Bulletin of Mathematical Analysis and Applications
ISSN: 1821-1291, URL: http://www.bmathaa.org
Volume 5 Issue 4 (2013), Pages 1-13
BEST APPROXIMATION OF FUNCTIONS OF GENERALIZED
ZYGMUND CLASS BY MATRIX-EULER SUMMABILITY
MEANS OF FOURIER SERIES
(COMMUNICATED BY HÜ SEIN BOR)
SHYAM LAL AND SHIREEN
(w)
Abstract. In this paper, new best approximations of the function f ∈ Zr ,
(r ≥ 1) class by Matrix-Euler means (∆.E1 ) of its Fourier Series have been
determined.
1. Introduction
The degree of approximation of a function f belonging to Lipschitz class by the
Cesàro mean and f ∈ Hα by the Fejér means has been studied by Alexits [4] and
Prössdorf [7] respectively. But till now no work seems to have been done to obtain
(w)
best approximation of functions belonging to generalized Zygmund class, Zr , (r ≥
(w)
1) by product summability means of the form (∆.E1 ). Zr class is a generalization
(w)
of Zα , Zα,r , Z
classes. The Matrix-Euler (∆.E1 ) summability means includes
(N, pn ).E1 , (N, pn , qn ).E1 and (C, 1).E1 means as particular cases. In attempt to
make an advance study in this direction, in this paper, best approximations of the
(w)
function belonging to generalized Zygmund class Zr , (r ≥ 1) have been obtained.
2. Definition and Notations
Let
∞
P
un be an infinite series having nth partial sum sn =
n=0
n
P
uν .
ν=0
Let f be a 2π periodic function, integrable in the Lebesgue sense over [0, 2π].
Let the Fourier series of f be given by
f (x) :=
∞
X
1
a0 +
(an cos nx + bn sin nx)
2
n=1
(1)
with nth partial sum sn (f ; x).
02010 Mathematics Subject Classification: Primary 42B05; Secondary 42B08.
Keywords and phrases. Best approximation, Fourier series, (E, 1) means, matrix summability
means and (∆.E1 )- summability means, generalized Minkowski’s inequality, generalized Zygmund
class.
c 2013 Universiteti i Prishtinës, Prishtinë, Kosovë.
Submitted August 14, 2013. Published September 15, 2013.
1
2
SHYAM LAL AND SHIREEN
Let T = (an,k ) be an infinite triangular matrix satisfying the (Silverman-Töeplitz
[8]) conditions of regularity, i.e.,
n
P
an,k = 1 as n → ∞ ,
(i).
k=0
(ii). an,k = 0 for k > n ,
n
P
|an,k | ≤ M , a finite constant.
(iii).
k=0
The sequence-to-sequence transformation
t∆
n :=
n
X
an,k sk =
k=0
n
X
an,n−k sn−k
k=0
defines the sequence t∆
n of triangular matrix means of the sequence {sn }, generated
by the sequence of coefficients (an,k ).
∞
P
If t∆
un is summable to s by triangular
n → s as n → ∞, then the series
n=0
matrix ∆- method (Zygmund [1], p.74).
n ∞
P
1 X n
(1)
sk . If En → s as n → ∞, then
Let En(1) := n
un is said to be
2
k
n=0
k=0
summable to s by the Euler’s method, E1 (Hardy [5]).
The triangular matrix ∆-transform of E1 transform defines the (∆.E1 ) transform
∞
P
∆E
tn of the partial sums sn of the series
un by
n=0
t∆E
:=
n
n
X
(1)
an,k Ek =
k=0
k=0
If t∆E
→ s as n → ∞, then the series
n
sn → s
⇒ En(1) =
n
X
∞
P
an,k
k 1X k
sν .
2k ν=0 ν
un is said to be summable (∆.E1 ) to s.
n=0
n 1 X n
sν → s as n → ∞, E1 method is regular,
2n ν=0 ν
(1)
∆E
⇒ t∆
→ s as n → ∞, ∆ method is regular,
n (En ) = tn
⇒ (∆.E1 ) method is regular.
Some important particular cases of triangular Matrix-Euler means (∆.E1 ) are
(i). (H,
1
n+1 ).(E1 )
means, when an,k =
(ii). (N, pn ).E1 means, when an,k =
1
(n−k+1) log(n+1) .
pn−k
Pn ,
(iii). (N, pn , qn ).E1 means, when an,k =
where Pn =
pn−k qk
Rn ,
n
P
pk 6= 0.
k=0
where Rn =
n
P
pk qn−k 6= 0.
k=0
Let C2π denote the Banach space of all 2π-periodic and continuous functions
defined on [0, 2π] under the supremum norm.
En (f ) := inf kf − tn k is the best n-order approximation of a function f ∈ C2π
tn
(Bernstein [6]), where tn (x) =
n
X
1
a0 +
(aν cos νx + bν sin νx).
2
ν=1
BEST APPROXIMATION OF FUNCTIONS
3
Zygmund modulus of continuity of f is defined by
w(f, h) :=
sup
|f (x + t) + f (x − t) − 2f ((x)| .
o≤t≤h,x∈R
For 0 < α ≤ 1, the function space
Z(α) := {f ∈ C2π : |f (x + t) + f (x − t) − 2f (x)| = O(|t|α )}
is a Banach space under the norm k.k(α) defined by
kf k(α)
:=
sup |f (x)| + sup
0≤x≤2π
Let
Lr [0, 2π] :=
x,t
t6=0
|f (x + t) + f (x − t) − 2f (x)|
.
α
|t|
Z
f : [0, 2π] → R :
0
2π
|f (x)|r dx < ∞ , r ≥ 1,
be the space of all 2π-periodic, integrable functions.
We define the norm k.kr by
 n
o r1
R 2π
r

