# The de la Vallée Poussin Mean and Polynomial Approximation for Exponential Weight

Authors: K. Itoh; R. Sakai; N. Suzuki
Journal: International Journal of Analysis
 (2015)

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## Abstract

We studyLp boundedness of the de la Vallée Poussin means vn(f) for exponential weight w(x)=exp(-Q(x)) on R. Our main result is vn(f)(w/T1/4)Lp(R)≤C∥fw∥Lp(R), for every n∈N and every 1≤p≤∞, where T(x)=xQ′(x)/Q(x). As an application, we obtain limn→∞(f-vn(f))w/T1/4Lp(R)=0 for fw∈Lp(R).


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## Body

## 1. Introduction

LetR=(-∞,∞) and N = {1,2,…}. We consider an exponential weight(1)w(x)=exp⁡-Qxon R, where Q is an even and nonnegative function on R. Throughout this paper we always assume that w belongs to a relevant class F(C2+) (see Section 2). We consider a function T defined by(2)Tx∶=xQ′xQx,x≠0.If T is bounded, then w is called a Freud-type weight, and otherwise, w is called an Erdős-type weight.Letw=exp⁡(-Q)∈F(C2+). By definition, Q′′>0, so that F(x)∶=∫01‍(xuQ′(xu)/(1-u2)1/2)du is an increasing and continuous function on [0,∞) with F(0)=0 and limx→∞F(x)=∞. By the intermediate value theorem, there exists a unique an>0 such that n=2/πF(an) for every n∈N; that is,(3)n=2π∫01‍anuQ′(anu)1-u21/2du,n∈Nholds. This an is called the Mhaskar-Rakhmanov-Saff number (MRS number). In this sequel, notation an always stands for the MRS number for a weight w defined by (3) (see [1, page 11], [2, page 180]).Let{pn} be orthogonal polynomials for a weight w; that is, pn(x)=pn(w,x) is the polynomial of degree n such that(4)∫R‍pn(x)pm(x)w2(x)dx=δmn.For 1≤p≤∞, we denote by Lp(R) the usual Lp space on R. For a function f with fw∈Lp(R), we set(5)snfx∶=∑k=0n-1‍ckfpkx,whereckf∶=∫R‍f(t)pk(t)w2(t)dtfor n∈N (the partial sum of Fourier series). The de la Vallée Poussin mean vn(f) of f is defined by(6)vnfx∶=1n∑j=n+12n‍sjfx.The main result of this paper is the following theorem.Theorem 1.
Let1≤p≤∞. We assume that w∈F(C2+) satisfies(7)T(an)≤cnan2/3for some c>0. Then there exists a constant C=C(w,p)>0 such that when T1/4fw∈Lp(R), then(8)vnfwLp(R)≤CT1/4fwLp(R),and when fw∈Lp(R), then(9)vnfwT1/4Lp(R)≤CfwLp(R).For Freud-type weights, the following inequality is known:(10)vnfwLp(R)≤CfwLpR(cf. [3, Lemma 6], [2, Theorem 3.4.2]). Note that when T is bounded, then (7) holds evidently, so (10) follows from (8) or (9) immediately. As for Erdős-type weights, Lubinsky and Mthembu [4] proved (8) for p=1, (9) for p=∞, and vn(f)wT-1/4Lp(R)≤CT1/4fwLp(R) for 1<p<∞ under the assumption T(x)=O(Q′(x)ε) for every ε>0. (They discussed on Cesaro means (=(C,1) means) mainly, but it easily ensures the result on de la Vallée Poussin means.) We note that their assumption implies our condition (7) (see Remark 16 in Section 7). Hence our results (8) and (9) improve their result. For more general weights, see [5–7].Using (9), we have the following result. There exists a constant C=C(w,p)>0 such that, for any wf∈Lp(R), (11)f-vnfwT1/4Lp(R)≤Cinf⁡P∈Pnwf-PLp(R),where Pn is the set of all polynomials of degree at most n. It is known that the right-hand side of (11) attains some Pn∈Pn; however, it is not easy to determine such Pn explicitly. Equation (11) means that the de la Vallée Poussin mean takes the place of Pn in some sense; that is, vn(f) is a good concrete approximation polynomial for given function f.This paper is organized as follows. The definition of classF(C2+) is given in Section 2. Recalling some estimates for exponential weights in Section 3, we will give a proof of Theorem 1 for the case of p=∞ in Section 4. Basic method of our proof is classic and well known ([3], see also [4, 6]), but we repeat it in order to make a role of T(x) clear. The complete proof of Theorem 1 is given in Section 5. The estimate (11) is shown in Section 6 together with another application. We discuss the condition (7) in Section 7.Throughout this paperC will denote a positive constant whose value is not necessarily the same at each occurrence; it may vary even within a line. When we write C=C(a,b,…), then C is a constant which depends on a,b,… only.


