|𝔼g^(2)(x-φ (x)hβ_2)+𝔼g^(2)(x+φ (x)hβ_2)|    ≤ h^2φ ^2 g^(2)φ ^2(x)/2(𝔼1/φ^2(x-φ(x)hβ_2)+𝔼1/φ^2(x+φ(x)hβ_2)).

Since

    1/φ^2(y)=1/y+1/1-y,   y∈ (0,1),

we get from (<ref>)

    φ^2(x)/2(𝔼1/φ^2(x-φ(x)hβ_2)+𝔼1/φ^2(x+φ(x)hβ_2))


    =1-x/2𝔼(1/1-r(x)β_2+1/1+r(x)β_2)+x/2𝔼(1/1-s(x)β_2+1/1+s(x)β_2).

On the other hand, we have for any 0≤ r≤ 1

    1/2𝔼(1/1-rβ_2+1/1+rβ_2)=𝔼1/1-(rβ_2)^2≤𝔼1/1-β_2^2=log 4,

where the last equality follows from (<ref>). This, together with (<ref>), (<ref>), and (<ref>), shows claim (<ref>).

Finally, observe that

    |Δ^2_hφ(x)f(x)|≤ 4B_nf-f+|Δ^2_hφ(x)B_nf(x)| .

Therefore, the result follows by applying claim (<ref>) to g=B_nf.
□





Let (U_k)_k≥ 1 be a sequence of independent identically distributed random variables having the uniform distribution on [0,1]. Define

    S_n(x)=∑_k=1^n1_[0,x](U_k),

where 1_A stands for the indicator function of the set A. Clearly,

    P(S_n(x)=k)=nkx^k(1-x)^n-k,   k=0,1,… n.

We consider the sequence L_n of positive linear operators  which associates to each function  ϕ:[0,∞ )→ℝ the function

    L_nϕ(x)=𝔼ϕ(S_n(x))=∑_k=0^nϕ(k)P(S_n(x)=k).

To obtain the derivatives of L_nϕ in closed form, consider first the mth forward differences of ϕ at step h≥ 0, i.e.,

    Δ_h^mϕ(y)=∑_j=0^mmj(-1)^m-jϕ (y+hj),   y≥ 0,   m∈ℕ_0.

If ϕ∈ C^m[0,∞ ), we have the representation (cf. <cit.>)

    Δ_h^mϕ(y)=h^m𝔼ϕ^(m)(y+h(U_1+⋯ +U_m)),   y≥ 0,   n∈ℕ,

where (U_k)_k≥ 1 is a sequence of independent copies of a random variable U having the uniform distribution on [0,1].

In second place, denote by K_m(x;y):=K_m(n,x;y), m=0,1,…,n, the orthogonal polynomials with respect to the probability measure defined in (<ref>), namely, the  Krawtchouk polynomials. Such polynomials are explicitly defined (see, for instance, Chihara <cit.>)) by

    K_m(x;y)=∑_j=0^mn-ym-jyj(-x)^m-j(1-x)^j,   m=0,1,…,n,   y∈ℝ,

and satisfy the orthogonality property

    𝔼K_r(x;S_n(x))K_m(x;S_n(x))=nmφ^2m(x)δ_r,m,   r,m∈{0,1,… n},

where δ_r,m is the Kronecker delta. In particular,

    K_2(x;y)=1/2((y-nx)^2-(1-2x)(y-nx)-nx(1-x)).


It was shown in <cit.> (see also Roos <cit.> and López-Blázquez and Salamanca <cit.>) that for m=0,1,…,n

    (L_nϕ)^(m)(x)=(n)_m𝔼Δ_1^mϕ(S_n-m(x))=m!/φ^2m(x)𝔼ϕ(S_n(x))K_m(x;S_n(x)),

where (n)_m=n(n-1)⋯ (n-m+1).

As follows from (<ref>) and (<ref>), the nth Bernstein polynomial can be written in probabilistic terms as

    B_nf(x)=𝔼f(S_n(x)/n).

From now on, we assume that the random variables S_n(x), (U_k)_k≥ 1, and (β_m)_m≥ 1, as defined in (<ref>), (<ref>), and (<ref>), respectively, are mutually independent.







Let m∈{1,2,…,n}. Then,

    (B_nf)^(m)(x)=(n)_m𝔼Δ_1/n^mf(S_n-m(x)/n)=m!/φ^2m(x)𝔼f(S_n(x)/n)K_m(x;S_n(x)).


In addition, if f∈ C^m[0,1], then

    (B_nf)^(m)(x)=(n)_m/n^m𝔼f^(m)(S_n-m(x)+U_1+⋯ +U_m/n).


Proof.Set ϕ(y)=f(y/n), 0≤ y≤ n. Observe that

    Δ_1^mϕ(y)=Δ _1/n^mf(y/n),   0≤ y≤ n.

Since B_nf(x)=L_nϕ (x), this formula and (<ref>) imply identity (<ref>). Formula (<ref>) is an immediate consequence of (<ref>) and the first equality in (<ref>). The proof is complete.
□

We point out that the first equality in (<ref>) is well known (see, for instance, Abel et al. <cit.>).  The right-hand side in (<ref>) is the probabilistic representation of the Bernstein-Kantorovich operators acting on f^(m) (see Acu et al. <cit.> and the references therein).









§ AN AUXILIARY RESULT FOR THE CENTRAL REGION




By X ℒ= Y, we mean that the random variables X and Y have the same law. It follows from (<ref>) that

    S_n(1-x)ℒ=n-S_n(x).

