Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 964101, 11 pages
doi:10.1155/2012/964101
Research Article
Trigonometric Approximation of Signals
(Functions) Belonging to WLr , ξt Class by
Matrix C1 · Np Operator
Uaday Singh, M. L. Mittal, and Smita Sonker
Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247667, India
Correspondence should be addressed to Uaday Singh, usingh2280@yahoo.co.in
Received 22 March 2012; Revised 24 April 2012; Accepted 3 May 2012
Academic Editor: Jewgeni Dshalalow
Copyright q 2012 Uaday Singh et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
Various investigators such as Khan 1974, Chandra 2002, and Liendler 2005 have determined
the degree of approximation of 2π-periodic signals functions belonging to Lipα, r class of
functions through trigonometric Fourier approximation using different summability matrices with
monotone rows. Recently, Mittal et al. 2007 and 2011 have obtained the degree of approximation
of signals belonging to Lipα, r- class by general summability matrix, which generalize some
of the results of Chandra 2002 and results of Leindler 2005, respectively. In this paper, we
determine the degree of approximation of functions belonging to Lip α and WLr , ξt classes
by using Cesáro-Nörlund C1 · Np summability without monotonicity condition on {pn }, which
in turn generalizes the results of Lal 2009. We also note some errors appearing in the paper of Lal
2009 and rectify them in the light of observations of Rhoades et al. 2011.
1. Introduction
For a given signal function f ∈ Lr : Lr 0, 2π, r ≥ 1, let
n
n
a0 sn f sn f; x ak cos kx bk sin kx uk f; x
2
k1
k0
1.1
denote the partial sum, called trigonometric polynomial of degree or order n, of the first
n 1 terms of the Fourier series of f. Let {pn } be a nonnegative sequence of real numbers
0 → ∞ as n → ∞ and P−1 0 p−1 .
such that Pn nk0 pk /
2
International Journal of Mathematics and Mathematical Sciences
Define
n
pn−k sk f; x ,
Nn f Nn f; x Pn −1
∀n ≥ 0,
1.2
k0
the Nörlund Np means of the sequence sn f or Fourier series of f. The Fourier series of
f is said to be Nörlund Np summable to sx if Nn f; x → sx as n → ∞. The Fourier
series of f is called Cesáro-Nörlund C1 · Np summable to Sx if
n
k
−1
−1
P
pk−i si f; x −→ Sx as n −→ ∞.
tCN
f
1
n
k
n
k0
1.3
i0
We note that Nn f and tCN
n f are also trigonometric polynomials of degree or order n.
Some interesting applications of the Cesáro summability can be seen in 1, 2.
The Lr -norm of signal f is defined by
f r
1
2π
2π
fxr dx
1/r
1 ≤ r < ∞,
0
f sup fx.
∞
1.4
x∈0,2π
A signal function f is approximated by trigonometric polynomials Tn f of degree n, and
the degree of approximation En f is given by
En f Minfx − Tn f r .
n
1.5
This method of approximation is called trigonometric Fourier approximation.
A signal function f is said to belong to the class Lip α if |fx t − fx| O|t|α , 0 <
α ≤ 1, and f ∈ Lipα, r if fx t − fxr O|t|α , 0 < α ≤ 1, r ≥ 1.
For a positive increasing function ξt and r ≥ 1, f ∈ Lipξt, r if fx t − fxr Oξt, and f ∈ WLr , ξt if fx t − fxsinβ x/2r Oξt, β ≥ 0.
If β 0, then WLr , ξt reduces to Lipξt, r, and if ξt tα 0 < α ≤ 1, then
Lipξt, r class coincides with the class Lipα, r. Lipα, r → Lip α for r → ∞.
We also write
φx, t φt fx t fx − t − 2fx,
Kn, t n
k
sink − i 1/2t
1
,
Pk −1 pi
2πn 1 k0
sint/2
i0
1.6
τ 1/t, the greatest integer contained in 1/t, Pτ P 1/t, Δpk ≡ pk − pk1 .
2. Known Results
Chandra 3 and Khan 4 have obtained the error estimates Nn f; x − fxr On−α in
Lipα, r class using monotonicity conditions on the means generating sequence {pn }, which
International Journal of Mathematics and Mathematical Sciences
3
was generalized by Leindler 5 to almost monotone weights {pn } and by Mittal et al. 6 to
general summability matrix. Further, Mittal et al. 7 have extended the results of Leindler
5 to general summability matrix, which in turn generalizes some results of Chandra 3 and
Mittal et al. 6. Recently, Lal 8 has determined the degree of approximation of the functions
belonging to Lip α and WLr , ξt classes using Cesáro-Nörlund C1 · Np summability with
nonincreasing weights {pn }. He proved the following theorem.