## 2. Definitions and Notation

We say that an exponential weightw=exp⁡(-Q) belongs to class F(C2+), when Q:R→[0,∞) is a continuous and even function and satisfies the following conditions.(a)
Q′(x) is continuous in R with Q(0)=0.(b)
Q′′(x) exists and is positive in R∖0.(c)
limx→∞Q(x)=∞.(d)
The functionT defined in (2) is quasi-increasing in (0,∞) (i.e., there exists C>0 such that T(x)≤CT(y) whenever 0<x<y), and there exists Λ∈R such that (12)T(x)≥Λ>1,x∈R∖{0}.(e)
There existsC>0 such that (13)Q′′(x)|Q′(x)|≤CQ′xQx,a.e.x∈Rand there also exist a compact subintervalJ(∋0) of R and C>0 such that (14)Q′′(x)|Q′(x)|≥C|Q′(x)|Q(x),a.e.x∈R∖J.Letw(x)=exp⁡(-Q(x))∈F(C2+). Suppose that Q∈C3(R) and λ>0. If there exist C>0 and K>0 such that for |x|≥K,(15)Q′′′(x)Q′′(x)≤CQ′′xQ′x,|Q′(x)|Qλ(x)≤C,then we write w∈Fλ(C3+). Note that if w is a Freud-type, the last inequality holds with λ=1.A typical example of Freud-type weight isw(x)=exp⁡(-|x|α) with α>1. Note that the Hermite polynomials are the orthogonal polynomials for the weight exp⁡(-|x|2). Let u≥0,  α>0,  α+u>1, and  l∈N, and we set(16)Qx∶=xuexp⁡lxα-exp⁡l0,where exp⁡l(x)=exp⁡(exp⁡(exp⁡(⋯(exp⁡x)))) (l-times). Then w(x)=exp⁡(-Q(x)) is an Erdős-type weight, which belongs to Fλ(C3+) with λ>1 (see [8]).We recall some notation which we use later (cf. [9]). By definition, the partial sum of Fourier series is given by(17)smfx=∫R‍Km(x,t)f(t)w2(t)dt,where(18)Km(x,t)=∑k=0m-1‍pk(x)pk(t).It is known that, by the Christoffel-Darboux formula,(19)Km(x,t)=γm-1γmpm(x)pm-1(t)-pm(t)pm-1(x)x-t,where γn is the leading coefficient of pn; that is, pn(x)=γnxn+⋯.The Christoffel functionλm(x)=λm(w,x) is defined by(20)λm(x)∶=1Km(x,x)=∑k=0m-1‍pk2(x)-1,and Lp type Christoffel function λm,p(x)=λm,p(w,x) is defined by(21)λm,px∶=inf⁡P∈Pm1Pxp∫R‍Ptwtpdt.If p=2, we know(22)λm,2(x)=λm(x).The MRS numbers{an} for weight w are monotonically increasing, and we see easily (23)limn→∞an=∞,limn→∞ann=0.Moreover, for n∈N, there exists C>0 independent of n, such that(24)anC≤γn-1γn≤Can(see [9, Lemma 13.9]). We also use the following functions: (25)Φnx∶=1-x/a2n1-x/an+δn,x≤an,Φnan,an<x,where δn∶=nTan-2/3 and (26)φnx∶=annΦnx.


