# L

Authors: Wei Xiao
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

Two important techniques to achieve the Jackson type estimation by Kantorovich type positive linear operators inLp spaces are introduced in the present paper, and three typical applications are given.


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## Body

## 1. Introduction

It is well known that Kantorovich type operators are usually used for approximation inLp spaces. Let f(x)∈L[0,1]p, the class of all p power integrable functions on the interval [0,1], and 𝒦n,j(x) (j=0,1,2,…,n, or j=1,2,…,n) be given kernels satisfying
(1)∑j=1n‍𝒦n,j(x)=1(or∑j=0n‍𝒦n,j(x)=1),𝒦n,j(x)≥0,x∈[0,1].This paper discusses how well the functionf(x)∈L[0,1]p can be approximated by the discrete Kantorovich-type operators such as
(2)Ln(f,x)=(n+1)∑j=0n‍𝒦n,j(x)∫j/(n+1)(j+1)/(n+1)‍f(t)dt
or
(3)Ln(f,x)=n∑j=1n‍𝒦n,j(x)∫(j-1)/nj/n‍f(t)dt,
and the Lp approximation is characterized by
(4)∥f-Ln(f)∥Lp≤Cω(f,ϵn)Lp,
where ϵn, a positive number depending on n only (such as n-1/2, n-1), is called Jackson order of Lp approximation; and C>0, a constant depending sometimes upon p as well as the kernels {𝒦n,j(x)} (e.g., the kernels of the Shepard operators; it contains a parameter λ; see (10)), is called Jackson constant. Since the kernel {𝒦n,j(x)} can have two types, decided by the summation indices, respectively (see (1)), the Kantorovich type operators are therefore defined by (2) or (3). However, with similar arguments, we can only investigate the positive linear operators of the form of (3).Sometimes, we need to writeQn(x,t)=𝒦n,j(x) for (j-1)/n≤t<j/n, j=1,2,…,n; then the operators (3) can be written as an integral form:
(5)Ln(f,x)=n∫01‍Qn(x,t)f(t)dt;
hence,
(6)∫01‍Qn(x,t)dt=n-1.
This means, for Kantorovich type operators, there does not exist any difference whether (3) or (5) is taken. In particular, we give the following examples.When the kernel functionQn(x,t) satisfies
(7)∂∂xQn(x,t)=nϕ(x)Qn(x,t)(t-x),
where ϕ(x)>0 satisfying ϕ(0)=ϕ(1)=0 is a polynomial with a degree at most 2, then the operators Ln(f,x) have exponential type, which have been studied deeply in [1–3], for example.When the kernel function𝒦n,j(x) in (3) is taken as
(8)Rj(x)=Pj(x)∑k=1nPk(x),Pk(x)=xλk∏l=1k‍(ln)-Δλl,
where {xλ0,xλ1,…,xλn,…} is a Müntz system satisfying that Δλn=λn-λn-1≥Mn (M is a given positive number), then the operator Ln(f,x) becomes the rational Müntz operator:
(9)Mn(f,x)=n∑k=1n‍Rk(x)∫(k-1)/nk/n‍f(t)dt;
readers can refer to [4, 5].When the kernel function𝒦n,j(x) of (2) is taken as
(10)rj(x)=|x-j/n|-λ∑k=0n|x-k/n|-λ(λ>1),
then the operator Ln(f,x) is the well-known Shepard operator
(11)Sn,λ(f,x)=(n+1)∑k=0n‍rk(x)∫k/(n+1)(k+1)/(n+1)‍f(t)dt;
one can check [6, 7], for example.When the kernel𝒦n,j(x) in (2) is taken as
(12)Pn,j(x)=(nj)xj(1-x)n-j,
then we obtain the Kantorovich-Bernstein operator
(13)Bn(f,x)=(n+1)∑k=0n‍Pn,k(x)∫k/(n+1)(k+1)/(n+1)‍f(t)dt,
which has been studied most widely among the positive linear operators of the form (2) (see [8–23]). Interested readers could also refer to the related papers for the other similar operators.This paper will take the above three typical operators (9), (11), and (13) as examples to illustrate two quantitative methods on Lp approximation. Over discussion, we find out that the Jackson order in Lp spaces to approximate f(x)∈L[0,1]p by the operators in (3) or (5) is decided completely by the kernels {𝒦n,j(x)}j=1n, or by the kernel function Qn(x,t). Therefore, on applying this idea, we need only to investigate the properties of the kernels to obtain the magnitude of the Jackson order of the corresponding operators, which seems to be a different approach from the past Lp approximating methods.