1
|f
(x)|
dx
for 1 ≤ r < ∞
2π 0
kf kr :=
 ess sup |f (x)| for r = ∞.
0<x<2π
For f ∈ Lr [0, 2π], r ≥ 1, the integral Zygmund modulus of continuity is defined by
 r1


 1 Z2π
r
, f orf ∈ Lr [0, 2π] where 1 ≤ r < ∞,
|f (x + t) + f (x − t) − 2f (x)| dx
wr (f, h) := sup

0<t≤h  2π
0
w(f, h) = w∞ (f, h) := sup max |f (x + t) + f (x − t) − 2f ((x)| , f orf ∈ C2π where r = ∞.
o<t≤h
x
It is known (Zygmund [1], p.45) that wr (f, h) → 0 as h → 0.
Define


 2π
 r1


Z


r
α
r


Z(α),r := f ∈ L [0, 2π] :
|f (x + t) + f (x − t) − 2f (x)| dx
= O (|t| ) .




0
The space Z(α),r , r ≥ 1, 0 < α ≤ 1 is a Banach space under the norm k.kα,r :
kf kα,r := kf kr + sup
t6=0
kf (. + t) + f (. − t) − 2f (.)kr
.
α
|t|
kf k0,r := kf kr .
The class of function Z
Z
(w)
(w)
is defined as
:= {f ∈ C2π : |f (x + t) + f (x − t) − 2f (x)| = O(w (t))}
where w is a Zygmund modulus of continuity, that is, w is a positive, non-decreasing
continuous function with the property: w(0) = 0, w(t1 + t2 ) ≤ w(t1 ) + w(t2 ).
Let w : [0, 2π] → R be an arbitrary function with w(t) > 0 for 0 < t ≤ 2π and
lim w(t) = w(0) = 0.
t→0+
We define
(
Zr(w)
:=
)
kf (. + t) + f (. − t) − 2f (.)kr
f ∈ L [0, 2π] : 1 ≤ r < ∞, sup
<∞
w (t)
t6=0
r
4
SHYAM LAL AND SHIREEN
and
(w)
kf kr
:= kf kr + sup
t6=0
kf (. + t) + f (. − t) − 2f (.)kr
, r ≥ 1.
w (t)
(w)
Zr .
kk(w)
r
Clearly
is a norm on
(w)
The completeness of the space Zr can be discussed considering the completeness of Lr (r ≥ 1).
Define
kf (. + t) + f (. − t) − 2f (.)kr
(v)
kf kr := kf kr + sup
, r ≥ 1.
v (t)
t6=0
be positive, non decreasing. Then
Let w(t)
v(t)
w(2π)
(v)
kf kr ≤ max 1,
kf k(w)
< ∞.
r
v(2π)
Thus,
Zr(w) ⊂ Zr(v) ⊂ Lr , r ≥ 1
Remarks.
(i). If we take w(t) = tα then Z (w) reduces to Zα class.
(w)
(ii). By taking w(t) = tα in Zr , it reduces to Zα,r .
(w)
(iii). If we take r → ∞ then Zr class reduces to Z (w)
We write,
φ(x, t) = f (x + t) + f (x − t) − 2f (x), ∆an,k = an,k − an,k+1 , 0 ≤ k ≤ n − 1.
n
Kn∆E =
sin (n − k + 1)( 2t )cosn−k ( 2t )
1 X
.
an,k
2π
sin ( 2t )
k=0
3. Theorems
In this paper, we prove the following theorems:
Theorem 3.1. Let the lower triangular matrix A = (an,k ) satisfying the following
conditions:
an,k ≥ 0(n = 0, 1, 2, ...; k = 0, 1, 2, ..n)
n
X
an,k = 1,
(2)
and (n + 1) an,n = O (1) .
(3)
,
k=0
n−1
X
|∆an,k | = O
k=0
1
n+1
If f : [0, 2π] → R be a 2π-periodic, Lebesgue integrable and belonging to the gener(w)
alized Zygmund class Zr , r ≥ 1; w, v be Zygmund modulus of continuity and w(t)
v(t)
be positive, non-decreasing then best approximation of f by triangular matrix-Euler
k
n
P
P
k
an,k 21k
means t∆E
=
n
ν sν of its Fourier series (1) is given by
k=0
ν=0


