Applied Mathematics, 2011, 2, 1448-1452
doi:10.4236/am.2011.212206 Published Online December 2011 (http://www.SciRP.org/journal/am)
Degree of Approximation of Conjugate of Signals
(Functions) by Lower Triangular Matrix Operator
Vishnu Narayan Mishra1, Huzoor H. Khan2, Kejal Khatri1
1Department of Mathematics, S. V. National Institute of Technology, Surat, India
2Department of Mathematics, Aligarh Muslim University, Aligarh, India
Received May 4, 2011; revised October 25, 2011; accepted November 5, 2011
Abstract
In the present paper, an attempt is made to obtain the degree of approximation of conjugate of functions
(signals) belonging to the generalized weighted W(LP, ξ(t)), (p ≥ 1)-class, by using lower triangular matrix
operator of conjugate series of its Fourier series.
Keywords: Conjugate Fourier Series, Generalized Weighted W(LP, ξ(t))-Class, Degree of Approximation and
Lower Triangular Matrix Means
1. Introduction
Let
f
be a -periodic signal (function) and let 2π


11
0, 2π
f
LL. Then the Fourier series of a function
(signal) f at any point
x
is given by

0
1
0
()cos sin
2
(;),
kk
k
k
k
a
f
xakxb
ufx




 



kx
(1.1)
with partial sums (;)
n
s
fx—a trigonometric polynomial
of degree (or order) n, of the first terms. (1)n
The conjugate series of Fourier series (1.1) of
f
is
given by
1
(cos sin )(;)
kk k
k
bkxakx vf



1k
x



(1.2)
with partial sums (;)
n
s
fx
.
If
f
is Lebesgue integrable and 1,(( ),)
p
fLiptp

 ,
then
 
ππ
0
0
2π()cot (2)dlimcot (2)d,
hh
f
xttt tt


 

t
exists for all x Zygmund [1, p. 131], ()
f
x
 is called the
conjugate function of ().
f
x
The matrix , in which ,nk a is the element
in n-th row and k-th column is usually called the matrix
of T. Matrices T such that for are
called lower triangular.
T( ),
nk
a
,nk
a0,,kn
Let , T( )
nk
a

be an infinite lower triangular matrix
satisfying Töeplitz [2] conditions of regularity, i.e.
,
0
n
nk
k
a
M


, where M is a finite constant independent
of n,
,
lim 0
nk
na 

, for each and kn,
0
lim 1
n
nk
nk
a
 

.
Let be an infinite series whose
partial sum
0n nu


(1)
th
k
0.
k
kn
n
s
u



The sequence-to-sequence transformation
,,
00
,0,1,2,
nnk knnk n k
kk
asas n 



 
 ,
defines the sequence


n

of lower triangular matrix
summability means of sequence


n
s
generated by the
sequence of coefficients ,nk The transforms n
(a).

are
called linear means or matrix means (determined by the
matrix T) of the sequence


.
n
s
An infinite series n
u

is said to be summable to s by
lower triangular matrix T-method, if lim n
n

 exists and
is equal to s Zygmund [1, p. 74] and we write (),
n
s
T


as The summability matrix T or the sequence-
to-sequence transformation
n
n

is said to be regular, if
li lim
nn
nn
m
s
ss

 

.
A function (signal) ()
f
xLip

, for 01,

 if
()() ().
f
xt fxt

 

V. N. MISHRA ET AL.1449
A function (signal) ()( , )
f
xLipp

 for
1, 01p

 , Fadden [3], if

1
2π
0
()()d ()
p
p,
f
xt fxxt

 

Given a positive increasing function ()t

and an in-
teger 1,()(( ),),pfxLiptp

 Khan [4], if

1
2π
0
()()d (()
p
p).
f
xt fxxt   

In case then () ,01,tt

   ((),)Lipt p

coin-
cides with the class (,)Lipp.

