Fourier Approximation of Functions Conjugate to
the Functions Belonging to Weighted Lipschitz
Class
Uaday Singh and Shailesh Kumar Srivastava
Abstract—The study of error estimates of periodic functions inLp(p≥1)-spaces through Fourier series, although is an old problem and known as Fourier approximation in the existing literature, has been of a growing interests over the last four decades. The most common methods used for the determination of the degree of approximation of periodic functions are based on the minimization of the Lp-norm of f(x)−T
n(x), where Tn(x) is a trigonometric polynomial of degree n, and called the approximant of f. The degree of approximation of f, so obtained depends heavily on p. In this paper, we obtain the degree of approximation of f˜, conjugate to the function f belonging to weighted Lipschitz class W(Lp, ξ(t))
by a trigonometric polynomial generated by the product matrix means of the conjugate Fourier series of f. The degree of approximation obtained in our theorems of this paper is sharper than others and free from p.Some corollaries have also been deduced from our theorems.
Index Terms—Fourier approximation, W(Lp, ξ(t)) -class, C1.T means, periodic functions,b
n,n−k≥0.
I. INTRODUCTION
F
OR a2π-periodic functionf ∈Lp:=Lp[0,2π], p≥1, integrable in the sense of Lebesgue, letsn(f;x) := a0
2 +
n
X
k=1
(akcoskx+bksinkx), n∈N
ands0(f;x) =
a0
2 , (1)
denote the(n+ 1)th partial sums, called trigonometric poly-nomials of degree (or order) n, of the Fourier series of f. The conjugate series of the Fourier series of f is defined by P∞
k=1(aksinkx−bkcoskx) and its n
th
partial sum is defined as
˜
sn(f;x) := n
X
k=1
(aksinkx−bkcoskx), n∈N
ands˜0(f;x) = 0. (2)
The conjugate of f denoted byf˜is defined as
˜
f(x) =− 1
2πǫlim→0
Z π
ǫ
ψx(t) cot(t/2)dt, (3)
Manuscript received March 23, 2013; revised April 09, 2013. This research was supported by the Council of Scientific and Industrial Research, (CSIR), New Delhi, India in the form of fellowship to the second author.
U. Singh and S. K. Srivastava are with the Department of Mathematics, In-dian Institute of Technology Roorkee, Roorkee -247667 (India) (e-mails: [email protected], ph.: +919453551769, [email protected], ph.: +919760197682).
whereψx(t) =f(x+t)−f(x−t)[1, p.131].
TheLp-norm of f ∈Lp[0,2π]is defined by
kfkp=
1
2π
Z 2π
0
|f(x)|pdx
1/p
(1≤p <∞)and
kfk∞= sup x∈[0,2π]
|f(x)|.
We determine the degree of approximation (error estimates)
En(f)off ∈Lp-space bynth degree trigonometric
polyno-mialsTn(x)given by
En(f) = min
Tn kf(x)−Tn(x)kp.
The Tn(x) is called Fourier approximant of f, and this
method of approximation is called Fourier approximation. In this paper , we consider the following function classes:
Lipα:={f : [0,2π]→R:|f(x+t)−f(x)|=O(tα)},
Lip(α, p) :={f ∈Lp[0,2π] :kf(x+t)−f(x)kp=O(tα)},
Lip(ξ(t), p) := {f ∈ Lp[0,2π] : kf(x+t)−f(x)kp =
O(ξ(t))}, W(Lp, ξ(t)) := {f ∈ Lp[0,2π] : k(f(x+t)−
f(x)) sinβ(x/2)kp =O(ξ(t))}, wherep ≥1, 0 < α ≤1,
β ≥0,t >0 andξ(t)is a positive increasing function of t
[2, 3].
