APPROXIMATION OF SIGNALS (FUNCTIONS)
BELONGING TO THE WEIGHTED W(L p ,ξ(t))-CLASS BY
LINEAR OPERATORS
M. L. MITTAL, B. E. RHOADES, AND VISHNU NARAYAN MISHRA
Received 23 May 2006; Accepted 2 October 2006
In memory of Professor Brian Kuttner (1908–1992)
Mittal and Rhoades (1999–2001) and Mittal et al. (2006) have initiated the studies of error
estimates En ( f ) through trigonometric Fourier approximations (TFA) for the situations
in which the summability matrix T does not have monotone rows. In this paper, we
determine the degree of approximation of a function f, conjugate to a periodic function
f belonging to the weighted W(L p ,ξ(t))-class (p ≥ 1), where ξ(t) is nonnegative and
increasing function of t by matrix operators T (without monotone rows) on a conjugate
series of Fourier series associated with f . Our theorem extends a recent result of Mittal
et al. (2005) and a theorem of Lal and Nigam (2001) on general matrix summability.
Our theorem also generalizes the results of Mittal, Singh, and Mishra (2005) and Qureshi
(1981-1982) for Nörlund (N p )-matrices.
Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.
1. Introduction
Let f be 2π-periodic function (signal) in L1 [−π,π]. The Fourier series associated with f
at a point x is defined by
∞
f (x) ∼
∞
a0 an cosnx + bn sinnx ≡
An (x)
+
2 n =1
n =0
(1.1)
with partial sums sn ( f ;x). The conjugate series of Fourier series (1.1) of f is given by
∞
bn cosnx − an sinnx ≡
n =1
∞
Bn (x)
(1.2)
n =1
with partial sums sn ( f ;x). Throughout this paper, we will call (1.2) as conjugate Fourier
series of function f .
Define for all n ≥ 0,
tn ( f ;x) =
n
an,k sk ( f ;x),
k =0
Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2006, Article ID 53538, Pages 1–10
DOI 10.1155/IJMMS/2006/53538
(1.3)
2
Approximation of W(L p ,ξ(t)) functions
where T ≡ (an,k ) is a linear operator represented by the infinite lower triangular matrix.
The series (1.1) is said to be T-summable to s if tn ( f ;x) → s as n → ∞. The T-operator
reduces to the Nörlund N p -operator if
an,k =
p n −k
,
Pn
= 0,
0 ≤ k ≤ n,
(1.4)
k > n,
where Pn = nk=0 pk = 0 and p−1 = 0 = P −1 . In this case, the transform tn ( f ;x) reduces to
the Nörlund transform Nn ( f ;x).
A linear operator T is said to be regular if it is limit-preserving over c, the space of
convergent sequences. Each matrix T in this paper has nonnegative entries with row sums
one. If limn→∞ an,k = 0, for each k, then T is regular.
The L p -norm is defined by
f p =
2π
0
f (x) p dx
f ∞ = supx∈[0,2π]
1/ p
,
p ≥ 1,
(1.5)
f (x),
and the degree of approximation En ( f ) is given by
En ( f ) = Minn f (x) − Tn (x) p ,
(1.6)
in terms of n, where Tn (x) is a trigonometric polynomial of degree n. This method of
approximation is called trigonometric Fourier approximation (TFA).
A function f ∈ Lip α if
f (x + t) − f (x) = O |t |α
for 0 < α ≤ 1,
(1.7)
and f (x) ∈ Lip(α, p), for 0 ≤ x ≤ 2π, if
ω p (t; f ) =
2π
0
f (x + t) − f (x) p dx
1/ p
= O |t |α ,
0 < α ≤ 1, p ≥ 1.
(1.8)
Given a positive increasing function ξ(t) and p ≥ 1, f (x) ∈ Lip(ξ(t), p) if
ω p (t; f ) =
2π
0
f (x + t) − f (x) p dx
1/ p
= O ξ(t) ,
(1.9)
and f (x) ∈ W(L p ,ξ(t)) if
2π
0
f (x + t) − f (x) sinβ x p dx
1/ p
= O ξ(t) ,
(β ≥ 0).
(1.10)
If β = 0, our newly defined class W(L p ,ξ(t)) coincides with the class Lip(ξ(t), p). We
observe that
Lipα ⊆ Lip(α, p) ⊆ Lip ξ(t), p ⊆ W L p ,ξ(t)
for 0 < α ≤ 1, p ≥ 1.