If in p
(,Lip p)

class then this class reduces to .Lip

For a given positive increasing function ()t

, an in-
teger 1,()(,( )),
p
pfxWLt

 Khan [4], if


1
2π
0
()()sin d(()),(0
p
p
fx tfxxxt

  
).
We note that, if 0


)
then the generalized weighted
(,( )),(1
p
WLtp


((), ).Lip t p
-class coincide with the class

Also we observe that
(, )((), )(,())
p
LipLippLiptpW Lt




for 01,p1,

  Mishra [5].
The
p
L-norm is defined by
1
2
0
()d, 1.
p
p
p
ffxxp






The norm of a function is defined
by
-L:fR R

sup() :,
f
fxxR

and the degree of approximation of a function
is given by
()
n
Ef
:fR R
()Min()(;) ,
nn
p
n
Effx fx  
in terms of n, where (;)
nfx

is a trigonometric poly-
nomial of degree (order) n. This method of approxima-
tion is called trigonometric Fourier Approximation (tfa)
Mishra [6]. Riesz-Hölder Inequality states that if p and q
be non-negative extended real numbers such that
111pq. If


,
p
fLab and


,,
q
g
Lab then


1
.fgLab, and
.
b
p
q
a
fgf g

Equality holds if and only if, for some non-zero con-
stants A and B, we have ..
pq
A
fBgae
Second Mean Value theorem for integration states that
if


:,Gab R is a positive monotonically decreasing
function and


:,ab R

,


is an integrable function,
then a number
() ()d(0)()d
bx
aa
Gttt Gatt  

.
Here (0Ga)

stands for , the existence of lim
aG 
which follows from the conditions. Note that it is
essential that the interval (a, b] contains b. A variant not
having this requirement is:
If


:,Gab R is a monotonic (not necessarily de-
creasing and positive) function and


:,ab R   is an
integrable function, then

a number


,
x
ab such
that
() ()d(0)()d(0)()d.
bx
aa
GtttGattGbtt   

b
x


We use the following notations:
()() (),tfxtfxt



,,,0
,1,
n
nknr n
rk
AaA n

0,



,
0
cos(12)
1,
() 2πsin( 2)
nnk
n
k
ak
Mt t


t


1t  —the greatest
integer not exceeding of 1t.
Furthermore C will denote an absolute positive con-
stant, not necessarily the same at each occurrence. Through-
out this paper, we take and
,0(0 ), nk
akn
,0 1. n
A
n


2. Main Result
It is well known that the theory of approximations i.e.,
tfa, which originated from a well known theorem of
Weierstrass, has become an exciting interdisciplinary
field of study for the last 130 years. These approxima-
tions have assumed important new dimensions due to
their wide applications in signal analysis, in general and
in digital signal processing [5] in particular, in view of
the classical Shannon sampling theorem.
This has motivated by various investigators such as
Qureshi ([7,8]), Khan ([4,9]) Chandra [10], Leindler [11]
Mishra [5] discussed the degree of approximation of
signals (functions) belonging to
,(,),((),LipLippLiptp)



and (,())
p
WL t

-classes by
using Cesàro and Nörlund means of an infinite series.
Qureshi ([12,13]) have determined the degree of ()
f
x
,
conjugate of a function ()fx Lip

 and (,)pLip

by Nörlund means of conjugate series of a Fourier series.
The purpose of this paper is to determine the degree of
approximation of()
f
x

()),(tp
, conjugate of a function
()(,1),
p
fx WL


 by lower triangular matrix
means.
We prove:

x
ab such that Theorem 2.1. Let ,
()
nk
Ta

be an infinite regular

V. N. MISHRA ET AL. 1450
lo ix such twer triangular matrhat the elements ,
()
nk
a
be non-negative, non-decreasing with k ≤ n. If :fR
is a 2π-periodic, Lebesgue integrable and bel
the gralized weighted (,()),1
p
WLtp
R
ging toon
ene

