International Scholarly Research Notices

International Scholarly Research Notices / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 725638 | https://doi.org/10.5402/2011/725638

Hee Sun Jung, Ryozi Sakai, Noriaki Suzuki, "On the Favard-Type Theorem and the Jackson-Type Theorem (II)", International Scholarly Research Notices, vol. 2011, Article ID 725638, 20 pages, 2011. https://doi.org/10.5402/2011/725638

On the Favard-Type Theorem and the Jackson-Type Theorem (II)

Academic Editor: E. Yee
Received30 Sep 2011
Accepted20 Oct 2011
Published14 Dec 2011

Abstract

Let R=(βˆ’βˆž,∞), and let π‘„βˆˆβ„‚1βˆΆβ„β†’[0,∞) be an even function. We consider the exponential weights 𝑀(π‘₯)=π‘’βˆ’π‘„(π‘₯), π‘₯βˆˆβ„. In this paper we investigate the relations between the Favard-type inequality and the Jackson-type inequality. We also discuss the equivalence of two K-functionals and the modulus of smoothness.

1. Introduction and Preliminaries

Let ℝ=(βˆ’βˆž,∞), and let π‘„βˆˆβ„‚1βˆΆβ„β†’[0,∞) be an even function. We consider the weights 𝑀(π‘₯)∢=exp(βˆ’π‘„(π‘₯)) satisfying for all 𝑛=0,1,2,…,ξ€œβˆž0π‘₯𝑛𝑀(π‘₯)𝑑π‘₯<∞.(1.1) First we need the following definition from [1]. We say that π‘“βˆΆβ„β†’[0,∞) is quasi-increasing if there exists 𝐢>0 such that 𝑓(π‘₯)⩽𝐢𝑓(𝑦), 0<π‘₯<𝑦.

Definition 1.1. One defines, 𝑀=exp(βˆ’π‘„)βˆˆβ„±(𝐢2+) as follows: Let π‘„βˆΆβ„β†’[0,∞) be continuous and an even function, and satisfy the following properties: (a)π‘„ξ…ž(π‘₯) is continuous in ℝ, with 𝑄(0)=0;(b)π‘„ξ…žξ…ž(π‘₯) exists and is positive in ℝ⧡{0};(c)limπ‘₯β†’βˆžπ‘„(π‘₯)=∞;(1.2)(d)the function 𝑇(π‘₯)∢=π‘₯π‘„ξ…ž(π‘₯)𝑄(π‘₯),π‘₯β‰ 0(1.3) is quasi-increasing in (0,∞) with 𝑇(π‘₯)β©ΎΞ›>1, π‘₯βˆˆβ„β§΅{0};(e)there exists 𝐢1>0 such that π‘„ξ…žξ…ž(π‘₯)||π‘„ξ…ž||(π‘₯)⩽𝐢1||π‘„ξ…ž||(π‘₯)𝑄(π‘₯),a.e.π‘₯βˆˆβ„β§΅{0}.(1.4) Moreover, there also exists a compact subinterval 𝐽(βˆ‹0) of ℝ, and 𝐢2>0 such that π‘„ξ…žξ…ž(π‘₯)||π‘„ξ…ž||(π‘₯)⩾𝐢2||π‘„ξ…ž||(π‘₯)𝑄(π‘₯),a.e.π‘₯βˆˆβ„β§΅π½.(1.5)

Example 1.2. Let 𝑀=exp(βˆ’π‘„)βˆˆβ„±(𝐢2+).(1) If 𝑇(π‘₯) is bounded, then we call the weight 𝑀=exp(βˆ’π‘„) the Freud-type weight. The following example is the Freud-type weight: 𝑀(π‘₯)=exp(βˆ’|π‘₯|𝛼),𝛼>1.(1.6)(2) If 𝑇(π‘₯) is unbounded, then we call the weight 𝑀=exp(βˆ’π‘„) the ErdΓΆs-type weight. The following examples give the ErdΕ‘s-type weight 𝑀=exp(βˆ’π‘„). (i)([1, Example  1.2], [2, Theorem  3.1]). For 𝛼>1, β„“=1,2,3,…,𝑄(π‘₯)=𝑄ℓ,𝛼(π‘₯)=expβ„“(|π‘₯|𝛼)βˆ’exp𝑙(0),(1.7)where expβ„“(π‘₯)=exp(exp(exp…expπ‘₯)…),(β„“times). More generally, we define for 𝛼+𝑒>1, 𝛼⩾0, 𝑒⩾0, and 𝑙⩾1, 𝑄𝑙,𝛼,𝑒(π‘₯)∢=|π‘₯|𝑒exp𝑙(|π‘₯|𝛼)βˆ’π›Όβˆ—exp𝑙(0),(1.8)where π›Όβˆ—=0 if 𝛼=0, otherwise π›Όβˆ—=1.(ii) For 𝛼>1, 𝑄(π‘₯)=𝑄𝛼(π‘₯)∢=(1+|π‘₯|)|π‘₯|π›Όβˆ’1.

We need the Mhaskar-Rakhmanov-Saff numbers π‘Žπ‘₯ defined by 2π‘₯=πœ‹ξ€œ10π‘Žπ‘₯π‘’π‘„ξ…žξ€·π‘Žπ‘₯𝑒1βˆ’π‘’2ξ€Έ1/2𝑑𝑒,π‘₯>0.(1.9) If π‘“βˆΆβ„β†’β„ is measurable, we define ‖𝑓𝑀‖𝐿𝑝(ℝ)⎧βŽͺ⎨βŽͺβŽ©ξ‚΅ξ€œβˆΆ=βˆžβˆ’βˆž||||𝑓(𝑑)𝑀(𝑑)𝑝𝑑𝑑1/𝑝,if0<𝑝<∞,esssuptβˆˆβ„||||𝑓(𝑑)𝑀(𝑑),if𝑝=∞,(1.10) where if 𝑝=∞, then we suppose that 𝑓 is continuous on ℝ and lim|π‘₯|β†’βˆž||||𝑓(π‘₯)𝑀(π‘₯)=0.(1.11) The class of all functions 𝑓 for which ‖𝑓𝑀‖𝐿𝑝(ℝ)<∞ will be denoted by 𝐿𝑝,𝑀(ℝ), with the usual understanding that two functions are identified if they are equal almost everywhere. For π‘“βˆˆπΏπ‘,𝑀(ℝ)(0<π‘β©½βˆž), the degree of weighted polynomial approximation is defined by 𝐸𝑝,𝑛(𝑀;𝑓)∢=infπ‘ƒβˆˆπ’«π‘›β€–π‘€(π‘“βˆ’π‘ƒ)‖𝐿𝑝(ℝ),(1.12) where 𝒫𝑛 denotes the class of all polynomials of degree at most 𝑛. There are various estimates for this degree. Among them, in our previous article [3], we discuss the relation between the Favard-type inequality, the Jackson-type inequality, and the estimates by two 𝐾-functionals π’¦π‘Ÿ,𝑝 and ξ‚‹π’¦π‘Ÿ,𝑝. We recall and summarize them in Section 2. In Section 3, we will discuss the equivalence of ξ‚‹π’¦π‘Ÿ,𝑝 and modulus of smoothness πœ”π‘Ÿ,𝑝. As the result, we see π’¦π‘Ÿ,π‘βˆΌξ‚‹π’¦π‘Ÿ,𝑝 for a weight in a subclass of β„±(𝐢2+). For any nonzero real valued functions 𝑓(π‘₯) and 𝑔(π‘₯), we write 𝑓(π‘₯)βˆΌπ‘”(π‘₯) if there exists a constant 𝐢>0 independent of π‘₯ such that for all π‘₯ξ‚€1𝐢𝑓(π‘₯)⩽𝑔(π‘₯)⩽𝐢𝑓(π‘₯).(1.13) Similarly, for any positive numbers {𝑐𝑛}βˆžπ‘›=1 and {𝑑𝑛}βˆžπ‘›=1 we define π‘π‘›βˆΌπ‘‘π‘›. Throughout this paper 𝐢,𝐢1,…,𝑐,𝑐1,… denote positive constants independent of 𝑛,π‘₯,𝑑, or 𝑃𝑛(π‘₯). The same symbols do not necessarily denote the same constants occurrences.