## 2. Notations and Terminologies

In this section, we give all preliminary notations and terminologies. Forf(x)∈L[0,1]p, the usual Lp norm is defined by
(14)∥f∥L[a,b]p={∫ab‍|f(x)|pdx}1/p,∥f∥Lp=∥f∥L[0,1]p,
and the Lp modulus of continuity of f∈L[0,1]p is defined by
(15)ω(f,δ)Lp=ω(f,δ)L[0,1-δ]p=sup0<t≤δ∥f(x+t)-f(x)∥L[0,1-δ]p.
To understand clearly Lp approximation by the positive linear operators, we need to make analysis of the kernels corresponding to the operators. Hence, some new terminologies on the kernels will be given and explained by some examples. For convenience, C always indicates an absolute positive constant and Cx indicates a positive constant depending upon at most x. C or Cx may have different values at different occurrences even at the same line. Sometimes we write 𝒦n=𝒦n(x)={𝒦n,j(x)}j=1n.Definition 1.
For anyx∈[0,1], if there exists a real sequence {Φ(k)}k=1∞ with 0<Φ(k)<1, k=1,2,…, satisfying
(16)𝒦n,r(x)≤CΦ(|[nx]-r|+1),r=1,2,…,n,
then {Φ(k)}k=1∞ is called a global domination of 𝒦n(x), or we say 𝒦n(x) is globally dominated by {Φ(k)}k=1∞, where [x] indicates the greatest integer not exceeding x. In particular, if Φ(k)=qk, 0<q<1, k=1,2,…, then the kernel sequence 𝒦n(x) is called a globally dominated geometrical sequence, or being dominated by a global geometrical sequence, or having (global) geometrical order. If Φ(k)=k-ρ, ρ>0, k=1,2,…, then the kernel sequence 𝒦n(x) is called a globally dominated arithmetic sequence with powerρ, or being dominated by a global arithmetic sequence with power ρ, or having (global) ρ-arithmetic order.Definition 2.
For anyx∈[0,1], if there exist a sequence {Φ(k)}k=1∞ with 0<Φ(k)<1, k=1,2,… and an integer subset Nx⊂{1,2,…,n} satisfying that
(17)𝒦n,r(x)≤CΦ(|[nx]-r|+1),r∈Nx,
then {Φ(k)}k=1∞ is called a local domination of 𝒦n(x). Like in Definition 1, a locally dominated geometrical sequence of 𝒦n(x) and a locally dominatedρ-arithmetic sequence can be defined similarly.The conceptions of Definitions1 and 2 will be illustrated by the following examples.Example 3.
The kernel functionsRk(x) of the rational Müntz operators defined by (9) satisfy
(18)Rk(x)≤CMexp(-CM|[nx]-k|),k=0,1,2,…,n
(see [4] or [5, Lemma 1]). The kernel functions Rk(x) then have geometrical order.Example 4.
The kernelsrk(x) of the Shepard operators Sn,λ(f,x)  (λ>1) defined by (11) satisfy
(19)rk(x)≤(2|[(n+1)x]-k|+1)λ,k=0,1,2,…,n
(see [12, Lemma 1]). The kernels rk(x) have λ-arithmetic order.Example 5.
The kernelsPn,k(x) of the Kantorovich-Bernstein operators (13) satisfy
(20)Pn,k(x)≈(2πnx(1-x))-1/2·exp(-n2x(1-x)(kn-x)2),
where x∈(0,1), and k satisfies
(21)|kn-x|≤n-α
for real number α>1/3 (see [15, Theorem 1.5.2]). That is to say, if the set of all k satisfying (21) is written as Nx, then, while k∈Nx, Pn,k(x) has asymptotic expression (20). Hence, the kernels Pn,k(x) of Bernstein operators, from Definition 2, have local geometrical order.Definition 6.
Forr=0,1,…, and any x∈[0,1], denote
(22)Mr(𝒦n)∶=supn≥1∑j=1n‍|nx-j|r𝒦n,j(x),
particularly,
(23)M(𝒦n)=M1(𝒦n),D(𝒦n)=M0(𝒦n).Definition 7.
Forr=1,2,…, write
(24)DMr(𝒦n)∶=supn≥1∑k=1n‍krΦ(k),
where {Φ(k)}k=1∞ is the global domination of 𝒦n(x) in Definition 1. In particular,
(25)DM(𝒦n)=DM1(𝒦n).Example 8.
For anyϵ∈(0,1), from Example 3, we have
(26)D(Rkϵ(x))≤CM∑k=0∞‍exp(-CMϵk)<∞,
where the kernels Rk(x) (8), as well as Φ(k)=e-CMk, from Definitions 6 and 7, satisfy Mr(𝒦nϵ)<∞ and DMr(𝒦nϵ)<∞ for all r=1,2,… and any given 0<ϵ≤1.Example 9.
Propose thatλ>1, write ϵλ=(λ+1)/2. In view of Example 4, we get
(27)D(rkϵλ(x))≤2λ(λ+1)/2∑k=1∞‍k-λ(λ+1)/2<∞.
Thus, from Definitions 6 and 7, the kernels rk(x) (10), as well as Φ(k)=k-λ, satisfy Mr(𝒦nϵλ)<∞ and DMr(𝒦nϵλ)<∞ for some r.Remark 10.
We make a brief discussion on the above definitions.(1)
Mr(𝒦nϵ)<∞ or DMr(𝒦nϵ)<∞ for some 0<ϵ<1 obviously yields that Mr(𝒦n)<∞ or DMr(𝒦n)<∞; the converse may not be true.(2)
The kernels having geometrical order satisfyMr(𝒦nϵ)<∞ and DMr(𝒦nϵ)<∞ for all r and 0<ϵ≤1.(3)
The kernels havingρ-arithmetic order satisfy D(𝒦nϵ)<∞ for ρ>1.