-class, then
the degree of approximation of ()
f
x
, conjugate of
()(, ()),
p
f
xWL t

 by lower trian matrix means gular
(;)fx
n

is given by

1/
(
n

ided
;)()fx
(1)n0,
p
p
fx Onn



  (2.1)
prov ()t

ollow
is positive increasinctiong fun of t satis-
fying the fing conditions
1
π() p ntψt



1)



0
sin d (
()
p
ptt On
ξt
 

 



 (2.2)
1
π
π
() d()
()
p
p
n
tψt) tOn
ξt










 


(2.3)
and ()is decreasing in
ξtt
t (2.4)
where

is an arbitrary number such that
(1) 10,qδ>+

 q the conjugate index
ld uniformly in x and 11
1p+q=
 .
Note 1. Condition (2.4) implies
of p and con-
ditions (2.2), (2.3) ho



π  π1,nn  
for

π1nn
Nrote 2. Also fo 0

 our Theorem (2.1)
f Lal
mas
o prove our Theorem 2.1, we requi
nder the conditions of our
reduces to
re the fol-
Theorem 2.1
on
. Lem
order t
on
e of the theorem oand Kushwaha [14].
3
In
lowing lemma.
Lemma 3.1. U
,
(),
nk
a we have
,
() ,
n
n
A
MtO t




for ππ.t
n
Proof. For 1 ππ,nt
 1
(sin )π2,for0 π2ttt,
,n

 we have
1
,,
0
1
,,
0
,1
01
0
,
,
()t
1cos(12)1cos(12)
2πsin( 2)2πsin( 2)
11
cos( 12)cos( 12)
22
12maxcos(1 2)
2
cos(12)
1
2
n
n
n
nk nk
kkn
nn
nk nk
kkn
r
nn rn k
n
nk
kn
n
M
kt kt
aa
tt
aktakt
tt
akt
t
akt
a
O
t










  


















1
,,
n
n
A
t











and
,,
,,1,
,1 ,1,1
,1
,1
,1
(1)
11
n
nnk
kn
nn nnnn
nn nnnn
nn
nn
nn
Aa
aa a
aa a
a
a
t
a
t











 




 
 












Therefore, , () n
n
A
MtO t




 This completes the
proof of Lemma 3.1.
4. Proof of Theorem 2.1
The partial sum of the conjugate series the Fou-
rier sees (1.2) is given by
th
k
ri
of
π
0
π
0
π
0
π
0
1
(;)cot(2)()d
2π
1cos(12)()d
2πsin( 2)
1
(;)cot(2)()d
2π
1cos(1 2)
2πsin( 2)
nt
t


() d
n
n
sfxt tt
nt
tt
t
sfxt tt
tt
















Then
π
,
00
π
,
0
0
1
(;)cot(2)()d
2π
1cos(1 2)()d
2πsin( 2)
n
nk n
k
n
nk
k
asfxt tt
nt
at
t




t







 












or,
12
(;)()
n
f
xfxII 



 (4.1)
Using Riesz-Hölder’s inequality, condition (2.2), (2.4),
note 1, the fact that 1π (sin),for 0π2
2
tt
t
,
1
pq
1
1


integrals, we fi
and the second mean value theorem for
nd

V. N. MISHRA ET AL.1451
1
π
1
0
1
π
0
1
π
0
1
π() cos( 12)
q
q
ntk
t 




0sin stt







2
() sin d
()
() () d
sin
() sin d
()
d
in (2)
sin
p
p
n
q
q
n
n
p
p
n
tt
Itt
t
tMt t
tt
tt tt
t
t
t
ntt




































 



1( )t
OO

  


1
πq
q
n



0
1
π
2
1
π
2
1
π
(2 )
1
d
1π() d; 0
sin(π)
1()
d; 0
1πd
1
q
qq
n
h
q
q
n
h
q
n
q
h
t
nt
OOth
nn t
t
Oth
nt
OOtt
nn
On n

















  


 

  



 




 





 


  

 











21
1/ 1
q
p
On
On n













 (4.2)
Now by Riesz-Hölder’s inequality, conditions (2.3),
(2.4), note 1, Lemma 3.1, the fact that
1π
(sin), for0π2
2
tt
t