2. Known Results and Summarization

Let π‘Ÿβ©Ύ1 be an integer (in this paper, we suppose that π‘Ÿ (or 𝑠) is an integer). The π‘Ÿth order 𝐾-functional of a function π‘“βˆˆπΏπ‘,𝑀(ℝ) is defined by the formula π’¦π‘Ÿ,𝑝‖(𝑀,𝑓,𝛿)∢=inf𝑀(π‘“βˆ’π‘”)‖𝐿𝑝(ℝ)+π›Ώπ‘Ÿβ€–β€–π‘€π‘”(π‘Ÿ)‖‖𝐿𝑝(ℝ),(2.1) for 𝛿>0, where the infimum is taken over all π‘Ÿβˆ’1 times continuously differentiable 𝑔 such that 𝑔(π‘Ÿβˆ’1) is absolutely continuous, and 𝑔(π‘Ÿ)βˆˆπΏπ‘,𝑀(ℝ) (this means 𝑀𝑔(π‘Ÿ)βˆˆπΏπ‘(ℝ)). Using this π‘Ÿth order 𝐾-functional, we can estimate the order of 𝐸𝑝,𝑛(𝑀;𝑓).

Theorem 2.1 (see [3]). Let 𝑀=exp(βˆ’π‘„)βˆˆβ„±(𝐢2+). Let π‘Ÿβ©Ύ1, 𝑠⩾0 be integers, and let 1β©½π‘β©½βˆž. Let 𝑓 be π‘ βˆ’1 times continuously differentiable, 𝑓(π‘ βˆ’1) be absolutely continuous on each compact interval, and 𝑓(𝑠)βˆˆπΏπ‘,𝑀(ℝ) (when 𝑠=0, these assumptions state merely that π‘“βˆˆπΏπ‘,𝑀(ℝ)). Then, for every integer π‘›β©Ύπ‘Ÿ+𝑠, 𝐸𝑝,𝑛(ξ‚€π‘Žπ‘€;𝑓)β©½πΆπ‘›π‘›ξ‚π‘ π’¦π‘Ÿ,𝑝𝑀,𝑓(𝑠),π‘Žπ‘›π‘›ξ‚.(2.2)

Theorem 2.1 was shown by using the following Favard-type inequalities (see [3]).

Theorem 2.2 ([4, Corollary 8]). Let 𝑀=exp(βˆ’π‘„)βˆˆβ„±(𝐢2+). Let 𝑓 be π‘ βˆ’1 times continuously differentiable, and let 𝑓(π‘ βˆ’1) for some integer 𝑠⩾1 be absolutely continuous on each compact interval. Let 1β©½π‘β©½βˆž and 𝑓(𝑠)βˆˆπΏπ‘,𝑀(ℝ). Then one has 𝐸𝑝,𝑛(ξ‚€π‘Žπ‘€;𝑓)⩽𝐢𝑛𝑛𝑠‖‖𝑀𝑓(𝑠)‖‖𝐿𝑝(ℝ),(2.3) equivalently, 𝐸𝑝,𝑛(ξ‚€π‘Žπ‘€;𝑓)⩽𝐢𝑛𝑛𝑠𝐸𝑝,π‘›βˆ’π‘ ξ€·π‘€;𝑓(𝑠)ξ€Έ.(2.4)

Remark 2.3. (1)([4, Remark 11]) Let π‘€βˆˆβ„±(𝐢2+) and let 0<𝑝<1. Then there exists a constant 𝐢0 such that for every absolutely continuous function 𝑓 with π‘“β€²βˆˆπΏβˆž,𝑀 and π‘€β€²π‘“βˆˆπΏπ‘(ℝ), and for every π‘›βˆˆβ„•, we have 𝐸𝑝,𝑛(𝑀;𝑓)⩽𝐢0ξ‚€π‘Žπ‘›π‘›β€–β€–ξ‚ξ‚†π‘€π‘“ξ…žβ€–β€–πΏβˆž(ℝ)+β€–β€–π‘€ξ…žπ‘“β€–β€–πΏπ‘(ℝ).(2.5)
(2)([3, Section 4]) As a by-product of the method of the proof for Theorem 2.2, we can obtain the degree of functions which satisfy the HΓΆlder-Lipschitz condition. Let π‘€βˆˆβ„±(𝐢2+),1β©½π‘β©½βˆž and 1/𝑝⩽𝛽⩽1. Let 𝑔 be absolutely continuous with |𝑔′|π›½βˆˆπΏπ‘,𝑀(ℝ) (and for 𝑝=∞, we require 𝑔 to be continuous, and 𝑔𝑀 to vanish at ±∞). Let π‘“βˆˆπΏπ‘,𝑀(ℝ) and set π‘“βˆ—π‘”,𝛽(π‘₯)∢=supπ‘¦βˆˆβ„||||𝑓(π‘₯)βˆ’π‘“(𝑦)||||𝑔(π‘₯)βˆ’π‘”(𝑦)𝛽.(2.6) Then we define 𝐅𝑔,𝛽(𝑀,𝑝)∢=π‘“βˆ£π‘“βˆˆπΏπ‘,𝑀‖‖𝑓(ℝ),βˆ—π‘”,π›½β€–β€–πΏβˆž(ℝ)<∞.(2.7)

Now, we have the following theorem:

Theorem see ([3, Theorem 4.2]). Let 1β©½π‘β©½βˆž, 1/𝑝⩽𝛽⩽1, and let π‘“βˆˆπ…π‘”,𝛽(𝑀,𝑝). Then one has 𝐸𝑝,π‘›ξ‚€π‘Ž(𝑀;𝑓)⩽𝐢𝑛𝑛𝛽‖‖𝑀||π‘”ξ…ž||𝛽‖‖𝐿𝑝(ℝ)β€–β€–π‘“βˆ—π‘”,π›½β€–β€–πΏβˆž(ℝ).(2.8)

We define the following class of weights from [5, Definition  1.1].

Definition 2.5. Let 𝑀(π‘₯)∢=exp(βˆ’π‘„(π‘₯)), where π‘„βˆΆβ„β†’β„ is even, continuous, and π‘„ξ…ž is positive in (0,∞). Then one writes 𝑀=exp(βˆ’π‘„)βˆˆβ„°1, if the following are satisfied: (a)π‘₯π‘„ξ…ž(π‘₯) is strictly increasing in (0,∞) with limπ‘₯β†’0+π‘₯π‘„ξ…ž(π‘₯)=0;(b)the function𝑇(π‘₯)∢=π‘₯π‘„ξ…ž(π‘₯)𝑄(π‘₯)(2.9)is quasi-increasing in (𝐢,∞) for some 𝐢>0 and limπ‘₯β†’βˆžπ‘‡(π‘₯)=∞;(c)assumeπ‘¦π‘„ξ…ž(𝑦)π‘₯π‘„ξ…ž(π‘₯)⩽𝐢1𝑄(𝑦)𝑄(π‘₯)𝐢3,𝑦⩾π‘₯⩾𝐢2,(2.10)for some positive constants 𝐢1,𝐢2, and 𝐢3.

Remark 2.6. Let 𝑀(π‘₯)∢=exp(βˆ’π‘„(π‘₯))βˆˆβ„±(𝐢2+) and 𝑇(π‘₯) is unbounded, then we see π‘€βˆˆβ„°1. In fact, we see this as follows.
From Definition 1.1(e) and (d), we have for 𝑦⩾π‘₯>0, π‘„ξ…ž(𝑦)π‘„ξ…žξ‚΅ξ€œ(π‘₯)=exp𝑦π‘₯π‘„ξ…žξ…ž(𝑑)π‘„ξ…žξ‚Ά,𝐢(𝑑)𝑑𝑑⩽exp1ξ€œπ‘¦π‘₯π‘„ξ…ž(𝑑)ξ‚Ά=𝑄(𝑑)𝑑𝑑𝑄(𝑦)𝑄(π‘₯)𝐢1,𝑦π‘₯ξ‚΅ξ€œ=exp𝑦π‘₯1𝑑1𝑑𝑑⩽expΞ›ξ€œπ‘¦π‘₯π‘„ξ…ž(𝑑)ξ‚Ά=𝑄(𝑑)𝑑𝑑𝑄(𝑦)𝑄(π‘₯)1/Ξ›.(2.11) Therefore we obtain (c) in Definition 2.5 for 𝐢3=𝐢1+1/Ξ›.

