Skip to content

Advertisement

  • Research
  • Open Access

Euler-Hausdorff matrix summability operator and trigonometric approximation of the conjugate of a function belonging to the generalized Lipschitz class

Journal of Inequalities and Applications20132013:59

https://doi.org/10.1186/1029-242X-2013-59

  • Received: 20 November 2012
  • Accepted: 17 January 2013
  • Published:

Abstract

In this paper, three new estimates for the degree of approximation of a function f ˜ , the conjugate of a function f belonging to classes Lipα and Lip ( ξ , r ) , r 1 , by E ( q ) Δ H summability operator of conjugate series of the Fourier series have been determined.

MSC:42A24, 41A25, 42B05, 42B08.

Keywords

  • trigonometric approximation
  • Fourier series
  • conjugate series of Fourier series
  • Hausdorff matrix summability means ( Δ H means)
  • ( E , q ) means and E ( q ) Δ H summability operator
  • generalized Lipschitz class
  • generalized Minkowski’s inequality

1 Introduction

The degree of approximation of the function f Lip α and Lip ( α , r ) , r 1 , classes have been determined by several investigators like Alexits [1], Sahney and Goel [2], Chandra [3], Qureshi [4, 5] and Qureshi and Nema [6]. After quite a good amount of work on the degree of approximation of functions by different summability means of its Fourier series, Lal and Singh [7] established the degree of approximation of conjugates of Lip ( α , r ) functions by ( C , 1 ) ( E , 1 ) product means of conjugate series of a Fourier Series in the following form.

Theorem 1.1 If f : R R is a 2π-periodic and Lip ( α , r ) function, then the degree of approximation of its conjugate function f ˜ by the ( C , 1 ) ( E , 1 ) product means of conjugate series of the Fourier series of f satisfies
M n ( f ˜ ) = min ( C E ) n 1 f ˜ r = O ( 1 ( n + 1 ) α 1 / p ) ,

where ( C E ) n 1 = 1 n + 1 k = 0 n 1 2 k j = 0 k ( k j ) s j is ( C , 1 ) ( E , 1 ) means of conjugate series of the Fourier series.

Recently Nigam and Sharma have obtained the degree of approximation by the Karmata summability method [8] and also by ( C , 1 ) ( E , q ) means of its Fourier series [9] as follows.

Theorem 1.2 If a function f is 2π-periodic, Lebesgue integrable on [ 0 , 2 π ] and belongs to Lip ( ξ ( t ) , r ) class, then its degree of approximation by ( C , 1 ) ( E , q ) summability means of its Fourier series is given by
C n 1 E n q f r = O [ ( n + 1 ) 1 r ξ ( 1 n + 1 ) ]
provided ξ ( t ) satisfies the following conditions:
{ 0 1 n + 1 ( t | ϕ ( t ) | ξ ( t ) ) r d t } 1 r = o ( 1 n + 1 )
(1.1)
and
{ 1 n + 1 π ( t δ | ϕ ( t ) | ξ ( t ) ) r d t } 1 r = O ( ( n + 1 ) δ ) ,
(1.2)

where δ is an arbitrary positive number such that s ( 1 δ ) 1 > 0 , 1 r + 1 s = 1 , 1 r , conditions (1.1) and (1.2) hold uniformly in x and C n 1 E n q is ( C , 1 ) ( E , q ) means of the Fourier series.

Working in a slightly different direction, in this paper, the degree of approximation of a function f ˜ , the conjugate of a function f belonging to classes Lipα and Lip ( ξ , r ) , r 1 , by the product summability operator E ( q ) Δ H of conjugate series of the Fourier series has been established. The results are new, sharper and better than previously known results. Furthermore, some interesting particular estimates have been also derived from the main theorems.

2 Preliminaries

Let f ( x ) be a 2π-periodic function, Lebesgue integrable on [ 0 , 2 π ] and belong to Lip ( ξ , r ) class. The Fourier series of f ( x ) is given by
f ( x ) = 1 2 a 0 + n = 1 ( a n cos n x + b n sin n x ) = 1 2 a 0 + n = 1 A n ( x )
(2.1)

with n th partial sum s n ( f ; x ) .

The series
n = 1 ( a n sin n x b n cos n x ) = n = 1 B n ( x )
(2.2)

is called ‘conjugate series’ of the Fourier series (2.1) with n th partial sum s ˜ n ( f ; x ) .

