Vol 7, No 4 (2016)

Available online throug

𝛍𝛍 −

Best One-Sided Approximation of Unbounded Functions in the Space

𝐋𝐋

(

𝐗𝐗

)

SAHEB K.AL-SAIDY

_{, NAGHAM ALI HUSSEN*}

1

_{Al-Mustansiriyah University,}

College of Science, Department of Mathematics, Baghdad, Iraq.

2

_{Tikrit University, College of Education, Department of Mathematics, Tikrit, Iraq.}

(Received On: 17-03-16; Revised & Accepted On: 12-04-16)

ABSTRACT

T

(1≤ 𝑝𝑝<∞) by Spline polynomials ,we consider the point wise estimes in terms of Ditzian-Totic modulus of smoothness are true for spline approximation in the space 𝐿𝐿_𝑝𝑝,µ[𝑎𝑎,𝑏𝑏].

1. INTRODUCTION

Let X= [a, b], Consider the space

ℙ𝑛𝑛 = {𝑝𝑝(𝑥𝑥):𝑝𝑝(𝑥𝑥) = ∑𝑛𝑛𝑖𝑖=0𝑐𝑐𝑖𝑖𝑥𝑥𝑖𝑖−1, 𝑐𝑐1,𝑐𝑐2, … ,𝑐𝑐𝑛𝑛 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑟𝑟𝑟𝑟}

of polynomials of order 𝑛𝑛 which has the attractive features [6]

Let 𝑎𝑎=𝑥𝑥₀<𝑥𝑥₁<⋯<𝑥𝑥_𝑘𝑘 <𝑥𝑥_𝑘𝑘+1=𝑏𝑏 and write ∆= {𝑥𝑥_𝑖𝑖}₀𝑘𝑘+1. The ∆ partitions of the interval [a, b] into k+1

subintervals 𝐼𝐼_𝑖𝑖 = [𝑥𝑥_𝑖𝑖,𝑥𝑥_𝑖𝑖+1], 𝑖𝑖= 0,1, … ,𝑘𝑘 −1 and 𝐼𝐼_𝑘𝑘= [𝑥𝑥_𝑘𝑘,𝑥𝑥_𝑘𝑘+1].

Let

𝒫𝒫𝑛𝑛(∆) =�𝑓𝑓:𝑡𝑡ℎ𝑎𝑎𝑎𝑎𝑎𝑎_{𝑓𝑓(𝑥𝑥}𝑎𝑎𝑥𝑥𝑖𝑖𝑟𝑟𝑡𝑡_{) =𝑝𝑝}𝑝𝑝𝑝𝑝𝑟𝑟𝑝𝑝𝑛𝑛𝑝𝑝𝑝𝑝𝑖𝑖𝑎𝑎𝑟𝑟𝑟𝑟𝑝𝑝0,𝑝𝑝1, … ,𝑝𝑝𝑘𝑘𝑖𝑖𝑛𝑛ℙ𝑤𝑤𝑖𝑖𝑡𝑡ℎ

𝑖𝑖(𝑥𝑥) 𝑓𝑓𝑝𝑝𝑎𝑎 𝑥𝑥 ∈ 𝐼𝐼𝑖𝑖, 𝑖𝑖= 0,1, … ,𝑘𝑘 � (1.1)

We call 𝒫𝒫_𝑛𝑛(∆) the space of piecewise polynomials of order 𝑛𝑛 with knots 𝑥𝑥₁,𝑥𝑥₂, … ,𝑥𝑥_𝑘𝑘 . The termmology in (1.1) is perfectly descriptive-an element 𝑓𝑓 ∈ 𝒫𝒫_𝑛𝑛(∆) consists of k+1 polynomial pieces [9].

Let ∆ be a partition of the interval [a, b] as in (1.1) and let 𝑛𝑛 be a positive integer.

Let 𝒮𝒮_𝑛𝑛(∆) =𝒫𝒫_𝑛𝑛(∆) ∩Cn−2[𝑎𝑎,𝑏𝑏]. We call 𝒮𝒮_𝑛𝑛(∆) the space of polynomial splines of order 𝑛𝑛with simple knots at the points 𝑥𝑥₁, 𝑥𝑥₂ , … , 𝑥𝑥_𝑘𝑘.

Let L_∞(X), (1≤ 𝑝𝑝<∞) be the space of all bounded measurable functions with usual norm [8].