On the other hand, to prove Theorem <ref> it is enough to show that
φ ^2(B_nf)^(2)≤ C_nB_nf-f, f∈𝔉, for some absolute constant C_n, as follows from Theorem <ref>. We claim that it suffices to show that

    sup_0<x≤ 1/2|φ ^2(x)(B_nf)^(2)(x)|≤ C_n sup_0<x≤ 1/2|B_nf(x)-f(x)|,   f∈𝔉.

Actually, if f∈𝔉, we define f̃∈𝔉 as f̃(y)=f(1-y), y∈ [0,1]. By (<ref>), the second equality in (<ref>), and (<ref>), it can be checked that

    B_nf(1-x)-f(1-x)=B_nf̃(x)-f̃(x),    φ^2(1-x)(B_nf)^(2)(1-x)=φ^2(x)(B_nf̃)^(2)(x),

thus showing claim (<ref>). For this reason, we assume from now on that 0<x≤ 1/2. For the sake of simplicity, we also denote by ‖·‖ the sup-norm on the interval (0,1/2].

To show (<ref>), we distinguish if the supremum ‖φ^2(B_nf)^(2)‖  is attained in the central region [a_n,1/2] or in the non-central region (0,a_n), for a suitable 0<a_n≤ 1/2. The estimation procedures, inspired by the work of Sangüesa <cit.>, are quite different in both regions.

From now on, we will use many times the inequality 1-y≤ e^-y, y≥ 0, without explicitly mentioning it. The crucial quantity to estimate in the central region is the inverse moment

    H_n(x)=φ(x)√(n)/n+2𝔼|S_n(x)-nx|/φ^2(S_n(x)+V/n+2),

where

    V=U_1+U_2,

and U_1 and U_2 are the random variables defined in (<ref>). Recall that the strong law of large numbers and the central limit theorem for S_n(x) respectively state that

    S_n(x)/n⟶ x, a.s.,    and    S_n(x)-nx/φ(x)√(n)ℒ⟶ Z, as n→∞ ,

where Z is a standard normal random variable. Thus, we have for a fixed x∈ (0,1/2]

    lim_n→∞H_n(x)=lim_n→∞𝔼|S_n(x)-nx|/φ(x)√(n)=𝔼|Z|=√(2/π)= 0.7978…

This means that any upper bound for H_n(x) cannot be better than √(2/π). In fact, H_n(x) takes bigger values when  λ=nx, for moderate values of λ, that is, when S_n(x) approaches to a Poisson random variable. In this regard, let N_λ be a random variable having the Poisson distribution with mean λ, i.e.,

    P(N_λ=k)=e^-λλ^k/k!,   k∈ℕ_0,  λ≥ 0.

For any λ≥ 0, denote by

    C(λ)=2log27/16ν (λ)+r(λ),

where

    ν (λ)=√(λ)(2P(N_λ=
    ⌈λ⌉)-P(N_λ=
    0))+1/√(λ)(2P(N_λ≤⌈λ⌉)-1-P(N_λ=
    0)),

⌈λ⌉ being the ceiling of λ, and

    r(λ)=(log 4-2log27/16)λ^3/2e^-λ.

Numerical computations, carried out with the software Mathematica 10.2,  show that

    sup _λ≥ 0 C(λ)=0.9827 … < 0.99.

Finally, let λ_0>0 be such that

    √(2/π)+1/√(λ_0)≤ c:=0.8.


We state the main result of this section.

Suppose that n≥ 2λ_0. Then,

    H_n:=sup_0<x≤ 1/2 H_n(x)≤ 0.99+2D(λ_0)/nlog27/16+n^3/2/2^n+1/2,

where D(λ_0) is the explicit constant defined in (<ref>), only depending upon λ_0.


To prove this result, some auxiliary lemmas will be needed.


Let V be as in (<ref>) and let h(y)=ylog y, y≥ 0. Then,

    𝔼1/y+V=1/y+1∑_j=0^∞𝔼(V-1/y+1)^2j=Δ_1^2h(y).

In particular,

    𝔼1/V=log 4,  𝔼1/1+V=log27/16.


Proof.By (<ref>), the random variable V is symmetric around 1 and therefore 𝔼(V-1)^2j+1=0, j∈ℕ_0. Hence,

    𝔼1/y+V=1/y+1𝔼1/1+(V-1)/(y+1)=1/y+1∑_j=0^∞𝔼(V-1/y+1)^2j.

On the other hand, let y>0. Since h^(2)(y)=1/y, we have from (<ref>)

    𝔼1/y+V=Δ_1^2h(y).

Letting y→ 0, this formula also holds for y=0. Finally, the second statement in Lemma <ref> follows by choosing y=0 and y=1 in (<ref>).
□


Using (<ref>), we rewrite the function H_n(x) defined in (<ref>) as

    H_n(x)=φ(x)√(n)(𝔼|S_n(x)-nx|/S_n(x)+V+𝔼|S_n(x)-nx|/n-S_n(x)+2-V)


    =φ(x)√(n)(𝔼|S_n(x)-nx|/S_n(x)+V+𝔼|S_n(1-x)-n(1-x)|/S_n(1-x)+V),

as follows from (<ref>), the fact that 2-Vℒ=V, and the independence between S_n(x) and V.

Observe that H_n(x) depends on inverse moments involving the random variables S_n(x) and V. In the following result, we estimate H_n(x) in terms of inverse moments only involving S_n(x).



Let H_n(x) be as in (<ref>). Then,

    H_n(x)≤ 2log27/16(I_n(x)+I_n(1-x))+(log 4-2log27/16)(nx)^3/2(1-x)^n+1/2+n^3/2/2^n+1/2,

where

    I_n(x)=φ (x)√(n) 𝔼|S_n(x)-nx|/S_n(x)+1.


