International Journal of Mathematics and Mathematical Sciences Volume 2012, Article ID 964101,11pages
doi:10.1155/2012/964101
Research Article
Trigonometric Approximation of Signals
(Functions) Belonging to
W
L
r
,
ξ
t
Class by
Matrix
C
1
·
N
p
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,[email protected]
Received 22 March 2012; Revised 24 April 2012; Accepted 3 May 2012
Academic Editor: Jewgeni Dshalalow
Copyrightq2012 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 Khan1974, Chandra2002, and Liendler2005have determined the degree of approximation of 2π-periodic signalsfunctions belonging to Lipα, r class of functions through trigonometric Fourier approximation using different summability matrices with monotone rows. Recently, Mittal et al.2007 and 2011have obtained the degree of approximation of signals belonging to Lipα, r- class by general summability matrix, which generalize some of the results of Chandra2002 and results of Leindler2005, respectively. In this paper, we determine the degree of approximation of functions belonging to LipαandWLr,ξt classes by using Ces´aro-N ¨orlundC1·N
psummability without monotonicity condition on{pn}, which in turn generalizes the results of Lal2009. We also note some errors appearing in the paper of Lal
2009and rectify them in the light of observations of Rhoades et al.2011.
1. Introduction
For a given signalfunctionf ∈Lr :Lr0,2π,r≥1, let
snfsnf;xa0
2
n
k1
akcoskxbksinkx
n
k0
ukf;x 1.1
denote the partial sum, called trigonometric polynomial of degreeor ordern, of the first
n1terms of the Fourier series off. Let{pn}be a nonnegative sequence of real numbers
Define
NnfNnf;xPn−1
n
k0
pn−kskf;x, ∀n≥0, 1.2
the N ¨orlundNpmeans of the sequencesnfor Fourier series of f. The Fourier series of
f is said to be N ¨orlundNpsummable tosxifNnf;x → sxasn → ∞. The Fourier
series offis called Ces´aro-N ¨orlundC1·Npsummable toSxif
tCNn
f n1−1
n
k0
Pk−1
k
i0
pk−isif;x−→Sx asn−→ ∞. 1.3
We note thatNnfandtCN
n fare also trigonometric polynomials of degreeor ordern.
Some interesting applications of the Ces´aro summability can be seen in1,2.
TheLr-norm of signalfis defined by
f
r
1 2π
2π
0
fxr
dx 1/r
1≤r <∞, f∞ sup
x∈0,2π
fx. 1.4
A signalfunctionfis approximated by trigonometric polynomialsTnfof degreen, and
the degree of approximationEnfis given by
EnfMin
n fx−Tn
fr. 1.5
This method of approximation is called trigonometric Fourier approximation.
A signalfunctionfis said to belong to the class Lipαif|fxt−fx|O|t|α,0<
α≤1, andf∈Lipα, riffxt−fxrO|t|α,0< α≤1, r≥1.
For a positive increasing functionξtandr≥1,f ∈Lipξt, riffxt−fxr
Oξt, andf ∈WLr, ξtiffxt−fxsinβx/2
r Oξt, β≥0.
Ifβ 0, then WLr, ξtreduces to Lipξt, r, and if ξt tα 0 < α ≤ 1, then
Lipξt, rclass coincides with the class Lipα, r. Lipα, r → Lipαforr → ∞.
We also write
φx, t φt fxt fx−t−2fx,
Kn, t 2πn11n
k0
Pk−1
k
i0
pisinksint/2−i1/2t, 1.6
τ 1/t, the greatest integer contained in 1/t,Pτ P1/t,Δpk≡pk−pk1.
2. Known Results
Chandra3and Khan4have obtained the error estimatesNnf;x−fxr On−αin
was generalized by Leindler5to almost monotone weights{pn}and by Mittal et al.6to
general summability matrix. Further, Mittal et al.7have extended the results of Leindler
5to general summability matrix, which in turn generalizes some results of Chandra3and
Mittal et al.6. Recently, Lal8has determined the degree of approximation of the functions
belonging to LipαandWLr, ξtclasses using Ces´aro-N ¨orlundC1·Npsummability with
nonincreasing weights{pn}. He proved the following theorem.