A Housdorff matrix is a lower triangular matrix with entries
h n , k = ( n k ) Δ n k μ k ,

where Δ is the forward difference operator defined by Δ μ k = μ k μ k + 1 and Δ ( Δ n μ k ) = Δ n + 1 μ k .

Let n = 0 u n be an infinite series with n th partial sum s n = k = 0 n u k .

If t n H = k = 0 n h n , k s k s as n , n = 0 u n is said to be summable to the sum s by the Hausdorff matrix summability method ( Δ H means) (Boos and Cass [10]).

The Hausdorff matrix H is regular, i.e., H preserves the limit of each convergent sequence if and only if
0 1 | d ( α ( z ) ) | < ,
where the mass function α B V [ 0 , 1 ] , α ( 0 + ) = α ( 0 ) = 0 , and α ( 1 ) = 1 . In this case, the μ n have the representation
μ n = 0 1 z n d α ( z ) .

Let E n ( q ) = 1 ( 1 + q ) n k = 0 n ( n k ) q n k s k , q > 0 .

If E n ( q ) s as n , n = 0 u n is said to be summable to s by the Euler method ( E , q ) (Hardy [11], p.180).

The ( E , q ) transform of the t n H transform defines the E ( q ) Δ H transform of ( s n ) . It is denoted by t n E H . Thus,
t n E H = 1 ( 1 + q ) n k = 0 n ( n k ) q n k t k Δ H = 1 ( 1 + q ) n k = 0 n ( n k ) q n k ν = 0 k h k , ν s ν .

If t n E H s as n , u n is said to be summable to s by the Euler-Hausdorff matrix summability means, i.e., E ( q ) Δ H means.

Thus, if the method of summability ( E , q ) is superimposed on the t n H method, another new method of summability E ( q ) Δ H is obtained.
s n s t n H s , as  n , Δ H  method is regular , E n q ( t n H ) = t n E H s , as  n , ( E , q )  method is regular , ( E q Δ H )  method is regular .
The important particular cases of E q Δ H means are as follows:
  1. 1.

    ( E , q ) ( C , δ ) if α ( z ) = k = 1 δ z k , δ 1 ,

     
  2. 2.

    ( E , 1 ) ( C , δ ) if q = 1 and α ( z ) = k = 1 δ z k , δ 1 ,

     
  3. 3.

    ( E , q ) ( C , 1 ) if α ( z ) = z ,

     
  4. 4.

    ( E , 1 ) ( C , 1 ) if q = 1 and α ( z ) = z .

     
Let of a function f : [ 0 , 2 π ] R be defined by
f = ess sup 0 x 2 π | f ( x ) |
and the degree of approximation of a function f : [ 0 , 2 π ] R by a trigonometric polynomial of order n, t n = 1 2 a 0 + ν = 1 n ( a ν cos ν x + b ν sin ν x ) , under esssup norm be defined by (Zygmund [12], p.114)
t n f = ess sup 0 x 2 π | t n ( x ) f ( x ) | .
We define the norm L r = r by f r = { 1 2 π 0 2 π | f ( x ) | r d r } 1 / r , r 1 , and let the degree of trigonometric approximation E n ( f ) be given by
E n ( f ) = min t n f t n r .
Define the norm r ( ξ ) on class L ( ξ ) r of functions by
f r ( ξ ) = sup t 0 f ( + t ) f ( ) r ξ ( | t | ) .
The degree of approximation E n ( f ; L r ( ξ ) ) of a function f belonging to Lip ( ξ , r ) class under the norm r ( u ) is given by
E n ( f ; L ( ξ ) r ) = t n f r ( u ) .

A function f Lip α if | f ( x + t ) f ( x ) | = O ( | t | α ) for 0 < α 1 (Titchmarsh [13], p.426).

f Lip ( α , r ) if { 0 2 π | f ( x + t ) f ( x ) | r d x } 1 r = O ( | t | α ) for 0 < α 1 , r 1 (def. 5.38 of McFadden [14]).

f Lip ( ξ , r ) if { 0 2 π | f ( x + t ) f ( x ) | r d x } 1 / r = O ( ξ ( t ) ) ( r 1 ), where ξ is a modulus of continuity, that is, ξ is a non-negative, non-decreasing, continuous function with the properties ξ ( 0 ) = 0 , ξ ( t 1 + t 2 ) ξ ( t 1 ) + ξ ( t 2 ) (Khan [15]).