Proof.Let us bound the first term on the right-hand side in (<ref>). Replace y by S_n(x) in (<ref>) and use the independence between S_n(x) and V to obtain

    φ (x)√(n) 𝔼|S_n(x)-nx|/S_n(x)+V=φ (x)√(n)(𝔼|S_n(x)-nx|/S_n(x)+1+∑_j=1^∞𝔼(V-1)^2j𝔼|S_n(x)-nx|/(S_n(x)+1)^2j+1).

Let j≥ 1. By considering the events {S_n(x)=0} and {S_n(x)≥ 1}, we get

    𝔼|S_n(x)-nx|/(S_n(x)+1)^2j+1≤ nxP(S_n(x)=0)+1/2^2j𝔼|S_n(x)-nx|/S_n(x)+11_{S_n(x)≥ 1}


    =nxP(S_n(x)=0)+1/2^2j(𝔼|S_n(x)-nx|/S_n(x)+1-nxP(S_n(x)=0)).

This and Lemma <ref> imply that the sum in (<ref>) is bounded above by

    (log 4-1)nxP(S_n(x)=0)+(2log27/16-1)(𝔼|S_n(x)-nx|/S_n(x)+1-nxP(S_n(x)=0)).

We therefore have from (<ref>) and (<ref>)

    φ (x)√(n) 𝔼|S_n(x)-nx|/S_n(x)+V≤ 2log27/16I_n(x)+(log 4-2log27/16)(nx)^3(1-x)^n+1/2.

To bound the second term on the right-hand side in (<ref>), we follow the same procedure, replacing x by 1-x and noting that

    (n(1-x))^3/2x^n+1/2≤n^3/2/2^n+1/2,

since, by assumption, 0<x≤ 1/2. This and (<ref>) show the result.
□

An exact expression for I_n(x), useful to perform computations, is given in the following result.


Let I_n(x) be as in (<ref>). Set λ=nx. Then,

    I_n(x)=n(1-x)^3/2/n+1(√(λ)(2P(S_n(x)=⌈λ⌉)-P(S_n(x)=0))+1/√(λ)(2P(S_n(x)≤⌈λ⌉ )-1-P(S_n(x)=0))).


Proof.Since |y|=2y_+-y, where y_+=max (0,y), y∈ℝ, we can write

    𝔼|S_n(x)-nx|/S_n(x)+1=2𝔼(S_n(x)-nx)_+/S_n(x)+1-𝔼S_n(x)-nx/S_n(x)+1.

Using the identity

    1/k+1P(S_n(x)=k)=1/(n+1)xP(S_n+1(x)=k+1),   k=0,1,… ,n,

we have

    𝔼S_n(x)-nx/S_n(x)+1=∑ _k=0^nk-λ/k+1P(S_n(x)=k)=1-λ+1/(n+1)xP(S_n+1(x)≥ 1)


    =-1-x/(n+1)x+λ+1/(n+1)xP(S_n+1(x)=0)=-1-x/(n+1)x(1-(λ+1)P(S_n(x)=0)).

Also, by (<ref>) and (<ref>), we get

    𝔼(S_n(x)-nx)_+/S_n(x)+1=∑ _k=⌈λ⌉^nk-λ/k+1P(S_n(x)=k)=P(S_n(x)≥⌈λ⌉)-λ+1/(n+1)xP(S_n+1(x)≥⌈λ⌉ +1)


    =P(S_n(x)≥⌈λ⌉ )-λ+1/(n+1)x(xP(S_n(x)≥⌈λ⌉)+(1-x)P(S_n(x)≥⌈λ⌉ +1))


    =1-x/(n+1)x(λ P(S_n(x)≥⌈λ⌉)-(λ +1)P(S_n(x)≥⌈λ⌉ +1))


    =1-x/(n+1)x(λ P(S_n(x)= ⌈λ⌉)-P(S_n(x)≥⌈λ⌉ +1)).

This, together with (<ref>), (<ref>), and some simple computations, shows the result.
□

We give two upper  bounds for I_n(x), according to whether or not λ=nx is large. In the first case, we use the well known Stirling's approximation stating that

    e^1/(2m+1)√(2π m)≤m!/m^me^m≤ e^1/(2m)√(2π m),   m∈ℕ.

Observe that this implies that

    P(S_n(m/n)=m)≤1/√(2π)√(n/m(n-m)),   m=1,…, n-1.

For the second case, recall that the total variation distance between two ℕ_0-valued random variables X and Y is defined as

    d_TV(X,Y)=sup _A⊆ℕ_0 |P(X∈ A)-P(Y∈ A)|=1/2∑_k=0^∞|P(X=k)-P(Y=k)|.

It was shown in <cit.> that

    d_TV(S_n(x),N_λ)≤λ/n(√(2)/  4+4/11(3λ+4)λ ^2/n),  λ=nx,   n≥ 10.

Other estimates can be found in Barbour and Hall <cit.> and Deheuvels et al. <cit.>.


Let λ_0 be as in (<ref>) and assume that n≥ 2λ_0. If λ:=nx≥λ_0, then

    I_n(x)≤ c(1-x),

where c is defined in (<ref>). If λ =nx < λ_0, then

    I_n(x)≤ (1-x)(ν (λ)+D(λ_0)/n),

where ν (λ) is defined in (<ref>) and

    D(λ_0)=3√(λ_0)(λ_0+1)(√(2)/4+2/11(3λ_0+4)λ_0).


Proof.  Assume that λ≥λ_0. From Lemma <ref> and (<ref>), we see that

    I_n(x)≤ (1-x)(2√(λ(n-λ)/n)P((S_n(x)=⌈λ⌉ )+1/√(λ_0)).

Differentiating with respect to x, it can be checked that

    P((S_n(x)=⌈λ⌉ )≤ P((S_n(⌈λ⌉ /n)=⌈λ⌉ ).