(1.11)
M. L. Mittal et al. 3
We write
Wn (r) =
r
(k + 1)Δk an,n−k ,
k =0
π f(x) =
π
Wn = Pn−1 0
ψ(t)cot
Vn,k =
τ=
0 ≤ r ≤ n,
J(n,t) =
π
t
dt = limh→0
2
h
n
k p k − p k −1 ,
ψ(t) = ψx (t) = 2−1 f (x + t) − f (x − t) ,
K(n,t) =
1
1
, the integral part of ,
t
t
k =1
an,n−k cos n − k +
k =0
t
dt,
2
ψ(t)cot
(n − k + 1)an,k
,
An,k
n
An,k =
n
1
t,
2
an,r ,
r =k
J(n,t)
,
sin(t/2)
Δk an,k = an,k − an,k+1 .
(1.12)
Furthermore, C denotes an absolute positive constant, not necessarily the same at each
occurrence.
2. Previous results
Qureshi [12] has proved a theorem on the degree of approximation of a function f(x),
conjugate to a periodic function f (x) with period 2π and belonging to the class Lipα,
for 0 < α < 1, by N p -means of its conjugate Fourier series. He has proved the following
theorem.
Theorem 2.1 [12]. If the sequence { pn } satisfies the conditions
(i)n pn < C Pn ,
(2.1)
(ii)Wn < C,
then the degree of approximation of a function f(x), conjugate to a periodic function f (x)
with period 2π and belonging to the class Lipα, 0 < α < 1, by N p -means of its conjugate
Fourier series, is given by
n
Pk
−1
f(x) − tn (x) = O Pn
,
α+1
k =1
k
(2.2)
where tn (x) is the N p -mean of conjugate Fourier series (1.2).
Remark 2.2. Qureshi [12] has taken pn ≥ 0, so conditions (2.1) can be stated without
modulus sign.
Generalizing Theorem 2.1 of Qureshi [12], many interesting results have been proved
by various investigators such as Qureshi [13, 14], Lal and Nigam [2], Mittal et al. [8, 9] for
functions of various classes (defined above) using N p -matrices and general summability
matrices.
4
Approximation of W(L p ,ξ(t)) functions
Qureshi [13, 14] has extended his Theorem 2.1 for the functions of classes Lip(α, p)
and Weighted, that is, W(L p ,ξ(t))(p ≥ 1) respectively, using monotonicity on the elements of N p -matrix. For the W(L p ,ξ(t))-class, he proved the following theorem.
Theorem 2.3 [14]. If a 2π-periodic function f belongs to the class W(L p ,ξ(t)), then its
degree of approximation by Nörlund means of a conjugate Fourier series is given by
1
β+1/ p
f(x) − tn (x) p = O n
ξ
n
(2.3)
provided that ξ(t) satisfies the conditions
π/n t |ψ(t)|
ξ(t)
0
π π/n
p
sinβp t dt
t −δ |ψ(t)|
ξ(t)
p
1/ p
1/ p
dt
=O
1
,
n
= O nδ ,
(2.4)
(2.5)
where δ is an arbitrary number such that q(1 − δ) − 1 > 0. Conditions (2.4) and (2.5) hold
uniformly in x and
π/n 0
ξ(t)
t 2+β
q
1/q
dt
1
= O nβ+1+1/ p ξ
,
n
(2.6)
where p−1 + q−1 = 1 and 1 ≤ p ≤ ∞.
Remark 2.4. Qureshi [12–14] has used monotonicity on the generating sequence { pn } in
the proof of his theorems but has not mentioned it explicitly in the statement of these
theorems.
Remark 2.5. Condition (2.1)(i) can be dropped in Theorem 2.1, as condition (2.1)(ii)
implies condition (2.1)(i) [1, page 16].
Recently Mittal et al. [8] have generalized Theorem 2.3 by taking semimonotonicity
on the generating sequence { pn } and also by dropping the condition (2.6). They proved
the following theorem.
Theorem 2.6 [8]. The degree of approximation of a function f, conjugate to a2π-periodic
function f belonging to weighted class W(L p ,ξ(t))(p ≥ 1) by N p -means of its conjugate
Fourier series (1.2), is given by
1
f(x) − tn (x) p = O nβ+1/ p ξ
,
n
(2.7)
provided that { pn } satisfies (2.1)(ii) and ξ(t) satisfies conditions (2.4) and (2.5) uniformly
in x and
ξ(t)
is nonincreasing in t,
t
(2.8)
where δ is an arbitrary number such that 0 = δq + 1 < q, such that p−1 + q−1 = 1, 1 ≤ p ≤
∞, and tn (x) is the same as in Theorem 2.1.
M. L. Mittal et al. 5
Recently, Mittal et al. [9] extended Theorem 2.1 [12] for functions of Lip(ξ(t), p)(p ≥
1)-class to matrix summability using semimonotonicity on the sequence {an,k }, which in
turn generalizes a result of Lal and Nigam [2]. They proved the following theorem.