## 3. Elementary Approximation Technique (I)

This section gives one of the approximation techniques inLp spaces by the Kantorovich type operators (3). We mainly apply K-functional and maximum principle to obtain the Jackson type estimation in Lp spaces.Theorem 11.
Let{δn} be a positive null sequence. For any f(x)∈L[0,1]p, 1<p<∞, the Kantorovich operators defined by (3) satisfy (i)
∥Ln(f)∥Lp which is uniformly bounded;(ii)
Ln(|t-x|,x)=O(δn).Then, the estimate
(28)∥Ln(f)-f∥Lp≤Cpω(f,δn)Lp
holds, where Cp is a positive constant depending upon p only.Proof.
For any functionf(x)∈L[0,1]p, from the definition of the K-functional
(29)K(f,h)Lp=infg∈AC[0,1],g′∈Lp(∥f-g∥Lp+h∥g′∥Lp),
where AC[0,1] indicates the set of all absolute continuous functions on the interval [0,1], we know that K(f,h)Lp and ω(f,h)Lp are equivalent (see [24, Theorem 2.1]); that is,
(30)ω(f,h)Lp≈K(f,h)Lp,
where Ah≈Bh we mean there is a constant C>0 independent of h such that C-1Ah≤Bh≤CAh.
For givenf(x)∈L[0,1]p, the Hardy-Littlewood maximum function is defined by
(31)M(f,x)=supδ>01δ∫0δ‍|f(x+u)|du,
where if x+u∉[0,1], we simply set f(x+u)=0. It is well known that (see [25])
(32)∥M(f)∥Lp≤Cp∥f∥Lp,p>1.
Since for any g(x)∈AC[0,1], g′(x)∈L[0,1]p, p>1, we have
(33)∥Ln(f)-f∥Lp≤∥Ln(f-g)-(f-g)∥Lp+∥Ln(g)-g∥Lp,
then the condition (i) of Theorem 11 induces that
(34)∥Ln(f)-f∥Lp≤C∥f-g∥Lp+∥Ln(g)-g∥Lp.
It is easy to deduce from definition (3) that
(35)|Ln(g,x)-g(x)|≤n∑j=1n‍𝒦n,j(x)∫(j-1)/nj/n‍|g(t)-g(x)|dt=n∑j=1n‍𝒦n,j(x)∫(j-1)/nj/n‍|∫xt‍g′(u)du|dt≤nM(g′,x)∑j=1n‍𝒦n,j(x)∫(j-1)/nj/n‍|t-x|dt=M(g′,x)Ln(|t-x|,x).
With the condition (ii) in Theorem 11, we obtain
(36)|Ln(g,x)-g(x)|≤CM(g′,x)δn.
Combining (32) and (36) leads to
(37)∥Ln(g)-g∥Lp≤Cpδn∥g′∥Lp.
Together with (34) and (37), we get
(38)∥Ln(f)-f∥Lp≤C∥f-g∥Lp+Cpδn∥g′∥Lp.
Finally, from the definition of the K-functional we achieve that
(39)∥Ln(f)-f∥Lp≤CpK(f,δn)Lp.
Therefore, in view of (30) and (39), we have completed Theorem 11.