We thus have from (<ref>)

    2√(λ(n-λ)/n)P(S_n(x)=⌈λ⌉ )≤√(2/π)√(λ(n-λ)/⌈λ⌉ (n-⌈λ⌉ ))≤√(2/π),

where the last inequality follows from two facts: first n/2≥λ≥λ_0, since 0<x≤ 1/2, and second, the function h(λ)=λ (n-λ) increases for n≥ 2λ. Therefore, (<ref>) follows from (<ref>), (<ref>), and (<ref>).

Since λ_0>5, inequality (<ref>) is an immediate consequence of Lemma <ref> and (<ref>).
□

Proof of Theorem <ref>
Let C(λ) be as in (<ref>) and denote

    C(λ)=2clog27/16+r(λ),  λ≥ 0,

where c=0.8 and r(λ) is defined in (<ref>). Numerical computations show that

    0.9792… =sup _λ≥ 0C(λ)<sup _λ≥ 0C(λ)=0.9827… <0.99.

Since n(1-x)≥ n/2≥λ_0, we have from (<ref>)

    I_n(1-x)≤ cx.

Set λ =nx. We distinguish the following two cases.

Case λ≥λ_0. We have from (<ref>), (<ref>), and Lemma <ref>

    H_n(x)≤ 2clog27/16+r(λ)+n^3/2/2^n+1/2=C(λ )+n^3/2/2^n+1/2.

Case λ < λ_0. We have from (<ref>), (<ref>), and Lemma <ref>

    H_n(x)≤ 2log27/16(ν (λ)(1-x)+cx)+r(λ)+2log27/16D(λ_0)/n+n^3/2/2^n+1/2


    ≤max (C(λ),C(λ))+2log27/16D(λ_0)/n+n^3/2/2^n+1/2.

In view of (<ref>), the conclusion follows from (<ref>) and (<ref>).
□













§ ESTIMATES IN THE CENTRAL REGION


We define the central region as

    R_n(a):={x∈ (0,1/2]: nφ ^2(x)>a },

where a>0 will be chosen later on. We make the following assumption

(H_1)        ‖φ ^2(B_nf)^(2)‖ =sup{|φ ^2(x)(B_nf)^(2)(x)| : x∈ R_n(a)}.

We denote by B_n^k, k∈ℕ, the kth iterate of the operator B_n, as well as

    g=B_nf,    τ_m(x)=x+(S_n(x)/n-x)β_m,   m∈ℕ.

Since

    𝔼(S_n(x)/n-x)^2=φ^2(x)/n,

we use Taylor's formula (<ref>) replacing y by S_n(x)/n to obtain the following starting identity for the central region

    B_n^3f(x)-B_n^2f(x)=B_n^2g(x)-B_ng(x)=𝔼B_ng(S_n(x)/n)-B_ng(x)


    =φ^2(x)/2ng^(2)(x)+φ^2(x)/2n((B_ng)^(2)(x)-g^(2)(x))+1/6𝔼(B_ng)^(3)(τ_3(x))(S_n(x)/n-x)^3.

The second term on the right-hand side in (<ref>) is bounded as follows.


We have

    1/2n‖φ^2((B_ng)^(2)-g^(2))‖≤1/√(2)‖ B_nf-f‖ .


Proof.  By (<ref>) and the second equality in (<ref>), we have

    φ ^2(x)/2n| (B_ng)^(2)(x)-g^(2)(x) | = φ ^2(x)/2n|( B_n^2f-B_nf
     )^(2)(x) |


    =1/nφ^2(x)|𝔼(B_nf-f)(S_n(x)/n)K_2(x;S_n(x))|≤1/nφ ^2(x)‖ B_nf-f‖𝔼| K_2(x;S_n(x)) | .

Therefore, the result follows from (<ref>) and Schwarz's inequality.
□

To estimate the last term in (<ref>), some auxiliary results will be needed.


Let H_n be as in Theorem <ref>. Then,

    ‖φ^3 (B_ng)^(3)‖≤√(n+1)H_n-2‖φ ^2g^(2)‖ .


Proof.  Applying (<ref>) with m=2, we have

    (B_ng)^(2)(x)=n-1/n𝔼g^(2)(S_n-2(x)+V/n),

where V is defined in (<ref>). We differentiate this formula using the second equality in (<ref>) and take into account that K_1(x;y)=y-nx to obtain

    (B_ng)^(3)(x)=n-1/n1/φ ^2(x)𝔼g^(2)(S_n-2(x)+V/n)(S_n-2(x)-(n-2)x).

By (<ref>), this implies that

    |φ ^3(x)(B_ng)^(3)(x)|≤n-1/n‖φ ^2g^(2)‖φ (x)𝔼(| S_n-2(x)-(n-2)x|/φ ^2(S_n-2(x)+V/n))=n-1/√(n-2)‖φ ^2g^(2)‖ H_n-2(x),

which shows the result, because n≥ 3.
□

Denote by

    μ_n(x)=𝔼|S_n(x)/n-x| ^k,   k∈ℕ,

the kth absolute central moment of the Bernstein polynomials. Explicit expressions for the first even central moments can be found, for instance, in <cit.>. Particularly,

    μ _4(x)=3φ^4(x)/n^2+φ^2(x)/n^3(1-6φ^2(x))≤φ^4(x)/n^2(3+1/nφ^2(x)),

and

    μ _6(x)=φ^2(x)/n^5(15φ^4(x)n^2+5φ^2(x)(5-26φ^2(x))n+1-30φ^2(x)(1-2x)^2)


    ≤φ^6(x)/n^3( 15+25/nφ^2(x)+1/(nφ^2(x))^2) .



