Direct and strong converse inequalities for approximation with Fejér means

Jorge Bustamante

doi:10.1515/dema-2020-0051

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Direct and strong converse inequalities for approximation with Fejér means

Jorge Bustamante

Published/Copyright: June 17, 2020

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From the journal Demonstratio Mathematica Volume 53 Issue 1

Abstract

We present upper and lower estimates of the error of approximation of periodic functions by Fejér means in the Lebesgue spaces L2πp. The estimates are given in terms of a K-functional for 1≤p≤∞ and in terms of the first modulus of continuity in the case 1<p<∞. We pay attention to the involved constants.

Keywords: Fejér operators; rate of convergence; direct and converse results

MSC 2010: 42A10; 42A20; 41A25; 41A27

1 Introduction

Let C2π denote the Banach space of all 2π-periodic, continuous functions f defined on the real line ℝ with the sup norm

∥f∥∞=maxx∈[−π,π]|f(x)|.

For 1≤p<∞, the Banach space Lp consists of all 2π-periodic, pth power Lebesgue integrable functions f on ℝ with the norm

∥f∥p=(12π∫−ππ|f(x)|pdx)1/p.

In order to simplify, we write Xp=Lp for 1≤p<∞ and X∞=C2π.

Recall that for f∈L1 and k∈ℕ0, the Fourier coefficients are defined by

ak(f)=1π∫−ππf(t)cos(kt)dtandbk(f)=1π∫−ππf(t)sin(kt)dt,

and the (formal) Fourier series is given by

(1)f(x)∼a0(f)2+∑k=1∞(ak(f)cos(kx)+bk(f)sin(kx))≔∑n=0∞An(f).

For n∈ℕ and f∈X1, the Fejér sum of order n is defined by

σn(f,x)=∑k=0n(1−kn+1)Ak(f,x).

Recall that for f∈Xp(1≤p≤∞) and t>0, the usual modulus of continuity of f is defined by

ω1(f,t)p=sup|h|≤t∥f(∘+h)−f(∘)∥p.

In [1], Ching and Chui proved that, for each f∈X2 and n∈ℕ,

1πω1(f,πn+1)2≤∥f−σn(f)∥2≤12ω1(f,πn+1)2.

In this note, we show that a similar relation holds for all Xp spaces with 1<p<∞. In the cases p=1 and p=∞, we provide another kind of characterization. At the end of the article, we compare our results with similar ones given in [2].

2 Main results

We need a result of Alexits [3].

Proposition 1

(Alexits, [3, Lemma 1]) Let X be a Banach space with norm∥⋅∥and let{xn}be a sequence in X. If there exists a constant K such that, for eachn∈ℕ,

||1n+1∑k=1n∑j=1kxj||=||∑k=1n(1−kn+1)xk||≤K,

then there existsx∈Xsuch that

||x−1n+1∑k=1n∑j=1k1jxj||≤3Kn.

For f∈X1, the conjugate function is defined by

f˜(x)=−12π∫0πf(x+t)−f(x−t)tan(t/2)dt=−limε→012π∫επf(x+t)−f(x−t)tan(t/2)dt,

whenever the limit exists. It is known that if f∈Xp with 1<p<∞, then f˜∈Xp, and that is not the case for p=1 and p=∞. It can be proved that if f∈Xp(1<p<∞) is absolutely continuous and f′∈Xp, then f˜′=(f˜)′.

The associated series S˜(f) is defined by [4, p. 49]

(2)S˜(f,x)=∑k=1∞(−bkcos(kx)+aksin(kx)).

Recall that the conjugate Fejér mean of order n of a function f∈L1 with Fourier series (1) is defined by [4, p. 85]

σ˜n(f,x)=∑k=1n(1−kn+1)(−bkcos(kx)+aksin(kx)).

Proposition 2

If1<p<∞, f∈Xpis absolutely continuous andf′∈Xp, thenf˜is absolutely continuous,(f˜)′∈Xpand(f˜)′=f˜′.

Proof

Since (see [4, p. 40])

(3)S(f′,x)=S′(f,x)∼∑k=1∞k(bkcos(jx)−aksin(kx)),

we have

S˜(f′,x)∼∑k=1∞k(akcos(jx)+bksin(kx)).

Therefore,

σ˜n(f′,x)=∑k=1n(1−kn+1)k(akcos(kx)+bksin(kx)).

From the Riesz theorem, we know that f˜ exists a.e. and f˜∈Xp. Since

f˜(x)∼∑k=1∞(−bkcos(kx)+aksin(kx)),

one has

σn(f˜,x)=∑k=1n(1−kn+1)(−bkcos(kx)+aksin(kx)),

and

σn′(f˜,x)=∑k=1n(1−kn+1)k(akcos(kx)+bksin(kx)).