## 3. Lemmas

We recall some basic estimates.Lemma 2 (infinite-finite range inequality [9, Theorem 1.9(a)]).
Letw∈F(C2+), 0<p≤∞ and P∈Pn(n≥1). Then (27)PwLp(R)≤2PwLp(|x|<an).Lemma 3 (see [10, Lemma 3.4]).
Letw∈F(C2+). There exists a constant C=C(w)>0 such that(28)ann1Txφnx≤C,and hence(29)1Tx≤CΦn(x)holds for every n∈N and x>0.Lemma 4 (see [9, Theorem 9.3]).
Letw∈F(C2+), and 0<p≤∞. Then there exists a constant C=C(w,p)>0 such that, for every n∈N and x∈R,(30)λn,p(x)≥Cφn(x)wp(x)and when |x|≤an, one has(31)1Cφn(x)wp(x)≤λn,p(x)≤Cφn(x)wp(x).Lemma 5.
Letw∈F(C2+). Then there exist 0<c0<1 and C=C(w)>0 such that(32)1CTx≤Tx±c0Tx≤CTxfor every x∈R.Proof.
It is known that there exist0<c0<1 and C=C(w)>0 such that(33)1CTx≤Tx1±c0Tx≤CTxfor every x∈R [9, Theorem 3.2(e)]. If |x|≤1, then(34)x±c0Tx≤x+c0≤2,and hence (32) holds evidently. If x>1, then(35)x1+c0Tx≥x±c0Tx≥x1-c0T(x).Since T is quasi-increasing, (33) shows (32).Lemma 6 (see [10, Theorem 3.2]).
Ifw∈F(C2+) is an Erdős-type weight, then for any η>0 there exists a constant C=C(w,η)>0 such that(36)an≤Cnηfor every n∈N.Lemma 7 (see [9, Lemma 3.5(a), (b) and Lemma 3.11(3.52)]).
Letw∈F(C2+). There exists a constant C=C(w)>0 such that(37)a2n≤Can,(38)Ta2n≤CTan,(39)1CanT(an)≤a2n-anfor every n∈N.


