An improvement of the constant in Videnskiĭ’s inequality for Bernstein polynomials

Ulrich Abel; Hartmut Siebert

doi:10.1515/gmj-2017-0065

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An improvement of the constant in Videnskiĭ’s inequality for Bernstein polynomials

Ulrich Abel und Hartmut Siebert

Veröffentlicht/Copyright: 7. Februar 2018

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Aus der Zeitschrift Georgian Mathematical Journal Band 27 Heft 1

Abstract

In this paper, we deal with improvements on the constant Mn⁢(γ) in the so-called Videnskiĭ inequality

|(Bn⁢f)⁢(x)-f⁢(x)-x⁢(1-x)2⁢n⁢f′′⁢(x)|≤Mn⁢(γ)⁢x⁢(1-x)n⁢ω⁢(f′′;γn)

for a fixed constant γ≥1 and x∈[0,1], where Bn⁢f is the Bernstein polynomial, f∈C2⁢[0,1] and ω is the first order modulus of continuity. Let M⁢(γ)=supn∈ℕ⁡Mn⁢(γ). We prove the Videnskiĭ inequality for arbitrary γ≥1. In particular, we improve the constant M⁢(2)=0.9 (Gonska and Ra şa [8], 2008) to M⁢(2)=0.6875. Finally, we consider Mn⁢(1) for small values of n.

Keywords: Approximation by positive operators; rate of convergence; degree of approximation

MSC 2010: 41A36; 41A25

1 Introduction

The classical Bernstein polynomials, defined for f∈C⁢[0,1] by

(Bnf)(x)=∑ν=0npn,ν(x)f(νn) (x∈[0,1])

with

pn,ν(x)=(nν)xν(1-x)n-ν (ν=0,…,n)

converge uniformly to f on [0,1]. Popoviciu [11] gave the rate of convergence

∥Bn⁢f-f∥∞≤c0⋅ω⁢(f;n-1/2),

estimated by the ordinary modulus of continuity with the constant c0=32. An elementary proof with constant c0=54 can be found in Lorentz’s book [10]. Sikkema [12] found the best (smallest) value of the constant c0. The asymptotic behaviour was described by Voronovskaja [17] in 1932 (see, e.g., [10, p. 22]). She showed that, for each bounded function f on [0,1] whose second derivative f′′⁢(x) exists at a certain point x∈[0,1], one has

limn→∞⁡n⁢[(Bn⁢f)⁢(x)-f⁢(x)]=12⁢x⁢(1-x)⁢f′′⁢(x),

the convergence being uniform, for f∈C2⁢[0,1] (see, e.g., [3, Chapter 10, Theorem 3.1] and cf. [5]). In that same year Bernstein [2] published a generalization. Gonska [6, Theorem 5.1] gave the following quantitative form:

|n⁢[(Bn⁢f)⁢(x)-f⁢(x)]-12⁢x⁢(1-x)⁢f′′⁢(x)|≤x⁢(1-x)2⁢ω~⁢(f′′;1n2+x⁢(1-x)n),

where ω~ is the least concave majorant of ω satisfying

ω⁢(f;δ)≤ω~⁢(f;δ)≤2⁢ω⁢(f;δ) for ⁢δ>0.

In 1985, Videnskiĭ [16] proved that, for all x∈[0,1],

(1.1)|(Bn⁢f)⁢(x)-f⁢(x)-x⁢(1-x)2⁢n⁢f′′⁢(x)|≤M⁢x⁢(1-x)n⁢ω⁢(f′′;2n)

with the constant M=1 being independent of x. By application of a general theorem due to Sikkema and van der Meer [13], Gonska and Raşa [8, p. 94] recently proved that the inequality is valid with constant M=0.9.

The purpose of the present note is to show by direct methods that the constant M can be diminished to the value

(1.2)M=1116=0.6875.

In 2005, Floater [5] generalized Voronovskaja’s theorem [17] to differentiated Bernstein polynomials. Quantitative results of Floater’s theorem have been given by Gonska and Raşa [9] and Gonska, Heilmann and Raşa [7] in 2009 and 2010, respectively. Further inequalities for the left-hand side of (1.1) with the aid of Ditzian–Totik moduli of first order and average moduli of smoothness were given by Tachev [15] and Finta [4].

2 Main result

We define the sequence of linear operators

(2.1)(Lnf)(x):=n(Bn⁢f)⁢(x)-f⁢(x)x⁢(1-x)-12f′′(x) (x∈(0,1)),

which allows to write (1.1) as

|(Ln⁢f)⁢(x)|≤M⋅ω⁢(f′′;2n).

Our main result is the following theorem which states a slightly more general inequality.

Theorem 2.1.

Let γ≥1 and f∈C2⁢[0,1]. For all n≥1 and all x∈[0,1],

|(Ln⁢f)⁢(x)|≤M⁢(γ)⋅ω⁢(f′′;γn),

where

M⁢(γ)=12+38⁢γ.