,11 1pq

, we obtain

π
2
π
11
π
ππ
1
π
π
1/
π
,
() d
sin
q
q
n
A
tOt
t
tt





()1()1
1
π
12
1π
1
π
2
1π
1
1
1
(1 )d
(π)d
π
1
1ππ
()1
1
qq
q
q
n
q
n
qq
q
yy
On yy
ny
On ny
n
On nq
On On
n
 










  













 


 
  


 


 

   

   















1
11 1
(1)(1). (4.3)
q
qp
OnnOnn




 

π
,
() ()d
() () ( )sindd
() sin
( )sind
()
()
n
n
pq
pq
n
nn
p
p
n
n
n
ItMtt
tM t tt t
tt
ttt
tt t
t
t
tA
On t
 





 








































 


1
π
1
π
d
q
q
n
t
 









Since A has non-negative entries and row sums one,
Combining 1
I
and 2
I
yields

1
(;)()(1).
p
fx fxOnn






n
Now, using the -norm, we get
p
L


1
2π
0
1
2π
1/
0
1
2π
1
0
1
(;) ()(;)()d
(1 d
(1 )d
(1) .
p
p
nn
p
p
p
p
p
p
p
fx fxfx fxx
On nx
Onnx
On n












 























This completes the proof of our Theorem 2.1.
5. Applications
The following corollaries can be derived fromheo-
rem 2.1.
If
our T
Corollary 5.1. 0


and
generalized weighted class
() ,01,tt

 

,()
p
WLt  re- then the
duces to class (,Lip )p

and
() Lip
the
tion of a functi
degree of approxima-
) on fx ( ,p


is given by


1p  
(;)().
np
fx fxOn  


Proof of corollary 5.1. From our Theorem 2.1 for
, we have 0


p

1
2π
0
1
1
(1 ), 1.
(;)()(;)()d
1
p
nn
p
p
p
f
xfxfxfx x 

 






e proof of corollary 5.1.
On nOp
n






This completes th
Corollary 5.2. If
p
 in corollary 5.1, then for

V. N. MISHRA ET AL. 1452
01,



(;)().
np
fx fxOn

Corolla





ry 5.3. If 1
,,0,()
nknknn
apPP tt




 then
the degree of approximation of (),
f
x
 conjugate o
(,)
f
f
Lip p

 by Nörlund means
1
(;)
nn
k0(;)
n
nkk
f
xP 



the conjugate series psfx

 of
of Fourier series is given by

1
(;)().
p
np
fx fxOn  

rollary 5.4.
 

Co If
then the degree
1
,,0,(),
nknknn
apPP ttp





of approximation of (),
f
x
 conjugate of
f
Lip

 by Nörlmeans und 1
0
n
nn nk
kk
P
ps 





ries is given by

of
ate series of Fourier sethe conjug


log(
nf





(1),0 1,On





Corollary 5.5. If such that
1)π(1), 1.Onen 


(5.1)
1
,nknkkn
apqR



0,qRy
0n
kk
k
n
n y
Rp

 is monot


g then the degre

onic non-decreas-
ine of approximation of (),
f
x
 con-
jugate of a function ,
f
Lip

generalized Nö
1
0
)n
by
means
rlund
(; (;)
nn nkkk
k
f
xR pqs



fx 

of the conju-
gate series (1.2) satisfies equatio
6. Remarks
R Kushwaha [14]. The degree o
proximation
n (5.1).
emark 6.1. Lal andf ap-

1. p  determined (;)()fx fxOn





if
val
estigate
oxim conjunctions be-
np
by Qureshs to  i [13, p. 561, L. 12] tend
1
031   and 2p and also for otherues.
Therefore, this deficiency has encouraged to inv


degree of appr
longing to Lip
ation of
)
ugate of f
(,p

considering
7.ents
for
uable sug
or the improvement of this paper.
of. Chris C
ings, University of Sheffield, UK and AM Editorial Assi-
Huang, Scientific Research Publishing,
SA for their kind cooperation during communication.
es
Cambridge
University
te zne JournaVol. 22,
. 113-119.
th.org/zmath/en/journals/searc
11.p
 
