# On the Approximation of Generalized Lipschitz Function by Euler Means of Conjugate Series of Fourier Series

Authors: Jitendra Kumar Kushwaha
Journal: The Scientific World Journal
 (2013)

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## Abstract

Approximation  theory is a very important field which has various applications in pure and applied mathematics. The present study deals with a new theorem on the approximation of functions of Lipschitz class by using Euler’s mean of conjugate series of Fourier series. In this paper, the degree of approximation by using Euler’s means of conjugate of functions belonging toLip(ξ(t),p) class has been obtained. Lipα and Lip(α,p) classes are the particular cases of Lip(ξ(t),p) class. The main result of this paper generalizes some well-known results in this direction.


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## Body

## 1. Introduction and Definitions

Letfbe periodic with period2πand integrable in the sense of Lebesgue. The Fourier series associated withfat the pointxis given by
(1)f(x)~12a0+∑n=1∞(ancosnx+bnsinnx)=∑n=0∞An(x)
with partial sumssn(f;x). The conjugate series of (1) is given by
(2)Bn(x)=∑n=1∞(ansinnx-bncosnx)
with partial sumss~n(f;x). Throughout this paper, we call (2) as conjugate series of Fourier series of functionf. Iffis Lebesgue integrable, then
(3)f~(x)=-12π∫0πψ(t)cot(t2)dt=-12πlimε→0∫επψ(t)cot(t2)dt
exists for almost allx(Hardy [1], page 131).f~(x)is called the conjugate function off(x).A functionf∈Lipαif
(4)|f(x+t)-f(x)|=O(|t|α)for0<α≤1.f
∈
Lip
(
α
,
p
),p>1consider that if
(5){∫02π|f(x+t)-f(x)|pdx}1/p=O(|t|α),0<α≤1,p≥1(Definition5.38of Chandra [2]).Given a positive increasing functionξ(t),f∈Lip(ξ(t),p),
(6)(∫02π|(f(x+t)-f(x))|pdx)1/p≤M(ξ(t)t-1/p),p>1,
whereMis a positive number independent ofxandt.In caseξ(t)=tα, thenLip(ξ(t),p)coincides withLip(α,p). Ifp→∞inLip(α,p), then it coincides withLipα.L
∞-norm of a functionf:R→Ris defined by
(7)∥f∥∞=sup{|f(x):x∈R|}.L
p-norm is defined by
(8)∥f∥p=(∫02π|f(x)|pdx)1/p,p≥1.The degree of approximation of a functionf:R→Rby a trigonometric polynomialtnof ordernunder sup norm ∥∥∞ is defined by ([1], page 114-115])
(9)∥tn-f∥∞=sup{|tn(x)-f(x)|:x∈R},
andEn(f)of a functionf∈Lpis given by
(10)En(f)=min∥tn-f∥p.Let{Sn}be the sequence of partial sums of the given series∑n=0∞un. Then, forq>0, the Euler (E,q) means of{Sn}are defined to be
(11)Wn=(1+q)-n∑k=0n(nk)qn-kSk.The series is said to be Euler (E,q) summable toSprovided that the sequence{Wn}converges toSasn→+∞.We write(12)ψ(t)=f(x+t)-f(x-t),σ~n(f;x)=(1+q)-n∑k=0n(nk)qn-kS~k,S~(t)=∑k=0n(nk)qn-kcos(k+12)t,R(t)=sin{t2+ntan-1(sintq+cost)}.


