# Direct and strong converse inequalities for approximation with Fejér means

Jorge Bustamante 1
Jorge Bustamante
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## 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. We pay attention to the involved constants.

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

We need a result of Alexits .

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 each$n∈ℕ$,

$||1n+1∑k=1n∑j=1kxj||=||∑k=1n(1−kn+1)xk||≤K,$
then there exists$x∈X$such 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, then $f˜∈Xp$, and that is not the case for $p=1$ and $p=∞$. It can be proved that if $f∈Xp$$(1 is absolutely continuous and $f′∈Xp$, then $f˜′=(f˜)′$.

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

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

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

If$1, $f∈Xp$is absolutely continuous and$f′∈Xp$, then$f˜$is absolutely continuous,$(f˜)′∈Xp$and$(f˜)′=f˜′$.

Proof

Since (see [4, p. 40])

$S(f′,x)=S′(f,x)∼∑k=1∞k(bkcos(jx)−aksin(kx)),$
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
But it is known that (see [4, p. 180]), for $1, $f˜∈Lip(1,p)$ 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$σ˜n(h,x)=σn(h˜,x)$ [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

Assume$1≤p≤∞$. If $f,f˜,(f˜)′∈Xp$(we assume that f and$f˜$are absolutely continuous) and$n∈ℕ$, 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

Assume$1≤p≤∞$. If$n∈ℕ$and$f∈Xp$, then

$12∥f−σn(f)∥p≤K˜(f,3n+1)p≤4∥f−σn(f)∥p,$
where
$K˜(f,t)p=inf{∥f−g∥p+t∥(g˜)′∥p:g,g˜∈Xp,g˜∈AC,(g˜)′∈Xp}.$

Moreover, if$1, there exists a constant$Cp$such that, for each$f∈Xp$,

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

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, $h∈Lp$ 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 ) that there exists a constant $Cp≥1$ such that

$∥g′˜∥p≤Cp∥g′∥pand∥g′∥p=∥h˜∥p≤Cp∥h∥p=Cp∥g′˜∥p.$
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 :

$P(p)={tanπ2pif1
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 , 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 .

## References

• 

C.-H. Ching and Ch. K. Chui, Some inequalities in trigonometric approximation, Bull. Austral. Math. Soc. 8 (1973), 393–395.

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• 

Z. Ditzian and K. Ivanov, Strong converse inequalities, J. Anal. Math. 61 (1993), 61–111.

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• 

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.

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• 

A. Zygmund, Trigonometric Series, 3rd edition, Vol. I and II combined, Cambridge University Press, 2002.

• 

P. L. Butzer and R. J. Nessel, Fourier Analysis and Approximation, Academic Press, New-York and London, 1971.

• 

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.

• 

R. A. DeVore and G. G. Lorentz, Constructive Approximation, Springer-Verlag, Berlin Heidelberg, 1993.

• 

G. G. Lorentz, Approximation of Functions, Holt, Rinehart and Winston, New York, 1966.

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• 

C.-H. Ching and Ch. K. Chui, Some inequalities in trigonometric approximation, Bull. Austral. Math. Soc. 8 (1973), 393–395.

• Crossref
• Export Citation
• 

Z. Ditzian and K. Ivanov, Strong converse inequalities, J. Anal. Math. 61 (1993), 61–111.

• Crossref
• Export Citation
• 

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.

• Crossref
• Export Citation
• 

A. Zygmund, Trigonometric Series, 3rd edition, Vol. I and II combined, Cambridge University Press, 2002.

• 

P. L. Butzer and R. J. Nessel, Fourier Analysis and Approximation, Academic Press, New-York and London, 1971.

• 

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.

• 

R. A. DeVore and G. G. Lorentz, Constructive Approximation, Springer-Verlag, Berlin Heidelberg, 1993.

• 

G. G. Lorentz, Approximation of Functions, Holt, Rinehart and Winston, New York, 1966.

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