‖𝑓𝑓‖𝐿𝐿∞ =‖𝑓𝑓‖∞= sup{|𝑓𝑓(𝑥𝑥)| , 𝑥𝑥 ∈ 𝑋𝑋}≤ ∞, (1.2) 𝐿𝐿𝑝𝑝(𝑋𝑋) be the space of all of all bounded measurable function f on X, for which [3]

‖𝑓𝑓‖𝐿𝐿𝑝𝑝 = ‖𝑓𝑓‖𝑝𝑝 =��∫𝑋𝑋|𝑓𝑓(𝑥𝑥)|𝑝𝑝𝑑𝑑𝑥𝑥� 1

𝑝𝑝 _{<∞�, (1.3)}

the locally global norm for δ > 0 and (1≤ 𝑝𝑝<∞) of f is defined by

‖𝑓𝑓‖𝛿𝛿,𝑝𝑝= (∫_𝑋𝑋(sup{ |𝑓𝑓(𝑝𝑝)|𝑝𝑝: 𝑝𝑝 ∈ �𝑥𝑥 −₂𝛿𝛿,𝑥𝑥+𝛿𝛿₂�}𝑑𝑑𝑝𝑝) 1

𝑝𝑝 _(1.4)

Let 𝐿𝐿_𝑝𝑝,µ(𝑋𝑋), (1≤ 𝑝𝑝<∞) be the space of all bounded µ-measurable functions f on X, for which [1 ]

‖𝑓𝑓‖𝐿𝐿𝑝𝑝,𝜇𝜇 = ‖𝑓𝑓‖𝑝𝑝,µ =��∫𝑋𝑋|𝑓𝑓(𝑥𝑥)|𝑝𝑝𝑑𝑑µ(𝑥𝑥)� 1

𝑝𝑝 _{<∞�, (1.5)}

where µ is the non-negative measureable function on set X.

For δ > 0, the modulus of continuity of the function f on X [10] is defined by

𝜔𝜔(𝑓𝑓,𝛿𝛿) =𝑟𝑟𝑠𝑠𝑝𝑝{|𝑓𝑓(𝑥𝑥1)− 𝑓𝑓(𝑥𝑥2)|: |𝑥𝑥1− 𝑥𝑥2| <𝛿𝛿,𝑥𝑥1,𝑥𝑥2∈ 𝑋𝑋} (1.6)

Corresponding Author: Nagham Ali Hussen*

The moduli of smoothness form a natural generalization of the modulus of continuity.

For every function f we define the kth difference with step h at a point x as follows:

∆ℎ𝑘𝑘𝑓𝑓(𝑥𝑥) = ∑𝑖𝑖𝑘𝑘=0(−1)𝑘𝑘+𝑖𝑖�𝑘𝑘𝑖𝑖�𝑓𝑓(𝑥𝑥+𝑖𝑖h),𝑥𝑥,𝑥𝑥+𝑖𝑖ℎ ∈ 𝑋𝑋 (1.7)

For δ > 0, the modulus of smoothness of order k of function it following function [10]

𝜔𝜔𝑘𝑘(𝑓𝑓,𝛿𝛿) =𝑟𝑟𝑠𝑠𝑝𝑝�

|ℎ|<𝛿𝛿

��∆ℎ𝑘𝑘𝑓𝑓(𝑥𝑥)�, 𝑥𝑥,𝑥𝑥+𝑘𝑘ℎ ∈ 𝑋𝑋� (1.8)

The 𝑘𝑘𝑡𝑡ℎ ordinary modulus of continuity for 𝑓𝑓 ∈ 𝐿𝐿_𝑝𝑝(𝑋𝑋) and 𝑓𝑓 ∈ 𝐿𝐿_𝑝𝑝,µ(𝑋𝑋) respectively by

𝜔𝜔𝑘𝑘(𝑓𝑓,𝛿𝛿)𝑝𝑝= 𝑟𝑟𝑠𝑠𝑝𝑝�

|ℎ|<𝛿𝛿

��∆ℎ𝑘𝑘𝑓𝑓(. )�_𝑝𝑝�, 𝛿𝛿> 0 (1.9)

𝜔𝜔𝑘𝑘(𝑓𝑓,𝛿𝛿)𝑝𝑝,µ= 𝑟𝑟𝑠𝑠𝑝𝑝�

|ℎ|<𝛿𝛿

��∆ℎ𝑘𝑘𝑓𝑓(. )�_𝑝𝑝,µ�, 𝛿𝛿> 0 (1.10)

The local modulus of smoothness of function f of order k at a point 𝑥𝑥 ∈ 𝑋𝑋 is following function of δ > 0 [10]

𝜔𝜔𝑘𝑘(𝑓𝑓,𝑥𝑥,𝛿𝛿) =𝑟𝑟𝑠𝑠𝑝𝑝 ��∆ℎ𝑘𝑘𝑓𝑓(𝑡𝑡)�, 𝑡𝑡,𝑡𝑡+𝑘𝑘ℎ ∈[𝑥𝑥 −₂𝛿𝛿,𝑥𝑥+𝛿𝛿₂]∩ 𝑋𝑋� (1.11)

The 𝑘𝑘𝑡𝑡ℎ averaged modulus of smoothness of function 𝑓𝑓of order k (or τ-modulus) of the function 𝑓𝑓 ∈ 𝐿𝐿_𝑝𝑝 (𝑋𝑋) is following function of δ > 0 is given by [11]:

𝜏𝜏𝑘𝑘(𝑓𝑓,𝛿𝛿)𝑝𝑝=‖𝜔𝜔𝑘𝑘(𝑓𝑓, . ,𝛿𝛿)‖𝑝𝑝 =�∫_𝑋𝑋|𝜔𝜔𝑘𝑘(𝑓𝑓,𝑥𝑥,𝛿𝛿)|𝑝𝑝𝑑𝑑𝑥𝑥� 1

𝑝𝑝 _(1.12)