## 4. Proof of Theorem1 for p=∞

We begin with the following proposition.Proposition 8.
Letw∈F(C2+). Then there exists a constant C=C(w)>0 such that(40)w2xTx∑k=0n-1‍pk2x≤Cnanfor every x∈R and every n∈N.Proof.
From Lemmas3 and 4 for p=2, the proposition follows immediately.The following estimate is known essentially (see [6, Theorem 4], [7, Theorem 1.2] and [5, Lemma 1]), but we give a proof for the sake of completeness.Theorem 9.
Letw∈F(C2+). Then there exists a constant C=C(w)>0 such that(41)vnfwΦ2n1/2L∞(R)≤CfwL∞Rfor any wf∈L∞(R) and n∈N.Proof.
Letx∈R and n,m∈N with n-1≤m≤2n. We set(42)g(t)∶=f(t)χ(x-an/n,x+an/n)(t),h(t)∶=f(t)-g(t),where χ(a,b) is the characteristic function of a set (a,b). Then(43)vnfx=vngx+vnhx.We first consider an estimate of vn(g). By the Schwarz inequality(44)smgx=∫x-an/nx+an/n‍f(t)Km(x,t)w2(t)dt≤fwL∞(R)∫x-an/nx+an/n‍Kmx,tw(t)dt≤fwL∞(R)2ann1/2∫R‍Km2x,tw2tdt1/2,and since (45)∫R‍Km2(x,t)w2(t)dt=∑i=0m-1‍pi2(x)≤∑i=02n-1‍pi2x=1λ2n(x),(26) and (30) give us (46)wxsmgxΦ2n1/2x≤fwL∞(R)2φ2n(x)λ2n(x)w2(x)1/2≤CfwL∞(R),so that(47)vngxwxΦ2n1/2x≤CfwL∞(R).
Next, for an estimate ofvn(h)(x), we set(48)H(t)∶=h(t)x-tand denote by {bj(H)} the Fourier coefficients of H; that is,(49)bj(H)∶=∫R‍H(t)pj(t)w2(t)dt.Using the Christoffel-Darboux formula (19), we have(50)smhx=∫R‍γm-1γmpm(x)pm-1(t)-pm(t)pm-1(x)x-t∫R‍γm-1γmpm(x)pm-1(t)-pm(t)pm-1(x)x-tppppppp·h(t)w2(t)dt=γm-1γmpm(x)∫R‍pm-1(t)H(t)w2(t)dt-pm-1(x)ppppppppp·∫R‍pm(t)H(t)w2(t)dt≤γm-1γmpmxbm-1H+pm-1(x)bmH.By (24) and the Schwarz inequality, we have (51)vnhx≤1nmax⁡m≤2nγm-1γm×2λ2n1/2(x)∑m=02n‍bm2(H)1/2≤Cann1λ2n1/2(x)∑m=02n‍bm2(H)1/2.Since (52)∑m=02n‍bm2H≤∑m=0∞‍bm2(H)≤∫R‍Htwt2dt=∫|t-x|≥an/n‍f(t)w(t)t-x2dt≤fwL∞R22nan,(26) and (30) imply(53)vnhxwx≤Cannw2(x)λ2n(x)1/2fwL∞(R)nan≤CfwL∞(R)1Φ2n1/2(x)w2(x)φ2n(x)λ2n(x)1/2≤CfwL∞(R)1Φ2n1/2(x).This together with (47) gives us (54)vnfxwxΦ2n1/2x≤CfwL∞(R),which completes the proof of (41).The following corollary contains (9) for p=∞.Corollary 10.
Letw∈F(C2+). Then there exists a constant C=C(w)>0 such that(55)vnfwT1/4L∞(R)≤CfwL∞Rfor any wf∈L∞(R).Proof.
From Theorem9 and Lemma 3, this corollary follows immediately.We give a proof of (8) for p=∞.Theorem 11.
Letw∈F(C2+) such that T(an)≤c(n/an)2/3 with some c>0. Then there exists a constant C=C(w)>0 such that if T1/4wf∈L∞(R), one has(56)vnfwL∞(R)≤CT1/4fwL∞(R).Proof.
Letx∈R and n,m∈N such that n-1≤m≤2n. As in the proof of Theorem 9, we set g(t)∶=f(t)χ(x-an/n,x+an/n)(t),  h(t)∶=f(t)-g(t). Then vn(f)(x)=vn(g)(x)+vn(h)(x). Since vn(g),vn(h)∈P2n, by Lemma 2, we may suppose that |x|≤a2n.
We first prove that, for anyn∈N,(57)Tx≤CTtwhenever |x|≤a2n and |x-t|≤an/n. In fact, we now denote by C1>0 a constant in Lemma 7. Since T is quasi-increasing, there exists a constant C2>0 such that T(x)≤C2T(y) whenever 0≤x≤y. Then by (7), there exists N0∈N such that n≥C12T(an) holds for every n≥N0, so that Lemma 7 gives us(58)t≤x+ann≤a2n+1C12anT(an)≤a2n+1C1a2nT(a2n)≤a4n.This together with (38) and (7) implies(59)Tt≤C2T(a4n)≤C12C2T(an)≤cC12C2nan2/3.Since n/an→∞ as n→∞, we may assume that(60)cC12C2nan2/3≤c0nanholds for every n≥N0 if we take N0 larger as needed, where c0 is the constant in Lemma 5. Then(61)x≤ann+t≤c0Tt+t,and hence Lemma 5 gives us (62)T(x)=Tx≤C2Tc0Tt+t≤C2CT(t)=CT(t).When n≤N0, then 1≤T(x)≤C for |x|≤a2N0, so that (57) holds immediately.
Now we estimatevn(g)(x). By the Schwarz inequality, (57), and Proposition 8,(63)smgxwx≤C∑k=0m-1‍w2(x)T1/2(x)pk2(x)1/2·∫x-an/nx+an/n‍|T1/4(t)w(t)f(t)|2dt1/2≤CmamT1/4wfL∞(R)∫x-an/nx+an/n‍dt1/2≤CT1/4wfL∞(R),which implies(64)vngxwx≤CT1/4fwL∞(R).
Next, we estimatevn(h)(x). Since(65)vnhx=1n∑m=n+12n‍γm-1γmbm-1Hpmx-bmHpm-1xas before, where H(t)=h(t)/(x-t), Proposition 8 gives us(66)wxvnhx≤2ann∑m=02n‍w2(x)∑m=n2n‍pm2(x)1/2∑m=02n‍bm2(H)1/2≤CannT1/2(x)∫|x-t|>an/n‍f(t)w(t)x-t2dt1/2.By Lemma 5, we can take c>0 small enough such that T(x)≥CT(x±c/T(x)) for C>0. Then we have (67)T1/2(x)∫c/T(x)>|x-t|>an/nftwt2x-t2dt‍≤CT1/2(x)∫c/T(x)>|x-t|>an/n1T1/2|x-c/T(x)|‍ppppppppppppppppppppppp·ftT1/4twt2x-t2dt≤CT1/4fwL∞(R)2∫c/T(x)>|x-t|>an/n‍1x-t2dt≤CnanT1/4fwL∞(R)2.Also since T(an)≤C(n/an)2/3 and |x|≤a2n, we have(68)T1/2(x)∫c/T(x)≤|x-t|‍ftwt2x-t2dt≤fwL∞(R)2T1/2(x)∫c/T(x)≤|x-t|‍1x-t2dt≤CfwL∞(R)2T3/2(x)≤CnanfwL∞(R)2.These estimates show (69)T1/2(x)∫|x-t|>an/n‍f(t)w(t)x-t2dt1/2≤CnanT1/4fwL∞(R),and hence (70)vn(h)wL∞(R)≤CT1/4fwL∞(R)follows. This together with (64) implies (56).


