# Strong Converse Results for Linking Operators and Convex Functions

Authors: Ana-Maria Acu; Margareta Heilmann; Ioan Rasa
Journal: Journal of Function Spaces
 (2020)

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## Abstract

We consider a familyBn,ρc of operators which is a link between classical Baskakov operators (for ρ=∞) and their genuine Durrmeyer type modification (for ρ=1). First, we prove that for fixed n,c and a fixed convex function f, Bn,ρcf is decreasing with respect to ρ. We give two proofs, using various probabilistic considerations. Then, we combine this property with some existing direct and strong converse results for classical operators, in order to get such results for the operators Bn,ρc applied to convex functions.


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## Body

## 1. Introduction

The Baskakov-type operators depending on a real parameterc were introduced by Baskakov in [1]. This class of operators includes the classical Bernstein, Szász-Mirakjan, and Baskakov operators as special cases for c=−1, c=0, and c=1, respectively.Letc∈ℝ, n∈ℝ, n>c for c≥0, and −n/c∈ℕ for c<0. Furthermore, let Ic=0,∞ for c≥0 and Ic=0,−1/c for c<0. Consider f:Ic→ℝ given in such a way that the corresponding integrals and series are convergent.The Baskakov-type operators are defined as follows:(1)Bn,∞cfx=∑j=0∞pn,jcxfjn,where
(2)pn,jx=njj!xje−nx,c=0,nc,j¯j!xj1+cx−n/c+j,c≠0,=1n−c−cj+1Bj+1,−n/c−j+1xj1+cx−n/c+j,c<0,njΓj+1xje−nx,c=0,1n−ccj+1Bj+1,n/c−1xj1+cx−n/c+j,c>0,and ac,j¯≔∏l=0j−1a+cl,ac,0¯≔1.We remark that (2) is well defined also for j∈ℝ, j≥0, which will be considered below.The genuine Baskakov-Durrmeyer-type operators are given by(3)Bn,1cfx=f0pn,0cx+f−1cpn,−n/ccx+∑j=1−n/c−1pn,jcxn+c∫0−1/cpn+2c,j−1ctftdt,forc<0,Bn,1cfx=f0pn,0cx+∑j=1∞pn,jcxn+c∫0∞pn+2c,j−1ctftdt,forc≥0.In the last years, a nontrivial link between classical Baskakov-type operators and their genuine Durrmeyer-type modification came into the focus of research. Depending on a parameterρ∈ℝ+, the linking operators are given by
(4)Bn,1cfx=∑j=0∞Fn,j,ρcfpn,jcx,where
(5)Fn,j,ρcf=f0,j=0,c∈ℝ,f−1c.j=−nc,c<0,∫Icμn,j,ρctftdt,otherwise,with μn,j,ρct=nρ+cpnρ+2c,jρ−1ct.Forc<0, Bn,ρcf is well defined if f∈L10,1 with finite limits at the endpoints of the interval 0,−1/c, i.e., f0=limx→0+fx and f−1/c=limx→−1/c−fx.Forc≥0, the operators Bn,ρc are well defined for functions f∈Wnρ having a finite limit f0=limx→0+fx where Wnρ denotes the space of functions f∈L1,loc0,∞ satisfying certain growth conditions, i. e., there exist constants M>0, 0≤q<nρ+c, such that a. e. on 0,∞.
(6)ft≤Meqtforc=0,ft≤Mtq/cforc>0.First, we prove that for fixedn,c and a fixed convex function f, Bn,ρcf is decreasing with respect to ρ. We give two proofs, using various probabilistic considerations. Then, we combine this property with some existing direct and strong converse results for classical operators, in order to get such results for the operators Bn,ρc applied to convex functions.


## 2. The Casec=−1

For the linking Bernstein operator, i.e.,c=−1, Rasa and Sta˘nila˘ [2], (10) proved that for a convex function f∈C0,1,
(7)Bn,ρ−1f;x≥Bn,σ−1f;x≥fx,1≤ρ<σ≤∞.For the proof, they used thatBn,ρ can be written as a combination of the classical Bernstein operator and Beta operator and some corresponding results for the Beta operator from Adell et al. [3], Theorem 1. For the case ρ=1 and the case ρ=∞, strong converse results are known [4], Theorem 1.1, [5], p.117 [6], and [7], Theorem 3.2, Theorem 5:
(8)Bn,ρ−1f−f∞~ωφ2f,n−1/2,ρ∈1,∞,where (see [5])
(9)ωφ2f,t≔sup0<h≤tΔhφ2f,with φ2 a weight function and Δh2fx=fx+h−2fx+fx−h.Thus, forf convex, 1≤ρ≤∞(10)0≤Bn,∞−1f,x−fx≤Bn,ρ−1f;x−fx≤Bn,1−1f;x−fx,leading to
(11)C1−1ωφ2f,n−1/2≤Bn,∞−1f−f≤Bn,ρ−1f−f≤Bn,1−1f−f≤C1ωφ2f,n−1/2,i.e.,(12)Bn,ρ−1f−f∞~ωφ2f,n−1/2,1≤ρ≤∞.