Further the kth averaged modulus of smoothness for 𝑓𝑓 ∈ 𝐿𝐿_𝑝𝑝,µ(𝑋𝑋) is given by

𝜏𝜏𝑘𝑘(𝑓𝑓,𝛿𝛿)𝑝𝑝,µ=‖𝜔𝜔𝑘𝑘(𝑓𝑓, . ,𝛿𝛿)‖𝑝𝑝,µ (1.13)

The 𝐾𝐾-functional for𝑓𝑓 ∈ 𝑋𝑋₀ and 𝑔𝑔 ∈ 𝑋𝑋₁ is given by

𝐾𝐾(𝑓𝑓,𝛿𝛿) =𝐾𝐾(𝑓𝑓,𝛿𝛿,𝑋𝑋0,𝑋𝑋1) =𝑖𝑖𝑛𝑛𝑓𝑓�

𝑔𝑔∈𝑋𝑋1

�‖𝑓𝑓 − 𝑔𝑔‖𝑋𝑋0+𝛿𝛿‖𝑔𝑔‖𝑋𝑋1,𝛿𝛿> 0� (1.14)

Where 𝑋𝑋₀ and 𝑋𝑋₁ be two Banach spaces with 𝑋𝑋₁⊂ 𝑋𝑋₀. [6]

The inequality 𝐾𝐾(𝑓𝑓,𝛿𝛿) <𝜖𝜖 for some 𝛿𝛿> 0, 𝜖𝜖 is a positive real number, implies that f has approximated with error

‖𝑓𝑓 − 𝑔𝑔‖<𝜖𝜖 in 𝑋𝑋₀ by an element𝑔𝑔 ∈ 𝑋𝑋₁, whose norm is not too large (‖𝑔𝑔‖_𝑋𝑋1<𝜖𝜖𝛿𝛿−1).

The K-functional in 𝐿𝐿_𝑝𝑝(𝑋𝑋) space is given by [6]

𝐾𝐾𝑎𝑎(𝑓𝑓,𝛿𝛿𝑎𝑎)𝑝𝑝 = 𝑖𝑖𝑛𝑛𝑓𝑓� 𝑔𝑔∈𝑊𝑊𝑝𝑝𝑎𝑎

�‖𝑓𝑓 − 𝑔𝑔‖𝑝𝑝+𝛿𝛿𝑎𝑎�𝑔𝑔(𝑎𝑎)�_𝑝𝑝 ,𝛿𝛿> 0� (1.15)

Where 𝑋𝑋₀= 𝐿𝐿_𝑝𝑝(𝑋𝑋) and 𝑋𝑋₁=𝑊𝑊_𝑝𝑝𝑎𝑎 and𝑋𝑋₁⊂ 𝑋𝑋₀,

Now, we introduce 𝐾𝐾-functional of a function𝑓𝑓 ∈ 𝐿𝐿_𝑝𝑝,µ(𝑋𝑋) such that [2]

𝐾𝐾𝑎𝑎(𝑓𝑓,𝛿𝛿𝑎𝑎)𝑝𝑝,µ = 𝑖𝑖𝑛𝑛𝑓𝑓� 𝑔𝑔∈𝑊𝑊𝑝𝑝𝑎𝑎

�‖𝑓𝑓 − 𝑔𝑔‖𝑝𝑝,µ+𝛿𝛿𝑎𝑎�𝑔𝑔(𝑎𝑎)�_𝑝𝑝,µ ,𝛿𝛿> 0� (1.16)

The Ditzian-Totic modulus of smoothness for 𝑓𝑓 ∈ 𝐿𝐿_𝑝𝑝(𝑋𝑋) as [4]

𝜔𝜔_𝑘𝑘𝜑𝜑 (𝑓𝑓.𝛿𝛿)𝑝𝑝 = 𝑟𝑟𝑠𝑠𝑝𝑝�

|ℎ|<𝛿𝛿

�∆ℎ𝜑𝜑𝑘𝑘 𝑓𝑓(. )�_𝑝𝑝 (1.17)

Where ∆_ℎ𝜑𝜑𝑘𝑘 𝑓𝑓(𝑥𝑥) = �∑𝑖𝑖𝑘𝑘=1(−1)𝑘𝑘+𝑖𝑖 �𝑘𝑘𝑖𝑖�𝑓𝑓(𝑥𝑥+𝑖𝑖𝜑𝜑ℎ),𝑥𝑥+𝜑𝜑ℎ ∈ 𝑋𝑋

0 𝑝𝑝𝑡𝑡ℎ𝑎𝑎𝑎𝑎𝑤𝑤𝑖𝑖𝑟𝑟𝑎𝑎 �

Also, the locally µ- Ditzian-Totic modulus of smoothness for 𝑓𝑓 ∈ 𝐿𝐿_𝑝𝑝,µ(𝑋𝑋) is defined by