Theorem 2.1. Let Np be a regular Nörlund method defined by a sequence {pn } such that
Pτ
n
Pk −1 On 1.
2.1
kτ
Let f ∈ L1 0, 2π be a 2π-periodic function belonging to Lip α 0 < α ≤ 1, then the
degree of approximation of f by C1 · Np means of its Fourier series is given by
⎧ ⎪
0 < α < 1,
⎨O n 1−α ,
CN
sup tCN
−
f
−
fx
x
t
logn
1πe
n
n
⎪
∞
, α 1.
⎩O
x∈0,2π
n 1
2.2
Theorem 2.2. If f is a 2π-periodic function and Lebesgue integrable on [0, 2π] and is belonging to
WLr , ξt class, then its degree of approximation by C1 · Np means of its Fourier series is given by
CN
tn − f O n 1β1/r ξ n 1−1 ,
2.3
r
provided ξt satisfies the following conditions:
ξt
be a decreasing sequence,
t
r 1/r
1/n1 tφtsinβ t
dt
O n 1−1 ,
ξt
0
π
1/n1
r 1/r
t−δ φt
dt
O n 1δ ,
ξt
2.4
2.5
2.6
where δ is an arbitrary number such that s1 − δ − 1 > 0, r −1 s−1 1, 1 ≤ r ≤ ∞, conditions 2.5
and 2.6 hold uniformly in x.
Remark 2.3. In the proof of Theorem 2.1 of Lal 8, page 349, the estimate for α 1 is obtained
as
O
1
n1
O
logn 1π
n1
O
log e
n1
O
logn 1π
n1
O
logn 1πe
.
n1
2.7
4
International Journal of Mathematics and Mathematical Sciences
Since 1/n 1 ≤ logn 1π/n 1, the e is not needed in 2.2 for the case α 1 cf. 9,
page 6870.
Remark 2.4. i The author has used monotonicity condition on sequence {pn } in the proof of
Theorem 2.1 and Theorem 2.2, but not mentioned it in the statements. Further in condition
2.4, {ξt/t} is a function of t not a sequence.
1/n1 −β1s
t
dt
ii The condition 2.5 of Theorem 2.2 leads to the divergent integral ε
as ε → 0 and β ≥ 0 8, page 349. Also in 8, pages 349-350, the author while writing the
proof of Theorem 2.2 has used sin t ≥ 2t/π in the interval 1/n 1, π, which is not valid
for t π.
3. Main Results
The observations of Remarks 2.3 and 2.4 motivated us to determine a proper set of conditions
to prove Theorems 2.1 and 2.2 without monotonocity on {pn }. More precisely, we prove the
following theorem.
Theorem 3.1. Let Np be the Nörlund summability matrix generated by the nonnegative sequence
{pn }, which satisfies
n 1pn OPn ,
∀n ≥ 0.
3.1
Then the degree of approximation of a 2π-periodic signal (function) f ∈ Lip α by C1 · Np means of its
Fourier series is given by
⎧
⎪
0 < α < 1,
⎨On−α ,
CN tn f − fx log n
⎪
∞
, α 1.
⎩O
n
3.2
Theorem 3.2. Let the condition 3.1 be satisfied. Then the degree of approximation of a 2π-periodic
signal (function) f ∈ WLr , ξt with 0 ≤ β ≤ 1 − 1/r by C1 · Np means of its Fourier series is given
by
1
CN β1/r
ξ
,
tn f − fx O n
r
n
3.3
provided positive increasing function ξt satisfies the condition 2.4 and
1/r
π/n φtsinβ t/2 r
dt
O1,
ξt
0
π
π/n
r 1/r
t−δ φt
dt
O nδ ,
ξt
3.4
3.5
International Journal of Mathematics and Mathematical Sciences
5
where δ is an arbitrary number such that sβ − δ − 1 > 0, r −1 s−1 1, r ≥ 1, conditions 3.4 and
3.5 hold uniformly in x.
Remark 3.3. For nonincreasing sequence {pn }, we have
Pn n
pk ≥ pn
k0
n
1 n 1pn ,
that is, n 1pn OPn .
3.6
k0
Thus condition 3.1 holds for nonincreasing sequence {pn }; hence our Theorems 3.1 and 3.2
generalize Theorems 2.1 and 2.2, respectively.
Note 1. Using condition 2.4, we getn/πξπ/n ≤ nξ1/n.
4. Lemmas
For the proof of our Theorems, we need the following lemmas.
Lemma 4.1 see 10, 5.11. If {pn } is nonnegative and nonincreasing sequence, then for 0 ≤ a <
b ≤ ∞, 0 ≤ t ≤ π and for any n
b
O P t−1 ,
in−kt pk e
ka
O t−1 pa ,
for any a,
4.1
for a ≥ t−1 .