(v)
En (f ) = inf kt∆E
n − f kr = O 
t∆E
n
1
(n + 1)
Zπ
1
n+1

w(t) 
dt .
t2 v(t)
(4)
BEST APPROXIMATION OF FUNCTIONS
5
Theorem 3.2. Let A = (an,k ) be the lower triangular matrix satisfying the conw(t)
is non-increasing. For
ditions (2) and (3) and in addition to Theorem 3.1, tv(t)
(w)
f ∈ Zr , r ≥ 1, its best approximation by triangular matrix-Euler means t∆E
n
satisfies

w
1
n+1

log(n + 1)π  .
En (f ) = O  1
v n+1
(5)
4. Lemmas
Following Lemmas are required to prove the theorems:
Lemma 4.1. Under our conditions on (an,k ),
Kn∆E (t) = O(n + 1), for 0 < t ≤ (n + 1)−1 .
Proof. For 0 < t ≤ (n + 1)−1 , sin 2t ≥
∆E Kn (t)
=
≤
t
π,
sin nt ≤ nt, |cos t| ≤ 1, we have
n
1 X
sin (n − k + 1)( 2t )cosn−k ( 2t ) an,k
2π
sin ( 2t )
k=0
n
(n − k + 1)( 2t ) cosn−k ( 2t )
1 X
an,k
2π
( πt )
k=0
n
X
1
an,k
(n + 1)
4
≤
k=0
≤
=
1
(n + 1)
4
O(n + 1).
Lemma 4.2. Kn∆E (t) = O
1
(n+1)t2
, for (n + 1)−1 < t < π.
6
SHYAM LAL AND SHIREEN
Proof. For (n + 1)−1 < t < π, sin 2t ≥ πt , using Abel’s lemma, we get
n
1 X
t
n−k t ∆E sin
(n
−
k
+
1)(
)cos
(
)
2
2 K (t) = an,k
n
t
2π
sin
(
)
2
k=0
n−1
k
X
t
1 X
n−ν t
cos
(a
−
a
)
sin
(n
−
ν
+
1)
≤
n,k
n,k+1
2t 2
2
ν=0
k=0
n
X
t
t +an,n
sin (n − k + 1)
cosn−k
2
2 k=0
"n−1
t
sin (2n − k + 2)( 4t ) sin (n + 1)( 4t ) 1 X
+ an,n sin (n + 2)( 4 ) sin(n + 1)
≤
|∆an,k | 2t
sin ( 4t )
sin 4t
k=0
"n−1
#
t t
π X
sin(n + 1)
|∆an,k | + an,n max sin (2n − k + 2)
≤ 2
0≤k≤n
t
2
2 k=o
#
"n−1
π X
= 2
|∆an,k | + an,n
t
k=o
π
1
1
= 2 O
+O
by (3)
t
n+1
n+1
1
.
= O
(n + 1)t2
5. Proof of the Theorem3.1
Following Titchmarsh [3], sk (f ; x) of Fourier series (1) is given by
Zπ
sin k + 12 t
1
dt, k = 0, 1, 2... .
sk (f ; x) − f (x) =
φ(x, t)
2π
sin 2t
0
Then
n 1 X n
(sk (f ; x) − f (x)) =
or
En1 (x) − f (x) =
2n
k=0
k
n 1 X n sin k + 21 t
dt
2n
k
sin 2t
0
k=0
)
(
Zπ
n X
1
1
n i(k+ 1 )t
2
dt
Im
φ(x, t) n
e
2π
k
2 sin ( 2t )
1
2π
Z
1
2π
Zπ
π
φ(x, t)
k=0
0
=
=
1
2π
0
Zπ
0
=
1
2π
Zπ
0
1
φ(x, t) n