## 5. Proof of Theorem1 for 1≤p<∞

In this section we complete the proof of Theorem1.Proof of (8).
TheL∞-norm case is Theorem 11. We prove the L1-norm case. By the duality of L1-norm,(71)vnfwL1(R)=sup⁡gwL∞R≤1∫R‍vnfxgxwx2dx.Since Km(x,t)=Km(t,x), we see(72)∫R‍smfxgxw2xdx=∫R‍fxsmgxw2xdx,and hence(73)∫R‍vnfxgxw2xdx=∫R‍fxvngxw2xdx.Therefore, using Corollary 10, we have(74)vnfwL1(R)=sup⁡∥gw∥L∞R≤1∫R‍vnfxgxw2(x)dx=sup⁡∥gw∥L∞R≤1∫R‍fxvngxw2(x)dx≤sup⁡∥gw∥L∞(R)≤1T1/4fwL1(R)1T1/4vngwL∞(R)≤sup⁡∥gw∥L∞(R)≤1T1/4fwL1(R)gwL∞(R)≤T1/4fwL1(R).Since the operator norms(75)F⟼wvnF1wT1/4for p=1 and p=∞ are bounded, the Riesz-Thorin interpolation theorem gives us(76)sup⁡∥F∥Lp(R)≤1wvnF1wT1/4Lp(R)<∞for every p with 1≤p≤∞. This implies (8).Proof of (9).
TheL∞-norm case is Corollary 10. Now, we show L1-norm case. Similar as above, (77)vnfwT1/4L1(R)=sup⁡∥wg∥L∞R≤1∫R‍vnfxgxTx1/4w2(x)dx=sup⁡∥wg∥L∞(R)≤1∫R‍f(x)vngT1/4(x)w2(x)dx≤sup⁡∥wg∥L∞(R)≤1wvngT1/4L∞(R)fwL1(R).Since (78)wvngT1/4L∞(R)≤CT1/4gT1/4wL∞(R)=CgwL∞(R)≤Cby (56), we see (79)vnfwT1/4L1(R)≤CfwL1(R).Hence by the Riesz-Thorin interpolation theorem for the operator(80)F⟼wT1/4vn(w-1F),we have(81)sup⁡FLp(R)≤1wT1/4vnw-1FLp(R)<∞for every 1≤p≤∞. This implies (9).