Theorem 2.7 [9]. Let T ≡ (an,k ) be an infinite regular triangular matrix with nonnegative
entries such that
Wn (r) = O An,n−r ,
0 ≤ r ≤ n,
(2.9)
then the degree of approximation of a function f(x), conjugate to a 2π-periodic function
f (x) belonging to Lip(ξ(t), p)-class, by using a matrix operator on its conjugate Fourier
series, is given by
1
f(x) − t n (x) p = O n1/ p ξ
,
n
(2.10)
provided that ξ(t) is nonnegative, increasing, and satisfies conditions (2.4), (2.5) uniformly
in x and (2.8), where δ is an arbitrary positive number such that q(1 − δ) − 1 > 0 and tn (x)
are the matrix means of the conjugate Fourier series (1.2).
3. Main result
Mittal [3], Mittal and Rhoades [4–6], and Mittal et al. [7] have obtained many interesting
results on TFA (these approximations have assumed important new dimensions due to
their wide application in signal analysis [10] in general, and in digital signal processing
[11] in particular, in view of the classical Shannon sampling theorem), using summability
methods without monotonicity on the rows of the matrix T. In this paper, we extend
Theorem 2.6 to matrix (linear) operators and generalize Theorem 2.7 for functions of
the weighted class W(L p ,ξ(t)). We prove the following theorem.
Theorem 3.1. Let T ≡ (an,k ) be an infinite regular triangular matrix with nonnegative
entries satisfying (2.9), then the degree of approximation of function f(x), conjugate to a
2π-periodic function f (x) belonging to class W(L p ,ξ(t)), p ≥ 1, by using a matrix operator
on its conjugate Fourier series, is given by
1
β+1/ p
f(x) − tn (x) = O n
ξ
n
(3.1)
provided that ξ(t) satisfies (2.4) and (2.5) uniformly in x, in which δ is an arbitrary positive
number with q(1 − δ) − 1 > 0, where p−1 + q−1 = 1, 1 ≤ p ≤ ∞ and condition (2.8) holds.
Note 3.2. In the case of the N p -transform, condition (2.9), for r = n, reduces to (2.1)(ii)
and thus Theorem 3.1 extends Theorem 2.6 to matrix summability for the weighted class
functions.
Note 3.3. Also for β = 0, Theorem 3.1 reduces to Theorem 2.7, and thus generalizes the
theorem of Lal and Nigam [2].
6
Approximation of W(L p ,ξ(t)) functions
4. Lemmas
Lemma 4.1 [3]. Let T ≡ (an,k ) be an infinite triangular matrix satisfying (2.9). Then
Vn,k = O(1),
0 ≤ k ≤ n.
(4.1)
For k = 0,
Vn,0 = O(1),
(4.2)
(n + 1)an,0 = O(1).
(4.3)
that is,
Lemma 4.2 [9]. Let T ≡ (an,k ) be an infinite triangular matrix satisfying (2.9). Then
−1
n
J(n,t) = O An,n−τ + O t −1
Δk an,n−k + an,0 .
(4.4)
k =τ
5. Proof of Theorem 3.1
It is well known that
1
sn−k (x) − f(x) = −
π
2π
f(x) − tn (x) =
n
0
cos(n − k + 1/2)t
dt,
sint/2
an,n−k f(x) − sn−k (x)
k =0
1
=
2π
ψ(t)
π
0
(5.1)
n
cos(n − k + 1/2)t
ψ(t)
an,n−k
dt.
sint/2
k =0
Therefore,
1
f(x) − tn (x) ≤
2π
π
0
ψ(t)K(n,t)dt
π/n π 1
ψ(t)K(n,t)dt = 1 I1 + I2 , say.
=
+
2π
0
π/n
2π
(5.2)
M. L. Mittal et al. 7
Since K(n,t) = O(1/t), using Hölder’s inequality, condition (2.4), and the fact that
(sint)−1 ≤ π/2t, for 0 < t ≤ π/2, and the second mean value theorem for integrals,
I1 =
π/n
0
=O
=O
ψ(t)K(n,t)dt
π/n 0
t |ψ(t)| sinβ t
ξ(t)
π/n 1
n
h
1
=O
n
=O
O
1
n
h
1/ p π/n dt
ξ(t)
βq ξ(t)
t 2+β
0
q
π/n
h
q
1/q
ξ(t)|K(n,t)|
t sinβ t
q
1/q
dt
dt
t 2 sinβ t
π/n
sinπ/n
π/n p
(5.3)
ξ(t)
O 2+β
t
1/q
q
dt
1/q
dt
.