𝜔𝜔_𝑘𝑘𝜑𝜑 (𝑓𝑓.𝛿𝛿)𝑝𝑝,µ= 𝑟𝑟𝑠𝑠𝑝𝑝�

|ℎ|<𝛿𝛿

�∆ℎ𝜑𝜑𝑘𝑘 𝑓𝑓(. )�_𝑝𝑝,µ, where 𝜑𝜑(𝑥𝑥) = (1− 𝑥𝑥2) 1

2 (1.18)

The degree of best approximation to a given continuous function with respect to a polynomial spline on interval X is given by [5]:

While the degree of best approximation of a function 𝑓𝑓 ∈ 𝐿𝐿_𝑝𝑝(𝑋𝑋) with respect to a polynomial spline of degree ≤ n on

𝑋𝑋 is given by

𝐸𝐸𝑛𝑛(𝑓𝑓)𝑝𝑝=𝑖𝑖𝑛𝑛𝑓𝑓�‖𝑓𝑓 − 𝑟𝑟‖𝑝𝑝; 𝑟𝑟 ∈ 𝒮𝒮𝑛𝑛(∆)�. (1.20)

Also, the degree of µ- best approximation to a given function 𝑓𝑓 ∈ 𝐿𝐿_𝑝𝑝,µ(𝑋𝑋) with respect to polynomial spline of degree

≤𝑛𝑛 on 𝑋𝑋 is defined by

𝐸𝐸𝑛𝑛(𝑓𝑓)𝑝𝑝,µ=𝑖𝑖𝑛𝑛𝑓𝑓�‖𝑓𝑓 − 𝑟𝑟‖𝑝𝑝,µ; 𝑟𝑟 ∈ 𝒮𝒮𝑛𝑛(∆)�. (1.21)

The degree of best one-sided approximation of function 𝑓𝑓 ∈ 𝐿𝐿_𝑝𝑝(𝑋𝑋)with respect to polynomial spline of degree ≤𝑛𝑛 on interval 𝑋𝑋 is given by

Ẽ𝑛𝑛(𝑓𝑓)𝑝𝑝=𝑖𝑖𝑛𝑛𝑓𝑓 �‖𝑟𝑟̅ − 𝑟𝑟̿‖_{𝑟𝑟̿(𝑥𝑥)}𝑝𝑝_{≤ 𝑓𝑓}; 𝑟𝑟̅,𝑟𝑟̿ ∈ 𝒮𝒮_{(𝑥𝑥)≤ 𝑟𝑟̅}𝑛𝑛(∆_(𝑥𝑥) 𝑎𝑎𝑛𝑛𝑑𝑑₎ �. (1.22)

The degree of µ-best one-sided approximation of function 𝑓𝑓 ∈ 𝐿𝐿_𝑝𝑝,µ(𝑋𝑋) with respect to polynomial spline of degree ≤

𝑛𝑛on interval 𝑋𝑋 is given by

Ẽ𝑛𝑛(𝑓𝑓)𝑝𝑝,µ=𝑖𝑖𝑛𝑛𝑓𝑓 �‖𝑟𝑟̅ − 𝑟𝑟̿‖_{𝑟𝑟̿(𝑥𝑥}𝑝𝑝₎,µ_{≤ 𝑓𝑓}; 𝑟𝑟,₍𝑟𝑟̿ ∈ 𝒮𝒮_{𝑥𝑥)≤ 𝑟𝑟̅}𝑛𝑛(₍∆_𝑥𝑥) ₎𝑎𝑎𝑛𝑛𝑑𝑑� . (1.23)

2. AUXILIARY LEMMAS

Lemma I [7]: For 𝑓𝑓 ∈ 𝐿𝐿_𝑝𝑝(𝑋𝑋), (0 <𝑝𝑝 ≤ ∞), we have

𝜔𝜔𝑘𝑘(𝑓𝑓,𝛿𝛿)𝑝𝑝≤ 𝑐𝑐(𝑝𝑝)𝜔𝜔𝑘𝑘𝜑𝜑(𝑓𝑓,𝛿𝛿)𝑝𝑝 (2.1)

where c is constant depending on p.

Lemma II [7]: For 𝑓𝑓 ∈ 𝐿𝐿_𝑝𝑝(𝑋𝑋), (0 <𝑝𝑝 ≤ ∞), we have

𝜔𝜔𝑘𝑘(𝑓𝑓,𝛿𝛿)𝑝𝑝≤ 𝜔𝜔𝑘𝑘(𝑓𝑓,𝛿𝛿)∞, 𝛿𝛿> 0. (2.2)

Lemma III [9]: If 𝑓𝑓 is a bounded measurable function on the interval [a,b],a,b∈ℝ , then

∫ 𝑓𝑓_𝑎𝑎𝑏𝑏 (𝑥𝑥)𝑑𝑑𝑥𝑥≈𝑏𝑏−𝑎𝑎_𝑛𝑛 ∑𝑛𝑛𝑖𝑖=1𝑓𝑓(𝑥𝑥𝑖𝑖) (2.3) 𝑤𝑤ℎ𝑎𝑎𝑎𝑎𝑎𝑎 𝑥𝑥𝑖𝑖 =𝑎𝑎+(𝑏𝑏−𝑎𝑎₂)(2_𝑛𝑛𝑖𝑖−1) .