Lemma 4.2 see 8, page 348. For 0 < t ≤ π/n, Kn, t On.
Lemma 4.3. If {pn } is nonnegative sequence satisfying 3.1, then for πn−1 < t ≤ π,
Kn, t O
t−2
n 1
O t−1 .
4.2
Proof. We have
n
k
1
1
−1
t
Kn, t Pk
pr sin k − r 2πn 1 sint/2 k0
2
r0
1
2πn 1 sint/2
τ
k0
n
Pk
kτ1
−1
1
t
pr sin k − r 2
r0
k
4.3
K1 n, t K2 n, t, say.
Now, using sin t/2−1 ≤ π/t, for 0 < t ≤ π, we get
−1 −1
|K1 n, t| O n 1 t
τ
Pk
k0
−1
k
pr
r0
O
τt−1
n 1
O
t−2
n 1
.
4.4
6
International Journal of Mathematics and Mathematical Sciences
Using sin t/2−1 ≤ π/t, for 0 < t ≤ π and changing the order of summation, we have
n
k
1
P −1 p sin k − r t
|K2 n, t| O t−1 n 1 kτ1 k r0 r
2 τ1
n
1
−1 −1
O t n 1 pr
t
Pk −1 sin k − r r0 kτ1
2
−1
n
pr
rτ1
n
Pk
−1
kr
4.5
1 sin k − r t.
2 Again using sin t/2−1 ≤ π/t, for 0 < t ≤ π, Lemma 4.1, in view of Pn being positive and
Pn1 −1 ≤ Pn −1 for all n ≥ 0 and t−1 < τ 1, we get
τ1
τ1 n
τ1
n
1 1
−1
−1 ik−rt t ≤
.
Pk sin k − r pr Pk e
pr O
pr
O t−1 Pτ1 −1
r0 kτ1
2
t
r0
r0
kτ1
4.6
Using Abel’s transformation, we get
n−1 k
1
1
ΔPk −1
Pk −1 sin k − r sin k − j t
t
2
2
j0
kr
kr
n
Pn −1
r−1
1
1
t − Pr −1 sin k − j t
sin k − j 2
2
j0
j0
n
4.7
n−1 1
1
−1 −1
−1
O
O
Pn −1 Pr −1 ,
ΔPk Pn Pr
t
t
kr
in view of sin t/2−1 ≤ π/t, for 0 < t ≤ π and Pn ≥ Pn−1 for all n ≥ 0.
Combining 4.5–4.7, we get
−2
|K2 n, t| O t n 1
−1
−2
−1
O
1
t−2
n1
n
1 Pn
1
−1
n
pr Pn
−1
Pr
−1
n n
pr
pr Pr
r0
rτ1
r 1
rτ1
n−τ
1O
τ 1
in view of 3.1 and τ ≤ 1/t < τ 1.
rτ1
O t n 1
−2
O t n 1
−1
−1
O t−2 n 1−1 O t−1 ,
4.8
International Journal of Mathematics and Mathematical Sciences
7
Finally collecting 4.3, 4.4 and 4.8, we get Lemma 4.3.
Proof of Theorem 3.1. We have
1
sn f − fx 2π
π
0
sinn 1/2t
φtdt.
sint/2
4.9
Denoting C1 · Np means of {sn f} by tCN
n f, we write
CN tn f − fx ≤
1
2πn 1
π/n
π
n
k
sink − i 1/2t
−1
dt
pi
φt Pk
0
sint/2
i0
k0
φtKn, tdt 0
π
φtKn, tdt I1 I2 , say.
4.10
π/n
Now, using Lemma 4.2 and the fact that f ∈ Lip α ⇒ φt ∈ Lip α 10, we have
I1 On
π/n
tα dt O n−α .
4.11
0
Using Lemma 4.3, we get
I2 O
π
t
π/n
α
t−2
−1
dt OI21 OI22 , say,
t
n 1
4.12
where
⎧
⎪
0 < α < 1,
⎨On−α ,
I21 n 1−1
tα−2 dt log n
⎪
, α 1,
⎩O
π/n
n
⎧
π
π
⎪
0 < α < 1,
⎨On−α ,
π α−1
α−1
t dt t dt O
.
log n
⎪
, α 1,
⎩O
π/n
π/n nt
n
π
I22
4.13
Collecting 4.10–4.13 and writing 1/n ≤ log n/n, for large values of n, we get
⎧
⎪
0 < α < 1,
⎨On−α ,
CN tn f − fx log n
⎪
, α 1.