## 4. Elementary Approximation Technique (II)

In Section3, by applying the K-functional, we obtain the Jackson type estimation in Lp spaces for p>1. However, the Jackson constant in that case must depend upon p, and thus we cannot establish corresponding result in L1 space! In this section, we will exhibit another efficient technique in Lp spaces which will be used to obtain Jackson constant independent of p!Theorem 12.
Letf(x)∈L[0,1]p, 1≤p<∞, an ϵ be given with 0<ϵ<1 and the positive linear operators Ln(f,x) defined by (3). If the kernels 𝒦n(x) with (1) are dominated globally by {Φ(k)}k=1∞, and for some 0<ϵ<1 satisfy the following conditions:(i)
D(𝒦n1-ϵ)<∞,(ii)
DM(𝒦nϵ)<∞,then, the estimate
(40)∥Ln(f)-f∥L[0,1]p≤Cω(f,1/n)L[0,1]p
holds, where C>0 is an absolute constant.To prove Theorem12, we first give two lemmas.Lemma 13.
Givenh with 0<h<1 and f(x)∈L[0,1]p, write the Steklov function of f(x) as
(41)fh(x)=h-1∫0h‍f(x+u)du,
where x+u∈[0,1]. Then, the following results hold:
(42)(i)fh′(x)=f(x+h)-f(x)h;(43)(ii)∥f-fh∥L[0,1-h]p≤ω(f,h)Lp.Proof.
Equations (42) and (43) can be directly verified from the definition of the Steklov function (41).Lemma 14.
Propose that0≤a, b≤1, 0<δ<1, a+δ≤1, and b+δ≤1, then for any f(x)∈L[0,1]p, one has
(44)∫aa+δ‍∫bb+δ‍|f(x)-f(y)|pdxdy≤2δωp(f,|a-b|+δ)Lp.Proof.
This Lemma is proved in [7]; we give a sketch here for self-completeness. Due to the symmetries on a and b, as well as on x and y, we need only to prove the lemma under b≥a. By calculations,
(45)∫aa+δ‍∫bb+δ‍|f(x)-f(y)|pdxdy=2∫aa+δ‍(∫b-a+xb+δ‍|f(x)-f(y)|pdy)dx=2∫aa+δ‍(∫b-ab+δ-x‍|f(x)-f(x+y)|pdy)dx=2∫b-ab-a+δ‍(∫ab+δ-y‍|f(x)-f(x+y)|pdx)dy≤2δωp(f,b-a+δ)Lp.
Lemma 14 is done.Proof of Theorem12.
Takeh=1/n in the Steklov function (41); check
(46)∥Ln(f)-f∥L[1-h,1]p={∫1-h1‍(n∑r=1n‍𝒦n,r(x)∫(r-1)/nr/n‍(f(t)-f(x))dt)pdx}1/p
by applying Minkowski inequality, we get
(47)∥Ln(f)-f∥L[1-h,1]p≤n∑r=1n‍{∫1-1/n1‍𝒦n,rp(x)(∫(r-1)/nr/n‍|f(t)-f(x)|dt)pdx}1/p.
Then, apply Hölder inequality:
(48)∥Ln(f)-f∥L[1-h,1]p≤n1/p∑r=1n‍{∫1-1/n1‍𝒦n,rp(x)∫(r-1)/nr/n‍|f(t)-f(x)|pdtdx}1/p.
When x∈[1-1/n,1], [nx]=n-1. From the definition of the domination, we know
(49)𝒦n,r(x)≤CΦ(|[nx]-r|+1)=CΦ(|n-r-1|+1).
Then,
(50)∥Ln(f)-f∥L[1-h,1]p≤Cn1/p∑r=1n‍Φ(|n-r-1|+1)≤Cn1/p∑r=1n‍×{∫1-1/n1‍∫(r-1)/nr/n‍|f(t)-f(x)|pdtdx}1/p.
Applying Lemma 14, we have
(51)∥Ln(f)-f∥L[1-h,1]p≤C∑r=1n‍Φ(|n-r-1|+1)ω(f,n-r+1n)Lp≤Cω(f,1n)Lp∑r=1n‍Φ(|n-r-1|+1)(n-r+1)≤Cω(f,1n)Lp∑r=1n‍(|n-r-1|+1)Φ(|n-r-1|+1).
Applying (ii), we see that
(52)∑r=1n‍(|n-r-1|+1)Φ(|n-r-1|+1)≤∑r=1n‍(|n-r-1|+1)Φϵ(|n-r-1|+1)=DM(𝒦nϵ)<∞,
so that we obtain
(53)∥Ln(f)-f∥L[1-1/n,1]p≤Cω(f,1n)Lp.
Now we verify the case whenx∈[0,1-1/n]. It is not difficult to see from Lemma 13 that n∫r/n(r+1)/n‍f(t)dt=fh(r/n). By the definition of Kantorovich type operators, rewrite
(54)Ln(f,x)-fh(x)=n∑r=1n‍𝒦n,r(x)∫(r-1)/nr/n‍f(t)dt-fh(x)=∑r=1n‍𝒦n,r(x)(fh(r-1n)-fh(x))=∑r=1n‍𝒦n,r(x)∫x(r-1)/n‍fh′(u)du.
This leads to
(55)∥Ln(f)-fh∥L[0,1-n-1]pp=∫01-1/n‍|Ln(f,x)-fh(x)|pdx=∫01-1/n‍|∑r=1n‍𝒦n,r(x)∫x(r-1)/n‍fh′(u)du|pdx≤∫01-1/n‍(∑r=1n‍𝒦n,rp(1-ϵ)/(p-1)(x))p-1≤∫01-1/n‍·∑j=1n‍𝒦n,jpϵ(x)|∫x(j-1)/n‍fh′(u)du|pdx≤∫01-1/n‍(∑r=1n‍𝒦n,r1-ϵ(x))p≤∫01-1/n‍·∑j=1n‍𝒦n,jpϵ(x)|j-1n-x|p-1≤∫01-1/n‍·∑j=1n‍·|∫x(j-1)/n‍|fh′(u)|pdu|dx.
Since D(𝒦n1-ϵ)<∞,
(56)∥Ln(f)-fh∥L[0,1-n-1]pp≤Cp∫01-1/n‍∑j=1n‍𝒦n,jpϵ(x)|j-1n-x|p-1≤Cp∫01-1/n∑j=1n‍‍×|∫x(j-1)/n‍|fh′(u)|pdu|dx=Cp∑l=1n-1‍∫(l-1)/nl/n‍∑j=1n‍𝒦n,jpϵ(x)|j-1n-x|p-1=Cp∑l=1n-1‍∫(l-1)/nl/n∑j=1n‍‍×|∫x(j-1)/n‍|fh′(u)|pdu|dx=Cp∑l=1n-1‍∑j=1n‍∫01/n‍𝒦n,jpϵ(t+l-1n)|j-ln-t|p-1=Cp∑l=1n-1‍∑j=1n‍∫01/n‍×|∫t+((l-1)/n)(j-1)/n‍|fh′(u)|pdu|dt≤Cpn-p∑l=1n-1‍∑j=1n‍Φpϵ(|l-j-1|+1)≤Cpn-p∑l=1n-1‍∑j=1n‍×(|l-j-1|+1)p-1≤Cpn-p∑l=1n-1‍∑j=1n‍×|∫τ(j-1)/n‍|fh′(u)|pdu|,
where
(57)τ={l-1n,forj>l,ln,forj≤l.
Note that
(58)∑l=1n-1‍∑j=1n‍Φpϵ(|l-j-1|+1)(|l-j-1|+1)p-1∑l=1n-1‍∑j=1n‍×|∫τ(j-1)/n‍|fh′(u)|pdu|=∑m=1n‍∑1≤l≤n-1,1≤j≤n|l-j-1|+1=m‍Φpϵ(m)mp-1=∑m=1n‍∑1≤l≤n-1,1≤j≤n|l-j-1|+1=m‍×|∫τ(j-1)/n‍|fh′(u)|pdu|≤∑m=1n‍Φpϵ(m)mp-1·2m·∫01-h‍|fh′(u)|pdu≤2∫01-h‍|fh′(u)|pdu·(∑m=1n‍mΦϵ(m))p.
Furthermore, condition (ii) implies that
(59)∑m=1n‍mΦϵ(m)≤DM(𝒦nϵ)<∞.
This means
(60)∥Ln(f)-fh∥L[0,1-1/n]pp≤Cpn-p∫01-1/n‍|fh′(u)|pdu.
However, from Lemma 13,
(61)∫01-1/n‍|fh′(u)|pdu=np∫01-1/n‍|f(u+1n)-f(u)|pdu≤npωp(f,1n)Lp,
which leads to
(62)∥Ln(f)-fh∥L[0,1-n-1]p≤Cω(f,1n)Lp.
Combining (62) with (43), we get
(63)∥Ln(f)-f∥L[0,1-n-1]p≤Cω(f,1n)Lp.
This, with (53), finishes Theorem 12.ForL1 space, we have the following result while conditions of Theorem 12 can be loosed.Theorem 15.
Letf(x)∈L[0,1]1, Ln(f,x) be defined by (3). If the kernels with (1) are dominated globally by {Φ(k)} and satisfy DM(𝒦n)<∞, then the estimate
(64)∥Ln(f)-f∥L1≤Cω(f,1n)L1
holds.Proof.
The argument of proof is similar, and we can just repeat the corresponding parts of the proof of Theorem12.Remark 16.
(1) If the kernels possess good properties, the conditions of Theorem12 can be easily verified on the terminology of domination. For instance, if the kernels have geometric order, then the corresponding conditions of Theorem 12 are obviously satisfied (see the next section).
(2) There exists essential difference between Theorem11 and Theorem 12. Theorem 11 requires weaker conditions than Theorem 12 does, but the latter obtains stronger result (the Jackson order is complete up to 1/n, and the Jackson constant is independent of p!); we will make further illustrations in the coming section.