If ξ ( t ) = t α , Lip ( ξ , r ) class coincides with the class Lip ( α , r ) , and if r , then Lip ( α , r ) reduces to Lipα class. Thus, it is obvious that Lip α Lip ( α , r ) Lip ( ξ , r ) .

Let u be a modulus of continuity such that ξ ( t ) u ( t ) is positive, non-decreasing.

Then f r ( u ) max ( 1 , ξ ( 2 π ) u ( π ) ) f r ( ξ ) . Thus, L ( ξ ) r L ( u ) r L r .

We write ψ ( x , t ) = f ( x + t ) f ( x t ) , f ˜ ( x ) = 1 2 π 0 π ψ ( x , t ) cot t 2 d t ,
( E H ˜ ) n ( t ) = 1 2 π ( 1 + q ) n k = 0 n ( n k ) q n k ν = 0 k 0 1 ( k ν ) z ν ( 1 z ) k ν d α ( z ) cos ( ν + 1 2 ) t sin t 2 .

3 Results

In this paper, three new estimates for the degree of approximation of a function f ˜ , the conjugate of a function f belonging to the classes Lipα and Lip ( ξ , r ) ( r 1 ) by E ( q ) Δ H summability operator have been determined in the following form.

Theorem 3.1 If f : [ 0 , 2 π ] R is 2π-periodic, Lebesgue integrable on [ 0 , 2 π ] and belongs to the class Lipα, then the degree of approximation of f ˜ , the conjugate of f, by E ( q ) Δ H means
t n E H = 1 ( 1 + q ) n k = 0 n ( n k ) q n k ν = 0 k h k , ν s ν
of conjugate series (2.2) of its Fourier series (2.1) satisfies, for n = 0 , 1 , 2 , 3 ,  ,
t ˜ n E H f ˜ = ess sup 0 x 2 π { t ˜ n E H ( x ) f ˜ ( x ) } = { O ( ( n + 1 ) α ) , 0 < α < 1 , O ( log ( n + 1 ) π ( n + 1 ) ) , α = 1 .
(3.1)
Theorem 3.2 Let ξ ( t ) be the modulus of continuity such that
0 v ξ ( t ) t d t = O ( ξ ( v ) ) , 0 < v < π .
(3.2)
If f : [ 0 , 2 π ] R is 2π-periodic, Lebesgue integrable on [ 0 , 2 π ] and belongs to the class Lip ( ξ , r ) ( r 1 ), then the degree of approximation of f ˜ , the conjugate of f, by E ( q ) Δ H means t ˜ n E H of conjugate series (2.2) of its Fourier series (2.1) is given by
t ˜ n E H f ˜ r = O ( 1 n + 1 1 n + 1 π ξ ( t ) t 2 d t ) for  n = 0 , 1 , 2 , 3 , .
(3.3)
Theorem 3.3 The degree of approximation E n ( f ; L r ( ξ ) ) of a function f ˜ , the conjugate of a function f belonging to Lip ( ξ , r ) class, by t ˜ n E H means of conjugate series (2.2) is given by
t ˜ n E H f ˜ r ( u ) = O ( 1 n + 1 1 n + 1 π ξ ( t ) t 2 u ( t ) d t ) ,
(3.4)
where ξ ( t ) and u ( t ) are the modulus of continuity such that
0 v ξ ( t ) t u ( t ) d t = O ( ξ ( v ) u ( v ) ) , 0 < v < π .
(3.5)

4 Lemmas

For the proof of our theorems, the following lemmas are required.

Lemma 4.1 ( E H ˜ ) n ( t ) = O ( 1 t ) for 0 < t 1 n + 1 .