Lemma IV [1]: Let 𝑓𝑓 be a bounded 𝜇𝜇 −measurablefunction and (1≤ 𝑝𝑝<∞) , then we have

‖𝑓𝑓‖𝑝𝑝≤ 𝑐𝑐(𝑝𝑝)‖𝑓𝑓‖𝑝𝑝,𝜇𝜇 (2.4)

3. MAIN RESULTS

In this section, we will get an estimation for Ẽ_𝑛𝑛(𝑓𝑓)_{𝑝𝑝,𝜇𝜇}. The estimation will be given in terms of kth local modulus of continuity and Ditzian-Totic modulus of smoothness.

Now, we need the following lemmas:

Lemma 1: Let 𝑓𝑓 ∈ 𝐿𝐿_𝑝𝑝,µ(𝑋𝑋), (1≤ 𝑝𝑝<∞) Then

𝜔𝜔𝑘𝑘(𝑓𝑓,𝛿𝛿)∞,µ≤ 𝑐𝑐(𝑝𝑝) 𝜔𝜔𝑘𝑘𝜑𝜑(𝑓𝑓,𝛿𝛿)𝑝𝑝,µ . (3.1)

Proof: From (2.3) and (2.1)

𝜔𝜔𝑘𝑘(𝑓𝑓,𝛿𝛿)∞𝑝𝑝,µ=𝑟𝑟𝑠𝑠𝑝𝑝|ℎ|<𝛿𝛿�∆ℎ𝑘𝑘𝑓𝑓(. )�_∞𝑝𝑝_,µ=𝑟𝑟𝑠𝑠𝑝𝑝|ℎ|<𝛿𝛿�∆ℎ𝑘𝑘𝑓𝑓(. )𝑑𝑑µ(. )�_∞𝑝𝑝

=𝑟𝑟𝑠𝑠𝑝𝑝_{|ℎ|<𝛿𝛿}�sup�∆_ℎ𝑘𝑘𝑓𝑓(𝑥𝑥)𝑑𝑑µ(𝑥𝑥)�𝑝𝑝,𝑥𝑥 ∈ 𝑋𝑋�

≤1

𝑛𝑛∑ �∆ℎ𝑘𝑘𝑓𝑓(𝑥𝑥𝑖𝑖)𝑑𝑑µ(𝑥𝑥𝑖𝑖)� 𝑝𝑝 𝑛𝑛

𝑖𝑖=1

≅ ∫ �∆_𝑋𝑋 𝑘𝑘_ℎ𝑓𝑓(𝑥𝑥_𝑖𝑖)�𝑝𝑝𝑑𝑑µ(𝑥𝑥_𝑖𝑖) Implies that

𝜔𝜔𝑘𝑘(𝑓𝑓,𝛿𝛿)∞,µ≤ �∫ �_𝑋𝑋 ∆ℎ𝑘𝑘𝑓𝑓(𝑥𝑥𝑖𝑖)�𝑝𝑝𝑑𝑑µ(𝑥𝑥𝑖𝑖)� 1 𝑝𝑝

=�∆_ℎ𝑘𝑘𝑓𝑓(. )𝑑𝑑µ(. )�

𝑝𝑝≤ 𝑟𝑟𝑠𝑠𝑝𝑝�∆ℎ𝑘𝑘𝑓𝑓(. )𝑑𝑑µ(. )�𝑝𝑝 =𝜔𝜔𝑘𝑘( 𝑓𝑓𝑑𝑑µ,𝛿𝛿)𝑝𝑝

Theorem 1: Let 𝑓𝑓 ∈ 𝐿𝐿_𝑝𝑝,µ(𝑋𝑋), (1≤ 𝑝𝑝<∞). Then

𝐸𝐸𝑛𝑛(𝑓𝑓)𝑝𝑝,µ≤ 𝑐𝑐(𝑝𝑝)𝜔𝜔𝑘𝑘(𝑓𝑓,𝛿𝛿)𝑝𝑝,µ. (3.2)

Proof: Consider𝑟𝑟 ∈ 𝒮𝒮_𝑛𝑛(∆)is a best approximation of a function f .

