Direct and strong converse inequalities for approximation with Fejér means

Jorge Bustamante 1
  • 1 Posgrado de Matemática, Facultad de Ciencias Físico-Matemáticas, Benemérita Universidad Autónoma de Puebla, Puebla, Mexico
Jorge Bustamante
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  • Posgrado de Matemática, Facultad de Ciencias Físico-Matemáticas, Benemérita Universidad Autónoma de Puebla, Puebla, Mexico
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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 1p and in terms of the first modulus of continuity in the case 1<p<. 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 with the sup norm

f=maxx[π,π]|f(x)|.

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

fp=(12πππ|f(x)|pdx)1/p.
In order to simplify, we write Xp=Lp for 1p< and X=C2π.

Recall that for fL1 and k0, the Fourier coefficients are defined by

ak(f)=1πππf(t)cos(kt)dtandbk(f)=1πππf(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=0An(f).
For n and fX1, the Fejér sum of order n is defined by
σn(f,x)=k=0n(1kn+1)Ak(f,x).

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

ω1(f,t)p=sup|h|tf(+h)f()p.

In [1], Ching and Chui proved that, for each fX2 and n,

1πω1(f,πn+1)2fσn(f)212ω1(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 normand let{xn}be a sequence in X. If there exists a constant K such that, for eachn,

||1n+1k=1nj=1kxj||=||k=1n(1kn+1)xk||K,
then there existsxXsuch that
||x1n+1k=1nj=1k1jxj||3Kn.

For fX1, the conjugate function is defined by

f˜(x)=12π0πf(x+t)f(xt)tan(t/2)dt=limε012πεπf(x+t)f(xt)tan(t/2)dt,
whenever the limit exists. It is known that if fXp with 1<p<, then f˜Xp, and that is not the case for p=1 and p=. It can be proved that if fXp(1<p<) is absolutely continuous and fXp, 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 fL1 with Fourier series (1) is defined by [4, p. 85]

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

Proposition 2

If1<p<, fXpis absolutely continuous andfXp, thenf˜is absolutely continuous,(f˜)Xpand(f˜)=f˜.

Proof

Since (see [4, p. 40])

S(f,x)=S(f,x)k=1k(bkcos(jx)aksin(kx)),
we have
S˜(f,x)k=1k(akcos(jx)+bksin(kx)).
Therefore,
σ˜n(f,x)=k=1n(1kn+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(1kn+1)(bkcos(kx)+aksin(kx)),
and
σn(f˜,x)=k=1n(1kn+1)k(akcos(kx)+bksin(kx)).

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

σn(f˜)p=σ˜n(f)pCpσn(f)pCpfp
(at the end of the article we explain the nature of the constant Cp). Since σn(f˜)f˜p0, and for any h>0 (we set Δhf(x)=f(x+h)f(x))
Δhf˜p=limnΔhσn(f˜)psupnσn(f˜)phCpfph,
we conclude that f˜Lip(1,p). This means that there is a constant C such that
sup0<ht(12πππ|f˜(x+h)f˜(x)|pdx)1/pCt.
But it is known that (see [4, p. 180]), for 1<p<, f˜Lip(1,p) if and only if there exists gXp 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=1k(akcos(kx)+bksin(kx)).

We know that, for any hXp(1<p<)σ˜n(h,x)=σn(h˜,x) [4, p. 92] and σn(h˜)h˜p0 as n. By taking h=f, this yields

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

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

Theorem 1

Assume1p. If f,f˜,(f˜)Xp(we assume that f andf˜are absolutely continuous) andn, then

(Iσn)(f)p3n(f˜)p.

Proof

It is known that if f is absolutely continuous, then

f(x)k=1k(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=fC, where C=a0(f)/2.

Note that (f˜)(x)=(g˜)(x)k=1k(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+1k=1nj=1kxkp=1n+1k=1n(n+1k)xkp=σn(f˜)pf˜p.

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

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

Now it follows from Proposition 2 (with K=f˜p) that there exists hXp satisfying

h1n+1k=1nj=1k1jxjp=hσn(g)p3f˜pn.

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

Remark 1

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

(f˜)(x)=k=0nk(akcos(kx)+bksin(kx)),
while
(Iσn)(f,x)=1n+1k=1nk(akcos(kx)+bksin(kx))=1n+1(f˜)(x).
Therefore,
n(Iσn)(f)p(f˜)p=nn+1
and
supnsupfXp,fconstn(Iσn)(f)p(f˜)p1.

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

Theorem 2

Assume1p. IfnandfXp, then

12fσn(f)pK˜(f,3n+1)p4fσn(f)p,
where
K˜(f,t)p=inf{fgp+t(g˜)p:g,g˜Xp,g˜AC,(g˜)Xp}.

Moreover, if1<p<, there exists a constantCpsuch that, for eachfXp,

12Cpfσn(f)pω1(f,3n+1)p8Cpfσn(f)p.

Proof

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

(σnI)(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)(Iσn)(σnf)p=(n+1)σn(fσn(f))p(n+1)fσn(f)p.
Hence,
K˜(f,3n+1)pfσn(f)p+3n+1(σ˜n(f))p4fσn(f)p.

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

fσn(f)pfgσn(fg)p+gσn(g)p2fgp+3n(g˜)p2fgp+6n+1(g˜)p
and this yields fσn(f)p2K˜(f,3/(n+1))p.

If 1<p<, hLp 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 Cp1 such that

g˜pCpgpandgp=h˜pCphp=Cpg˜p.
Therefore,
1CpK˜(f,t)pinf{fgp+tgp:gXp,gAC,gXp}CpK˜(f,t)p.
But it is known that (see [7, pp. 175–177])
12ω1(f,t)pinf{fgp+tgp:gXp,gAC,gXp}2ω1(f,t)p.
Hence,
14Cpfσn(f)p12CpK˜(f,3n+1)pω1(f,3n+1)p2CpK˜(f,3n+1)p8Cpfσn(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<p2,cotπ2pif2<p<.
In particular, when p=2 and Cp=1, equation (5) can be written as
14fσn(f)2ω1(f,3n+1)28fσn(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(1p) 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.

    • Crossref
    • Export Citation
  • [2]

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

    • Crossref
    • Export Citation
  • [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.

    • Crossref
    • Export Citation
  • [4]

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

  • [5]

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

  • [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.

  • [7]

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

  • [8]

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

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

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

    • Crossref
    • Export Citation
  • [2]

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

    • Crossref
    • Export Citation
  • [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.

    • Crossref
    • Export Citation
  • [4]

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

  • [5]

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

  • [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.

  • [7]

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

  • [8]

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

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