It is important to note that the increasing function ξ(t) in the definition ofW(Lp, ξ(t))-class is not the same as in the
definition ofLip(ξ(t), p)-class. Theξ(t)inLip(ξ(t), p)-class depends ont only, whereas inW(Lp, ξ(t))-class it depends
on t and β [3]. In particular, if we take ξ(t) =tβψ(t) for
β ≥ 0 and some positive increasing function ψ(t), then
W(Lp, ξ(t))-class defined above reduces to W′
(Lp, ψ(t))
-class defined by Khan [3]. We also note that
Lipα⊆Lip(α, p)⊆Lip(ξ(t), p)⊆W(Lp, ξ(t))[2,4].
Let T ≡ (an,k) be a lower triangular matrix with
non-negative entries such thatan,−1= 0, An,k=Pnr=kan,r and
An,0= 1, n∈N0.The sequence-to-sequence transformation
˜
tn(f;x) := n
X
k=0
an,k˜sk(f;x), n∈N0,
defines the matrix means of {˜sn(f;x)}. The conjugate
Fourier series of the functionf is said to beT-summable to
s, if˜tn(f;x)→sas n→ ∞.
By superimposingC1-summability (Ces`aro summability of
Thus theC1.T means of{˜s
n(f;x)}denoted by˜tC
1.T
n (f;x)
are given by
˜
tCn1.T(f;x) := (n+ 1)−1 n
X
r=0
r
X
k=0
ar,k˜sk(f;x)
, n∈N0.
(4) If ˜tC1.T
n (f;x)→s1 as n→ ∞, then the conjugate Fourier
series of f is said to be C1.T - summable to the sum s 1.
The regularity of methods C1 and T implies regularity of
methodC1.T.
We also write
(C1.T)n(t) =
1 2π(n+ 1)
n
X
r=0 r
X
k=0
ar,r−k
cos(r−k+ 1/2)t
sin(t/2) ,
bn,n−k = ∆nan,n−k=an,n−k−an+1,n+1−k andτ= [1/t],
the integral part of1/t.
In the last four decades, many researchers have been approx-imated the functionf ,˜ conjugate of a functionf belonging to
Lipα, Lip(α, p), Lip(ξ(t), p)andW(Lp, ξ(t))-classes with
p ≥ 1, by different summability means of the conjugate Fourier series of f and obtained the error of approximation
En( ˜f), which depends heavily on p [2, 5–7]. A detailed
review of the previous work done in this direction can be seen in our recent paper [2], in which authors have determined the degree of approximation off ,˜ conjugate off ∈W(Lp, ξ(t))
by the Hausdorff means of the conjugate Fourier series of
f and improved previous results in the light of Kranz et al. [8], Łenski and Szal [9] and Mishra et al. [4]. The degree of approximation of f˜ obtained in [2] is of order
(n+ 1)β+1/pξ(1/(n+ 1)), which clearly depends onp, and
leads to an open question whether this error of approximation can be made independent ofp.
II. MAINRESULTS
The importance of Fourier approximation discussed in [5] and the observation mentioned above motivate us to study further the degree of approximation of f˜. In this paper, we obtain the degree of approximation of conjugate of functions belonging to the Lipschitz class W(Lp, ξ(t)) (p ≥ 1), by
a general summability method, i.e., C1.T means of their
conjugate Fourier series. This work is an attempt to make degree of approximation free from p. More precisely, we prove the following.