2 sin ( 2t )
φ(x, t)
(
)
n X
n ikt i t
Im
e e 2 dt
k
k=0
o
n
1
it n i 2t
dt
.e
1
+
e
I
m
2n sin ( 2t )
sin (n + 1) ( 2t ) cosn ( 2t )
dt.
φ(x, t)
sin ( 2t )
t
4
#
BEST APPROXIMATION OF FUNCTIONS
7
Now
t∆E
n (x) − f (x)
=
=
n
1
1 X
(x) − f (x)
an,k En−k
Pn
k=0
Zπ
n
X
sin (n − k + 1)( 2t ) cosn−k ( 2t )
1
φ(x, t)
an,k
dt.
2π
sin ( 2t )
k=0
0
Let
ln (x) =
t∆E
n (x)
− f (x) =
Zπ
φ(x, t)Kn∆E (t)dt.
0
Then
ln (x + y) + ln (x − y) − 2ln (x)
=
Zπ
(φ(x + y, t) + φ(x − y, t) − 2φ(x, t))Kn∆E (t)dt.
0
By generalized Minkowski’s inequality (Chui[2], p.37), we get
kln (. + y) + ln (. − y) − 2ln (.)kr
≤
Zπ
0
kφ(. + y, t) + φ(. − y, t) − 2φ(., t)kr Kn∆E (t) dt
1
=
Zn+1
0
+
Zπ
1
n+1
=
kφ(. + y, t) + φ(. − y, t) − 2φ(., t)kr Kn∆E (t) dt
kφ(. + y, t) + φ(. − y, t) − 2φ(., t)kr Kn∆E (t) dt
I1 + I2 , say
(6)
Clearly
|φ(x + y, t) + φ(x − y, t) − 2φ(x, t)|
≤ |f (x + y + t) + f (x + y − t) − 2f (x + y)|
+ |f (x − y + t) + f (x − y − t) − 2f (x − y)|
+2 |f (x + t) + f (x − t) − 2f (x)| .
Applying Minkowski’s inequality, we have
kφ(. + y, t) + φ(. − y, t) − 2φ(., t)kr
≤
kf (. + y + t) + f (. + y − t) − 2f (. + y)kr
+ kf (. − y + t) + f (. − y − t) − 2f (. − y)kr
+2 kf (. + t) + f (. − t) − 2f (.)kr
=
O (w(t)) .
(7)
8
SHYAM LAL AND SHIREEN
Also
kφ(. + y, t) + φ(. − y, t) − 2φ(., t)kr
≤
=
kf (. + t + y) + f (. + t − y) − 2f (. + t)kr
+ kf (. − t + y) − f (. − t − y) − 2f (. − t)kr
+2 kf (. + y) + f (. − y) − 2f (.)kr
O (w(y)) .
(8)
For v is positive, non decreasing, t ≤ y, we obtained
kφ(. + y, t) + φ(. − y, t) − 2φ(., t)kr
Since
w(t)
v(t)
= O (w(t))
w(t)
= O v(t)
v(t)
w(t)
= O v(y)
.
v(t)
is positive, non-decreasing, if t ≥ y, then
kφ(. + y, t) + φ(. − y, t) − 2φ(., t)kr
w(t)
v(t)
≥
w(y)
v(y) ,
so that
= O (w(y))
w(t)
.
= O v(y)
v(t)
Using lemma (4.1) and (9) we obtain
1
Z n+1
I1 =
kφ(. + y, t) + φ(. − y, t) − 2φ(., t)kr Kn∆E (t) dt
0
!
1
Z n+1
w(t)
v(y)
= O
(n + 1)dt
v(t)
0
!
1
Z n+1
w(t)
dt
= O (n + 1)v(y)
v(t)
0