Theorem 2.1. LetNpbe a regular N¨orlund method defined by a sequence{pn}such that
Pτ n
kτ
Pk−1 On1. 2.1
Letf ∈ L10,2πbe a 2π-periodic function belonging to Lipα0 < α ≤ 1, then the
degree of approximation offbyC1·Npmeans of its Fourier series is given by
sup
x∈0,2π
tCN
n x−fxtCNn −f∞
⎧ ⎪ ⎨ ⎪ ⎩
On1−α, 0< α <1,
O
logn1πe
n1
, α1. 2.2
Theorem 2.2. Iff is a 2π-periodic function and Lebesgue integrable on [0, 2π] and is belonging to
WLr, ξtclass, then its degree of approximation byC1·Npmeans of its Fourier series is given by
tCN
n −fr O
n1β1/rξn1−1, 2.3
providedξtsatisfies the following conditions:
ξt
t
be a decreasing sequence, 2.4
1/n1
0
tφtsinβt
ξt
r
dt 1/r
On1−1, 2.5
π
1/n1
t−δφt
ξt
r
dt 1/r
On1δ, 2.6
whereδis an arbitrary number such thats1−δ−1>0,r−1s−11, 1≤r≤ ∞, conditions2.5 and2.6hold uniformly inx.
Remark 2.3. In the proof of Theorem 2.1 of Lal8, page 349, the estimate forα1 is obtained as
O
1
n1
O
logn1π
n1
O
log
e
n1
O
logn1π
n1
O
logn1πe
n1
.
Since 1/n1≤ logn1π/n1, theeis not needed in2.2for the caseα 1cf.9, page 6870.
Remark 2.4. iThe author has used monotonicity condition on sequence{pn}in the proof of
Theorem2.1and Theorem2.2, but not mentioned it in the statements. Further in condition
2.4,{ξt/t}is a function oftnot a sequence.
iiThe condition2.5of Theorem2.2leads to the divergent integralε1/n1t−β1sdt
asε → 0 andβ ≥ 08, page 349. Also in8, pages 349-350, the author while writing the
proof of Theorem2.2has used sint≥ 2t/π in the interval1/n1, π, which is not valid
fortπ.
3. Main Results
The observations of Remarks2.3and2.4motivated us to determine a proper set of conditions
to prove Theorems2.1and2.2without monotonocity on{pn}. More precisely, we prove the
following theorem.
Theorem 3.1. LetNp be the N¨orlund summability matrix generated by the nonnegative sequence
{pn}, which satisfies
n1pnOPn, ∀n≥0. 3.1
Then the degree of approximation of a 2π-periodic signal (function)f ∈LipαbyC1·Npmeans of its
Fourier series is given by
tCN
n
f−fx
∞
⎧ ⎪ ⎨ ⎪ ⎩
On−α, 0< α <1,
O
log
n n
, α1. 3.2
Theorem 3.2. Let the condition3.1be satisfied. Then the degree of approximation of a 2π-periodic
signal (function)f ∈WLr, ξtwith 0≤β≤1−1/rbyC1·Npmeans of its Fourier series is given
by
tCN
n
f−fx
r O
nβ1/rξ
1 n
, 3.3
provided positive increasing functionξtsatisfies the condition2.4and
π/n
0
φtsinβt/2
ξt
r
dt 1/r
O1, 3.4
π
π/n
t−δφt
ξt r
dt 1/r
whereδis an arbitrary number such thatsβ−δ−1>0,r−1s−1 1,r≥1, conditions3.4and
3.5hold uniformly inx.
Remark 3.3. For nonincreasing sequence{pn}, we have
Pn
n
k0
pk≥pn
n
k0
1 n1pn, that is,n1pnOPn. 3.6
Thus condition3.1holds for nonincreasing sequence{pn}; hence our Theorems3.1and3.2
generalize Theorems2.1and2.2, respectively.
Note 1. Using condition2.4, we getn/πξπ/n≤nξ1/n.
4. Lemmas
For the proof of our Theorems, we need the following lemmas.