Theorem 1. Let T ≡(an,k)be a lower triangular regular
matrix with non-negative and non-decreasing (with respect
tok) entries which satisfy
bn,n−k≥0f or 0≤k≤n. (5)
Then the degree of approximation of f˜, conjugate of a2π
-periodic functionf belonging to the weighted Lipschitz class
W(Lp, ξ(t)), withp >1and 0≤β <1/pby C1.T means
of its conjugate Fourier series is given by
k˜tCn1.T(f;x)−f˜(x)kp=O (n+ 1)βξ(1/(n+ 1))
, (6)
provided a positive increasing function ξ(t) satisfies the
following conditions:
ξ(t)/tβ+1−σ is non-decreasing,
(7)
(
Z π/(n+1)
0
t−σ|ψ
x(t)|sinβ(t/2)
ξ(t)
!p
dt
)1/p
=O((n+ 1)σ−1/p),
(8)
forβ < σ <1/p,
ξ(t)/t is non-increasing, (9)
( Z π
π/(n+1)
t−δ |ψ x(t)|
ξ(t)
p
dt
)1/p
=O((n+ 1)δ−1/p),
(10)
where δ is an arbitrary number such that 1/p < δ < β+
1/p and p−1+q−1 = 1. The conditions (8) and (10) hold
uniformly inx.
The conditions (8) and (10) can be verified by using the fact thatψx(t)∈W(Lp, ξ(t))andψx(t)/ξ(t)is a bounded
function. The condition (10) above is improved version of condition (14) of [2].
Note 1: Condition (9) implies that ξ(π/(n+ 1))/(π/(n+ 1))≤ξ(1/(n+1))/(1/(n+1)),i.e.,(n+1)/πξ(π/(n+1))≤
(n+ 1)ξ(1/(n+ 1)).
Lemmas
We need the following lemmas for the proof of our theorem.
Lemma 1. For0< t≤π/(n+ 1),(C1.T)
n(t) =O(1/t).
Proof. Using |cost| ≤ 1 andsin(t/2) ≥ t/π for 0 < t ≤
π/(n+ 1),we have
|(C1.T)n(t)|
= (2π(n+ 1))−1×
n
X
r=0 r
X
k=0
ar,r−k(cos(r−k+ 1/2)t)/(sint/2)
≤ (2π(n+ 1))−1× n
X
r=0 r
X
k=0
ar,r−k|(cos(r−k+ 1/2)t)/(sint/2)|
≤ (2π(n+ 1))−1 n
X
r=0 r
X
k=0
ar,r−k 1/(t/π)
= O((n+ 1)t)−1 n
X
r=0 r
X
k=0
ar,r−k
!
=O(1/t).
Lemma 2. If {an,k} is non-negative and non-decreasing
(with respect tok)sequence satisfying(5), then
|(C1.T)
n(t)|=O t−2/(n+ 1), forπ/(n+ 1)< t≤π.
have
|(C1.T)n(t)|
= (2π(n+ 1))−1×
n X r=0 r X k=0
ar,r−k(cos(r−k+ 1/2)t)/sin(t/2)
= O(t(n+ 1))−1
Re n X r=0 r X k=0
ar,r−kei(r−k+1/2)t
= O(t(n+ 1))−1
n X r=0 r X k=0
ar,r−kei(r−k)t
.
Following [10, pp. 445-446], we have
| n X r=0 r X k=0
ar,r−kei(r−k)t|
≤ τ X r=0 r X k=0
ar,r−kei(r−k)t
+ n X r=τ+1 τ X k=0
ar,r−kei(r−k)t
+ n X r=τ+1 r X k=τ+1
ar,r−kei(r−k)t
≤ K1+K2+K3, say.
Now
K1 ≤ τ X r=0 r X k=0
ar,r−k
e
i(r−k)t
≤ τ
X
r=0
Ar,0= (τ+ 1) =O(t−1).
Using Abel’s transformation after changing the order of summation in K2, we have
K2 =
τ X k=0 n X r=τ+1
ar,r−kei(r−k)t
= τ X k=0
n−1 X
r=τ+1
br,r−k r
X
v=0
ei(v−k)t
+an,n−k n
X
v=0
ei(v−k)t−a
τ+1,τ+1−k τ
X
v=0
ei(v−k)t
=O(t−1) τ
X
k=0 n−1
X
r=τ+1
br,r−k+an,n−k+aτ+1,τ+1−k
!