## 2. Main Theorem

Hardy [1] established a theorem on (C,α), (α>0) summability of the series. Harmonic summability is weaker than (C,α) summability. Iyengar [3] proved a theorem on harmonic summability of a Fourier series. The result of Iyengar [3] has been generalized by several researchers like Siddiqi [4], Pati [5], Lal and Kushwaha [6], and Rajagopal [7], for Nörlund means.Alexits [8] proved the following theorem concerning the degree of approximation of a functionf∈Lipαby the (C,δ) means of its Fourier series.Theorem A.
If a periodic functionf∈Lipα,0<α≤1, then the degree of approximation of the (C,δ) means of its Fourier series for0<α<δ≤1is given by
(13)max0≤x≤2π|f(x)-σnδ(x)|=O(1nα)
and for0<α≤δ≤1is given by
(14)max0≤x≤2π|f(x)-σnδ(x)|=O(lognnα),
whereσnδ(x)are the (C,δ) means of the partial sums of (2).Later on, Hölland et al. [9] extended Theorem A to functions belonging toC*[0,2π], the class of2π-periodic continuous functions on [0,2π], using Nörlund means of Fourier series. Their theorem is as follows.Theorem B.
Ifw(t)is the modulus of continuity off∈C*[0,2π], then the degree of approximation offby the Nörlund means of the Fourier series for f is given by
(15)En=max0≤t≤2π|f(t)-Tn(t)|=O(1pn∑k=1npkw(1/k)k),
whereTnare the(N,pn)means of Fourier series off.Hölland et al. [9] have shown that Theorem B reduces to Theorem A if we deal with Cesàro means of orderδand consider a functionf∈Lipα,0<α≤1. Working in same direction we prove the following theorem.Theorem 1.
Iff:R→Ris a2πperiodic, Lebesgue integrable and belonging toLip(ξ(t),p)for>1and if
(16){∫01/n(t|ξ(t)|t1/p)pdt}1/p=O(ξ(1n)),(17){∫1/nπ(|ξ(t)|t1/p+2)pdt}1/p=O(ξ(1n)n)
conditions (16) and (17) hold uniformly inx, then degree of approximation off~(x), conjugate off∈Lip{ξ(t),p}, by Euler (E,q) mean
(18)σ~n(f;x)=(1+q)-n∑k=0n(nk)qn-kS~k,
of the conjugate series (2) is given by
(19)∥σ~n(f;x)-f~(x)∥p=O(ξ(1n)(n)1/2p).In order to prove our theorem, we need the following lemma.Lemma 2.
If0<t≤π, then
(20)(1+q)-n(1+q2+2qcost)n/2≤e-2qt2n/{π(1+q)}2.Proof.
We have(21)(1+q)-2(1+q2+2qcost)=1-4qsin2(t/2)(1+q)2≤1-4qt2π2(1+q)2≤e-4qt2/{π(1+q)}2,
sinceex(1-x)<1when0<x<1. Therefore,
(22)(1+q)-n(1+q2+2qcost)n/2≤e-2qt2n/{π(1+q)}2.Proof of Theorem1.
Thekthpartial sum of the conjugate series of the Fourier series (2) is given by
(23)S~k(f;x)=-12π∫0πcot(t2)ψ(t)dt+12π∫0πcos(k+1/2)tsin(t/2)ψ(t)dtS~k(f;x)-(-12π∫0πcot(t2)ψ(t)dt)=12π∫0πcos(k+1/2)tsin(t/2)ψ(t)dt.
Taking Euler (E,q) means, we get
(24)[(1+q)-n∑k=0n(nk)qn-k{S~k(f;x)-f~(x)}]dt=1π(1+q)n∫0πψ(t)sin(t/2)×{∑k=0n(nk)qn-kcos(k+12)t}σ~n(f;x)-f~(x)=1π(1+q)n∫0πψ(t)sin(t/2)S~(t)dt=1π(1+q)n[∫01/n+∫1/nπ]ψ(t)sin(t/2)S~(t)dt=K~1(x)+K~2(x),(25)S~(t)(1+q)n≤1(1+q)n×|∑k=0n(nk)qn-kei(k+1/2)t|=|q+eit|n(1+q)n=(1+q2+2qcost)n/2(1+q)n≤e-2qt2n/{π(1+q)}2,
using Lemma 2.
Clearly,(26)|ψ(x+t)-ψ(x)|≤|f(u+x+t)-f(u+x)|+|f(u+x)-f(u-x-t)|.
Hence, by Minkowski’s inequality,
(27){∫02π|(ψ(x+t)-ψ(x))|pdx}1/p≤{∫02π|(f(u+x+t)-f(u+x))|pdx}1/p+{∫02π|f(u+x)-(f(u-x-t))|pdx}1/p=O(ξ(t)).
Then,f∈Lip(ξ(t),p)⇒ψ∈Lip(ξ(t),p).
Using Hölder’s inequality,ψ(t)∈Lip(ξ(t),p), condition (16),sint≥(2t/π), lemma, and second mean value theorem for integrals, we have
(28)|K~1(x)|=O{∫01/n(|ψ(t)|)pdt}1/p×{∫01/n((1+q)-nS~(t)sin(t/2))p′dt}1/p′,
where
(29)p′=pp-1|K~1(x)|=O{∫01/n(|ξ(t)t1/p|)pdt}1/p×{∫01/n(e-2qt2n/{π(1+q)}2|sin(t/2)|)p′dt}1/p′=O(ξ(1n)){∫01/nt-p′dt}1/p′=O(ξ(1n)(n)1/2p).
Now,
(30)|K~2(x)|=O[∫1/nπ|ψ(t)|sin(t/2)(1+q)-n(1+q2+2qcost)n/2|R(t)|dt]=O{∫1/nπ|ψ(t)|sin(t/2)∫1/nπ|ψ(t)|sin(t/2)(1+q)-n×(1+q2+2qcost)n/2dt∫1/nπ|ψ(t)|sin(t/2)}=O{∫1/nπ|ψ(t)|sin(t/2)e-2qt2n/{π(1+q)}2dt}=O[{∂∂t(e-2qt2n/{π(1+q)}2)}1n∫1/nπψ(t)sin(t/2)×{∂∂t(e-2qt2n/{π(1+q)}2)}dt{∂∂t(e-2qt2n/{π(1+q)}2)}].
Using Hölder’s inequality,ψ(t)∈Lip(ξ(t),p), and condition (17), we have
(31)|K~2(x)|=O[1n∫1/nπ(ξ(t)t1/p+2)pdt]1/p×[∫1/nπ{∂∂t(e-2qt2n/{π(1+q)}2)}p′dt]1/p′=O(ξ(1n)(n)1/2p).
Combining (24) with (31), we have
(32)∥σ~n(f;x)-f~(x)∥p=O(ξ(1n)(n)1/2p),
which completes the proof of the theorem.


## 3. Corollaries

The following corollaries may be derived from our theorem.Corollary 3.
Ifξ(t)=tα, then the degree of approximation of a functionf~(x), conjugate off∈Lip(α,p),1/p<α<1, by Euler’s means(E,q)of the conjugate series of the Fourier series (2) is given by
(33)∥σ~n(f;x)-f~(x)∥p=O(1n(αp-1)/2p).Corollary 4.
Ifp→∞in Corollary 3, then, for0<α<1,
(34)∥σ~n(f;x)-f~(x)∥∞=O(1nα/2).



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