We have proved that σn′(f˜,x)=σ˜n(f′,x). Hence,

∥σn′(f˜)∥p=∥σ˜n(f′)∥p≤Cp∥σn(f′)∥p≤Cp∥f′∥p

(at the end of the article we explain the nature of the constant Cp). Since ∥σn(f˜)−f˜∥p→0, and for any h>0 (we set Δhf(x)=f(x+h)−f(x))

∥Δhf˜∥p=limn→∞∥Δhσn(f˜)∥p≤supn∈ℕ∥σn′(f˜)∥ph≤Cp∥f′∥ph,

we conclude that f˜∈Lip(1,p). This means that there is a constant C such that

sup0<h≤t(12π∫−ππ|f˜(x+h)−f˜(x)|pdx)1/p≤Ct.

But it is known that (see [4, p. 180]), for 1<p<∞, f˜∈Lip(1,p) if and only if there exists g∈Xp such that

f˜(x)=∫−πxg(s)ds.

Therefore, (f˜)′(x) exists a.e. and (f˜)′(x)=g(x)∈Xp. In this case, equation (3) can be written as

S((f˜)′,x)=S′(f˜,x)=∑k=1∞k(akcos(kx)+bksin(kx)).

We know that, for any h∈Xp(1<p<∞)σ˜n(h,x)=σn(h˜,x) [4, p. 92] and ∥σn(h˜)−h˜∥p→0 as n→∞. By taking h=f′, this yields

S˜(f′,x)=S(f′˜,x)=∑k=1∞k(akcos(kx)+bksin(kx)).

We have proved that (f˜)′(x)=f˜′(x) a.e.□

Theorem 1

Assume1≤p≤∞. If f,f˜,(f˜)′∈Xp(we assume that f andf˜are absolutely continuous) andn∈ℕ, then

∥(I−σn)(f)∥p≤3n∥(f˜)′∥p.

Proof

It is known that if f is absolutely continuous, then

f′(x)∼∑k=1∞k(bk(f)cos(kx)−ak(f)sin(kx)),

see [4, p. 41]. If f˜ exists and f˜∈L1, then S˜(f,x)=S(f˜,x) a.e. [4, p. 51].

Set g=f−C, where C=a0(f)/2.

Note that (f˜)′(x)=(g˜)′(x)∼−∑k=1∞k(ak(f)cos(kx)+bk(f)sin(kx)).

We will apply Proposition 2 with xk=k(ak(f)cos(kx)+bk(f)sin(kx)). Take into account that

∥1n+1∑k=1n∑j=1kxk∥p=∥1n+1∑k=1n(n+1−k)xk∥p=∥σn(f˜′)∥p≤∥f˜′∥p.

On the other hand, with the notations given above one has

1n+1∑k=1n∑j=1k1jxj=1n+1∑j=1n∑k=1j(ak(f)cos(kx)+bk(f)sin(kx))=1n+1∑j=1n∑k=1jAk(f)=σn(g).

Now it follows from Proposition 2 (with K=∥f˜′∥p) that there exists h∈Xp satisfying

∥h−1n+1∑k=1n∑j=1k1jxj∥p=∥h−σn(g)∥p≤3∥f˜′∥pn.

Since σn(g)→g, g=h a.e. But σn(g)−g=σn(f)−f.□

Remark 1

Note that, if f∈Tn is not a constant polynomial, say f(x)=∑k=0n(akcos(kx)+bksin(kx)), then

(f˜)′(x)=∑k=0nk(akcos(kx)+bksin(kx)),

while

(I−σn)(f,x)=1n+1∑k=1nk(akcos(kx)+bksin(kx))=1n+1(f˜)′(x).

Therefore,

n∥(I−σn)(f)∥p∥(f˜)′∥p=nn+1

and

supn∈ℕsupf∈Xp,f≠constn∥(I−σn)(f)∥p∥(f˜)′∥p≥1.

This provides a lower bound for the constant in Theorem 1.

Theorem 2

Assume1≤p≤∞. Ifn∈ℕandf∈Xp, then

(4)12∥f−σn(f)∥p≤K˜(f,3n+1)p≤4∥f−σn(f)∥p,

where

K˜(f,t)p=inf{∥f−g∥p+t∥(g˜)′∥p:g,g˜∈Xp,g˜∈AC,(g˜)′∈Xp}.

Moreover, if1<p<∞, there exists a constantCpsuch that, for eachf∈Xp,

(5)12Cp∥f−σn(f)∥p≤ω1(f,3n+1)p≤8Cp∥f−σn(f)∥p.

Proof

A direct computation shows that, for any trigonometric polynomial T of degree not greater than n,

(σn−I)(T)=−1n+1(T˜)′.