1
1
Z n+1
w n+1
dt
= O (n + 1)v(y) 1
0
v n+1


1
w n+1
.
= O v(y) 1
v n+1
(9)
(10)
Also,using Lemma (4.2) and (9) we get
I2
=
Zπ
1
n+1
kφ(. + y, t) + φ(. − y, t) − 2φ(., t)kr Kn∆E (t) dt


= O
Zπ
v(y)
1
n+1
= O
1
n+1
Z

1
w(t)

dt
v(t) (n + 1)t2
π
1
n+1
!
w(t)
dt .
v(y) 2
t v(t)
(11)
BEST APPROXIMATION OF FUNCTIONS
9
By (6), (10) and (11), we have

kln (. + y) + ln (. − y) − 2ln (.)kr
w
1
n+1


= O v(y) 1
v n+1


Zπ
w(t) 
 1
+O 
dt .
v(y) 2
n+1
t v(t)
1
n+1
Thus,
kln (. + y) + ln (. − y) − 2ln (.)kr
sup
v(y)
y6=0
=

w
1
n+1


O 1
v n+1
!
Z π
1
w(t)
+O
dt . (12)
1
n + 1 n+1
t2 v(t)
Clearly
|φ(x, t)|
= |f (x + t) + f (x − t) − 2f (x)|
Applying Minkowski’s inequality, we have
kφ(., t)kr
≤ kf (x + t) + f (x − t) − 2f (x)kr
= O (w(t)) .
(13)
Using(13), Lemma (4.1),Lemma (4.2) we obtain
1
Z n+1
Z π !
∆E
kln (.)kr = tn − f r ≤
kφ(., t)kr Kn∆E (t) dt
+
1
n+1
0
=
Z
1
n+1
0
=
=
kφ(., t)kr (Kn∆E (t) dt +
Z
π
1
n+1
kφ(., t)kr Kn∆E (t) dt
!
Z π
1
w(t)
dt
w(t)dt + O
O (n + 1)
1
(n + 1) n+1
t2
0
!
Z π
1
w(t)
1
O w
dt .
+O
1
(n + 1)
(n + 1) n+1
t2
Z
!
1
n+1
Now, By (12) and (14)
(v)
kln (.)kr
kln (. + y) + ln (. − y) − 2ln (.)kr
v(y)
y6=0


Zπ
w(t) 
1
 1
= O w
+O
dt
n+1
(n + 1)
t2
= kln (.)kr + sup
1
n+1

w
1
n+1


1
+O
+O  
1
(n + 1)
v n+1
Zπ
1
n+1

w(t) 
dt .
v(t)t2
(14)
10
SHYAM LAL AND SHIREEN
Using the fact that w(t) =
w(t)
v(t) .v(t)

w
≤ v(π) w(t)
v(t) , 0 < t ≤ π, we get
1
n+1


1
+O
= O 
1
(n
+
1)
v n+1
(v)
kln (.)kr
Zπ
1
n+1

w(t) 
dt .
v(t)t2
Since w and v are zygmund modulus of continuity such that
decreasing, therefore
1
n+1
Zπ
1
n+1
w(t)
v(t)
(15)
is positive, non
1
1
Zπ
w n+1
w n+1
w(t)
dt
1
.
dt ≥ ≥
1
1
v(t)t2
n+1
t2
v n+1
2v
1
n+1
n+1
Then
w
v
1
n+1
1
n+1
=


O
Zπ
1
(n + 1)
1
n+1

w(t) 
dt .
t2 v(t)
By (15) and (16), we have
∆E
tn − f (v)
r
En (f ) = inf
t∆E
n
kt∆E
n
−
=
O
f k(v)
r
1
(n + 1)
=O
Z
π
1
n+1
1
(n + 1)
Z
!
w(t)
dt .
t2 v(t)
π
1
n+1
!
w(t)
dt .
t2 v(t)
This completes the proof of theorem 3.1.
6. Proof of the Theorem 3.2
Following the proof of the theorem 3.1,


En (f ) == O 
1
(n + 1)
Zπ
1
n+1

w(t) 
dt .
t2 v(t)
(16)
BEST APPROXIMATION OF FUNCTIONS
11
w(t)
is positive, non increasing, therefore by second mean value theorem of
Since tv(t)
integral calculus,