In the particular case n=1, the constant M⁢(γ) can be taken equal to 12.

Remark 2.2.

Notice that for γ=2 the theorem provides inequality (1.1) with the constant M announced in (1.2).

Remark 2.3.

In the convenient case γ=1, we have the constant M⁢(1)=78=0.875.

Proof of Theorem 2.1.

Let γ≥1 and f∈C2⁢[0,1]. By taking into account the endpoint interpolation property of the Bernstein polynomials, the assertion is obvious for x∈{0,1}. Now let x∈(0,1). By Taylor’s formula we have, for 0≤ν≤n,

f⁢(νn)=f⁢(x)+f′⁢(x)⁢(νn-x)+12⁢f′′⁢(x)⁢(νn-x)2+12⁢(f′′⁢(ξν/n,x)-f′′⁢(x))⁢(νn-x)2

with certain intermediate points ξν/n,x between νn and x. Since Bernstein polynomials preserve linear functions and Bn⁢e2=e2+e1n⁢(1-e1), where ek⁢(x)=xk, we conclude that

(Bn⁢f)⁢(x)=f⁢(x)+12⁢f′′⁢(x)⁢x⁢(1-x)n+12⁢∑ν=0npn,ν⁢(x)⁢(νn-x)2⁢(f′′⁢(ξν/n,x)-f′′⁢(x)).

This implies that

(2.2)x⁢(1-x)n⁢|(Ln⁢f)⁢(x)|≤12⁢∑ν=0npn,ν⁢(x)⁢(νn-x)2⁢ω⁢(f′′;|ξν/n,x-x|).

The case n=1 will be considered separately. In this case the right-hand side of the latter inequality becomes

12⁢x2⁢(1-x)⁢ω⁢(f′′;|ξ0,x-x|)+12⁢x⁢(1-x)2⁢ω⁢(f′′;|ξ1,x-x|)

and we obtain

x⁢(1-x)⁢|(L1⁢f)⁢(x)|≤12⁢x⁢(1-x)⁢ω⁢(f′′;1).

Taking advantage of the well-known obvious inequality

(2.3)ω⁢(g;λ⁢h)≤⌈λ⌉⁢ω⁢(g;h),

which is valid for any function g and λ,h>0, we obtain, for δ>0,

x⁢(1-x)⁢|(L1⁢f)⁢(x)|≤12⁢x⁢(1-x)⁢⌈δ-1⌉⁢ω⁢(f′′;δ).

In particular, for δ=γ≥1, it follows that

x⁢(1-x)⁢|(L1⁢f)⁢(x)|≤12⁢x⁢(1-x)⁢ω⁢(f′′;γ),

which proves the assertion for n=1.

We continue with the case n≥2. A consequence of (2.3) is

ω⁢(f′′;|ξν/n,x-x|)≤ω⁢(f′′;|νn-x|)≤⌈δ-1⁢|νn-x|⌉⁢ω⁢(f′′;δ)≤(1+δ-1⁢|νn-x|)⁢ω⁢(f′′;δ).

Hence, by inequality (2.2),

2⁢x⁢(1-x)n⁢|(Ln⁢f)⁢(x)|≤∑0≤ν≤n,|νn-x|<δpn,ν⁢(x)⁢(νn-x)2⁢ω⁢(f′′;δ)+∑0≤ν≤n,|νn-x|≥δpn,ν⁢(x)⁢(νn-x)2⁢(1+δ-1⁢|νn-x|)⁢ω⁢(f′′;δ).

If |νn-x|≥δ, then δ-1⁢|νn-x|≤δ-2⁢|νn-x|2, so we conclude that

2⁢x⁢(1-x)n⁢|(Ln⁢f)⁢(x)|≤∑ν=0npn,ν⁢(x)⁢(νn-x)2⁢ω⁢(f′′;δ)+δ-2⁢∑0≤ν≤n,|νn-x|≥δpn,ν⁢(x)⁢(νn-x)4⁢ω⁢(f′′;δ)

≤[(Bn⁢ψx2)⁢(x)+δ-2⁢(Bn⁢ψx4)⁢(x)]⁢ω⁢(f′′;δ),

where the function ψx is defined by ψx⁢(t)=t-x. Using (Bn⁢ψx2)⁢(x)=xn⁢(1-x) and

(Bn⁢ψx4)⁢(x)=x⁢(1-x)n3⁢(1-6⁢x⁢(1-x)+3⁢n⁢x⁢(1-x))

(see, e.g., [10, p. 14]), we have

|(Ln⁢f)⁢(x)|≤(12+1-6⁢x⁢(1-x)+3⁢n⁢x⁢(1-x)2⁢δ2⁢n2)⁢ω⁢(f′′;δ).