Lemma 4.1see10, 5.11. If{pn}is nonnegative and nonincreasing sequence, then for 0 ≤a <
b≤ ∞,0≤t≤πand for anyn
b
ka
pkein−kt
OPt−1, for anya,
Ot−1pa, fora≥t−1. 4.1
Lemma 4.2see8, page 348. For 0< t≤π/n,Kn, t On.
Lemma 4.3. If{pn}is nonnegative sequence satisfying3.1, then forπn−1< t≤π,
Kn, t O t
−2
n1
Ot−1. 4.2
Proof. We have
Kn, t 1
2πn1sint/2
n
k0
Pk−1
k
r0
prsin
k−r1
2
t
2πn11sint/2 τ
k0
n
kτ1
Pk−1
k
r0
prsin
k−r1
2
t
K1n, t K2n, t,say.
4.3
Now, usingsint/2−1≤π/t, for 0< t≤π, we get
|K1n, t|On1−1t−1
τ
k0
Pk−1
k
r0
pr
O τt−1
n1
O t−2
n1
Usingsint/2−1≤π/t, for 0< t≤πand changing the order of summation, we have
|K2n, t|Ot−1n1−1
n
kτ1
Pk−1
k
r0
prsin
k−r1
2 t
Ot−1n1−1
τ1
r0
pr n
kτ1
Pk−1sin
k−r1
2
t
n
rτ1
pr n
kr
Pk−1sin
k−r1
2
t.
4.5
Again usingsint/2−1 ≤ π/t, for 0 < t ≤ π, Lemma4.1,in view ofPn being positive and
Pn1−1≤Pn−1for alln≥0andt−1< τ1, we get
τ1
r0
pr n
kτ1
Pk−1sin
k−r1
2
t≤
τ1
r0
pr
n
kτ1
Pk−1eik−rt
Ot−1Pτ1−1
τ1
r0
pr O
1 t . 4.6
Using Abel’s transformation, we get
n
kr
Pk−1sin
k−r1
2
t
n−1
kr
ΔPk−1k
j0
sin
k−j1
2 t Pn−1 n
j0
sin
k−j1
2
t−Pr−1
r−1
j0
sin
k−j1
2 t O 1 t n−1
kr
ΔPk−1Pn−1Pr−1
O 1 t
Pn−1Pr−1,
4.7
in view ofsint/2−1≤π/t, for 0< t≤πandPn≥Pn−1 for alln≥0.
Combining4.5–4.7, we get
|K2n, t|Ot−2n1−1 1
n
rτ1
prPn−1Pr−1
Ot−2n1−1 1Pn−1
n
r0
pr
n
rτ1
pr Pr
Ot−2n1−1 1
n
rτ1
r1−1
O t
−2
n1
1O
n−τ
τ1
Ot−2n1−1Ot−1,
4.8
Finally collecting4.3,4.4and4.8, we get Lemma4.3.
Proof of Theorem3.1. We have
snf−fx 1
2π
π
0
sinn1/2t
sint/2
φtdt. 4.9
DenotingC1·Npmeans of{snf}bytCN
n f, we write
tCN
n
f−fx
1
2πn1
π
0
φtn
k0
Pk−1
k
i0
pi
sink−
i1/2t
sint/2
dt
≤
π/n
0
φtKn, tdtπ
π/n
φtKn, tdtI1I2,say. 4.10
Now, using Lemma4.2and the fact thatf∈Lipα⇒φt∈Lipα10, we have
I1On
π/n
0
tαdtOn−α. 4.11
Using Lemma4.3, we get
I2O
π
π/n
tα t
−2
n1t−1
dt
OI21 OI22,say, 4.12
where
I21 n1−1
π
π/n
tα−2dt
⎧ ⎪ ⎨ ⎪ ⎩
On−α, 0< α <1,
O
logn
n
, α1,
I22
π
π/n
tα−1dtO
π
π/n π
ntt
α−1dt
⎧ ⎪ ⎨ ⎪ ⎩
On−α, 0< α <1,
O
log
n n
, α1, .