If π‘“βˆΆβ„β†’β„, and β„Ž>0, then we define the differences of 𝑓 inductively by the formula Ξ”0β„ŽΞ”π‘“(π‘₯)=𝑓(π‘₯),1β„Žπ‘“(π‘₯)=Ξ”β„Žξ‚€β„Žπ‘“(π‘₯)=𝑓π‘₯+2ξ‚ξ‚€β„Žβˆ’π‘“π‘₯βˆ’2,Ξ”π‘˜β„Žπ‘“(π‘₯)=Ξ”β„Žπ‘˜βˆ’1ξ€·Ξ”β„Žπ‘“ξ€Έ(π‘₯)=π‘˜ξ“π‘—=0(βˆ’1)π‘—ξ‚΅π‘˜π‘—ξ‚Άπ‘“ξ‚€π‘₯+π‘˜β„Ž2ξ‚βˆ’π‘—β„Ž,π‘˜=2,3,..,π‘₯βˆˆβ„.(2.12) We set ξ‚†π‘ŽπœŽ(𝑑)∢=infπ‘’βˆΆπ‘Žπ‘’π‘’ξ‚‡Ξ¦β©½π‘‘,𝑑>0,π‘‘ξƒŽ(π‘₯)∢=||||1βˆ’|π‘₯|𝜎||||(𝑑)+𝑇(𝜎(𝑑))βˆ’1/2,π‘₯βˆˆβ„.(2.13) By [6], when 𝑀=exp(βˆ’π‘„) is the ErdΕ‘s-type weight we define for π‘“βˆˆπΏπ‘,𝑀(ℝ),0<π‘β©½βˆž, πœ”π‘Ÿ,𝑝(𝑀,𝑓,𝑑)∢=sup0<β„Žβ©½π‘‘β€–β€–π‘€Ξ”π‘Ÿβ„ŽΞ¦π‘‘(π‘₯)β€–β€–(𝑓)𝐿𝑝(|π‘₯|⩽𝜎(2𝑑))+infπ‘…βˆˆπ’«π‘Ÿβˆ’1‖‖𝑀(π‘₯)(π‘“βˆ’π‘…)(π‘₯)𝐿𝑝(|π‘₯|⩾𝜎(4𝑑)).(2.14) If 𝑀=exp(βˆ’π‘„) is the Freud-type weights, then we define πœ”π‘Ÿ,𝑝(𝑀,𝑓,𝑑)∢=sup0<β„Žβ©½π‘‘β€–β€–π‘€Ξ”π‘Ÿβ„Žβ€–β€–(𝑓)𝐿𝑝(|π‘₯|⩽𝜎(β„Ž))+infπ‘…βˆˆπ’«π‘Ÿβˆ’1‖𝑀(π‘₯)(π‘“βˆ’π‘…)(π‘₯)‖𝐿𝑝(|π‘₯|⩾𝜎(𝑑)).(2.15)

The following Jackson-type inequality is known.

Theorem 2.7 (see [5, Theorem  1.2], [6, Corollary  1.4]). Let π‘€βˆˆβ„°1. Let 0<π‘β©½βˆž. Then for π‘“βˆΆβ„β†’β„ for which π‘“βˆˆπΏπ‘,𝑀(ℝ) (and for 𝑝=∞, we require 𝑓 to be continuous, and 𝑓𝑀 to vanish at ±∞), one has for 𝑛⩾𝐢3, 𝐸𝑝,𝑛(𝑓;𝑀)⩽𝐢1πœ”π‘Ÿ,𝑝𝑀,𝑓,𝐢2π‘Žπ‘›π‘›ξ‚,(2.16) where 𝐢𝑗, 𝑗=1,2,3, do not depend on 𝑓 and 𝑛.

We also consider the following class of weights which are called the Freud weights.

Definition 2.8 ([7, Definition 3.3]). Let 𝑀=exp(βˆ’π‘„), where π‘„βˆΆβ„β†’β„ is even, and 𝑄′ exists and is positive on (0,∞). Moreover, assume that π‘₯π‘„ξ…ž(π‘₯) is strictly increasing, with right limit 0 at 0, and for some πœ†,𝐴,𝐡>1, 𝐢>0, π‘„π΄β©½ξ…ž(πœ†π‘₯)π‘„ξ…ž(π‘₯)⩽𝐡,π‘₯⩾𝐢.(2.17) Then we call 𝑀 Freud weight, and write π‘€βˆˆβ„±βˆ—.

Remark 2.9. Let 𝑀=exp(βˆ’π‘„)βˆˆβ„±(𝐢2+). Then we see the following.
(1)If 𝑇(π‘₯)=(π‘₯𝑄′(π‘₯))/𝑄(π‘₯) is bounded, then we say that 𝑀=exp(βˆ’π‘„) is the Freud-type weight, and we write π‘€βˆˆβ„±. Then we see β„±βˆ—βˆ©β„±(𝐢2+)=β„±. In fact, when π‘€βˆˆβ„±, 1<𝜈 and π‘₯>0 large enough, by Definition 1.1(e), π‘„ξ…ž(𝜈π‘₯)π‘„ξ…žξ‚΅ξ€œ(π‘₯)=expπ‘₯𝜈π‘₯π‘„ξ…žξ…ž(𝑒)π‘„ξ…žξ‚Άξ‚΅πΆ(𝑒)𝑑𝑒⩽exp1ξ€œπ‘₯𝜈π‘₯π‘„ξ…ž(𝑒)𝑄(𝑒)𝑑𝑒,(2.18) and then since 𝑇(π‘₯) is bounded, there exists 𝐢>0 such that π‘„ξ…ž(𝜈π‘₯)π‘„ξ…žξ‚΅πΆ(π‘₯)β©½exp1πΆξ€œπ‘₯𝜈π‘₯1𝑒𝑑𝑒=𝜈𝐢1𝐢.(2.19) Similarly, π‘„ξ…ž(𝜈π‘₯)π‘„ξ…žξ‚΅ξ€œ(π‘₯)=expπ‘₯𝜈π‘₯π‘„ξ…žξ…ž(𝑒)π‘„ξ…žξ‚Άξ‚΅πΆ(𝑒)𝑑𝑒⩾exp2ξ€œπ‘₯𝜈π‘₯π‘„ξ…ž(𝑒)𝐢𝑄(𝑒)𝑑𝑒⩾exp2Ξ›ξ€œπ‘₯𝜈π‘₯1𝑒𝑑𝑒=𝜈𝐢2Ξ›.(2.20) Therefore, if we take πœ†βˆΆ=𝜈>1 large enough, then we have (2.17).
Conversely, to show β„±βˆ—βˆ©β„±(𝐢2+)βŠ‚β„± we suppose that there exists 𝑀=exp(βˆ’π‘„)βˆˆβ„±(𝐢2+) such that 𝑇(π‘₯)=(π‘₯𝑄′(π‘₯))/𝑄(π‘₯) is unbounded. Then since 𝑇(π‘₯) is quasi-increasing in (0,∞), we see that 𝑇(π‘₯)β†’βˆž as π‘₯β†’βˆž. So, for any 𝑀>0 there exists 𝐿>0 (large enough) such that 𝑇(π‘₯)>𝑀 for π‘₯>𝐿. Therefore, we have for π‘₯>𝐿 and any πœ†>1, π‘„ξ…ž(πœ†π‘₯)π‘„ξ…žξ‚΅ξ€œ(π‘₯)=expπ‘₯πœ†π‘₯π‘„ξ…žξ…ž(𝑒)π‘„ξ…žξ‚Άξ‚΅πΆ(𝑒)𝑑𝑒⩾exp3ξ€œπ‘₯πœ†π‘₯π‘„ξ…ž(𝑒)𝐢𝑄(𝑒)𝑑𝑒=exp3ξ€œπ‘₯πœ†π‘₯𝑇(𝑒)𝑒𝑑𝑒=πœ†πΆ3𝑀,(2.21) where 𝐢3>0 is a constant, that is, (1) does not hold. Hence we have β„±βˆ—βˆ©β„±(𝐢2+)βŠ‚β„±. Consequently, we have β„±=β„±βˆ—βˆ©β„±(𝐢2+).
(2)If 𝑇(π‘₯)=(π‘₯𝑄′(π‘₯))/𝑄(π‘₯) is bounded, then for π‘₯β©Ύ1, there exists 𝑐,𝐢>0 such that π‘₯Λ⩽𝑄(π‘₯)⩽𝐢π‘₯𝑐.(2.22)

Theorem 2.10  2.10 ([7, Theorem 3.5]). Let π‘€βˆˆβ„±βˆ—. Let 0<π‘β©½βˆž. Then for π‘“βˆΆβ„β†’β„ for which π‘“βˆˆπΏπ‘,𝑀(ℝ) (and for 𝑝=∞, one requires 𝑓 to be continuous, and 𝑓𝑀 to vanish at ±∞), one has for 𝑛⩾𝐢3, 𝐸𝑝,𝑛(𝑓;𝑀)⩽𝐢1πœ”π‘Ÿ,𝑝𝑀,𝑓,𝐢2π‘Žπ‘›π‘›ξ‚,(2.23) where 𝐢𝑗, 𝑗=1,2,3, do not depend on 𝑓 and 𝑛.