We have

    n√(n)/φ^3(x)(μ _3(x)+3/8μ_4(x)/φ^2(x)+3/16μ_5(x)/φ^4(x) +7/16μ_6(x)/φ^6(x)) ≤ K(nφ^2(x)),

where

    K(s)=√(3+1/s)+3/8√(s)(3+1/s)+3/16s√((3+1/s)(15+25/s+1/s^2))


    +7/16s√(s)(15+25/s+1/s^2),   s>0.


Proof.  By Schwarz's inequality, we see that

    μ _3(x)=𝔼|S_n(x)/n-x||S_n(x)/n-x| ^2 ≤√(μ_2(x)μ _4(x)).

Similarly,

    μ_5(x)≤√(μ_4(x)μ _6(x)).

Thus, the result follows from (<ref>), (<ref>), (<ref>), and some simple computations.
□


Let β _m be as in (<ref>), m∈ℕ. For any 0≤ z≤ 1, we have

    𝔼φ^m(x)/φ^m(x+(z-x)β_m)≤ 1+m/2(m+1)|z-x|/φ^2(x)+m/4(m+1)|z-x|^2/φ^4(x)+m+4/4(m+1)|z-x|^3/φ^6(x).


Proof.  We claim that

    𝔼1/(1-rβ_m)^m/2≤ 1+m/2(m+1)r+m/4(m+1)r^2+m+4/4(m+1)r^3,   0≤ r≤ 1.

Indeed, we see from (<ref>) that

    𝔼1/(1-β_m)^m/2=m∫ _0^1(1-θ)^m/2-1dθ =2.

Using the binomial expansion and the fact that 𝔼β_m=1/(m+1) and 𝔼β_m^2=2/((m+1)(m+2)), we have

    𝔼1/(1-rβ_m)^m/2≤ 1+m/2(m+1)r+m/4(m+1)r^2+r^3∑_k=3^∞(-1)^k-m/2  k𝔼β_m^k


    =1+m/2(m+1)r+m/4(m+1)r^2+r^3(-1-m/2(m+1)-m/4(m+1)+∑_k=0^∞(-1)^k-m/2  k𝔼β_m^k).

This shows claim (<ref>), since the series in (<ref>) equals 2, as follows from (<ref>).

Assume first that 0≤ z≤ x. As noted in Sangüesa <cit.>, a useful property of the weight function φ is

    φ^2(rx)≥ rφ^2(x),   0≤ r≤ 1.

As a consequence,

    φ^m(x+(z-x)β_m)=φ^m(x(1-x-z/xβ_m))≥(1-x-z/xβ_m)^m/2φ^m(x).

We therefore have from (<ref>)

    𝔼φ^m(x)/φ^m(x+(z-x)β_m)≤𝔼1/(1-x-z/xβ_m)^m/2


    ≤ 1+m/2(m+1)|z-x|/φ^2(x)(1-x)+m/4(m+1)|z-x|^2/φ^4(x)(1-x)^2+m+4/4(m+1)|z-x|^3/φ^6(x)(1-x)^3,

thus showing the result in this case. If x≤ z≤ 1, the proof is similar by observing that φ^2(x)=φ^2(1-x) and

    φ^2(x+(z-x)β_m)=φ^2(1-x-(z-x)β_m)=φ^2((1-x)(1-z-x/1-xβ_m)).

This completes the proof.
□

We are in a position to estimate the last term in (<ref>).











Let H_n and K(s) be as in Theorem <ref> and (<ref>), respectively. Then,

    1/6|𝔼(B_ng)^(3)(τ_3(x))(S_n(x)/n-x)^3|≤√(n+1/n)H_n-2K(nφ^2(x))/3‖φ^2g^(2)‖/2n.


Proof.  Fix z=S_n(x)/n. Applying Lemma <ref> and Lemma <ref> with m=3, we get

    |𝔼(B_ng)^(3)(x+(z-x)β_3)|≤‖φ^3(B_ng)^(3)‖/φ^3(x)𝔼φ^3(x)/φ^3(x+(z-x)β _3)


    ≤√(n+1)H_n-2‖φ^2g^(2)‖/φ^3(x)(1+3/8|z-x|/φ^2(x)+3/16|z-x|^2/φ^4(x)+7/16|z-x|^3/φ^6(x)).

We multiply both sides of this inequality by |z-x|^3/6 and replace z by S_n(x)/n. Recalling (<ref>), we obtain

    1/6|𝔼(B_ng)^(3)(τ_3(x))(S_n(x)/n-x)^3|


    ≤√(n+1/n)H_n-2/3‖φ^2g^(2)‖/2nn√(n)/φ^3(x)(μ_3(x)+3/8μ_4(x)/φ^2(x)+3/16μ_5(x)/φ^4(x)+7/16μ_6(x)/φ^6(x)).

Thus, the conclusion follows from Lemma <ref>.
□

We state the main result of this section.






Assume (H_1). Then,

    (1-√(n+1/n)H_n-2K(a)/3)‖φ^2g^(2)‖/2n≤√(2)+1 /√(2)‖ B_nf-f ‖ .


Proof.  Let x∈ R_n(a). Starting from (<ref>) and applying Lemma <ref>, we obtain

    |φ^2(x)g^(2)(x)|/2n≤‖ B_nf-f‖+1/√(2)‖ B_nf-f‖ +1/6|𝔼(B_ng)^(3)(τ_3(x))(S_n(x)/n-x)^3|


    ≤√(2)+1/√(2)‖ B_nf-f‖ +√(n+1/n)H_n-2K(a)/3‖φ^2g^(2)‖/2n,

where the last inequality follows from Lemma <ref>. Thus, the result follows from assumption (H_1).
□





Theorem <ref> makes sense only when the term in front of ‖φ^2g^(2)‖/(2n)  is positive. This, in turn, implies that the parameter a cannot be close to 0, because of the definition of K(a) given in (<ref>).