For n≥2, the nominator 1+(3⁢n-6)⁢x⁢(1-x) can be estimated by 14⁢(3⁢n-2)<34⁢n, since 3⁢n-2 is nonnegative and 0≤x⁢(1-x)≤14 on [0,1]. Hence, we obtain, for each n≥2,

|(Ln⁢f)⁢(x)|≤(12+38⁢δ2⁢n)⁢ω⁢(f′′;δ).

Inserting the special value δ=γn proves Theorem 2.1. ∎

3 Best constants in the case δ=1n for some small n

Let us now consider the special choice δ=1n in the inequality

supx∈(0,1)|(Ln⁢f)⁢(x)|ω⁢(f′′;δ)≤Mn (n∈ℕ).

We are interested in the best (smallest) value Mn[opt] of the constant Mn, i.e.,

Mn[opt]:=supf∈Xn⁡supx∈(0,1)⁡|(Ln⁢f)⁢(x)|,

where Xn consists of all functions f∈C2⁢[0,1] with ω⁢(f′′;1n)=1. By the interpolation property of Bn we have to consider only the case x∈(0,1).

3.1 The case n=1

We prove that M1[opt]=12. For n=1, by Theorem 2.1 we know that M1[opt]≤12. Put bk=1-1k (k∈ℕ) and define a sequence of functions fk⁢(x)=k6⁢(x-bk)+3, where u+=u if u≥0 and u+=0 if u<0. Obviously, fk∈C2⁢[0,1] and fk′′⁢(x)=k⁢(x-bk)+ and we see that ω⁢(fk′′;1)=fk′′⁢(1)-fk′′⁢(bk)=1. We show that

limk→∞⁡supx∈(0,1)⁡|(L1⁢fk)⁢(x)|=12.

Since

x⁢(1-x)⁢(L1⁢f)⁢(x)=(1-x)⁢f⁢(0)+x⁢f⁢(1)-f⁢(x)-12⁢x⁢(1-x)⁢f′′⁢(x)

and fk⁢(x)=k6⁢(x-bk)+3, we have

(L1fk)(x)=fk⁢(1)1-x=16⁢(1-x)⁢k2≤16⁢k (0<x<bk).

Now let x∈(bk,1]. We obtain

(L1⁢fk)⁢(x)=f⁢(1)-f⁢(x)x⁢(1-x)-fk⁢(1)x-12⁢fk′′⁢(x).

By the mean value theorem we conclude that

|f⁢(1)-f⁢(x)x⁢(1-x)|=x-1⁢|fk′⁢(ξ)|≤bk-1⁢k2⁢(1-bk)2=12⁢(k-1).

Furthermore,

0≤fk⁢(1)x≤16⁢k⁢bk=16⁢(k-1)

and

0≤12⁢fk′′⁢(x)=12⁢k⁢(x-bk)≤12.

Hence, limk→∞⁡(L1⁢fk)⁢(1)=-12 and the claim follows by continuity.

3.2 The cases n≥2

For the considerations we use the following auxiliary results.

Lemma 3.1 (Stancu [14, equation (11.5)]).

For each function f:[0,1]→R there holds

(3.1)(Bn⁢f)⁢(x)-f⁢(x)=x⁢(1-x)n⁢∑ν=0n-1pn-1,ν⁢(x)⁢[x,νn,ν+1n;f]

provided that x∈[0,1]∖{νn∣ν=0,…,n}.

A new interesting proof of Stancu’s formula is given in [5, p. 132]. In case when x coincides with νn or 1n⁢(ν+1), an extension of Stancu’s formula to the whole interval [0,1] can be obtained using a limit process if f possesses a derivative on [0,1]. For instance, the divided difference on multiple knots [νn,νn,ν+1n;f] is equal to n2⁢(f⁢(ν+1n)-f⁢(νn))-n⁢f′⁢(νn).

Lemma 3.2 (see, e.g., [10, Chapter 4, Section 7, equation (7.16), p. 123]).

Let m∈N and f∈Cm⁢[a,b]. If all points x0,…,xm∈[a,b] are distinct, there holds

[x0,…,xm;f]=1(m-1)!∫abf(m)(t)[x0,…,xm;(⋅-t)+m-1]dt.

Because of ∑ν=0n-1pn-1,ν⁢(x)=1, Stancu’s formula (3.1) implies

(Bn⁢f)⁢(x)-f⁢(x)-x⁢(1-x)2⁢n⁢f′′⁢(x)=x⁢(1-x)n⁢∑ν=0n-1pn-1,ν⁢(x)⁢([x,νn,ν+1n;f]-12⁢f′′⁢(x)).

By Lemma 3.2,

[x,νn,ν+1n;f]=∫01f′′(t)[x,νn,ν+1n;(⋅-t)+]dt (x∈(0,1)).