From (1.21), we have

𝐸𝐸𝑛𝑛(𝑓𝑓)𝑝𝑝,µ=‖𝑓𝑓 − 𝑟𝑟‖𝑝𝑝,µ

=�∫| (𝑓𝑓 − 𝑟𝑟)(𝑥𝑥)|𝑝𝑝

𝑋𝑋 𝑑𝑑µ(𝑥𝑥)�

1 𝑝𝑝

≤ �∫ 𝑟𝑟𝑠𝑠𝑝𝑝|(𝑓𝑓 − 𝑟𝑟)(𝑥𝑥)|𝑝𝑝

𝑋𝑋 𝑑𝑑µ(𝑥𝑥)�

1 𝑝𝑝

≤ 𝑐𝑐(𝑝𝑝)𝑟𝑟𝑠𝑠𝑝𝑝�∫| 𝑓𝑓(𝑥𝑥)− 𝑟𝑟(𝑥𝑥) |𝑝𝑝

𝑋𝑋 𝑑𝑑µ(𝑥𝑥)�

1 𝑝𝑝

=𝑐𝑐(𝑝𝑝) 𝑟𝑟𝑠𝑠𝑝𝑝�∫ �∆_{𝑋𝑋 ℎ}𝑘𝑘𝑓𝑓(𝑥𝑥)�𝑝𝑝 𝑑𝑑µ(𝑥𝑥)� 1 𝑝𝑝

=𝑐𝑐(𝑝𝑝)𝑟𝑟𝑠𝑠𝑝𝑝�∆_ℎ𝑘𝑘𝑓𝑓(. )�

𝑝𝑝,µ =𝑐𝑐(𝑝𝑝)𝜔𝜔𝑘𝑘(𝑓𝑓,𝛿𝛿)𝑝𝑝,µ.

Now, we want to find a relation between 𝜇𝜇-best approximation and 𝜇𝜇-best one-sided approximation.

Theorem 2: Let 𝑓𝑓 ∈ 𝐿𝐿_𝑝𝑝,µ(𝑋𝑋), (1≤ 𝑝𝑝<∞) and ∆= {0 =𝑥𝑥₀<𝑥𝑥₁<⋯<𝑥𝑥_𝑛𝑛 = 1}. Then

𝐸𝐸𝑛𝑛(𝑓𝑓)𝑝𝑝,µ≤ 𝑐𝑐(𝑝𝑝)Ẽ𝑛𝑛(𝑓𝑓)𝑝𝑝,µ ≤ 𝑐𝑐(𝑝𝑝) 𝐸𝐸𝑛𝑛(𝑓𝑓)𝑝𝑝,µ (3.3)

where c constant depending on p.

Proof: Consider 𝑟𝑟 ∈ 𝒮𝒮_𝑛𝑛(∆) is the best approximation of𝑓𝑓 ∈ 𝐿𝐿_𝑝𝑝,µ(𝑋𝑋) and 𝑟𝑟̅,𝑟𝑟̿ ∈ 𝒮𝒮_𝑛𝑛(∆) are the best one-sided approximation of𝑓𝑓 ∈ 𝐿𝐿_𝑝𝑝,µ(𝑋𝑋) such that 𝑟𝑟̿(𝑥𝑥)≤ 𝑓𝑓(𝑥𝑥)≤ 𝑟𝑟̅(𝑥𝑥);𝑥𝑥 ∈ 𝑋𝑋

We want to prove

𝐸𝐸𝑛𝑛(𝑓𝑓)𝑝𝑝,µ≤ 𝑐𝑐(𝑝𝑝)Ẽ𝑛𝑛(𝑓𝑓)𝑝𝑝,µ

𝐸𝐸𝑛𝑛(𝑓𝑓)𝑝𝑝,µ=‖𝑓𝑓 − 𝑟𝑟‖𝑝𝑝,µ

= �∫_𝑋𝑋| (𝑓𝑓 − 𝑟𝑟)(𝑥𝑥)|𝑝𝑝𝑑𝑑µ(𝑥𝑥)� 1 𝑝𝑝

=�∫_𝑋𝑋|𝑓𝑓(𝑥𝑥)− 𝑟𝑟(𝑥𝑥)|𝑝𝑝𝑑𝑑µ(𝑥𝑥)� 1 𝑝𝑝

=�∫_𝑋𝑋|𝑓𝑓(𝑥𝑥)− 𝑟𝑟̅(𝑥𝑥) +𝑟𝑟̅(𝑥𝑥)− 𝑟𝑟̿(𝑥𝑥) +𝑟𝑟̿(𝑥𝑥)− 𝑟𝑟(𝑥𝑥)|𝑝𝑝𝑑𝑑µ(𝑥𝑥)� 1 𝑝𝑝

≤ �∫_𝑋𝑋|𝑓𝑓(𝑥𝑥)− 𝑟𝑟̅(𝑥𝑥)|𝑝𝑝𝑑𝑑µ(𝑥𝑥)� 1

𝑝𝑝_{+�∫|𝑟𝑟̅(𝑥𝑥)− 𝑟𝑟̿(𝑥𝑥)|}𝑝𝑝_{𝑑𝑑µ(𝑥𝑥)}

𝑋𝑋 �

1

𝑝𝑝_{+�∫|𝑟𝑟̿(𝑥𝑥)− 𝑟𝑟(𝑥𝑥)|}𝑝𝑝_{𝑑𝑑µ(𝑥𝑥)}

𝑋𝑋 �

1 𝑝𝑝

=‖𝑓𝑓 − 𝑟𝑟̅‖_𝑝𝑝,µ+‖𝑟𝑟̅ − 𝑟𝑟̿‖_𝑝𝑝,µ+‖𝑟𝑟̿ − 𝑟𝑟‖_𝑝𝑝,µ ≤ ‖𝑓𝑓 − 𝑟𝑟‖_𝑝𝑝,µ+ c(p)Ẽ_𝑛𝑛(𝑓𝑓)_𝑝𝑝,µ