Zπ
w(t) 
 1
dt
En (f ) = O 
(n + 1)
t2 v(t)
1
n+1
=
=


1
Zπ
(n
+
1)w
n+1
dt 
 1
O

1
(n + 1)
t
v n+1
1
n+1

 1
w n+1
log(n + 1)π  .
O 1
v n+1
This completes the proof of theorem 3.2.
7. Applications
Following corollaries can be obtained from the Theorem 3.1.
Corollary 7.1. Let f ∈ Z(α),r , r ≥ 1, 0 < α ≤ 1 then
 1
 O
, 0 ≤ β < α < 1,
∆E
α−β
(n+1)
En (f ) = inf tn − f (β),r =
log(n+1)π
∆E
 O
tn
, β = 0, α = 1.
n+1
Proof. Taking w(t) = tα , v(t) = tβ in Theorem 3.1, proof of this corollary can be
obtained.
(w)
1
Corollary 7.2. The best approximation of a function f ∈ Zr by (H, n+1
).E1
means
n
k X
1 X k
1
1
HE
sν
tn
=
log(n + 1)
n − k + 1 2k ν=0 ν
k=0
of the Fourier series (1) is given by
(v)
En (f ) = inf tHE
− f r
n
tHE
n
Corollary 7.3. If we take an,k =
=
pn−k
Pn ,


O
n
P
w(t) 
dt .
t2 v(t)
pk 6= 0 in Theorem 3.1,
by (N, pn ).E1 means
n
k 1 X k
1 X
sν
pn−k k
Pn
2 ν=0 ν
k=0

k=0
(w)
=
1
n+1
where Pn =
then best approximation of a function f ∈ Zr
E
tN
n
1
(n + 1)
Zπ
12
SHYAM LAL AND SHIREEN
of the fourier series (1) is given by
E
(v)
En (f ) = inf tN
− f r
n
E
tN
n
Corollary 7.4. If we take an,k =
=
pn−k qk
Rn ,


O
1
(n + 1)
1
n+1
where Rn =
(w)
E
tN
n
=
1
Rn
of the fourier series (1) is given by
k=0
E
(v)
En (f ) = inf tN
− f r
n
E
tN
n
n
P

w(t) 
dt .
t2 v(t)
pk qn−k 6= 0 in Theorem
k=0
3.1, then best approximation of a function f ∈ Zr
n
X
Zπ
by (N, p, q).E1 means
1
k
sν
pn−k qk k
2 ν=0 ν
k X
=


O
1
(n + 1)
Zπ
1
n+1

w(t) 
dt .
t2 v(t)
Acknowledgement. The authors would like to express their gratitude to the
referee for his suggestions to improved the presentation of the paper.
Shyam Lal, one of the author, is thankful to DST-CIMS, Banaras Hindu University for encouragement to this work.
References
[1] A. Zygmund, Trigonometric series, 2nd rev.ed., I, Cambridge Univ. Press, Cambridge, 51(1968).
[2] C.K. Chui, An introduction to wavelets (Wavelet analysis and its applications),
Vol. 1, Academic Press, USA, 1992.
[3] E.C. Titchmarsh, The Theory of functions, Second Edition, Oxford University
Press, 1939, p. 403.
[4] G. Alexxits, Uber die Aannaherung einr stetigen Function durh die Cesaroechen
Mittel in hrer Fourier-reihe, Math. Anal. 100 (1928) 264-277.
[5] G.H. Hardy, Divergent series, first edition, Oxford Press, 1949.
[6] S.N. Bernshtein, On the best approximation of continuous functions by polynomials of given degree, in: Collected Works [in Russian], Vol. 1, Izd. Akad. Nauk
SSSR, Moscow (1952), pp. 11-104.
[7] S. Prössdorff, Zur konvergenzder Fourier reihen Hölder steliger funktionen,
Math. Nachr. 69(1975), 7-14.
[8] O. Töeplitz, Ub̈erallgemeine Lineare Mittelbuildungen, Press Mathematyezno
Fizyezne, 22 (1911), 113-119.
Shyam Lal
Department of Mathematics, Faculty of Science, Banaras Hindu University
Varanasi-221005, India
E-mail address: shyam lal@rediffmail.com
BEST APPROXIMATION OF FUNCTIONS
Shireen
Department of Mathematics, Faculty of Science, Banaras Hindu University
Varanasi-221005, India.
E-mail address: shrn.maths.bhu@gmail.com
13