Damelin [6] introduces the following 𝐾-functional: let π‘“βˆˆπΏπ‘,𝑀(ℝ), 0<π‘β©½βˆž and π‘Ÿβ©Ύ1 be an integer, then we define ξ‚‹π’¦π‘Ÿ,𝑝(𝑀,𝑓,𝛿)∢=infπ‘ƒβˆˆπ’«π‘›ξ‚†β€–π‘€(π‘“βˆ’π‘ƒ)‖𝐿𝑝(ℝ)+π›Ώπ‘Ÿβ€–β€–π‘€π‘ƒ(π‘Ÿ)‖‖𝐿𝑝(ℝ),(2.24) where 𝛿>0 are chosen in advance and ξ‚†π‘Žπ‘›=𝑛(𝛿)∢=infπ‘˜βˆΆπ‘˜π‘˜ξ‚‡β©½π›Ώ.(2.25)

Then Damelin gives the following.

Theorem 2.11 ([6, Theorem 1.3 (b)]). Let π‘€βˆˆβ„°1, π‘Ÿβ©Ύ1, 0<π‘β©½βˆž, and let π‘“π‘€βˆˆπΏπ‘(ℝ) (and for 𝑝=∞, one requires 𝑓 to be continuous, and 𝑓𝑀 to vanish at ±∞). Then one has πœ”π‘Ÿ,𝑝𝒦(𝑀,𝑓,𝑑)βˆΌπ‘Ÿ,𝑝(𝑀,𝑓,𝛿).(2.26)

For the Freud-type weights we have also the followings.

Theorem 2.12 ([7, Theorem 3.9, 3.10]). Let π‘€βˆˆβ„±βˆ—, π‘Ÿβ©Ύ1, 0<π‘β©½βˆž, and let π‘“π‘€βˆˆπΏπ‘(ℝ) (and for 𝑝=∞, we require 𝑓 to be continuous, and 𝑓𝑀 to vanish at ±∞). Then we have π’¦π‘Ÿ,𝑝(𝑀,𝑓,𝛿)βˆΌπœ”π‘Ÿ,𝑝𝒦(𝑀,𝑓,𝑑)βˆΌπ‘Ÿ,𝑝(𝑀,𝑓,𝛿).(2.27)

For π‘€βˆˆβ„±(𝐢2+), we see easily that π’¦π‘Ÿ,𝑝𝒦(𝑀,𝑓,𝛿)β©½π‘Ÿ,𝑝(𝑀,𝑓,𝛿).(2.28)

So, from Theorem 2.1 we obtain the following corollary.

Corollary 2.13. Let 𝑀=exp(βˆ’π‘„)βˆˆβ„±(𝐢2+). Let π‘Ÿβ©Ύ1,𝑠⩾0 be integers, and let 1β©½π‘β©½βˆž. Let 𝑓 be π‘ βˆ’1 times continuously differentiable, 𝑓(π‘ βˆ’1) be absolutely continuous, and 𝑓(𝑠)βˆˆπΏπ‘,𝑀(ℝ) (when 𝑠=0, these assumptions state merely that π‘“βˆˆπΏπ‘,𝑀(ℝ)). Then, for every integer π‘›β©Ύπ‘Ÿ+𝑠, 𝐸𝑝,𝑛(ξ‚€π‘Žπ‘€;𝑓)β©½πΆπ‘›π‘›ξ‚π‘ ξ‚‹π’¦π‘Ÿ,𝑝𝑀,𝑓(𝑠),π‘Žπ‘›π‘›ξ‚.(2.29)

The main theme in [3] is to summarize the above theorems. Let π‘€βˆˆβ„±(𝐢2+),π‘Ÿβ©Ύ1 be an integer, and let 1β©½π‘β©½βˆž. We have the following succession of the theorems. We use the constant 𝐢𝑖>0,𝑖=1,2,3,… which do not depend on 𝑓 and 𝑛.(a)[Theorem 2.7 with π‘Ÿ=1 (the ErdΕ‘s-type case)], [Theorem 2.10 with π‘Ÿ=1 (the Freud case)]:let π‘“βˆΆβ„β†’β„ and if 1⩽𝑝<∞, assume that π‘“π‘€βˆˆπΏπ‘(ℝ). If 𝑝=∞, assume in addition that 𝑓 is continuous and that 𝑓𝑀 has limit 0 at ±∞. Then we have 𝐸𝑝,𝑛(𝑀;𝑓)⩽𝐢1πœ”1,𝑝𝑀,𝑓,𝐢2π‘Žπ‘›π‘›ξ‚.(2.30)(b) [Theorem 2.2]: let 𝑓 be π‘ βˆ’1 times continuously differentiable, and let for some integer 𝑠⩾1, 𝑓(π‘ βˆ’1)(π‘₯) be absolutely continuous on each compact interval. Let 1β©½π‘β©½βˆž and 𝑓(𝑠)βˆˆπΏπ‘,𝑀(ℝ). Then we have 𝐸𝑝,𝑛(𝑀;𝑓)⩽𝐢3ξ‚€π‘Žπ‘›π‘›ξ‚π‘ β€–β€–π‘€π‘“(𝑠)‖‖𝐿𝑝(ℝ),(2.31) equivalently, 𝐸𝑝,𝑛(𝑀;𝑓)⩽𝐢4ξ‚€π‘Žπ‘›π‘›ξ‚π‘ πΈπ‘,𝑛𝑀;𝑓(𝑠)ξ€Έ.(2.32)(c) [Theorem 2.1]:let 𝑠⩾0 be an integer, and let 𝑓 be π‘ βˆ’1 times continuously differentiable, 𝑓(π‘ βˆ’1) be absolutely continuous, and 𝑓(𝑠)βˆˆπΏπ‘,𝑀(ℝ) (when 𝑠=0, these assumptions state merely that π‘“βˆˆπΏπ‘,𝑀(ℝ)). Then, for every integer π‘Ÿβ©Ύ1 and π‘›β©Ύπ‘Ÿ+𝑠, 𝐸𝑝,𝑛(𝑀;𝑓)⩽𝐢5ξ‚€π‘Žπ‘›π‘›ξ‚π‘ π’¦π‘Ÿ,𝑝𝑀,𝑓(𝑠),π‘Žπ‘›π‘›ξ‚.(2.33)(d) [Theorem 2.7, 2.11 (the ErdΕ‘s-type case)], [Theorem 2.10, 2.12 (the Freud case)]:for π‘“βˆΆβ„β†’β„ with π‘“βˆˆπΏπ‘,𝑀(ℝ) (and for 𝑝=∞ we require 𝑓 to be continuous, and 𝑓𝑀 to vanish at ±∞), for every 𝑛 large enough and for every integer π‘Ÿβ©Ύ1 we have 𝐸𝑝,𝑛(𝑀;𝑓)⩽𝐢6ξ‚‹π’¦π‘Ÿ,π‘ξ‚€π‘Žπ‘€,𝑓,𝑛𝑛.(2.34)

Consequently we find an interesting fact as follows:

(b) is shown by the use of (a) (see [4]). Using (b), we proved (c). If we use [6, Theorem 1.3 (b)] and [7, Theorems 3.9 and 3.10], we see easily that (c) means (d) with Theorem 2.11. Trivially, we have (a) from (d).

3. Equivalence of 𝐾-Functional and Modulus of Smoothness

In this section we consider a subclass of β„±(𝐢2+).

Assumption 3.1. Let π‘Ÿβ©Ύ2 be an integer. Let 𝑀(π‘₯)=exp(βˆ’π‘„(π‘₯))βˆˆβ„±(𝐢2+). Then we suppose that π‘„βˆˆπΆπ‘Ÿβˆ’1(ℝ), 𝑄(π‘Ÿ)(π‘₯) exists in ℝ⧡{0}, furthermore, the following inequalities hold ||𝑄(𝑗)||(π‘₯)⩽𝐢𝑗||𝑄(π‘—βˆ’1)||||||(π‘₯)𝑄′(π‘₯)||||𝑄(π‘₯),𝑗=1,2,…,π‘Ÿ,(3.1) where for 𝑗=π‘Ÿ we suppose that (3.1) hold almost everywhere on π‘₯βˆˆβ„β§΅{0}. Then we write π‘€βˆˆβ„±(πΆπ‘Ÿ+).

Remark 3.2. Example 1.2, (i), (ii) satisfy Assumption 3.1, moreover 𝑄(π‘₯)=π‘₯2π‘š, π‘š: a positive integer, satisfies Assumption 3.1.