## 5. Applications

This section illustrates how to apply Theorems11 and 12 to estimate Lp approximation. To check the efficiency of two techniques on Lp approximation by Kantorovich type positive linear operators, three examples will be exhibited. Those positive linear operators come from three different categories: rational Müntz operators from rational Müntz systems; the Shepard operators from general real rational function systems; and Bernstein polynomials from the polynomial system. Moreover, in our point of view, they represent three different types: positive linear operators with kernels of geometric order, positive linear operators of arithmetical order, and positive linear operators of local geometric order. It is because the kernels have different domination properties or different speeds of {Φ(n)} that the Lp approximations by the corresponding positive linear operators possess different Jackson orders.To show the key role of the global (or local) domination on the kernels, the condition (ii) of Theorem11 will be further explicated to the following lemma.Lemma 17.
For any kernel𝒦n(x), one has
(65)Ln(|t-x|,x)=O(n-1)(M(𝒦n)+1).
Especially, if the kernels 𝒦n(x) are globally dominated, then
(66)Ln(|t-x|,x)=O(n-1)DM(𝒦n).Proof.
From definition (3),
(67)Ln(|t-x|,x)=n∑r=1n‍𝒦n,r(x)∫(r-1)/nr/n‍|t-x|dt.
For anyx∈[0,1], there exists an integer m such that (m-1)/n≤x<m/n; then ∫(r-1)/nr/n‍|t-x|dt will be calculated according to the following three cases, respectively.(a)
When1≤r<m,
(68)∫(r-1)/nr/n‍|t-x|dt=∫(r-1)/nr/n‍(x-t)dt=n-2(nx-r+12).(b)
Whenm+1≤r≤n,
(69)∫(r-1)/nr/n‍|t-x|dt=∫(r-1)/nr/n‍(t-x)dt=n-2(r-nx-12).(c)
Whenr=m,
(70)∫(m-1)/nm/n‍|t-x|dt≤∫(m-1)/nm/n‍n-1dt=n-2.Combining (a), (b), and (c), we obtain that
(71)∫(r-1)/nr/n‍|t-x|dt≤n-2(|[nx]-r|+1).
From Definition 6, (67) and (71) yield (65). At the same time, since the kernels of Ln(f,x) are globally dominated, we obtain by Definition 1 that
(72)𝒦n,r(x)≤CΦ(|[nx]-r|+1),r=1,2,…,n.
Thus,
(73)Ln(|t-x|,x)≤Cn-1∑r=1n‍(|[nx]-r|+1)Φ(|[nx]-r|+1);
that is, from Definition 7, (66) holds.
Lemma17 is done.