= 𝐸𝐸_𝑛𝑛(𝑓𝑓)_𝑝𝑝,µ+ c(p) Ẽ_𝑛𝑛(𝑓𝑓)_𝑝𝑝,µ ≤ Ẽ_𝑛𝑛(𝑓𝑓)_𝑝𝑝,µ+c (p)Ẽ_𝑛𝑛(𝑓𝑓)_𝑝𝑝,µ = c (p)Ẽ_𝑛𝑛(𝑓𝑓)_𝑝𝑝,µ

Hence 𝐸𝐸_𝑛𝑛(𝑓𝑓)_𝑝𝑝,µ≤ 𝑐𝑐(𝑝𝑝)Ẽ_𝑛𝑛(𝑓𝑓)_𝑝𝑝,µ

Now to prove Ẽ_𝑛𝑛(𝑓𝑓)_𝑝𝑝,µ ≤ 𝑐𝑐(𝑝𝑝) 𝐸𝐸_𝑛𝑛(𝑓𝑓)_𝑝𝑝,µ

Ẽ𝑛𝑛(𝑓𝑓)𝑝𝑝,µ= ‖𝑟𝑟̅ − 𝑟𝑟̿‖𝑝𝑝,µ =�∫_𝑋𝑋|(𝑟𝑟̅ − 𝑟𝑟̿)(𝑥𝑥)|𝑝𝑝𝑑𝑑µ(𝑥𝑥)� 1 𝑝𝑝

= �∫|𝑟𝑟̅(𝑥𝑥)− 𝑟𝑟̿(𝑥𝑥)|𝑝𝑝𝑑𝑑µ(𝑥𝑥)

𝑋𝑋 �

1 𝑝𝑝

≤ �∫ �𝑟𝑟(𝑥𝑥) +�𝑓𝑓(𝑥𝑥)− 𝑟𝑟(𝑥𝑥)�)−(𝑟𝑟(𝑥𝑥)− �𝑓𝑓(𝑥𝑥)− 𝑟𝑟(𝑥𝑥)�)�𝑝𝑝𝑑𝑑µ(𝑥𝑥)

𝑋𝑋 �

1 𝑝𝑝

=�∫|(𝑓𝑓(𝑥𝑥)−2𝑟𝑟(𝑥𝑥) +𝑓𝑓(𝑥𝑥))|𝑝𝑝𝑑𝑑µ(𝑥𝑥)

𝑋𝑋 �

1 𝑝𝑝

= 2�∫_𝑋𝑋|𝑓𝑓(𝑥𝑥)− 𝑟𝑟(𝑥𝑥)|𝑝𝑝𝑑𝑑µ(𝑥𝑥)� 1 𝑝𝑝

≤ 𝑐𝑐(𝑝𝑝)‖𝑓𝑓 − 𝑟𝑟‖_𝑝𝑝,µ =𝑐𝑐(𝑝𝑝) 𝐸𝐸_𝑛𝑛(𝑓𝑓)_𝑝𝑝,µ.

Then we get

𝐸𝐸𝑛𝑛(𝑓𝑓)𝑝𝑝,µ≤ 𝑐𝑐(𝑝𝑝)Ẽ𝑛𝑛(𝑓𝑓)𝑝𝑝,µ≤ 𝑐𝑐(𝑝𝑝) 𝐸𝐸𝑛𝑛(𝑓𝑓)𝑝𝑝,µ

Theorem 3: If 𝑓𝑓 ∈ 𝐿𝐿_𝑝𝑝,µ(𝑋𝑋), (1≤ 𝑝𝑝<∞), then

Ẽ𝑛𝑛(𝑓𝑓)𝑝𝑝,µ≤ 𝜏𝜏𝑘𝑘(𝑓𝑓,∆𝑛𝑛)𝑝𝑝,𝜇𝜇.

Proof: Let ∆= {0 =𝑥𝑥₀<𝑥𝑥₁<⋯<𝑥𝑥_𝑛𝑛 = 1}, ∆_𝑛𝑛=𝑝𝑝𝑎𝑎𝑥𝑥|𝑥𝑥_𝑖𝑖− 𝑥𝑥_𝑖𝑖−1|, 𝑖𝑖= 0,1, … ,𝑛𝑛

Set 𝑟𝑟̅_𝑘𝑘(𝑥𝑥) =𝑟𝑟𝑠𝑠𝑝𝑝𝑓𝑓(𝑡𝑡), 𝑥𝑥 ∈[𝑥𝑥_𝑖𝑖−1,𝑥𝑥_𝑖𝑖), 𝑡𝑡 ∈[𝑥𝑥_𝑖𝑖−1,𝑥𝑥_𝑖𝑖]