In Section 2, we know π’¦π‘Ÿ,𝑝𝒦(𝑀,𝑓,𝑑)β©½π‘Ÿ,𝑝(𝑀,𝑓,𝑑)βˆΌπœ”π‘Ÿ,𝑝(𝑀,𝑓,𝑑).(3.2) The equivalences mentioned in the last of Section 2 give a certain suggestion, that is, π’¦π‘Ÿ,𝑝(𝑀,𝑓,𝑑)βˆΌπœ”π‘Ÿ,𝑝(𝑀,𝑓,𝑑).(3.3) In fact, this is true.

Theorem 3.3. Let π‘Ÿβ©Ύ2 be a positive integer, and let π‘€βˆˆβ„±(πΆπ‘Ÿ+). Let π‘“βˆˆπΏπ‘,𝑀(ℝ). Then one has (3.3). Similarly, when π‘€βˆˆβ„±(𝐢2+) and π‘Ÿ=1, one has (3.3).

To prove Theorem 3.3, we need some lemmas.

Lemma 3.4. Let π‘Ÿβ©Ύ1 be an integer. If 𝑓(π‘Ÿβˆ’1)(π‘₯) is absolutely continuous on ℝ, then for π‘˜=1,2,…,π‘Ÿ, one has the following representation: Ξ”π‘˜Ξ¦π‘‘(π‘₯)β„Žξ€·Ξ¦π‘“(π‘₯)=𝑑(ξ€Έπ‘₯)π‘˜ξ‚»ξ€œβ„Ž/2βˆ’β„Ž/2ξ‚Όπ‘˜π‘“(π‘˜)ξ€·π‘₯+Φ𝑑(𝑒π‘₯)1+𝑒2+β‹―+π‘’π‘˜ξ€Έξ€Έπ‘‘π‘’1𝑑𝑒2β‹―π‘‘π‘’π‘˜.(3.4)

Proof. ξ€œβ„Ž/2β„Ž/2𝑓(1)ξ€·π‘₯+Φ𝑑(π‘₯)𝑒1𝑑𝑒1=Φ𝑑(π‘₯)βˆ’1ξ€œπ‘₯+Φ𝑑(π‘₯)β„Ž/2π‘₯βˆ’Ξ¦π‘‘(π‘₯)β„Ž/2𝑓(1)𝑣1𝑑𝑣1=Φ𝑑(π‘₯)βˆ’1□Φ𝑑(π‘₯)β„Žπ‘“(π‘₯).(3.5) Therefore, for π‘Ÿ=1 we have (3.4). For some π‘˜β©Ύ1 we suppose (3.4). Then we have ξ‚»ξ€œβ„Ž/2βˆ’β„Ž/2ξ‚Όπ‘˜+1𝑓(π‘˜+1)(𝑒)𝑑𝑒1𝑑𝑒2β‹―π‘‘π‘’π‘˜+1=ξ€œβ„Ž/2βˆ’β„Ž/2ξƒ¬ξ‚»ξ€œβ„Ž/2βˆ’β„Ž/2ξ‚Όπ‘˜ξ€·π‘“ξ…žξ€Έ(π‘˜)(̃𝑒)𝑑𝑒1𝑑𝑒2β‹―π‘‘π‘’π‘˜ξƒ­π‘‘π‘’π‘˜+1=Φ𝑑(π‘₯)βˆ’π‘˜ξ€œβ„Ž/2βˆ’β„Ž/2Ξ”π‘˜Ξ¦π‘‘(π‘₯)β„Žπ‘“ξ…žξ€·π‘₯+Φ𝑑(π‘₯)π‘’π‘˜+1ξ€Έπ‘‘π‘’π‘˜+1=Φ𝑑(π‘₯)βˆ’(π‘˜+1)Ξ”π‘˜Ξ¦π‘‘(π‘₯)β„Žξ€·Ξ”Ξ¦π‘‘(π‘₯)β„Žπ‘“ξ€Έ=ξ€·Ξ¦(π‘₯)𝑑(π‘₯)βˆ’(π‘˜+1)Ξ”Ξ¦π‘˜+1𝑑(π‘₯)β„Žπ‘“(π‘₯),(3.6) where 𝑒=𝑒π‘₯;𝑒1,…,π‘’π‘˜+1ξ€Έ=π‘₯+Φ𝑑𝑒(π‘₯)1+𝑒2+β‹―+π‘’π‘˜+1ξ€Έ,̃𝑒=̃𝑒π‘₯;𝑒1,…,π‘’π‘˜+1ξ€Έ=π‘₯+Φ𝑑(π‘₯)π‘’π‘˜+1+Φ𝑑(𝑒π‘₯)1+𝑒2+β‹―+π‘’π‘˜ξ€Έ.(3.7) Hence we have (3.4) for π‘Ÿ=1,2,3,….

Lemma 3.5 (see [1, Lemma  3.4 (3.17)]). Uniformly for 𝑑>0, one has π‘„ξ…žξ€·π‘Žπ‘‘ξ€ΈβˆΌπ‘‘ξ”π‘‡ξ€·π‘Žπ‘‘ξ€Έπ‘Žπ‘‘.(3.8)

Lemma 3.6. Let π‘€βˆˆβ„±(πΆπ‘Ÿ+) and π‘Ÿβ©Ύ2 be an integer. Then for any integer π‘˜, 1β©½π‘˜β©½π‘Ÿβˆ’1, there exist 𝑐1,𝑐2>0 and 𝐴>0 such that for |π‘₯|⩾𝐴, 𝑐1β©½(βˆ’1)π‘˜π‘€(π‘˜)(π‘₯)ξ€·1+π‘„ξ…ž(π‘₯)2ξ€Έπ‘˜/2𝑀(π‘₯)⩽𝑐2.(3.9) Also, there exist 𝑐3,𝑐4>0 and 𝐡>0 such that for |π‘₯|⩾𝐡, 𝑐3β©½(βˆ’1)π‘˜ξ€·π‘€βˆ’1ξ€Έ(π‘˜)(π‘₯)ξ€·1+π‘„ξ…ž(π‘₯)2ξ€Έπ‘˜/2π‘€βˆ’1(π‘₯)⩽𝑐4.(3.10) Furthermore, for π‘˜=π‘Ÿ, (3.9) and (3.10) hold almost everywhere on |π‘₯|⩾𝐴 and |π‘₯|⩾𝐡, respectively. When π‘€βˆˆβ„±(𝐢2+) and π‘Ÿ=1, one also has (3.9) and (3.10).