= O(t−1) τ
X
k=0 n−1
X
r=τ+1
(ar,r−k−ar+1,r+1−k)
+an,n−k+aτ+1,τ+1−k)
= O(t−1) τ
X
k=0
(2aτ+1,τ+1−k+an,n−k+an+1,n+1−k)
= O(t−1) τ
X
k=0
(aτ,τ−k+an,n−k)
= O(t−1) τ
X
k=0
aτ,τ−k+ n
X
k=0
an,n−k
!
= O(t−1) (A
τ,0+An,0) =O(t−1),
in view ofbr,r−k =ar,r−k−ar+1,r+1−k≥0for0≤k≤r.
Again using Abel’s transformation inK3, we have
K3 =
n X r=τ+1
r−1 X
k=τ+1
∆kar,r−k k
X
v=0
ei(r−v)t
+ar,0 r
X
v=0
ei(r−v)t−a r,r−τ−1
τ
X
v=0
ei(r−v)t
=O(t−1) n
X
r=τ+1
r−1 X k=τ+1
ar,r−k−ar,r−k−1
+ar,0+ar,r−τ−1
=O(t−1) n
X
r=τ+1
(ar,r−τ−1)
=O(t−1)
aτ+1,0+aτ+2,1+aτ+3,2+...+an,n−τ−1
=O(t−1)a
τ+1,0+aτ+1,1+aτ+1,2+...+aτ+1,n−τ−1
=O(t−1)
n−τ−1
X
r=0
aτ+1,r=O(t−1)(1) =O(t−1),
in view ofbr,r−k =ar,r−k−ar+1,r+1−k≥0for 0≤k≤r
andan,k ≤an,k+1.
CollectingK1, K2 andK3, we get
|(C1.T)
n(t)|=O t−2/(n+ 1).
Proof of Theorem 1.The integral representation of˜sn(f;x)
defined in (2) is given by
˜
sn(f;x) =−
1
π
Z π
0
ψx(t)
cos(t/2)−cos(n+ 1/2)t
2 sin(t/2)
dt
and,
˜
sn(f;x)−f˜(x) =
1 2π
Z π
0
ψx(t)
cos(n+ 1/2)t
sin (t/2) dt.
Therefore,
˜
tCn1.T(f;x)−f˜(x)
= 1
n+ 1
n X r=0 r X k=0 ar,k h ˜
sk(f;x)−f˜(x)
i
=
Z π
0
ψx(t)(2π(n+ 1))−1× n X r=0 r X k=0
ar,r−k
cos(r−k+ 1/2)t
sin(t/2) dt =
Z π/(n+1)
0
ψx(t)(C1.T)n(t)dt
+
Z π
π/(n+1)
ψx(t)(C1.T)n(t)dt
= I1+I2,say. (11)
Using H¨older’s inequality, conditions (7), (8), sin(t/2) ≥
t/π, Lemma 1, the mean value theorem for integrals and
|I1|= lim
ǫ→0
Z π/(n+1)
ǫ
t−σ(ψ
x(t) sinβ(t/2)/ξ(t)) ×
(ξ(t)(C1.T)n(t)/t−σsinβ(t/2))
dt
≤
"
Z π/(n+1)
0
t−σ|ψ
x(t)|sinβ(t/2)/ξ(t)
p
dt
#1/p
×
"
lim
ǫ→0
Z π/(n+1)
ǫ
ξ(t)|(C1.T)
n(t)|/t−σsinβ(t/2) q
dt
#1/q
= O((n+ 1)σ−1/p)×
"
lim
ǫ→0
Z π/(n+1)
ǫ
ξ(t)/(t1−σ.sinβ(t/2))
q
dt
#1/q
= O((n+ 1)σ−1/p)×
"
lim
ǫ→0
Z π/(n+1)
ǫ
ξ(t)/(tβ+1−σ)
q
dt
#1/q
= O((n+ 1)σ−1/p)(n+ 1)β+1−1/q−σξ(π/(n+ 1))
= O((n+ 1)β ξ(π/(n+ 1)). (12)
Again using Lemma 2, H¨older’s inequality and
(sin(t/2))−1≤π/tfor 0< t≤π, we have |I2|=
" Z π
π/(n+1)
|ψx(t)|O t−2/(n+ 1)dt
#
=O
" Z π
π/(n+1)
t−2|ψ
x(t)|/(n+ 1)dt
#
=O
Z π
π/(n+1)
t−δ|ψ
x(t)|sinβ(t/2)
(n+ 1)ξ(t)
t−1ξ(t)
t−δtsinβ(t/2)dt
!