## 3. The Casec=0

Consider the classical Szász-Mirakjan operators(13)Bn,∞0f;x≔∑j=0∞pn,j0xfjnand also the operators
(14)Srf;x=rrxΓrx∫0∞trx−1e−rtftdt,x>0,f0,x=0,where r>0.Moreover, forr>0, let (see [8]).
(15)Grf;x=r/xrΓr∫0∞tr−1e−rt/xftdt,x>0,f0,x=0.Theorem 1 (see [8], Theorem 5 and Remark 6).
Letf and x be fixed and f convex, such that Grf;x<∞ for all r>0. Then, Grf;x is nonincreasing with respect to r.
Then,(16)Srf;x=Grxf;x.
Forc=0,
(17)Bn,ρ0=Bn,∞0∘Snρ.
Letf be convex and n and x be fixed, such that Grf;x<∞, for all r>0.
Let1≤ρ≤σ. Then, by (16) and Theorem 1,
(18)Snσf;x=Gnσxf;x≤Gnρxf;x=Snρf;x.
Now by (17),
(19)Bn,σ0f=Bn,∞0Snσf≤Bn,∞0Snρf=Bn,ρ0f.
Thus,(20)Bn,σ0f≤Bn,ρ0f.
Strong converse results are known also in this case (see [6], Theorem 1.2 and [9], Theorem 5.1, Theorem 5.2):
(21)Bn,ρ0f−f∞~ωφ2f,n−1/2,ρ∈1,∞.
Thus, forf convex,
(22)Bn,ρ0f−f∞~ωφ2f,n−1/2,1≤ρ≤∞.


## 4. The Casec=1

To treat this case, we need some preliminaries.IfX and Y are independent random variables with densities fθ, gθ, and θ>0, then X/Y has density
(23)wt=∫0∞ftθgθθdθ.LetUττ≥0 and Vττ≥0 be two independent gamma processes (see [8], p.129), i. e., Uτ has density 1/Γτθτ−1e−θ, θ>0. Let x>0 and r>0. Then, Zrx≔Urx/Vr+1 has density
(24)wr,xt=∫0∞1Γrxtθrx−1e−tθ1Γr+1θr+1e−θdθ=trx−1ΓrxΓr+1∫0∞θrx+re−θt+1dθ,fort>0.Substitutes=θt+1. Then,
(25)wr,xt=trx−1ΓrxΓr+1∫0∞srx+rt+1rx+re−sdst+1=trx−1t+1rx+r+1Γrx+r+1ΓrxΓr+1=1Brx,r+1trx−1t+1rx+r+1.LetBrf;x=∫0∞wr,xtftdt.Consequently,(26)Brf;x=EfUrxVr+1=EfZrx,compare with [8], (9).As in [10], Proof of Lemma 2, let 1≤r≤s, x>0. Since the random vectors Urx,Usx and Vr+1,Vs+1 are independent, we have
(27)EZrxUsx,Vs+1=EUrx∣UsxEVr+1−1∣Vs+1=rsUsxEVr+1−1∣Vs+1,with [10], Lemma 1. Moreover, as in [10], (19), we get
(28)EVr+1−1∣Vs+1=Vs+1−1·sr.Thus,(29)EZrxUsx,Vs+1=UsxVs+1=Zsx.As at the end of [10], Proof of Lemma 2, taking here the conditional expectation w.r.t. Zsx, we get
(30)EZrx∣Zsx=Zsx,a.s.,1≤r≤s,x>0.Now (30) is exactly the assumption of [8], Theorem 5 (a). Accordingly, [8], Theorem 5 (a) and Remark 6 show that
(31)Brf≥Bsf,1≤r≤s,fconvex.This implies(32)Bnρf≥Bnσf,1≤ρ≤σ,fconvex,Bn,∞1Bnρf≥Bn,∞1Bnσf,1≤ρ≤σ,fconvex,where Bn,∞1 are the classical Baskakov operators.Since,Bn,ρ1=Bn,∞1∘Bnρ, we infer that
(33)Bn,σ1f≤Bn,ρ1f,fconvex.The direct and strong converse results are known also in this case (see [11], Theorem 1.2, Theorem 1.3 and [12], Theorem 1.1):
(34)Bn,ρ1f−f∞~ωφ2f,n−1/2,ρ∈1,∞.Thus, forf convex,
(35)Bn,ρ1f−f∞~ωφ2f,n−1/2,1≤ρ≤∞.