Proof. First, we will see that for πœ‡=Β±1 the following equations hold: lim|π‘₯|β†’(π‘€πœ‡)(π‘˜)(π‘₯)π‘„ξ…ž(π‘₯)π‘˜π‘€πœ‡(π‘₯)=(βˆ’πœ‡)π‘˜,π‘˜=1,…,π‘Ÿβˆ’1.(3.11) For π‘˜=π‘Ÿ we take 𝑄(π‘₯) as follows; 𝑄(𝑗)(π‘₯)=𝑄(𝑗)𝑄(π‘₯),π‘₯βˆˆβ„,𝑗=1,2,…,π‘Ÿβˆ’1,(π‘Ÿ)(π‘₯)=𝑄(π‘Ÿ)(π‘₯),a.e.π‘₯βˆˆβ„β§΅{0},(3.12) and for all 𝑗=1,2,…,π‘Ÿ, 𝑄(π‘₯) satisfies (3.1) for all π‘₯β‰ 0. Then we obtain that exchanging 𝑄 with 𝑄, (3.11) also holds.
Let πœ‡=1, and let 1β©½π‘˜β©½π‘Ÿβˆ’1. π‘€ξ…ž(π‘₯)=βˆ’π‘„ξ…žπ‘€(π‘₯)𝑀(π‘₯),ξ…žξ…žξ€·(π‘₯)=βˆ’π‘„ξ…žξ…ž(π‘₯)+π‘„ξ…ž(π‘₯)2𝑀𝑀(π‘₯),ξ…žξ…žξ…žξ€·(π‘₯)=βˆ’π‘„ξ…žξ…žξ…ž(π‘₯)+3π‘„ξ…ž(π‘₯)π‘„ξ…žξ…ž(π‘₯)βˆ’π‘„ξ…ž(π‘₯)3𝑀(π‘₯),(3.13) and we continue this manner, so 𝑀(π‘˜)βŽ›βŽœβŽœβŽœβŽξ“(π‘₯)=𝑖1+2𝑖2+β‹―+π‘˜π‘–π‘˜π‘–=π‘˜,1<π‘˜π‘π‘–1,𝑖2,...,π‘–π‘˜π‘„ξ…ž(π‘₯)𝑖1π‘„ξ…žξ…ž(π‘₯)𝑖2⋯𝑄(π‘˜)(π‘₯)π‘–π‘˜+(βˆ’1)π‘˜π‘„ξ…ž(π‘₯)π‘˜βŽžβŽŸβŽŸβŽŸβŽ π‘€(π‘₯),(3.14) where 𝑐𝑖1,𝑖2,...,π‘–π‘˜ are coefficients. Here, from (3.1), for |π‘₯|⩾𝐴>0 large enough and 𝑖𝑗≠0, 2β©½π‘—β©½π‘˜,||𝑄(𝑗)||(π‘₯)𝑖𝑗⩽𝐢𝑗||||𝑄′(π‘₯)||||𝑄(π‘₯)π‘—βˆ’1||π‘„ξ…ž||ξƒͺ(π‘₯)𝑖𝑗=𝐢𝑗||π‘„ξ…ž||(π‘₯)𝑗𝑄(π‘₯)π‘—βˆ’1ξƒͺ𝑖𝑗.(3.15) Hence, from (3.14) and (3.15) we have ||||lim|π‘₯|β†’βˆžπ‘€(π‘˜)(π‘₯)π‘„ξ…ž(π‘₯)π‘˜βˆ’π‘€(π‘₯)(βˆ’1)π‘˜||||⩽𝐢lim|π‘₯|β†’βˆž1𝑄(π‘₯)=0,π‘˜=1,…,π‘Ÿβˆ’1,(3.16) where 𝐢 is a positive constant. Therefore, we have (3.11) with πœ‡=1. Similarly, for πœ‡=βˆ’1, lim|π‘₯|β†’βˆžξ€·π‘€βˆ’1ξ€Έ(π‘˜)(π‘₯)π‘„ξ…ž(π‘₯)π‘˜π‘€βˆ’1(π‘₯)=1,π‘˜=1,…,π‘Ÿβˆ’1.(3.17) Therefore, we have (3.11) for π‘˜=1,2,…,π‘Ÿβˆ’1, and hence we also have (3.9) and (3.10) for |π‘₯| large enough. If in (3.11), we replace 𝑄 with 𝑄, then repeating the above proof we also obtain (3.11), so for π‘˜=π‘Ÿ we conclude (3.9) and (3.10) with a.e.π‘₯ large enough, that is, 𝑐1β©½(βˆ’1)π‘Ÿπ‘€(π‘Ÿ)(π‘₯)ξ€·1+π‘„ξ…ž(π‘₯)2ξ€Έπ‘Ÿ/2𝑀(π‘₯)⩽𝑐2𝑐,a.e.|π‘₯|⩾𝐴,3β©½(βˆ’1)π‘Ÿξ€·π‘€βˆ’1ξ€Έ(π‘Ÿ)(π‘₯)ξ€·1+π‘„ξ…ž(π‘₯)2ξ€Έπ‘Ÿ/2π‘€βˆ’1(π‘₯)⩽𝑐4,a.e.|π‘₯|⩾𝐡.(3.18) For π‘Ÿ=1, the lemma is trivial.

Lemma 3.7. Let π‘Ÿβ©Ύ1 be an integer. There exists 𝐢>0 such that for every integer π‘˜=1,2,…,π‘Ÿ, ||π‘„ξ…ž||(π‘₯)π‘˜π‘€(π‘₯)⟢0,π‘Žπ‘ |π‘₯|⟢∞,(3.19)π‘€βˆ’1(π‘₯)ξ€·1+π‘„ξ…ž(π‘₯)2ξ€Έ(π‘˜βˆ’1)/2ξ€œβˆž|π‘₯|ξ€·1+π‘„ξ…ž(𝑑)2ξ€Έπ‘˜/2𝑀(𝑑)𝑑𝑑⩽𝐢,π‘₯βˆˆβ„,(3.20) and for π‘₯βˆˆβ„, ξ€·1+π‘„ξ…ž(π‘₯)2ξ€Έπ‘˜/2ξ€œπ‘€(π‘₯)0|π‘₯|ξ€·1+π‘„ξ…ž(𝑑)2ξ€Έβˆ’(π‘˜βˆ’1)/2π‘€βˆ’1(𝑑)𝑑𝑑⩽𝐢,π‘₯βˆˆβ„.(3.21)

Proof. From Definition 1.1(e) we see that there exist 𝐢>0 and πœ†>0 such that ||π‘„ξ…ž||(π‘₯)⩽𝐢𝑄(π‘₯)πœ†,π‘₯βˆˆβ„,(3.22) so we have ||π‘„ξ…ž||(π‘₯)π‘˜π‘€(π‘₯)β©½πΆπ‘˜π‘„(π‘₯)πœ†π‘˜π‘€(π‘₯)⟢0,as|π‘₯|⟢∞.(3.23) Hence, we have (3.19). From (3.9) with π‘˜βˆ’1, we see that for |π‘₯|⩾𝐴>0 and constants 𝑐1,𝑐2,𝑐3>0, π‘€βˆ’1(π‘₯)ξ€·1+π‘„ξ…ž(π‘₯)2ξ€Έ(π‘˜βˆ’1)/2ξ€œβˆž|π‘₯|ξ€·1+π‘„ξ…ž(𝑑)2ξ€Έπ‘˜/2𝑀(𝑑)𝑑𝑑⩽𝑐1π‘€βˆ’1(π‘₯)ξ€·1+π‘„ξ…ž(π‘₯)2ξ€Έ(π‘˜βˆ’1)/2ξ€œβˆž|π‘₯|(βˆ’1)π‘˜π‘€(π‘˜)(𝑑)𝑑𝑑⩽𝑐2π‘€βˆ’1(π‘₯)ξ€·1+π‘„ξ…ž(π‘₯)2ξ€Έ(π‘˜βˆ’1)/2(βˆ’1)π‘˜βˆ’1𝑀(π‘˜βˆ’1)(π‘₯)⩽𝑐3.(3.24) Especially, if we use (3.24) with |π‘₯|=𝐴, then we have π‘€βˆ’1(𝐴)ξ€·1+π‘„ξ…ž(𝐴)2ξ€Έ(π‘˜βˆ’1)/2ξ€œβˆžπ΄ξ€·1+π‘„ξ…ž(𝑑)2ξ€Έπ‘˜/2𝑀(𝑑)𝑑𝑑⩽𝑐3.(3.25) Then, for |π‘₯|⩽𝐴 there exists a constant 𝐢=𝐢(𝐴) such that π‘€βˆ’1(π‘₯)ξ€·1+π‘„ξ…ž(π‘₯)2ξ€Έ(π‘˜βˆ’1)/2π‘€β©½πΆβˆ’1(𝐴)ξ€·1+π‘„ξ…ž(𝐴)2ξ€Έ(π‘˜βˆ’1)/2.(3.26) In fact, for |π‘₯|⩽𝐴 we have 1+π‘„ξ…ž(𝐴)2β©½ξ€·1+π‘„ξ…ž(𝐴)2ξ€Έξ€·1+π‘„ξ…ž(π‘₯)2ξ€Έ,(3.27) that is, ξ€·1+π‘„ξ…ž(𝐴)2ξ€Έ(π‘˜βˆ’1)/2⩽𝐢(𝐴)1+π‘„ξ…ž(π‘₯)2ξ€Έ(π‘˜βˆ’1)/2,(3.28) where 𝐢(𝐴)=(1+𝑄′(𝐴)2)(π‘˜βˆ’1)/2. Hence we have (3.26).
Now, by (3.26), if |π‘₯|⩽𝐴, then we have π‘€βˆ’1(π‘₯)ξ€·1+π‘„ξ…ž(π‘₯)2ξ€Έ(π‘˜βˆ’1)/2ξ€œβˆžπ΄ξ€·1+π‘„ξ…ž(𝑑)2ξ€Έπ‘˜/2𝑀(𝑑)𝑑𝑑⩽𝑐4(𝐴).(3.29) When |π‘₯|<𝐴, we see easily that π‘€βˆ’1(π‘₯)ξ€·1+π‘„ξ…ž(π‘₯)2ξ€Έ(π‘˜βˆ’1)/2ξ€œπ΄|π‘₯|ξ€·1+π‘„ξ…ž(𝑑)2ξ€Έπ‘˜/2𝑀(𝑑)𝑑𝑑⩽𝑐5=𝑐5(𝐴).(3.30) Hence, with (3.29) we have for |π‘₯|⩽𝐴, π‘€βˆ’1(π‘₯)ξ€·1+π‘„ξ…ž(π‘₯)2ξ€Έ(π‘˜βˆ’1)/2ξ€œβˆž|π‘₯|ξ€·1+π‘„ξ…ž(𝑑)2ξ€Έπ‘˜/2𝑀(𝑑)𝑑𝑑⩽𝑐6(𝐴).(3.31) Consequently, from (3.24) and (3.31) we have (3.20). We need to show (3.21). For |π‘₯| large enough, we see π‘„ξ…žξ…žπ‘„(π‘₯)βˆΌξ…ž(π‘₯)2𝑄(π‘₯),a.e.π‘₯βˆˆβ„,(3.32) (see Definition 1.1(e)) so we can select |π‘₯|⩾𝐴>0 large enough such that π‘„ξ…žξ…ž1(π‘₯)<𝑄2π‘˜ξ…ž(π‘₯)2,a.e.π‘₯βˆˆβ„.(3.33)
We show (3.21) for ∫𝐴|π‘₯|, |π‘₯|⩾𝐴>0. For |π‘₯|⩾𝐴, we have by (3.33), ξ€œπΌβˆΆ=𝐴|π‘₯|π‘€βˆ’1(𝑑)π‘„ξ…ž(𝑑)π‘˜βˆ’1ξ€œπ‘‘π‘‘=𝐴|π‘₯|π‘„ξ…ž(𝑑)π‘€βˆ’1(𝑑)π‘„ξ…ž(𝑑)π‘˜β©½π‘€π‘‘π‘‘βˆ’1(π‘₯)π‘„ξ…ž(|π‘₯|)π‘˜ξ€œ+π‘˜π΄|π‘₯|π‘€βˆ’1𝑄(𝑑)ξ…žξ…ž(𝑑)π‘„ξ…ž(𝑑)π‘˜+1β©½π‘€π‘‘π‘‘βˆ’1(π‘₯)π‘„ξ…ž(|π‘₯|)π‘˜+12𝐼.(3.34) Hence, we have for a certain constant 𝑐7>0, ξ€·1+π‘„ξ…ž(π‘₯)2ξ€Έπ‘˜/2ξ€œπ‘€(π‘₯)𝐴|π‘₯|π‘€βˆ’1(𝑑)ξ€·1+π‘„ξ…ž(𝑑)2ξ€Έ(π‘˜βˆ’1)/2𝑑𝑑⩽𝑐7.(3.35) For ∫𝐴0 we have (3.21) by the use of (3.19). In fact, there exists 𝑐8=𝑐8(𝐴) such that ξ€·1+π‘„ξ…ž(π‘₯)2ξ€Έπ‘˜/2π‘€ξ€œ(π‘₯)𝐴0π‘€βˆ’1(𝑑)ξ€·1+π‘„ξ…ž(𝑑)2ξ€Έ(π‘˜βˆ’1)/2𝑑𝑑⩽𝑐8.(3.36) So, with (3.35) we conclude (3.21).