§ ESTIMATES IN THE NON-CENTRAL REGION


We define the non-central region as

    R_n(a):={x∈ (0,1/2]: nφ ^2(x)<a },   a>0,

and make the alternative assumption

(H_2)        ‖φ ^2g^(2)‖=‖φ ^2(B_nf)^(2)‖ =sup{|φ ^2(x)(B_nf)^(2)(x)| : x∈R_n(a)}.

In this section, we assume that x∈R_n(a). The starting point is the following identity involving the second derivatives of the iterates of the Bernstein polynomials

    φ^2(x)g^(2)(x)=∑_k=0^mφ^2(x)( (B_n^k+1g)^(2)(x)-(B_n^kg)^(2) (x))+φ^2(x)(B_n^m+1g)^(2)(x),   m∈ℕ.


We recall here the following probabilistic representation of the terms in (<ref>) given in Sangüesa <cit.>. Denote

    W_n(θ)=S_n-2(θ)+V/n,   0≤θ≤ 1,

where V is defined in (<ref>). Let (S_n-2,i(θ), 0≤θ≤ 1)_i≥ 1 and (V_i)_i≥ 1 be sequences of independent copies of (S_n-2(θ), 0≤θ≤ 1) and V, respectively. Define

    W_n,i(θ)=S_n-2,i(θ)+V_i/n,   0≤θ≤ 1,   i∈ℕ.

Clearly, (W_n,i(θ), 0≤θ≤ 1)_i≥ 1 is a sequence of independent copies of (W_n(θ), 0≤θ≤ 1), as defined in (<ref>). We consider the subordinated stochastic processes (W_n^(m)(θ), 0≤θ≤ 1)_m≥ 1 inductively defined as follow

    W_n^(1)(θ)=W_n,1(θ),     W_n^(m+1)(θ)=W_n,m+1(W_n^(m)(θ)),   0≤θ≤ 1,   m∈ℕ.


The following result was shown  by Sangüesa <cit.>. We give here a short proof of it for the sake of completeness.


Let m∈ℕ_0. Then,

    (B_n^m+1g)^(2)(x)=(n-1/n)^m+1𝔼g^(2)(W_n^(m+1)(x)).


Proof.  For m=0, the result follows from (<ref>) and (<ref>). Let 0≤θ≤ 1. Suppose that

    (B_n^mg)^(2)(θ)=(n-1/n)^m𝔼g^(2)(W_n^(m)(θ)),

for some m∈ℕ. Again by (<ref>) and (<ref>), we have

    (B_n^m+1g)^(2)(θ)=(n-1/n)^m𝔼(B_ng)^(2)(W_n^(m)(θ))=(n-1/n)^m+1𝔼g^(2)(  W_n,m+1(W_n^(m)(θ))   ),


    =(n-1/n)^m+1𝔼g^(2)(  W_n^(m+1)(θ)  ),

thus completing the proof.
□

Observe that if x∈R_n(a), then

    nx<b_n:=2a/1+√(1-4a/n),   n>4a.

For this reason, we assume in this section that n>4a. The main quantity to be estimated is the inverse moment.

    1/n𝔼1/φ^2(W_n^(m)(x)),   m∈ℕ.

To bound (<ref>), we will proceed by induction on m. In this respect, we define the iterates

    α_0(θ)=θ,  α_1(θ)=1-e^-θ,  α_m+1(θ)=α_1(α_m(θ)),   0≤θ≤ 1,   m∈ℕ.

Note that α_m(θ)→ 0, as m→∞, for any 0<θ≤ 1. However, the speed of convergence is very slow, since α_1^(1)(0)=1. The reason for these iterates comes from the following inequality, which follows from (<ref>):

    𝔼e^-θ S_n-2(x)=(1+x(1-e^-θ))^n-2≤ e^-(n-2)xα_1(θ)≤ e^2b_n/ne^-nxα_1(θ),   0≤θ≤ 1.

Also, we have from (<ref>)

    𝔼e^-θ V=(𝔼e^-θ U_1)^2=(1-e^-θ/θ)^2=(α_1(θ)/θ)^2,   0≤θ≤ 1.




For any m∈ℕ, we have

    1/n𝔼1/φ^2(W_n^(m)(x))   ≤ e^2b_n/n(2log27/16∫_0^1(α_m-1(θ)/θ)^2e^-nxα_m-1(θ)dθ
    +(log 4-2log27/16)α_m-1^2(1)e^-nxα_m-1(1)+ϵ _n),

where b_n is defined in (<ref>) and

    ϵ_n=4/nlog27/16+e^-n/2.



Proof.  We start with the case m=1. Using (<ref>), we have as in (<ref>)

    1/n𝔼1/φ^2(W_n^(1)(x))=1/n𝔼1/φ^2(S_n-2(x)+V/n)=𝔼1/S_n-2(x)+V+𝔼1/n-2-S_n-2(x)+2-V


    =𝔼1/S_n-2(x)+V+𝔼1/S_n-2(1-x)+V.

We claim that

    𝔼1/S_n-2(x)+V≤ e^2b_n/n(2log27/16∫_0^1e^-nxθdθ +(log 4-2log27/16)e^-nx).

In fact, following the lines of the proof of Lemma <ref>, it can be seen that

    𝔼1/S_n-2(x)+V≤ 2log27/16𝔼1/S_n-2(x)+1+(log 4-2log27/16)P(S_n-2(x)=0).

On the other hand, it follows from (<ref>) that

    𝔼1/S_n-2(x)+1=1/(n-1)xP(S_n-1(x)≥ 1)=1/x∫_0^x(1-u)^n-2du


    ≤∫_0^1e^-(n-2)xθdθ≤ e^2b_n/n∫_0^1e^-nxθdθ,

where the last inequality follows from (<ref>). This and (<ref>) show claim (<ref>).