In the particular case f⁢(x)=x2, we see that

(3.2)1=[x,νn,ν+1n;f]=2∫01[x,νn,ν+1n;(⋅-t)+]dt

and therefore, by (2.1),

(3.3)(Lnf)(x)=∑ν=0n-1pn-1,ν(x)∫01(f′′(t)-f′′(x))[x,νn,ν+1n;(⋅-t)+]dt.

Note that [x0,x1,x2;(⋅-t)+]≥0 for all pairwise different points x0,x1,x2 with 0≤x0<x1<x2≤1, and that [x0,x1,x2;(⋅-t)+]=0 if t<x0 or t>x2.

Taking advantage of inequality (2.3), we obtain, for δ>0,

|f′′⁢(t)-f′′⁢(x)|≤ω⁢(f′′;|t-x|)≤⌈δ-1⁢|t-x|⌉⋅ω⁢(f′′;δ).

By (3.2) and (3.3), we conclude that, for δ>0,

|(Ln⁢f)⁢(x)|≤cn⁢(δ,x)⋅ω⁢(f′′;δ),

where

(3.4)cn(δ,x)=∑ν=0n-1pn-1,ν(x)∫01⌈δ-1|t-x|⌉[x,νn,ν+1n;(⋅-t)+]dt.

We define

Cn:=supx∈(0,1)⁡cn⁢(1n,x).

A list giving the exact values of Cn for small n can be found in [1, p. 31]:

C1=12=0.500000,

C2=18⁢(97+140⁢2-10⁢2+1)≈0.504142,

C3=1162⁢(4⁢5340+6678⁢3-196⁢3-99)≈0.503828,

C4=132⁢(78⁢3-119)≈0.503124.

We conjecture that Mn[opt]=Cn (n∈ℕ). Since the proofs are rather long and tedious, we omit the calculations for n=3 and n=4. In order to illustrate the method, we restrict ourselves in proving that

C2=18⁢(97+140⁢2-10⁢2+1)

by estimating c2⁢(12,x).

3.3 The case n=2

Put δ=δ2=12≈0.707. By (3.4), we have c2⁢(δ2,x)=(1-x)⁢I1+x⁢I2 with

I1=∫01⌈δ-1|t-x|⌉[x,0,12;(⋅-t)+]dt,

I2=∫01⌈δ-1|t-x|⌉[x,12,1;(⋅-t)+]dt.

If 1-δ2≤x≤δ2, we have |t-x|≤δ2 for all t∈[0,1], which implies I1=I2=12 such that c2⁢(δ2,x)=12. Now we consider the case 0<x<1-δ2≈0.293. Since [x,0,12;(⋅-t)+]=0 for t∉[0,12],

I1=∫012[x,0,12;(⋅-t)+]dt=∫012[x,0,12;(⋅-t)+]dt=12.

Since [x,12,1;(⋅-t)+]=0 for t∉[x,1],

I2=∫xx+δ2[x,12,1;(⋅-t)+]dt+2∫x+δ21[x,12,1;(⋅-t)+]dt

=∫x1[x,12,1;(⋅-t)+]dt+∫x+δ21[x,12,1;(⋅-t)+]dt.

By (3.2), the first integral is equal to 12. Furthermore, [x,12,1;(⋅-t)+]=(1-t)+(1-x)⁢(1-12) if x+δ2≤t≤1. Hence,

I2=12+11-x⁢(1-x-δ2)2.

Finally,

c2⁢(δ2,x)=x3-(2-2)⁢x2-(2-1)⁢x+121-x.

The critical points of c2⁢(δ2,x) are given by

x1=14⁢(3-1+4⁢2)≈0.104978,

x2=12⁢(2-2)≈0.292893,

x3=14⁢(3+1+4⁢2)≈1.39502.

Note that c2⁢(δ2,0)=c2⁢(δ2,1-δ2)=12. Since x2,x3∉(0,1-δ2), we conclude that

c2⁢(δ2,x)≤c2⁢(δ2,x1)=18⁢(97+140⁢2-10⁢2+1)≈0.504142

for all x∈(0,1-δ2). The symmetry c2⁢(δ2,x)=c2⁢(δ2,1-x) implies this estimate also for x∈(δ2,1).

This completes the proof of

C2=18⁢(97+140⁢2-10⁢2+1).

Acknowledgements

The authors are grateful to the anonymous reviewer for valuable suggestions which improved the final outcome of the manuscript. In particular, he encouraged us to a deeper study of the inequality in the case of small n.

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Received: 2016-08-18

Revised: 2016-12-19

Accepted: 2016-12-29

Published Online: 2018-02-07

Published in Print: 2020-03-01

Artikel in diesem Heft

https://doi.org/10.1515/gmj-2017-0065

Schlagwörter für diesen Artikel

Approximation by positive operators; rate of convergence; degree of approximation