### 5.1. Rational Müntz Approximation

Rational Müntz Approximation has been researched in [5], shows the application of Theorem 12, and simplifies the proof of [5].Write(74)Λn={xλ1,xλ2,…,xλn},
where span{xλk}, λk∈Λn, is the class of all linear combinations of {xλk}k=1n. For f(x)∈L[0,1]p, define
(75)Rn(f,Λ)=minr∈R(Λn)∥f-r∥Lp.Corollary 18.
Givenf(x)∈L[0,1]p, 1≤p<∞, the rational Müntz operators are defined by (9). If Δλn≥Mn, n=1,2,3,…, where M>0 is an absolute constant, one has
(76)Rn(f,Λ)≤∥Mn(f)-f∥Lp≤CMω(f,1n)Lp,
where CM is an absolute constant depending only upon M (independent of p!).Proof.
We verify thatMn(f,x) satisfies all the requirements of Theorem 12. Take Φ(k)=e-CMk, where CM>0 is the constant appeared in [4], or [5]; see the following inequality (77). Given a fixed ϵ with 0<ϵ<1, we verify that
(i)D(Rk1-ϵ(x))<∞.
From [4], or [5],
(77)Rk(x)≤CMexp(-CM|[nx]-k|).
Then,
(78)∑k=1n‍Rk1-ϵ(x)≤CM∑k=1n‍exp(-CM|[nx]-k|(1-ϵ))≤CM∑k=0∞‍exp(-CMk(1-ϵ))<∞.
(ii)DM(Rkϵ(x))<∞.
Due to Example3, we have Φ(k)=exp(-CMk). From Definition 7,
(79)DM(Rkϵ)=supn>1∑k=1n‍kexp(-CMkϵ)≤∑k=0∞‍kexp(-CMkϵ)<∞.
Therefore, Corollary 18 can be deduced from Theorem 12.By the same argument of (ii), we obviously can obtainDM(Rk)<∞. From (66), we know Mn(|t-x|,x)=O(n-1). Moreover, it is simple to verify ∥Mn(f)∥Lp<∞ (see, e.g., [7]). Hence, applying Theorem 11, we getCorollary 19.
For rational Müntz operatorMn(f,x) (9), one has
(80)Rn(f,Λ)Lp≤CM,pω(f,1n)Lp,
where CM,p is depending on M and p.Remark 20.
Since the kernelsRk(x) (8) have geometrical order (see Example 3), they certainly satisfy the conditions of both Theorems 11 and 12. Hence, Lp approximation by these rational Müntz operators can always reach the Jackson order by applying both techniques of Theorems 12 and 11. However, for these operators, the conclusion of Corollary 18 surely contains Corollary 19 and makes the latter trivial.


### 5.2. The Shepard Operators

The approximation of the Shepard operators in the continuous function spaceC[0,1] has been studied very deeply (see [13, 26–32]). The Lp approximation by the Shepard operators is investigated in [6, 7].Corollary 21.
Proposed thatf(x)∈L[0,1]p, p>1, the Shepard operators are defined by (11). Then,
(81)∥Sn,λ(f)-f∥Lp≤Cp,λω(f,ϵn)Lp,
where
(82)ϵn={n-1,ifλ>2,n-1logn,ifλ=2,n1-λ,if1<λ<2.Proof.
By applying Theorem11, we verify this result.(i)
∥Sn,λ(f)∥Lp are uniformly bounded (see [6]).(ii)
Sn,λ(|t-x|,x)=ϵn (Lemma 17, Inequality (66)).
Due to Example4, we know that the global dominated sequence is Φ(k)=k-λ. Then, from Definition 7,
(83)DM(rk(x))=supn>1∑k=1n‍kΦ(k)=supn>1∑k=1n‍k1-λ.
Whenλ>2, ∑k=1∞‍k1-λ<∞.
Whenλ=2, ∑k=1n+1‍k1-λ≤2logn.
When1<λ<2, ∑k=1n+1‍k1-λ≤Cλn2-λ.
By Theorem11, Corollary 21 holds.Corollary 22.
Propose thatf(x)∈L[0,1]p, 1≤p<∞, the Shepard operators are defined by (11). If λ>3, then
(84)∥Sn,λ(f)-f∥Lp≤Cλω(f,1n+1)Lp,
where Cλ depends on λ only.Proof.
Theorem12 will be applied to prove this result. The details can be referred in [7]. When λ>3, let ϵ=(λ+1)/2λ, then 0<ϵ<1, λϵ>2 and λ(1-ϵ)>1. Evidently, the Shepard kernels have λ-arithmetic order (see Example 4). We check the corresponding conditions of Theorem 12:
(i)D(rk1-ϵ(x))<∞.
From Definition6 and Example 4,
(85)∑k=1n‍rk1-ϵ(x)≤Cλ∑k=0n‍(|[(n+1)x]-k|+1)-λ(1-ϵ)≤Cλ∑k=1∞‍k-λ(1-ϵ)<∞.
Then,
(ii)DM(rkϵ(x))<∞.
Note that the present dominated sequenceΦ(k)=k-λ; then from Definition 7(86)DM(rkϵ(x))=supn>1∑k=1n‍kΦϵ(k)≤supn>1∑k=1∞‍k1-λϵ<∞.
Corollary22 is completed from Theorem 12 as (i) and (ii) hold.Remark 23.
The difference betweenLp approximation techniques (I) and (II) with respect to Theorems 11 and 12 is fully exhibited on the Shepard operators by Corollaries 21 and 22. Stronger requirements by applying technique (II) than by applying technique (I) are needed. However, if the conditions are satisfied, the former can obtain essentially better result (the Jackson constant is independent of p!). On this particular case, we can obtain the Jackson type estimation by applying technique (I) for λ>1, while achieve the corresponding result by applying technique (II) only for λ>3. We still do not know how to deal with the cases when 1<λ≤3 by applying technique (II).