## 5. An Application of Ohlin’s Lemma

For more details about the techniques used in this section, the reader is referred to [13] and the references therein.Lemma 2 (Ohlin’s Lemma) (see [14]).
LetX and Y be two random variables on the same probability space such that EX=EY. If the distribution functions FX and FY cross exactly one time, i.e., for some x0 holds
(36)FXx≤FYxifx<x0andFXx≥FYxifx>x0,then EfX≤EfY, for all convex functions f:ℝ→ℝ.We have∫Icμn,j,ρctdt=1 and ∫Ictμn,j,ρtdt=j/n.Therefore,μn,j,ρc is the probability density function of a random variable Xn,j,ρc with expectation EXn,j,ρc=j/n and probability distribution function Gn,j,ρcx=∫0xμn,j,ρctdt.Letρ<σ. We will apply Ohlin’s Lemma to the random variables Xn,j,ρc and Xn,j,σc. Since their expectation is equal, we have to prove that Gn,j,ρc and Gn,j,σc cross exactly ones.Let(37)gx=Gn,j,ρcx−Gn,j,σcx,x∈Ic.Then,g′x=μn,j,ρcx−μn,j,σcx.In what follows, we supposec≠0; the case c=0 can be treated similarly or we can consider c→0 in the computations below. For c≠0, we have
(38)g′x=K1xjρ−11+cx−np/c+jρ+1−K2xjσ−11+cx−nσ/c+jσ+1=K2xjρ−11+cx−np/cnρc+jρ+1K1K2−xj1+cx−n/c+jσ−ρwith positive constants K1 and K2.OnintIc, the equation g′x=0 is equivalent to K1/K21/σ−ρ1+cxn/c+j=xj.First, suppose thatj>0 and, if c<0, j<−n/c. Then, on intIc, the equation g′x=0 is equivalent to hx=x, where
(39)hx≔K1K21/jσ−ρ1+cx1+n/cj,x∈intIcis a strictly convex function. The equation hx=x has at most two roots in intIc, and so the derivative g′ has at most two zeroes in intIc. If c<0, g0=g−1/c=0; if c>0, g0=limx→∞gx=0. Therefore, g′ has at least one zero in intIc.Suppose thatg′ has exactly one zero in intIc, let it be x0. Then, g′ has opposite signs in the two intervals determined by x0. But (38) shows that g′ is positive near the endpoints of Ic. This contradiction leads us to the conclusion that g′ has exactly two zeroes x1<x2 in intIc; they are also roots of the equation x=hx, hx being a strictly convex function. Moreover, g′ is positive outside of x1,x2 and negative inside it. Therefore, gx is strictly increasing for x<x1 and for x>x2, and strictly decreasing for x1<x<x2, with gx1>0 and gx2<0.We conclude that there existsx0∈x1,x2 with gx0=0, gx≥0 for x<x0, and gx≤0 for x>x0. Remembering that gx=Gn,j,ρcx−Gn,j,σcx, we see that
(40)Gn,j,σcx≤Gn,j,ρcx,x<x0,Gn,j,σcx≥Gn,j,ρcx,x>x0.Now Ohlin’s Lemma shows that(41)EfXn,j,σc≤EfXn,j,ρc,i.e.,(42)∫Icμn,j,σctftdt≤∫Icμn,j,ρctftdt,for f convex, j>0, and if c<0,j<−n/c.With notation from (5), this means that
(43)Fn,j,σcf≤Fn,j,ρcf,fconvex.Moreover, (5) shows that the above relation is an equality if j=0 and, for c<0, if j=−n/c. Now according to (4),
(44)Bn,σcf≤Bn,ρcf,fconvex,0<ρ<σ.Lettingσ→∞, we see that the above inequality is valid for 0<ρ<σ≤∞. This was proved with other methods, for c∈−1,0,1, in the preceding sections.



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*Source: 4049167-2020-12-07.xml*