To prove Theorem 3.3 we further need the following lemma.

Lemma 3.8. For π‘Ÿ=1, one lets π‘€βˆˆβ„±(𝐢2+), and for integer π‘Ÿβ©Ύ2 one lets π‘€βˆˆβ„±(πΆπ‘Ÿ+). Let 1β©½π‘β©½βˆž, and 1β©½π‘˜β©½π‘Ÿ be an integer. If π‘”βˆΆβ„β†’β„ is absolutely continuous, 𝑔(0)=0, and |π‘„ξ…ž(π‘₯)|π‘˜βˆ’1π‘”ξ…žβˆˆπΏπ‘,𝑀(ℝ), then β€–β€–ξ€·1+π‘„ξ…ž(π‘₯)2ξ€Έπ‘˜/2‖‖𝑀𝑔𝐿𝑝(ℝ)‖‖⩽𝐢1+π‘„ξ…ž(π‘₯)2ξ€Έ(π‘˜βˆ’1)/2π‘€π‘”ξ…žβ€–β€–πΏπ‘(ℝ).(3.37)

Proof. We will prove (3.37) for 𝑝=1 and 𝑝=∞, and then we use the Riesz-Thorin interpolation theorem. Let ξ€·πœ“(𝑑)∢=1+π‘„ξ…ž(𝑑)2ξ€Έ(π‘˜βˆ’1)/2𝑀(𝑑)π‘”ξ…ž(𝑑),π‘‘βˆˆβ„.(3.38) Then for almost all π‘₯β©Ύ0, ||||+||||β©½ξ€œπ‘”(π‘₯)𝑔(βˆ’π‘₯)π‘₯0π‘€βˆ’1(𝑑)ξ€·1+π‘„ξ…ž(𝑑)2ξ€Έ(π‘˜βˆ’1)/2ξ€½||||+||||ξ€Ύπœ“(𝑑)πœ“(βˆ’π‘‘)𝑑𝑑.(3.39) Denoting ||||+||||Ξ¨(𝑑)∢=πœ“(𝑑)πœ“(βˆ’π‘‘),(3.40) we get β€–β€–β€–ξ‚€1+π‘„ξ…ž2ξ‚π‘˜/2‖‖‖𝑀𝑔𝐿1(ℝ)=ξ€œβˆž0ξ€·1+π‘„ξ…ž(π‘₯)2ξ€Έπ‘˜/2ξ€½||||+||||ξ€Ύβ©½ξ€œπ‘€(π‘₯)𝑔(π‘₯)𝑔(βˆ’π‘₯)𝑑π‘₯∞0ξ€·1+π‘„ξ…ž(π‘₯)2ξ€Έπ‘˜/2ξ€œπ‘€(π‘₯)0|π‘₯|π‘€βˆ’1(𝑑)ξ€·1+π‘„ξ…ž(𝑑)2ξ€Έ(π‘˜βˆ’1)/2=ξ€œΞ¨(𝑑)𝑑𝑑𝑑π‘₯∞0ξƒ―π‘€βˆ’1(π‘₯)ξ€·1+π‘„ξ…ž(π‘₯)2ξ€Έ(π‘˜βˆ’1)/2ξ€œβˆž|π‘₯|ξ€·1+π‘„ξ…ž(𝑑)2ξ€Έπ‘˜/2𝑀(𝑑)𝑑𝑑Ψ(π‘₯)𝑑π‘₯,(3.41) by changing of the integral order. Hence, from (3.20) we have β€–β€–(1+π‘„ξ…ž2)π‘˜/2‖‖𝑀𝑔𝐿1(ℝ)ξ€œβ©½πΆβˆž0Ξ¨(π‘₯)𝑑π‘₯=β€–πœ“β€–πΏ1(ℝ).(3.42) By the definition of (3.38), we have (3.37) for 𝑝=1.
Next, we show (3.37) for 𝑝=∞. From (3.39) we see that|||ξ€·1+π‘„ξ…ž(π‘₯)2ξ€Έπ‘˜/2|||⩽𝑀(π‘₯)𝑔(π‘₯)1+π‘„ξ…ž(π‘₯)2ξ€Έπ‘˜/2ξ€½||||+||||⩽𝑀(π‘₯)𝑔(π‘₯)𝑔(βˆ’π‘₯)1+π‘„ξ…ž(π‘₯)2ξ€Έπ‘˜/2ξ€œπ‘€(π‘₯)0|π‘₯|π‘€βˆ’1(𝑑)ξ€·1+π‘„ξ…ž(𝑑)2ξ€Έ(π‘˜βˆ’1)/2ξ€½||||+||||ξ€Ύπœ“(𝑑)πœ“(βˆ’π‘‘)𝑑𝑑⩽2β€–πœ“β€–πΏβˆž(ℝ)ξ€·1+π‘„ξ…ž(π‘₯)2ξ€Έπ‘˜/2ξ€œπ‘€(π‘₯)0|π‘₯|π‘€βˆ’1(𝑑)ξ€·1+π‘„ξ…ž(𝑑)2ξ€Έ(π‘˜βˆ’1)/2π‘‘π‘‘β©½πΆβ€–πœ“β€–πΏβˆž(ℝ),(3.43) by (3.21). So we have (3.37) for 𝑝=∞. Let πœ™βˆˆπΏ1(ℝ)∩𝐿∞(ℝ)βŠ‚πΏπ‘(ℝ), 1β©½π‘β©½βˆž, then we set ξ€œπ‘”(π‘₯)∢=π‘₯0πœ™(𝑑)ξ€·1+π‘„ξ…ž(𝑑)2ξ€Έ(π‘˜βˆ’1)/2𝑀(𝑑)𝑑𝑑.(3.44) So we have ξ€·1+π‘„ξ…ž(π‘₯)2ξ€Έ(π‘˜βˆ’1)/2𝑀(π‘₯)π‘”ξ…ž(π‘₯)=πœ™(π‘₯)∈𝐿1(ℝ)∩𝐿∞(ℝ)βŠ‚πΏπ‘(ℝ),1β©½π‘β©½βˆž.(3.45) Since 𝐿1(ℝ)∩𝐿∞(ℝ) is dense in 𝐿𝑝(ℝ), 1β©½π‘β©½βˆž, using the Riesz-Thorin interpolation theorem for the linear operator, ξ€·πœ™βŸΌ1+π‘„ξ…ž(π‘₯)2ξ€Έπ‘˜/2ξ€œπ‘€(π‘₯)π‘₯0π‘€βˆ’1(𝑑)ξ€·1+π‘„ξ…ž(𝑑)2ξ€Έ(π‘˜βˆ’1)/2πœ™(𝑑)𝑑𝑑,πœ™βˆˆπΏπ‘(ℝ),(3.46) we have the result.