Acknowledgem
The authors are grateful to his beloved parents their
encouragement to his work. The authors are grateful to
the referee for his valgestions and useful com-
ments fThe authors are
also thankful to the AM Editor in chief Pran-
n
stant Ms. Tian
U
8. Referenc
[1] A. Zygmund, “Trigonometric Series Vol. I,”
Press, Cambridge, 1959.
[2] O. Töeplitz, “Über Allgemeine Lineare Mittelbildun-
gen,” Prace Mamatyczno—Fizyc l,
1913, pp
http://www.zentralblatt-ma
h/?an=00003590
[3] L. McFadden, “Absolute Nörlund Summability,” Duke
Mathematical Journal, Vol. 9, 1942, pp. 168-207.
doi:10.1215/S0012-7094-42-00913-X
[4] H. H. Khan, “On the Degree of Approximation to a Func-
tion Belonging to Weighted (LP, ξ(t)) Class,” Aligarh
Bulletin of Mathematics, Vol. 3-4, 1973-1974, pp. 83-88.
[5] V. N. Mishra, “Some Problems on Approximations of
Functions in Banach Spaces,” Ph.D. Thesis, Indian Insti-
tute of Technology, Roorkee, 2007.
[6] V. N. Mishra, “
nals (Function
On the Degree of Approximation of Sig-
s) Belonging to the Weighted W(LP, ξ(t)),
(p ≥ 1)-Class by Almost Matrix Summability Method of
, Vol. 5, No. 7, 2009,
α,” Indian Journal of Pure and Ap-
he Degree of Approximation of a Func-
2, pp.
cal Analysis and Ap-
Its Conjugate Fourier Series,” International Journal of
Applied Mathematics and Mechanics
pp. 16-27.
[7] K. Qureshi, “On the Degree of Approximation of a Func-
tion Belonging to Lip
plied Mathematics, Vol. 13, No. 8, 1982, pp. 898-903.
[8] K. Qureshi, “On t
tion Belonging to the Class Lip(α,p),” Indian Journal of
Pure and Applied Mathematics, Vol. 13, No. 4, 198
466-470.
[9] H. H. Khan, “On the Degree of Approximation to a Func-
tion by Triangular Matrix of Its Conjugate Fourier Series
II,” Indian Journal of Pure and Applied Mathematics,
Vol. 6, 1975, pp. 1473-1478.
[10] P. Chandra, “Trigonometric Approximation of Functions
in LP-Norm,” Journal of Mathemati
plications, Vol. 275, 2002, pp. 13-26.
doi:10.1016/S0022-247X(02)00211-1
[11] L. Leindler, “Trigonometric Approximation in LP-Norm,”
Journal of Mathematical Analysis and Applications, Vol.
302, 2005, pp. 129-136.
doi:10.1016/j.jmaa.2004.07.049
nd
unction Belonging to the Class Lip(α,p) by
the Generalized Lipschitz
[12] K. Qureshi, “On the Degree of Approximation of Conju-
gate of Function Belonging to the Lipschitz Class by
Means of Conjugate Series,” Indian Journal of Pure a
Applied Mathematics, Vol. 12, No. 9, 1981, pp. 1120-
1123.
[13] K. Qureshi, “On the Degree of Approximation of Conju-
gate of F
Means of Conjugate Series,” Indian Journal of Pure and
Applied Mathematics, Vol. 13, No. 5, 1982, pp. 560-563.
[14] S. Lal, and J. K. Kushwaha, “Approximation of Conju-
gate of Functions Belonging to
class by Lower Triangular Matrix Means,” International
Journal of Mathematical Analysis, Vol. 3, No. 21, 2009,
pp. 1031-1041.