4.13
Collecting4.10–4.13and writing 1/n≤logn/n, for large values of n, we get
tCN
n
f−fx
⎧ ⎪ ⎨ ⎪ ⎩
On−α, 0< α <1,
O
log
n n
Hence,
tCN
n
f−fx
∞x∈sup0,2π
tCN
n
f−fx
⎧ ⎪ ⎨ ⎪ ⎩
On−α, 0< α <1,
O
log
n n
, α1. 4.15
This completes the proof of Theorem3.1.
Proof of Theorem3.2. Following the proof of Theorem3.1, we have
tCNn
f−fx
π/n
0
φtKn, tdt
π
π/n
φtKn, tdtI3I4,say. 4.16
Using H ¨older’s inequality,φt ∈ WLr, ξt, condition3.4, Lemma4.2, andsint/2−1 ≤
π/t, for 0< t≤π, we have
|I3|
π/n
0
φtsinβt/2
ξt ·
ξtKn, t
sinβt/2 dt
≤ π/n
0
φtsinβt/2
ξt r
dt 1/r
lim
ε→0
π/n
ε
ξtKn, tsinβt/2
sdt
1/s
O1 lim
ε→0
π/n
ε
ξtn
sinβt/2
s dt
1/s
O
nξ
π
n εlim→0
π/n
ε
t−βsdt
1/s
O
ξ
1 n
nβ1−1/s
O
nβ1/rξ
1 n
,
4.17
in view of the mean value theorem for integrals,r−1s−11 and Note1.
Similarly, using H ¨older’s inequality, Lemma4.3,|sint/2| ≤ 1, sint/2−1 ≤ π/t,
condition3.5, and the mean value theorem for integrals, we have
|I4|O
π
π/n
φtt−2n1−1t−1dt
where
I41
π
π/n
φtt−2n1−1dtOn1−1 π
π/n
t−δφtsinβt/2
ξt
ξt
t2−δsinβt/2dt
On−1
π
π/n
t−δφt
ξt r dt 1/rπ π/n ξt
t2−δβ
s dt
1/s
Onδ−1
n/π
1/π
ξ1/y
yδ−2−β
s
y−2dy
1/s O nδ−1 n π ξ π n n/π ε1
y1−δβs−2dy
1/s
O
nδξ
1 n
n1−δβ−1/s
O ξ 1 n
nβ1/r
,
I42
π
π/n
φtt−1dtπ
π/n
t−δφtsinβt/2
ξt
ξt
t1−δsinβt/2dt
O π π/n
t−δφtsinβt/2
ξt r dt
1/rπ
π/n
t1−δsinξtβt/2
sdt 1/s O π π/n
t−δφt
ξt r dt 1/rπ π/n
t1−δξtβ
sdt
1/s
Onδ
n/π
1/π
ξ1/y
yδ−1−β
s
y−2dy
1/s
O
nδ1
π ξ π n n/π ε2
yβ−δs−2dy
1/s
O
nδ1ξ
1 n
n−δβ−1/s
O ξ 1 n
nβ1/r
,
4.19
in view of increasing nature ofyξ1/y,r−1s−1 1, whereε1, ε2lie inπ−1, nπ−1, and Note
1.
Collecting4.16–4.19, we get
tCN
n
f−fxO
nβ1/rξ
1 n
Hence,
tCN
n
f−fx
r
1 2π
2π
0
tCN
n
f−fxrdx
1/r
O
nβ1/rξ
1 n
. 4.21
This completes the proof of Theorem3.2.
5. Corollaries
The following corollaries can be derived from Theorem3.2.
Corollary 5.1. Ifβ0, then forf∈Lipξt, r,tCN
n f−fxr On1/rξ1/n.
Corollary 5.2. Ifβ0, ξt tα 0< α≤1, then forf∈Lipα, r α >1/r,
tCN
n f−fxr O
n1/r−α. 5.1
Corollary 5.3. Ifr → ∞in Corollary5.2, then forf∈Lipα0< α <1,5.1gives
tCN
n
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 Remarks2.3and2.4.
Acknowledgment
The authors are very grateful to the reviewer for his kind suggestions for the improvement of this paper.
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