=O
(
1
n+ 1
Z π
π/(n+1)
t−δ|ψ x(t)|
ξ(t)
p
dt
)1/p
×
( Z π
π/(n+1)
t−1ξ(t)
t−δ+β+1
q
dt
)1/q
=O
(n+ 1)δ−1−1/pξ
π
n+ 1
n+ 1
π
Z π
π/(n+1)
t−(β+1−δ)qdt
!1/q
=Oh(n+ 1)δ−1/pξ(π/(n+ 1))(n+ 1)β+1−δ−1/qi =O[(n+ 1)βξ(π/(n+ 1))], (13) in view of (9), (10), the mean value theorem for integrals,
1/p < δ < β+ 1/pandp−1+q−1= 1.
Collecting (11)-(13), we get
|˜tC1.T
n (f;x)−f˜(x)|=O[(n+ 1)βξ(π/(n+ 1))]. (14)
Fianally from (14), we easily get
kt˜Cn1.T(f;x)−f˜(x)kp=O (n+ 1)βξ(1/(n+ 1))
, (15)
in view of Note 1. This completes the proof of Theorem 1.
As mentioned in Remark 2 of [2], the above proof is not valid forp= 1.Therefore, forp= 1,we have the following theorem:
Theorem 2. Let T ≡ (an,k) be the same as in Theorem
1. Then the degree of approximation of f˜, conjugate of a
2π-periodic function f belonging to the weighted Lipschitz
class W(L1, ξ(t)), with 0 ≤β < 1 by C1.T means of its
conjugate Fourier series is given by
k˜tC1.T
n (f;x)−f˜(x)k1=O (n+ 1)βξ(1/(n+ 1))
, (16)
provided a positive increasing functionξ(t)satisfies
condi-tions (7) to (10) of Theorem 1 for p= 1, β < σ <1 and
1< δ < β+ 1.
Proof of Theorem 2.Following the proof of Theorem 1, for
p= 1, i.e., q=∞, we have
I1 =
Z π/(n+1)
0
t−σ|ψ
x(t)|sinβ(t/2)
ξ(t)
!
dt×
ess sup
0<t≤π/(n+1)
ξ(t)|(C1.T) n(t)|
t−σsinβ(t/2)
=
Z π/(n+1)
0
t−σ|ψ
x(t)|sinβ(t/2)
ξ(t)
!
dt×
ess sup
0<t≤π/(n+1)
ξ(t)
t−σ+1sinβ(t/2)
= O((n+ 1)σ−1)ess sup 0<t≤π/(n+1)
ξ(t)
tβ−σ+1
= O((n+ 1)σ−1)
ξ(π/(n+ 1)) (π/(n+ 1))β−σ+1
= O((n+ 1)β ξ(π/(n+ 1)). (17) in view of conditions (7) and (8) forp= 1.