## 6. Corollaries

We begin with the following corollary.Corollary 12.
Let1≤p≤∞ and w∈F(C2+) satisfy (7). Then there exists a constant C=C(w,p)>0 such that, for every wf∈Lp(R) and n∈N,(82)vnfwLp(R)≤CT1/4(an)fwLp(R).Proof.
By Lemma2 and (9) in Theorem 1,(83)vnfwLp(R)≤CvnfwLp([-a2n,a2n])≤CT1/4(an)vnfwT1/4Lp(R)≤CT1/4(an)fwLp(R)holds for every 1≤p≤∞.To discuss polynomial approximations, we define the degree of weighted polynomial approximation forwf∈Lp(R) by(84)Ep,nw,f∶=inf⁡P∈Pnwf-PLp(R).We quote two results from our previous papers.Proposition 13 (see [11, Theorem 1] and [10, Theorem 6.1]).
Let1≤p≤∞ and w∈F(C2+). Then there exists a constant C=C(w,p)>0 such that for every n∈N, if f is absolutely continuous and f′w∈Lp(R), then (85)Ep,nw,f≤Cannf′wLpR,and if P∈Pn, then(86)1TP′wLp(R)≤CnanPwLp(R)holds true.Corollary 14.
Let1≤p≤∞ and w∈F(C2+) satisfy (7). Then there exists a constant C=C(w,p)>0 such that, for every n∈N and every wf∈Lp(R),(87)f-vnfwT1/4Lp(R)≤CEp,nw,f,(88)f-vnfwLp(R)≤CT1/4anEp,nw,f,and when T1/4wf∈Lp(R),(89)f-vnfwLp(R)≤CEp,nT1/4w,f.Moreover if f is absolutely continuous and f′w∈Lp(R), then(90)f-vnfwT1/4Lp(R)≤Cannf′wLp(R).Proof.
Sincevn(P)=P for every P∈Pn, we have vn(f)=P+vn(f-P), and hence (9) gives us(91)f-vnfwT1/4Lp(R)≤f-PwT1/4Lp(R)+vnf-PwT1/4Lp(R)≤f-PwT1/4Lp(R)+f-PwLp(R)≤Cf-PwLp(R),which shows (87). By using (82) and (8), we also obtain (88) and (89), respectively. Favard-type inequality (90) follows from (85) and (87).Combining (8) with (86), we obtain the following estimate of the derivative of vn(f).Corollary 15.
Let1≤p≤∞ and w∈F(C2+) satisfy (7). Then there exists a constant C=C(w,p)>0 such that for every n∈N(92)1Tvn′fwLp(R)≤CnanT1/4fwLp(R).


## 7. Remarks

In this final section we make three remarks. First one is concerning the condition (7) in Theorem 1.Remark 16.
Letw=exp⁡(-Q)∈F(C2+).
(1) Let0<λ<2. If w satisfies (93)Q′xQλx≤C(x≥1)for some C>0. Then the condition (7) holds true. In particular, all w∈Fλ(C3+) satisfy (7).
(2) IfT(x)=O((Q′(x))ε) for some 0<ε<1/2, then (93) holds for some 0<λ<2. In particular, all the weights discussed in [4] satisfy (7).
In fact, ifw is Freud-type, then (7) holds clearly, so that we may assume that w is Erdős-type. In [10, Lemma 3.2], we showed that when w satisfies (93), then T(an)≤Cn2/3-δ holds for some 0<δ<2/3. Hence this and (36) imply (7). Since |xQ′(x)|/Qx=T(x)≤C|Q′(x)|ε, λ∶=1/(1-ε)<2 satisfies (93).
(3) The condition (7) is equivalent to(94)ann21/3≤CQanwith some constant C>0, because Q(an)~n/T(an) (see [1, Lemma 3.4(3.18)]).Next we remark on the degree of weighted polynomial approximationEp,n(w,f).Remark 17.
Let1≤p≤∞. It is known that if w∈F(C2+), then limn→∞Ep,n(w,f)=0 for wf∈Lp(R) (when p=∞ we further assume that lim|x|→∞f(x)w(x)=0) (e.g., [1, Theorem 1.4]). Hence (87) implies (95)limn→∞f-vnfwT1/4Lp(R)=0.This is a concrete polynomial approximation for a given function f. Similar argument can not apply to (89), because T1/4w may not belong to F(C2+). To overcome this difficulty, we use a mollification of a weight ([10, Theorem 4.1]): let w∈Fλ(C3+) with 0<λ≤3/2. Then we can construct a new weight w*∈F(C2+) which satisfies T1/4w~w*, an~an*, and T~T*, where an* and T* are the MRS number and a function defined (2) with respect to w*, respectively. Hence Ep,n(T1/4w,f)≤CEp,n(w*,f) and (89) show that if w∈Fλ(C3+) with 0<λ≤3/2 and if T1/4wf∈Lp(R), then (96)limn→∞f-vnfwLp(R)=0(when p=∞ we further assume that lim|x|→∞T1/4(x)f(x)w(x)=0).Remark 18.
Equation (92) suggests that the following inequality would be true: (97)vn′fwLp(R)≤CnanT3/4fwLp(R).We will discuss this estimate elsewhere.



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