### 5.3. Bernstein Operators

There are many results onLp approximation by Bernstein polynomials; interested readers may refer to [8–23]. These results can be classified into the following categories:(1)
uniform convergence; see [8, 12, 15, 18];(2)
quantitative estimations; see [10, 11, 14, 19];(3)
equivalence theorems; see [16, 17];(4)
saturation problems; see [9, 16, 20–23].Here, we apply our technique (I) to test the corresponding Jackson type estimation as an example. The following proof is simpler than the past proof.Corollary 24.
Givenf(x)∈L[0,1]p, p>1, one has the following estimation:
(87)∥Bn(f)-f∥Lp≤Cpω(f,1n)Lp.Proof.
∥
B
n
(
f
)
∥
L
p is uniformly bounded which is known from [33]. We only need to evaluate Bn(|t-x|,x) from Theorem 11. It is known from [15, page 15] and Definition 6 that
(88)M(Pn,k(x))=supn≥1∑k=0n‍|nx-k|Pn,k(x)≤Cn.
From (65) of Lemma 17 and (88),
(89)Bn(|t-x|,x)=O(n-1)M(Pn,k(x))=O(n-1/2)
which, from Theorem 11, leads to Corollary 24.Remark 25.
The approximation order of Corollary24 is sharp which shows that (88) cannot be improved. In other words, M(Pn,k(x)) is unbounded. Hence, the approximation technique (II) or Theorem 12 cannot be applied in this case. That is to say, we can never obtain the Jackson constant independent of p for Bernstein polynomials in Lp spaces. Furthermore, Corollary 24 also exhibits that Kantorovich type operators (2) or (3) cannot reach Jackson type estimation with the Jackson constant independent of p unless their kernels possess good properties such as having globally geometric domination (rational Müntz approximation case) or having global ρ-arithmetic domination for sufficiently large ρ (the Shepard operators with λ>3).


## 5.1. Rational Müntz Approximation

Rational Müntz Approximation has been researched in [5], shows the application of Theorem 12, and simplifies the proof of [5].Write(74)Λn={xλ1,xλ2,…,xλn},
where span{xλk}, λk∈Λn, is the class of all linear combinations of {xλk}k=1n. For f(x)∈L[0,1]p, define
(75)Rn(f,Λ)=minr∈R(Λn)∥f-r∥Lp.Corollary 18.
Givenf(x)∈L[0,1]p, 1≤p<∞, the rational Müntz operators are defined by (9). If Δλn≥Mn, n=1,2,3,…, where M>0 is an absolute constant, one has
(76)Rn(f,Λ)≤∥Mn(f)-f∥Lp≤CMω(f,1n)Lp,
where CM is an absolute constant depending only upon M (independent of p!).Proof.
We verify thatMn(f,x) satisfies all the requirements of Theorem 12. Take Φ(k)=e-CMk, where CM>0 is the constant appeared in [4], or [5]; see the following inequality (77). Given a fixed ϵ with 0<ϵ<1, we verify that
(i)D(Rk1-ϵ(x))<∞.
From [4], or [5],
(77)Rk(x)≤CMexp(-CM|[nx]-k|).
Then,
(78)∑k=1n‍Rk1-ϵ(x)≤CM∑k=1n‍exp(-CM|[nx]-k|(1-ϵ))≤CM∑k=0∞‍exp(-CMk(1-ϵ))<∞.
(ii)DM(Rkϵ(x))<∞.
Due to Example3, we have Φ(k)=exp(-CMk). From Definition 7,
(79)DM(Rkϵ)=supn>1∑k=1n‍kexp(-CMkϵ)≤∑k=0∞‍kexp(-CMkϵ)<∞.
Therefore, Corollary 18 can be deduced from Theorem 12.By the same argument of (ii), we obviously can obtainDM(Rk)<∞. From (66), we know Mn(|t-x|,x)=O(n-1). Moreover, it is simple to verify ∥Mn(f)∥Lp<∞ (see, e.g., [7]). Hence, applying Theorem 11, we getCorollary 19.
For rational Müntz operatorMn(f,x) (9), one has
(80)Rn(f,Λ)Lp≤CM,pω(f,1n)Lp,
where CM,p is depending on M and p.Remark 20.
Since the kernelsRk(x) (8) have geometrical order (see Example 3), they certainly satisfy the conditions of both Theorems 11 and 12. Hence, Lp approximation by these rational Müntz operators can always reach the Jackson order by applying both techniques of Theorems 12 and 11. However, for these operators, the conclusion of Corollary 18 surely contains Corollary 19 and makes the latter trivial.