⎩O
n
4.14
8
International Journal of Mathematics and Mathematical Sciences
Hence,
⎧
⎨On−α ,
⎪
CN
f
−
fx
tn f − fx sup tCN
log n
n
⎪
∞
,
O
⎩
x∈0,2π
n
0 < α < 1,
α 1.
4.15
This completes the proof of Theorem 3.1.
Proof of Theorem 3.2. Following the proof of Theorem 3.1, we have
tCN
n
f − fx π/n
φtKn, tdt 0
π
φtKn, tdt I3 I4 , say.
4.16
π/n
Using Hölder’s inequality, φt ∈ WLr , ξt, condition 3.4, Lemma 4.2, and sin t/2−1 ≤
π/t, for 0 < t ≤ π, we have
π/n φtsinβ t/2 ξtKn, t ·
dt
|I3 | β
0
ξt
sin t/2 1/r π/n π/n 1/s
φtsinβ t/2 r
ξtKn, t s
≤
lim
dt
sinβ t/2 dt
ε→0 ε
ξt
0
1/s
π/n
s 1/s
π
ξtn
−βs
lim
O1 lim
dt
O nξ
t dt
ε→0 ε
ε→0 ε
n
sinβ t/2
1
1
nβ1−1/s O nβ1/r ξ
,
O ξ
n
n
π/n 4.17
in view of the mean value theorem for integrals, r −1 s−1 1 and Note 1.
Similarly, using Hölder’s inequality, Lemma 4.3, | sint/2| ≤ 1, sint/2−1 ≤ π/t,
condition 3.5, and the mean value theorem for integrals, we have
|I4 | O
π
π/n
φt t−2 n 1−1 t−1 dt OI41 OI42 , say,
4.18
International Journal of Mathematics and Mathematical Sciences
9
where
I41 π
−1
−2
φtt n 1 dt
O n 1
−1
π
π/n
π/n
π
O n−1
π/n
O n
δ−1
r
t−δ φt
ξt
yδ−2−β
dt
π/n
n/π ξ1/y s
1/π
1/r π
t−δ φtsinβ t/2
ξt
dt
ξt
t2−δ sinβ t/2
s 1/s
dt
2−δβ
ξt
t
1/s
−2
y dy
1/s
n/π
π
n
1−δβs−2
ξ
O n
y
dy
π
n
ε1
1
1
n1−δβ−1/s O ξ
nβ1/r ,
O nδ ξ
n
n
π
π −δ
t φtsinβ t/2
ξt
φtt−1 dt dt
1−δ sinβ t/2
ξt
t
π/n
π/n
1/r π s 1/s
π t−δ φtsinβ t/2 r
ξt
dt
O
dt
β
1−δ
ξt
sin t/2 π/n π/n t
δ−1
I42
O
1/r 1/s
π t−δ φt r
ξt s
dt
1−δβ dt
π/n ξt π/n t
π
O n
δ
n/π ξ1/y s
1/π
yδ−1−β
1/s
−2
y dy
n/π
1/s
π
nδ1
β−δs−2
ξ
O
y
dy
π
n
ε2
1
1
δ1
−δβ−1/s
β1/r
n
n
O n ξ
O ξ
,
n
n
4.19
in view of increasing nature of yξ1/y, r −1 s−1 1, where ε1 , ε2 lie in π −1 , nπ −1 , and Note
1.
Collecting 4.16–4.19, we get
1
CN β1/r
.
ξ
tn f − fx O n
n
4.20
10
International Journal of Mathematics and Mathematical Sciences
Hence,
CN tn f − fx r
1
2π
1/r
2π r
1
CN .
O nβ1/r ξ
tn f − fx dx
n
0
4.21
This completes the proof of Theorem 3.2.
5. Corollaries
The following corollaries can be derived from Theorem 3.2.
1/r
Corollary 5.1. If β 0, then for f ∈ Lipξt, r, tCN
ξ1/n.
n f − fxr On
Corollary 5.2. If β 0, ξt tα 0 < α ≤ 1, then for f ∈ Lipα, r α > 1/r,
CN
tn f − fx O n1/r−α .
r
5.1
Corollary 5.3. If r → ∞ in Corollary 5.2, then for f ∈ Lipα0 < α < 1, 5.1 gives
CN tn f − fx O n−α .
∞
5.2
6. Conclusion
Various results pertaining to the degree of approximation of periodic functions signals
belonging to the Lipchitz classes have been reviewed and the condition of monotonocity
on the means generating sequence {pn } has been relaxed. Further, a proper set of conditions
have been discussed to rectify the errors pointed out in Remarks 2.3 and 2.4.
Acknowledgment
The authors are very grateful to the reviewer for his kind suggestions for the improvement
of this paper.
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