Since ξ(t) is nondecreasing with t and also using condition (2.8),
π/n
1/q
1
π
O ξ
t −(2+β)q dt
n
n
h
1
1
= O n −1 ξ
O n2+β−1/q = O nβ+1/ p ξ
.
n
n
I1 = O
(5.4)
Using Hölder’s inequality, condition (2.5), Lemma 4.2, Minkowski’s inequality, and condition (2.8),
I2 =
π
π/n
ψ(t)K(n,t)dt
1/ p q 1/q
β p
π −δ
π t ψ(t)sin t ξ(t)K(n,t) dt
dt
≤
β
ξ(t)
π/n π/n t −δ sin t =
π
π/n
=O n
δ
p
t −δ ψ(t)sinβ t ξ(t)
1/ p dt
π
ξ(t) q
π/n
t −δ+1+β
q 1/q
ξ(t)J(n,t) dt
β
−
δ
π/n t sin t sin(t/2) π
O An,n−τ + t
−1
an,0 +
n
−1
Δk an,n−k k =τ
1/q
q
dt
π
1/q π
1/q
δ −β
q
δ −1−β
q
π
= O nδ+1 ξ
O
t An,n−τ dt
+
t
an,0 dt
n
π/n
π/n
π
+
π/n
t
δ −1−β
n
−1
k =τ
1 I2,1 + I2,2 + I2,3 , say.
= O nδ+1 ξ
n
Δk an,n−k 1/q q
dt
(5.5)
8
Approximation of W(L p ,ξ(t)) functions
Since A has nonnegative entries and row sums one,
I2,1 =
π
q
t δ −β An,n−τ dt
π/n
1/q
=O
π
t (δ −β)q dt
π/n
1/q
= O nβ−δ −1/q .
(5.6)
Using Lemma 4.1,
I2,2 = O
π
π/n
=O n
−1
q
t δ −1−β an,0 dt
n
β−δ+1−1/q
1/q
=O n
1/q
π (δ −1−β)q
= O an,0
t
dt
β−δ −1/q
(5.7)
π/n
.
Finally from condition (2.9),
I2,3 =
π
π/n
=O
=O
t
π
π/n
(τ + 1)
n
−1
Δk an,n−k δ −β
n
−1
k =τ
t
δ −β
n
(k + 1)Δk an,n−k t
δ −β
dt
1/q
q
(5.8)
dt
q 1/q
(k + 1) Δk an,n−k
dt
k =0
1/q
q
k =τ
t
π/n
π
δ −β
π/n
=O
π
q 1/q
Δk an,n−k dt
k =τ
π/n
=O
π
n
−1
t δ −1−β
1/q
q
=O
Wn (n) dt
π
π/n
t
δ −β
1/q
q
= O nβ−δ −1/q .
An,0 dt
Combining (5.6), (5.7), and (5.8),
I2 = O nδ+1 ξ
1
n
nβ−δ −1/q = O nβ+1/ p ξ(1/n) .
(5.9)
Combining I1 and I2 yields
1
f(x) − tn (x) = O nβ+1/ p ξ
.
(5.10)
n
Now, using the L p -norm, we get
f(x) − tn (x) p =
2π
0
p
f(x) − tn (x) dx
=O n
β+1/ p
1
ξ
n
2π
0
1/ p
1/ p dx
=O
2π =O n
0
β+1/ p
nβ+1/ p ξ
1
ξ
n
.
p
1
n
1/ p
dx
(5.11)
M. L. Mittal et al. 9
6. Applications
The following corollaries can be derived from Theorem 3.1.
Corollary 6.1 [13]. If ξ(t) = t α , 0 < α ≤ 1, then the weighted class W(L p ,ξ(t)), p ≥ 1,
reduces to the class Lip(α,p) and the degree of approximation of a function f(x), conjugate
to a 2π-periodic function f belonging to the class Lip(α, p), is given by
tn (x) − f(x) = O
1
nα−1/ p
.
(6.1)
Proof. The result follows by setting β = 0 in (3.1).
Corollary 6.2 [12]. If ξ(t) = t α for 0 < α < 1 and p = ∞ in Corollary 6.1, then f ∈ Lip α.
In this case, using (6.1), one has Theorem 2.1.
Proof. For p = ∞, we get
f(x) − tn (x)∞ = sup0≤x≤2π f(x) − tn (x) = O
1 Pk
=O
Pn k=1 kα+1
n
1
nα
(6.2)
for pn ≥ pn+1 ,
that is,
n
Pk
f(x) − tn (x) = O Pn−1
.
α+1
k =1
k
(6.3)
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M. L. Mittal: Department of Mathematics, IIT Roorkee, Roorkee 247667, Uttaranchal, India
E-mail address: mymitfma@iitr.ernet.in
B. E. Rhoades: Department of Mathematics, Indiana University, Bloomington, IN 47405, USA
E-mail address: rhoades@indiana.edu
Vishnu Narayan Mishra: Department of Mathematics, IIT Roorkee, Roorkee 247667,
Uttaranchal, India
E-mail address: vishnu narayanmishra@yahoo.co.in