Finally, since 0<x≤ 1/2, we see from (<ref>) that S_n-2(1-x)≥ S_n-2(1/2). We thus have from (<ref>)

    𝔼1/S_n-2(1-x)+V≤𝔼1/S_n-2(1/2)+V≤ e^2b_n/n(2log27/16∫_0^1e^-nθ /2dθ +(log 4-2log27/16)e^-n/2)


    ≤ e^2b_n/n(4/nlog27/16 +e^-n/2),

which, in conjunction with (<ref>) and (<ref>), shows the result for m=1.

For m>1, the proof follows by induction on m, taking into account (<ref>), (<ref>), and the fact that for any 0≤θ≤ 1 and j∈ℕ_0, we have

    𝔼e^-nα_j(θ)W_n(x)=𝔼e^-α_j(θ)(S_n-2(x)+V)=𝔼e^-α_j(θ)S_n-2(x)𝔼e^-α_j(θ)V≤ e^2b_n/ne^-nα_j+1(θ)x(α_j+1(θ)/α_j(θ))^2,

as follows from (<ref>)-(<ref>). This completes the proof.
□

Denote by

    J_n(m,a)=sup _x∈R_n(a)𝔼φ^2(x)/φ^2(W_n^(m)(x)),   m∈ℕ.



Let m∈ℕ. Assume that b_n≤ 1/α_m-1(1). Then,

    J_n(m,a)
       ≤ b_ne^2b_nm/n(2log27/16∫_0^1(α_m-1(θ)/θ)^2e^-b_nα_m-1(θ)dθ
    +(log 4-2log27/16)α_m-1^2(1)e^-b_nα_m-1(1)+ϵ _n),

where b_n and ϵ_n are defined in (<ref>) and (<ref>), respectively.

Proof.  Set λ=nx. The result readily follows from Lemma <ref> after observing the following. First, (<ref>) implies that λ <b_n. Second, the function h(λ)=λ e^-λα_m-1(1), λ≥ 0, increases in λ < b_n≤ 1/α_m-1(1).
□

The following is the main result of this section.


    Let m∈ℕ and let b_n be as in (<ref>). Suppose that b_n≤ 1/α_i-1(1), for some 1≤ i ≤ m. Under assumption (H_2), we have
    
    ‖φ^2g^(2)‖/n(1-J_n(m+1,a))≤√(2)(i+∑_k=i^mJ_n(k,a))‖ B_nf-f‖ ,

    where J_n(m,a) is defined in (<ref>).

Proof.  Our starting point is identity (<ref>). Let k=0,1,… ,i-1. By the second equality in (<ref>), we have

    |φ^2(x)((B_n^k+1g)^(2)(x)-(B_n^kg)^(2)(x) )| =
    |φ^2(x)( B_n ( B_n^k+1f-B_n^kf   ))^(2)(x) |


    =2/φ^2(x)|𝔼(B_n^k+1f-B_n^kf)(S_n(x)/n)K_2(x;S_n(x))|


    ≤‖ B_nf-f‖2/φ^2(x)𝔼| K_2(x;S_n(x))|≤ n√(2)‖ B_nf-f‖ ,

where the last inequality follows from (<ref>) and Schwarz's inequality.

Let k=i,… ,m and set h=B_ng-g. By Lemma <ref> and (<ref>), we get

    |φ^2(x)((B_n^k+1g)^(2)(x)-(B_n^kg)^(2)(x) )| =
    |φ^2(x) (B_n^kh)^(2)(x) |
    =(n-1/n)^k|φ^2(x)𝔼h^(2)(W_n^(k)(x))|


    ≤‖φ^2(B_ng-g)^(2)‖ J_n(k,a)
    ≤ n√(2)‖ B_nf-f‖ J_n(k,a),

where the last inequality follows from Lemma <ref>. Similarly,

    |φ^2(x)(B_n^m+1g)^(2)(x)| =(n-1/n)^m+1|φ^2(x)𝔼g^(2)(W_n^(m+1)(x))|≤‖φ^2g^(2)‖ J_n(m+1,a).

In view of (<ref>) and (<ref>), the result follows from (<ref>) and assumption (H_2).
□


Obviously, Theorem <ref> makes sense only when J_n(m+1,a)<1. This can be achieved by choosing m large enough, as follows from the upper bound in Lemma <ref> and the fact that α _m(θ)→ 0, as m→∞, 0<θ≤ 1.




§ PROOF OF THEOREM <REF>






Upper bound.
Let a>0. From Theorem <ref>, we have

    _n→∞sup_f∈𝔉ω_2^φ(f;1/√(n))/B_nf-f≤  4+log 4  _n→∞sup_f∈𝔉n^-1‖φ^2(B_nf)^(2)‖/B_nf-f.

Assume (H_1). Applying Theorems <ref> and <ref>, the right-hand side in (<ref>) is bounded above by

    4+√(2)(√(2)+1)/1-0.99 K(a)/3log 4 ,

where K(a) is defined in (<ref>).

Assume (H_2). Let m∈ℕ and k=1,… ,m. Define

    J(k,a)=a(2 L_k(a)log27/16+(log 4-2log27/16)α_k-1^2(1)e^-aα_k-1(1)),

where

    L_k(a)=∫_0^1(α_k-1(θ)/θ)^2e^-aα_k-1(θ)dθ.

By (<ref>) and Lemma <ref>, we see that

    _n→∞J_n(k,a)≤ J(k,a).

