1. Introduction and Definitions
Let f be periodic with period 2π and integrable in the sense of Lebesgue. The Fourier series associated with f at the point x is given by
(1)f(x)~12a0+∑n=1∞(ancosnx+bnsinnx)=∑n=0∞An(x)
with partial sums sn(f;x). The conjugate series of (1) is given by
(2)Bn(x)=∑n=1∞(ansinnx-bncosnx)
with partial sums s~n(f;x). Throughout this paper, we call (2) as conjugate series of Fourier series of function f. If f is Lebesgue integrable, then
(3)f~(x)=-12π∫0πψ(t)cot(t2)dt=-12πlimε→0∫επψ(t)cot(t2)dt
exists for almost all x (Hardy [1], page 131). f~(x) is called the conjugate function of f(x).
A function f∈
Lip
α if
(4)|f(x+t)-f(x)|=O(|t|α) for 0<α≤1.
f
∈
Lip
(
α
,
p
)
, p>1 consider that if
(5){∫02π|f(x+t)-f(x)|pdx}1/p=O(|t|α), 0<α≤1, p≥1
(Definition 5.38 of Chandra [2]).
Given a positive increasing function ξ(t), f∈
Lip
(ξ(t),p),
(6)(∫02π|(f(x+t)-f(x))|pdx)1/p≤M( ξ(t)t-1/p), p>1,
where M is a positive number independent of x and t.
In case ξ(t)=tα, then
Lip
(ξ(t),p) coincides with
Lip
(α,p). If p→∞ in
Lip
(α,p), then it coincides with
Lip
α.
L
∞
-norm of a function f:R→R is 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 function f:R→R by a trigonometric polynomial tn of order n under sup norm ∥ ∥∞ is defined by ([1], page 114-115])
(9)∥tn-f∥∞=sup{|tn(x)-f(x)|:x∈R},
and En(f) of a function f∈Lp is 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, for q>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 to S provided that the sequence {Wn} converges to S as n→+∞.
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 function f∈
Lip
α by the (C,δ) means of its Fourier series.
Theorem A.
If a periodic function f∈
Lip
α, 0<α≤1, then the degree of approximation of the (C,δ) means of its Fourier series for 0<α<δ≤1 is given by
(13)max0≤x≤2π|f(x)-σnδ(x)|=O(1nα)
and for 0<α≤δ≤1 is given by
(14)max0≤x≤2π|f(x)-σnδ(x)|=O(lognnα),
where σnδ(x) are the (C,δ) means of the partial sums of (2).
Later on, Hölland et al. [9] extended Theorem A to functions belonging to C*[0,2π], the class of 2π-periodic continuous functions on [0,2π], using Nörlund means of Fourier series. Their theorem is as follows.
Theorem B.
If w(t) is the modulus of continuity of f∈C*[0,2π], then the degree of approximation of f by 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),
where Tn are the (N,pn) means of Fourier series of f.
Hölland et al. [9] have shown that Theorem B reduces to Theorem A if we deal with Cesàro means of order δ and consider a function f∈
Lip
α, 0<α≤1. Working in same direction we prove the following theorem.
Theorem 1.
If f:R→R is a 2π periodic, Lebesgue integrable and belonging to
Lip
(ξ(t),p) for >1 and 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 in x, then degree of approximation of f~(x), conjugate of f∈
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.
If 0<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 ≤1-4qt2 π2(1+q)2 ≤e-4qt2/{π(1+q)}2,
since ex(1-x)<1 when 0<x<1. Therefore,
(22)(1+q)-n(1+q2+2qcost)n/2≤e-2qt2n/{π(1+q)}2.
Proof of Theorem 1.
The kth partial 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.