Corollary 3.9. For π‘Ÿ=1, one lets π‘€βˆˆβ„±(𝐢2+), and for integer π‘Ÿβ©Ύ2 one lets π‘€βˆˆβ„±(πΆπ‘Ÿ+). Let 1β©½π‘β©½βˆž, and 1β©½π‘˜ be an integer. If π‘”βˆΆβ„β†’β„ is absolutely continuous on ℝ, 𝑔(𝑗)(0)=0, 𝑗=0,…,π‘˜βˆ’1, and 𝑔(π‘˜)βˆˆπΏπ‘,𝑀(ℝ), then β€–β€–ξ€·1+π‘„ξ…ž(π‘₯)2ξ€Έπ‘˜/2‖‖𝑀𝑔𝐿𝑝(ℝ)‖‖⩽𝐢𝑀𝑔(π‘˜)‖‖𝐿𝑝(ℝ).(3.47)

Lemma 3.10 ([4, Lemma  7]). Let π‘€βˆˆβ„±(𝐢2+). For a certain constant 𝐢1>0, let 𝑒 satisfy 0<𝐢1π‘Žβ©½π‘’,𝑑=𝑒𝑒.(3.48) Then there exists a constant 𝐢2>1 such that for every π‘₯ and 𝑦 which satisfy |π‘₯|⩽𝜎(2𝑑) and |π‘₯βˆ’π‘¦|⩽𝑑Φ𝑑(π‘₯)/2, one has 1𝐢2𝑀(𝑦)⩽𝑀(π‘₯)⩽𝐢2𝑀(𝑦).(3.49)

Proof of Theorem 3.3. For a given πœ€>0, we can select 𝑔, where 𝑔(π‘Ÿβˆ’1)(π‘₯) is absolutely continuous on ℝ, and 𝑔(π‘Ÿ)βˆˆπΏπ‘,𝑀(ℝ) such that ‖𝑀(π‘“βˆ’π‘”)‖𝐿𝑝(ℝ)+π‘‘π‘Ÿβ€–β€–π‘€π‘”(π‘Ÿ)‖‖𝐿𝑝(ℝ)βˆ’πœ€β©½π’¦π‘Ÿ,𝑝(𝑀,𝑓,𝑑).(3.50) Let 𝑓=β„Ž+𝑔, where β„ŽβˆˆπΏπ‘,𝑀(ℝ). Then we have πœ”π‘Ÿ,𝑝(β„Ž,𝑑)β©½πΆβ€–π‘€β„Žβ€–πΏπ‘(ℝ).(3.51) Let 0<𝑠⩽𝑑/π‘Ÿ. From Lemma 3.4, 3.10 and the HΕ‘lder-Minkowski inequality, we have β€–β€–π‘€Ξ”π‘Ÿπ‘ Ξ¦π‘‘(π‘₯)‖‖𝑔(π‘₯)𝐿𝑝(|π‘₯|⩽𝜎(2𝑑)),πΆβ€–β€–β€–ξ‚»ξ€œπ‘€(π‘₯)𝑠/2βˆ’π‘ /2ξ‚Όπ‘Ÿ||𝑔(π‘Ÿ)||(𝑒)𝑑𝑒1𝑑𝑒2β€¦π‘‘π‘’π‘Ÿβ€–β€–β€–πΏπ‘(|π‘₯|⩽𝜎(2𝑑))ξ€·noteΞ¦π‘‘ξ€Έβ€–β€–β€–ξ‚»ξ€œ(π‘₯)⩽𝐢⩽𝐢𝑠/2βˆ’π‘ /2ξ‚Όπ‘Ÿ||𝑀𝑔(π‘Ÿ)ξ€Έ||(𝑒)𝑑𝑒1𝑑𝑒2β‹―π‘‘π‘’π‘Ÿβ€–β€–β€–πΏπ‘(|π‘₯|⩽𝜎(2𝑑))ξƒ¬ξ‚»ξ€œβ©½πΆπ‘ /2βˆ’π‘ /2ξ‚Όπ‘Ÿβ€–β€–(𝑀𝑔(π‘Ÿ)β€–β€–)(𝑒)𝐿𝑝(|π‘₯|⩽𝜎(2𝑑))𝑑𝑒1𝑑𝑒2β‹―π‘‘π‘’π‘Ÿξƒ­β©½πΆπ‘ π‘Ÿβ€–β€–π‘€π‘”(π‘Ÿ)‖‖𝐿𝑝(ℝ)β©½πΆπ‘‘π‘Ÿβ€–β€–π‘€π‘”(π‘Ÿ)‖‖𝐿𝑝(ℝ),(3.52) where 𝑒=π‘₯+Φ𝑑(π‘₯)(𝑒1+𝑒2+β‹―+π‘’π‘Ÿ). We estimate infπ‘ƒβˆˆπ’«π‘Ÿβˆ’1β€–(𝑔(π‘₯)βˆ’π‘ƒ(π‘₯))𝑀(π‘₯)‖𝐿𝑝(|π‘₯|⩾𝜎(4𝑑)).(3.53) Then we may suppose 𝑔(𝑗)(0)βˆ’π‘ƒ(𝑗)(0)=0,𝑗=0,1,…,π‘Ÿβˆ’1.(3.54) Using Lemma 3.5, we see ||π‘„ξ…ž||(𝜎(4𝑑))βˆ’1⩽𝐢𝑑,(3.55) (see [7, page 12]). By Corollary 3.9 with π‘˜=π‘Ÿ and (3.55) we have ‖𝑀(π‘”βˆ’π‘ƒ)‖𝐿𝑝(|π‘₯|⩾𝜎(4𝑑))β©½||π‘„ξ…ž||(𝜎(4𝑑))βˆ’π‘Ÿβ€–β€–ξ€·1+π‘„ξ…ž(π‘₯)2ξ€Έπ‘Ÿ/2‖‖𝑀(π‘”βˆ’π‘ƒ)𝐿𝑝(|π‘₯|⩾𝜎(4𝑑))β©½πΆπ‘‘π‘Ÿβ€–β€–π‘€π‘”(π‘Ÿ)‖‖𝐿𝑝(ℝ).(3.56) Hence, from (3.51), (3.52), and (3.56) we have πœ”π‘Ÿ,𝑝‖(𝑀,𝑓,𝑑)βˆ’πΆπœ€β©½πΆπ‘€β„Žβ€–πΏπ‘(ℝ)+π‘‘π‘Ÿβ€–β€–π‘€π‘”(π‘Ÿ)‖‖𝐿𝑝(ℝ)ξ‚βˆ’πœ€β©½πΆπ’¦π‘Ÿ,𝑝(𝑀,𝑓,𝑑).(3.57) Consequently, we have πœ”π‘Ÿ,𝑝(𝑀,𝑓,𝑑)β©½πΆπ’¦π‘Ÿ,𝑝(𝑀,𝑓,𝑑).(3.58) Therefore, from (3.2) we have (3.3).

Corollary 3.11. For π‘Ÿ=1, one lets π‘€βˆˆβ„±(𝐢2+), and for integer π‘Ÿβ©Ύ2 one lets π‘€βˆˆβ„±(πΆπ‘Ÿ+). Let π‘“βˆˆπΏπ‘,𝑀(ℝ). Then one has πœ”π‘Ÿ,𝑝(𝑀,𝑓,𝑑)βˆΌπ’¦π‘Ÿ,𝑝𝒦(𝑀,𝑓,𝑑)βˆΌπ‘Ÿ,𝑝(𝑀,𝑓,𝑑).(3.59)

Acknowledgments

The authors thank the referees for many kind suggestions and comments. H. S. Jung was supported by SEOK CHUN Research Fund, Sungkyunkwan University, 2010.

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Copyright © 2011 Hee Sun Jung et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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