I2 = O
(
1
n+ 1
Z π
π/(n+1)
t−δ|ψ
x(t)|sinβ(t/2)
ξ(t) dt
)
×
ess sup
π/(n+1)≤t≤π
ξ(t)
t−δ+β+2
= O
(n+ 1)δ−2ξ π
n+ 1
(n+ 1)2+β−δ
π2+β−δ
= O[(n+ 1)βξ(π/(n+ 1))], (18) in view of (9), i.e., decreasing nature ofξ(t)/t−δ+β+2 and
(10). Collecting (17) and (18), we get
|˜tCn1.T(f;x)−f˜(x)|=O[(n+ 1)βξ(π/(n+ 1))]. (19)
Finally from (19), we easily get
k˜tC1.T
n (f;x)−f˜(x)k1=O (n+ 1)βξ(1/(n+ 1))
, (20)
in view of Note 1. This completes the proof of Theorem 2.
III. COROLLARIES
The following corollaries can be derived from our theo-rems:
1. Ifβ = 0, then for f ∈Lip(ξ(t), p)withp≥1,
k˜tCn1.T(f;x)−f˜(x)kp=O(ξ(1/(n+ 1))).
2. If β = 0, ξ(t) = tα(0 < α ≤ 1), then for f ∈
Lip(α, p)(0< α≤1),
kt˜Cn1.T(f;x)−f˜(x)kp =O (n+ 1)−α
3. Ifp→ ∞in Corollary 2, then forf ∈Lipα(0< α <1),
k˜tCn1.T(f;x)−f˜(x)k∞=O((n+ 1)−α).
Forα= 1, we can write an independent proof to obtain
k˜tCn1.T(f;x)−f˜(x)k∞=O(log(n+ 1)/(n+ 1)).
If we replace matrixT by N¨orlund matrix (Np), i.e.,an,k=
pn−k/Pn for 0 ≤ k ≤ n and an,k = 0 for k > n, where
Pn = Pnk=0pk → ∞ as n → ∞, then we get C1.Np
analogues of our theorems and corollaries of this paper.
REFERENCES
[1] A. Zygmund, Trigonometric Series, Third Edition. Cambridge University Press, London, 2002.
[2] U. Singh and S. K. Srivastava, “Approximation of con-jugate of functions belonging to weighted lipchitz class
w(lp, ξ(t))by hausdorff means of conjugate Fourier
se-ries,”J. Comput. Appl. Math., accepted for publication (in press).
[3] H. H. Khan, “A note on a theorem of Izumi,”Comm.
Fac. Sci. Math. Ankara (TURKEY), vol. 31, pp. 123–
127, 1982.
[4] V. N. Mishra, K. Khatri, and L. N. Mishra, “Product summability transform of conjugate series of Fourier series,” Int. J. Math. Math. Sci., vol. 2012, pp. 1–13, 2012.
[5] E. Z. Psarakis and G. V. Moustakides, “AnL2−based method for the design of 1-D zero phase FIR digital filters,” IEEE Transactions on Circuits and SystemsI:
Fundamental Theory And Applications, vol. 44, no. 7,
pp. 551–601, 1997.
[6] V. N. Mishra, K. Khatri, and L. N. Mishra, “Approx-imation of functions belonging to Lip(ξ(t), r)-class by (N, pn)(E, q) summability of conjugate series of
fourier series,” J. Inequal. Appl., vol. 296, pp. 1–10, 2012.
[7] S. Sonkar and U. Singh, “Degree of approximation of the conjugate of signals (functions) belonging to
Lip(α, r)-class by (C,1)(E, q) means of conjugate trigonometric Fourier series,” J. Inequal. Appl., vol. 128, pp. 1–12, 2012.
[8] R. Kranz, W. Łenski, and B. Szal, “On the degrees of approximation of functions belonging toLp( ˜ω)
β class
by matrix means of conjugate Fourier series,” Math.
Inequal. Appl., vol. 12, no. 3, pp. 717–732, 2012.
[9] W. Łenski and B. Szal, “Pointwise approximation of functions from Lp(ω)
β by linear operators of their
Fourier series,” J. Funct. Spaces Appl., vol. 2012, pp. 1–16, 2012.
[10] M. L. Mittal, “A sufficient condition for (F1)
-effectiveness of the C1.T method,” J. Math. Anal.