## 5.2. The Shepard Operators

The approximation of the Shepard operators in the continuous function spaceC[0,1] has been studied very deeply (see [13, 26–32]). The Lp approximation by the Shepard operators is investigated in [6, 7].Corollary 21.
Proposed thatf(x)∈L[0,1]p, p>1, the Shepard operators are defined by (11). Then,
(81)∥Sn,λ(f)-f∥Lp≤Cp,λω(f,ϵn)Lp,
where
(82)ϵn={n-1,ifλ>2,n-1logn,ifλ=2,n1-λ,if1<λ<2.Proof.
By applying Theorem11, we verify this result.(i)
∥Sn,λ(f)∥Lp are uniformly bounded (see [6]).(ii)
Sn,λ(|t-x|,x)=ϵn (Lemma 17, Inequality (66)).
Due to Example4, we know that the global dominated sequence is Φ(k)=k-λ. Then, from Definition 7,
(83)DM(rk(x))=supn>1∑k=1n‍kΦ(k)=supn>1∑k=1n‍k1-λ.
Whenλ>2, ∑k=1∞‍k1-λ<∞.
Whenλ=2, ∑k=1n+1‍k1-λ≤2logn.
When1<λ<2, ∑k=1n+1‍k1-λ≤Cλn2-λ.
By Theorem11, Corollary 21 holds.Corollary 22.
Propose thatf(x)∈L[0,1]p, 1≤p<∞, the Shepard operators are defined by (11). If λ>3, then
(84)∥Sn,λ(f)-f∥Lp≤Cλω(f,1n+1)Lp,
where Cλ depends on λ only.Proof.
Theorem12 will be applied to prove this result. The details can be referred in [7]. When λ>3, let ϵ=(λ+1)/2λ, then 0<ϵ<1, λϵ>2 and λ(1-ϵ)>1. Evidently, the Shepard kernels have λ-arithmetic order (see Example 4). We check the corresponding conditions of Theorem 12:
(i)D(rk1-ϵ(x))<∞.
From Definition6 and Example 4,
(85)∑k=1n‍rk1-ϵ(x)≤Cλ∑k=0n‍(|[(n+1)x]-k|+1)-λ(1-ϵ)≤Cλ∑k=1∞‍k-λ(1-ϵ)<∞.
Then,
(ii)DM(rkϵ(x))<∞.
Note that the present dominated sequenceΦ(k)=k-λ; then from Definition 7(86)DM(rkϵ(x))=supn>1∑k=1n‍kΦϵ(k)≤supn>1∑k=1∞‍k1-λϵ<∞.
Corollary22 is completed from Theorem 12 as (i) and (ii) hold.Remark 23.
The difference betweenLp approximation techniques (I) and (II) with respect to Theorems 11 and 12 is fully exhibited on the Shepard operators by Corollaries 21 and 22. Stronger requirements by applying technique (II) than by applying technique (I) are needed. However, if the conditions are satisfied, the former can obtain essentially better result (the Jackson constant is independent of p!). On this particular case, we can obtain the Jackson type estimation by applying technique (I) for λ>1, while achieve the corresponding result by applying technique (II) only for λ>3. We still do not know how to deal with the cases when 1<λ≤3 by applying technique (II).


## 5.3. Bernstein Operators

There are many results onLp approximation by Bernstein polynomials; interested readers may refer to [8–23]. These results can be classified into the following categories:(1)
uniform convergence; see [8, 12, 15, 18];(2)
quantitative estimations; see [10, 11, 14, 19];(3)
equivalence theorems; see [16, 17];(4)
saturation problems; see [9, 16, 20–23].Here, we apply our technique (I) to test the corresponding Jackson type estimation as an example. The following proof is simpler than the past proof.Corollary 24.
Givenf(x)∈L[0,1]p, p>1, one has the following estimation:
(87)∥Bn(f)-f∥Lp≤Cpω(f,1n)Lp.Proof.
∥
B
n
(
f
)
∥
L
p is uniformly bounded which is known from [33]. We only need to evaluate Bn(|t-x|,x) from Theorem 11. It is known from [15, page 15] and Definition 6 that
(88)M(Pn,k(x))=supn≥1∑k=0n‍|nx-k|Pn,k(x)≤Cn.
From (65) of Lemma 17 and (88),
(89)Bn(|t-x|,x)=O(n-1)M(Pn,k(x))=O(n-1/2)
which, from Theorem 11, leads to Corollary 24.Remark 25.
The approximation order of Corollary24 is sharp which shows that (88) cannot be improved. In other words, M(Pn,k(x)) is unbounded. Hence, the approximation technique (II) or Theorem 12 cannot be applied in this case. That is to say, we can never obtain the Jackson constant independent of p for Bernstein polynomials in Lp spaces. Furthermore, Corollary 24 also exhibits that Kantorovich type operators (2) or (3) cannot reach Jackson type estimation with the Jackson constant independent of p unless their kernels possess good properties such as having globally geometric domination (rational Müntz approximation case) or having global ρ-arithmetic domination for sufficiently large ρ (the Shepard operators with λ>3).


## 6. Conclusions

On the above discussions, the positive linear operators used inLp approximation can be classified according to the properties of their kernels. We have three categories: kernels with geometrical order (such as the rational Müntz operators), kernels with arithmetic order (such as the Shepard operators), and kernels with local arithmetic order (such as the Bernstein operators). In another word, for characterizing the Jackson type estimate in Lp spaces by the Kantorovich type operators (3) or (5), it always plays an essential role how well the kernels of the operators under study behave.



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