Since σ˜n(f) is a trigonometric polynomial of degree not greater than n and ∥σn∥=1, we know that

∥(σ˜n(f))′∥p=(n+1)∥(I−σn)(σnf)∥p=(n+1)∥σn(f−σn(f))∥p≤(n+1)∥f−σn(f)∥p.

Hence,

K˜(f,3n+1)p≤∥f−σn(f)∥p+3n+1∥(σ˜n(f))′∥p≤4∥f−σn(f)∥p.

On the other hand, it follows from Theorem 1 that, for each n∈ℕ and every g∈Xp such that (g˜)′∈Xp,

∥f−σn(f)∥p≤∥f−g−σn(f−g)∥p+∥g−σn(g)∥p≤2∥f−g∥p+3n∥(g˜)′∥p≤2∥f−g∥p+6n+1∥(g˜)′∥p

and this yields ∥f−σn(f)∥p≤2K˜(f,3/(n+1))p.

If 1<p<∞, h∈Lp and S(h) is the Fourier series of h, then h(x)=S(h,x) a.e. In particular, if h=g′˜, then h˜=−g′ a.e. and it follows from the Riesz theorem (see [5, p. 336] or [6]) that there exists a constant Cp≥1 such that

(6)∥g′˜∥p≤Cp∥g′∥pand∥g′∥p=∥h˜∥p≤Cp∥h∥p=Cp∥g′˜∥p.

Therefore,

1CpK˜(f,t)p≤inf{∥f−g∥p+t∥g′∥p:g∈Xp,g∈AC,g′∈Xp}≤CpK˜(f,t)p.

But it is known that (see [7, pp. 175–177])

12ω1(f,t)p≤inf{∥f−g∥p+t∥g′∥p:g∈Xp,g∈AC,g′∈Xp}≤2ω1(f,t)p.

Hence,

14Cp∥f−σn(f)∥p≤12CpK˜(f,3n+1)p≤ω1(f,3n+1)p≤2CpK˜(f,3n+1)p≤8Cp∥f−σn(f)∥p,

and this yields the result.□

Remark 2

The best constant Cp in inequality (6) was obtained by Pichorides in [6]:

P(p)={tanπ2pif1<p≤2,cotπ2pif2<p<∞.

In particular, when p=2 and Cp=1, equation (5) can be written as

14∥f−σn(f)∥2≤ω1(f,3n+1)2≤8∥f−σn(f)∥2.

Thus, we obtain a result similar to the one of Ching and Chui.

Remark 3

Ditzian and Ivanov [2, Theorem 2.1] proved (4) in the case of continuous functions. They used some ideas of Lorentz in [8], instead of the Alexits result recalled in Proposition 2. The general case of Xp(1≤p≤∞) is included in [2, Theorem 2.2]. In both theorems, no information is given concerning the constants. Hence, equation (4) is an improvement of the Ditzian-Ivanov results. We also remark that there is not an analogous of (5) in [2].

References

[1] C.-H. Ching and Ch. K. Chui, Some inequalities in trigonometric approximation, Bull. Austral. Math. Soc. 8 (1973), 393–395.10.1017/S0004972700042684Search in Google Scholar

[2] Z. Ditzian and K. Ivanov, Strong converse inequalities, J. Anal. Math. 61 (1993), 61–111.10.1007/BF02788839Search in Google Scholar

[3] G. Alexits, Sur l’ordre de grandeur de l’approximation d’une fonction périodique par les sommes de Fejér, Acta Math. Acad. Sci. Hungar. 3 (1952), 29–42.10.1007/BF02146066Search in Google Scholar

[4] A. Zygmund, Trigonometric Series, 3rd edition, Vol. I and II combined, Cambridge University Press, 2002.Search in Google Scholar

[5] P. L. Butzer and R. J. Nessel, Fourier Analysis and Approximation, Academic Press, New-York and London, 1971.10.1007/978-3-0348-7448-9Search in Google Scholar

[6] S. K. Pichorides, On the best value of the constant in the theorems of M. Riesz, Zygmund and Kolmogorov, Studia Math. XLIV (1972), 165–179.10.4064/sm-44-2-165-179Search in Google Scholar

[7] R. A. DeVore and G. G. Lorentz, Constructive Approximation, Springer-Verlag, Berlin Heidelberg, 1993.10.1007/978-3-662-02888-9Search in Google Scholar

[8] G. G. Lorentz, Approximation of Functions, Holt, Rinehart and Winston, New York, 1966.Search in Google Scholar

Received: 2019-12-06

Revised: 2020-04-24

Accepted: 2020-04-28

Published Online: 2020-06-17

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https://doi.org/10.1515/dema-2020-0051

Keywords for this article

Fejér operators; rate of convergence; direct and converse results

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