Therefore, by Theorem <ref>, the right-hand side in (<ref>) can be bounded above by

    4+√(2)(i+∑_k=i^mJ(k,a))/1-J(m+1,a)log 4,

where i∈{1,… ,m} is the first integer such that a<1/α_i-1(1).




After numerically experimenting with expressions (<ref>)-(<ref>), with the aid of Mathematica 10.2, we choose a=7.2, m=20, and i=13. Notice that this software system  allows to handle the involved iterates α _k(θ), k=0,… , m, defined in (<ref>), and evaluate the integrals L_k(a) with the command NIntegrate. As a result, both expressions (<ref>) and (<ref>) turn out to be less than 74.8. This shows the upper bound in Theorem <ref>.





Lower bound.
For each n∈ℕ, define the function f_n∈ C[0,1] as follows

    f_n(2-√(2)/n)=1,   f_n(1/n)=-0.8,   f_n(2/n)=-1,   f_n(3/n)=0.04,   f_n(2+√(2)/n)=1.

If y∈ [(2-√(2))/n,(2+√(2))/n], then f_n(y) is piecewise linear interpolating the values in (<ref>), whereas if y∈ [0,1]∖ [(2-√(2))/n,(2+√(2))/n], then f_n is constantly 1.

Taking the weighted second order difference of f at the point 2/n, it can be checked that

    lim _n→∞ω _2^φ(f_n;1/√(n))=4.


On the other hand, set λ=nx and denote g(λ)=f_n(λ /n), 0≤λ≤ n. It turns out that

    B_nf_n(x)-f_n(x)=𝔼f_n(S_n(x)/n)-f_n(λ/n)=𝔼g(S_n(λ/n))-g(λ),   0≤λ≤ n.


Let λ _0>2+√(2) and assume that n≥λ _0 is large enough. Using (<ref>), (<ref>), and (<ref>), we have for λ > λ _0

    |𝔼g(S_n(λ /n))-g(λ)|
    =|𝔼(g(S_n(λ /n))-g(λ))1_{S_n(λ/n)≤ 2+√(2)}|≤ 2P(S_n(λ /n)≤ 3)


    ≤ 2P(S_n(λ _0 /n)≤ 3)   ≤  2P(N_λ_0≤ 3)+2d_TV(S_n(λ_0 /n),N_λ_0)≤  2P(N_λ_0≤ 3)+2D(λ_0)/n.


Recalling (<ref>), denote by

    G(λ)=𝔼g(N_λ)   =P(N_λ=0)-0.8P(N_λ=1) 
        -P(N_λ=2)+0.04P(N_λ=3)+1-P(N_λ≤ 3),     0≤λ≤λ_0.

By (<ref>), (<ref>), and (<ref>), we have for λ≤λ _0

    |𝔼g(S_n(λ /n))-g(λ)|≤|𝔼g(N_λ)-g(λ)| +|𝔼g(S_n(λ /n))-𝔼g(N_λ)|


    ≤| G(λ)-g(λ) |
    +d_TV(S_n(λ /n),N_λ)  ≤| G(λ)-g(λ) |  +D(λ_0)/n.


Choose λ_0 and n large enough so that the upper bound in (<ref>) is less than 1/2, say. On the other hand, numerical computations, performed again with Mathematica 10.2, show that

    sup _0≤λ≤λ _0| G(λ)-g(λ)|≤ 0.79 …

We therefore have from (<ref>), (<ref>), and (<ref>)

    ‖ B_nf_n-f_n‖≤ 0.8,

for large enough n. By (<ref>), this implies that

    _n→∞sup _f∈𝔉ω_2^φ(f;1/√(n))/B_nf-f≥_n→∞ ω_2^φ(f_n;1/√(n))/B_nf_n-f_n≥4/0.8=5.

This shows the lower bound in Theorem <ref>, and concludes the proof.
□









§ PROOF OF THEOREM <REF>


Let f∈𝔊. By the first equality in (<ref>), we have

    |φ^2(x)(B_nf)^(2)(x)|≤ n^2φ^2(x)ω_2 ( f;1/n).

Let a>0. Define

    N_1:=N_1(a,f)={n∈ℕ:‖φ^2 (B_nf)^(2)‖ is attained on R_n(a)},     N_2:=ℕ∖ N_1

Suppose that N_1 is not finite. By (<ref>) and (<ref>), we see that

    lim_n→∞
     n∈ N_1‖φ^2 (B_nf)^(2)‖≤ a lim_n→∞
     n∈ N_1    n ω_2 (f;1/n)=0.

We thus have from Theorem <ref>

    _n→∞
     n∈ N_1ω_2^φ(f;1/√(n))/B_nf-f≤ 4+ log 4_n→∞
     n∈ N_1 ‖φ ^2(B_nf)^(2)‖/nB_nf-f=4,

where we have used that f∈𝔉 together with the well known fact that ‖ B_nf-f‖ =o(1/n) if and only if f is affine.

Suppose that N_2 is not finite. By (<ref>), we have

    _n→∞
     n∈ N_2ω_2^φ(f;1/√(n))/B_nf-f≤ 4+√(2)(√(2)+1)/1-0.99 K(a)/3log 4 .

Since a is arbitrary, and k(s)→√(3), as s→∞ (see (<ref>)), inequality (<ref>) follows from (<ref>) and (<ref>).

Finally, let f∈𝔉∩ C^1[0,1] and 0 <h ≤δ. We have from (<ref>)

    | f(x+h)-2f(x)+f(x-h)| = h |𝔼(f^(1)(x+hβ _1)-f^(1)(x-hβ _1))   |≤ hω _1(f^(1);2h).

By (<ref>), this readily implies that f∈𝔊. This concludes the proof.
□


Funding: The first author is supported by Research Project DGA (E48_23R). The second author is supported by Junta de Andalucía (Research Group FQM-0178).

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