𝑟𝑟̿_𝑘𝑘(𝑥𝑥) =𝑖𝑖𝑛𝑛𝑓𝑓 𝑓𝑓(𝑡𝑡), 𝑥𝑥 ∈[𝑥𝑥_𝑖𝑖−1,𝑥𝑥_𝑖𝑖), 𝑡𝑡 ∈[𝑥𝑥_𝑖𝑖−1,𝑥𝑥_𝑖𝑖]

and 𝑆𝑆̅(𝑓𝑓,𝑥𝑥,𝛿𝛿) = sup𝑓𝑓(𝑡𝑡) 𝑤𝑤ℎ𝑎𝑎𝑎𝑎𝑎𝑎 |𝑡𝑡 − 𝑥𝑥|≤ 𝛿𝛿/2

𝑆𝑆�(𝑓𝑓,𝑥𝑥,𝛿𝛿) = inf𝑓𝑓(𝑡𝑡) 𝑤𝑤ℎ𝑎𝑎𝑎𝑎𝑎𝑎 |𝑡𝑡 − 𝑥𝑥|≤ 𝛿𝛿/2

then, 𝑆𝑆�(𝑓𝑓,𝑥𝑥,𝛿𝛿)≤ 𝑟𝑟̿(𝑥𝑥)≤ 𝑟𝑟̅(𝑥𝑥)≤ 𝑆𝑆̅(𝑓𝑓,𝑥𝑥,𝛿𝛿)

we have

Ẽ𝑛𝑛(𝑓𝑓)𝑝𝑝,µ =‖𝑟𝑟̅𝑘𝑘− 𝑟𝑟̿𝑘𝑘‖𝑝𝑝,µ=�∫_𝑋𝑋|(𝑟𝑟̅𝑘𝑘− 𝑟𝑟̿𝑘𝑘)(𝑥𝑥)|𝑝𝑝𝑑𝑑µ(𝑥𝑥)� 1 𝑝𝑝

=�∫_𝑋𝑋|(𝑟𝑟̅_𝑘𝑘− 𝑟𝑟̿_𝑘𝑘)(𝑥𝑥)|𝑝𝑝𝑑𝑑µ(𝑥𝑥)� 1 𝑝𝑝

≤ �∫ �𝑆𝑆̅(𝑓𝑓,𝑥𝑥,𝛿𝛿)− 𝑆𝑆�(𝑓𝑓,𝑥𝑥,𝛿𝛿)�𝑝𝑝

𝑋𝑋 𝑑𝑑µ(𝑥𝑥)�

1 𝑝𝑝

≤ 𝑟𝑟𝑠𝑠𝑝𝑝 �∫ �𝑆𝑆̅_𝑋𝑋 (𝑓𝑓,𝑥𝑥,𝛿𝛿)− 𝑆𝑆�(𝑓𝑓,𝑥𝑥,𝛿𝛿)�𝑝𝑝𝑑𝑑µ(𝑥𝑥)� 1 𝑝𝑝

= 𝜏𝜏_𝑘𝑘(𝑓𝑓,∆_𝑛𝑛)_{𝑝𝑝,𝜇𝜇}.

Theorem 4: Let 𝑓𝑓 ∈ 𝐿𝐿_𝑝𝑝,µ(𝑋𝑋), (1≤ 𝑝𝑝<∞). Then

Ẽ𝑛𝑛(𝑓𝑓)𝑝𝑝,µ≤ 𝑐𝑐(𝑝𝑝)𝜔𝜔𝑘𝑘𝜑𝜑 (𝑓𝑓.𝛿𝛿)𝑝𝑝,µ.

Proof: By using (2.1), (2.2), (3.1), (2.4) and (3.2) we get

Ẽ𝑛𝑛,𝑘𝑘(𝑓𝑓)𝑝𝑝,µ =𝑖𝑖𝑛𝑛𝑓𝑓‖𝑟𝑟̅ − 𝑟𝑟̿‖𝑝𝑝,µ

≤ 𝑖𝑖𝑛𝑛𝑓𝑓‖𝑟𝑟̅ − 𝑟𝑟̿‖_𝑝𝑝 =Ẽ_𝑛𝑛(𝑓𝑓)_𝑝𝑝 ≤ 𝜔𝜔_𝑘𝑘𝜑𝜑 (𝑓𝑓.𝛿𝛿)_𝑝𝑝 =𝑟𝑟𝑠𝑠𝑝𝑝�∆_ℎ𝜑𝜑𝑘𝑘 𝑓𝑓(. )�

𝑝𝑝

≤ 𝑐𝑐(𝑝𝑝)𝑟𝑟𝑠𝑠𝑝𝑝�∆_ℎ𝜑𝜑𝑘𝑘 𝑓𝑓(. )�

𝑝𝑝,𝜇𝜇

=𝑐𝑐(𝑝𝑝)𝜔𝜔_𝑘𝑘𝜑𝜑 (𝑓𝑓.𝛿𝛿)_𝑝𝑝,µ.

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