Proof For 0 < t 1 n + 1 , | cos t | 1 , sin ( t / 2 ) ( t / π ) and sup 0 z 1 | α ( z ) | = N , we have
| ( E H ˜ ) n ( t ) | = | 1 2 π ( 1 + q ) n k = 0 n ( n k ) q n k ν = 0 k 0 1 ( k ν ) z ν ( 1 z ) k ν d α ( z ) cos ( ν + 1 2 ) t sin t 2 | 1 2 π ( 1 + q ) n k = 0 n ( n k ) q n k ν = 0 k 0 1 ( k ν ) z ν ( 1 z ) k ν d α ( z ) | cos ( ν + 1 2 ) t | | sin t 2 | 1 2 π ( 1 + q ) n k = 0 n ( n k ) q n k ν = 0 k 0 1 ( k ν ) z ν ( 1 z ) k ν d α ( z ) 1 t / π N 2 t ( 1 + q ) n k = 0 n ( n k ) q n k 0 1 ν = 0 k ( k ν ) z ν ( 1 z ) k ν d z = N 2 t ( 1 + q ) n k = 0 n ( n k ) q n k 0 1 ( z + 1 z ) k d z = N 2 t ( 1 + q ) n k = 0 n ( n k ) q n k = O ( 1 t ) .

 □

Lemma 4.2 ( E H ˜ ) n ( t ) = O ( 1 ( n + 1 ) t 2 ) for 1 n + 1 < t π .

Proof For 1 n + 1 < t π , sin ( n + 1 ) t 1 , sin ( t / 2 ) ( t / π ) and sup 0 z 1 | α ( z ) | = N
k = 0 n 0 1 ( n k ) z k ( 1 z ) n k e i ( k + 1 2 ) t d α ( z ) = 0 1 e i t / 2 k = 0 n ( n k ) z k ( 1 z ) n k e i k t d α ( z ) = 0 1 e i t / 2 ( 1 z ) n k = 0 n ( n k ) ( z 1 z ) k e i k t d α ( z ) = 0 1 e i t / 2 ( 1 z + z e i t ) n d α ( z ) N 0 1 e i t / 2 ( 1 z + z e i t ) n d z = N e i t / 2 ( n + 1 ) ( e i ( n + 1 ) t 1 e i t 1 ) = N ( n + 1 ) ( e i ( n + 1 ) t 1 2 i sin t 2 ) = N ( n + 1 ) ( ( cos ( n + 1 ) t 1 ) + i sin ( n + 1 ) t 2 i sin t 2 ) .
Equating real parts from both sides, we get
k = 0 n 0 1 ( n k ) z k ( 1 z ) n k cos ( k + 1 2 ) d α ( z ) = N ( n + 1 ) ( sin ( n + 1 ) t 2 sin t 2 ) = O ( 1 ( n + 1 ) t ) .
Therefore, using ν = 0 n ( n ν ) q n ν ν + 1 = O ( ( 1 + q ) n n + 1 ) , we get
( E H ˜ ) n ( t ) = 1 2 π ( 1 + q ) n k = 0 n ( n k ) q n k ν = 0 k 0 1 ( k ν ) z ν ( 1 z ) k ν d α ( z ) cos ( ν + 1 2 ) t sin t 2 1 2 t ( 1 + q ) n k = 0 n ( n k ) q n k ν = 0 k 0 1 ( k ν ) z ν ( 1 z ) k ν d α ( z ) cos ( ν + 1 2 ) t = 1 2 t ( 1 + q ) n k = 0 n ( n k ) q n k { O ( 1 ( k + 1 ) t ) } = O ( 1 2 t 2 ( 1 + q ) n k = 0 n ( n k ) q n k k + 1 ) = O ( 1 ( n + 1 ) t 2 ) .

 □

Lemma 4.3 If f Lip α , then
| ψ ( x , t ) | = O ( t α ) .

Proof

Clearly,

 □

Lemma 4.4 If f Lip ( ξ , r ) , r 1 , then
[ 0 2 π | ψ ( x , t ) | r d x ] 1 / r = O ( ξ ( t ) ) .

Proof

Using Lemma 4.3 and Minkowski’s inequality, we have

 □

Lemma 4.5
ψ ( + y , t ) ψ ( , t ) r = O ( u ( | y | ) ξ ( t ) u ( t ) ) .
Proof
| ψ ( x + y , t ) ψ ( x , t ) | = | { f ( x + y + t ) f ( x + y t ) } { f ( x + t ) f ( x t ) } | | f ( x + y + t ) f ( x + t ) | + | f ( x + y t ) f ( x t ) | .
Applying Minkowski’s inequality, we have
ψ ( + y , t ) ψ ( , t ) r f ( + y + t ) f ( + t ) r + f ( + y t ) f ( + t ) r = O ( ξ ( | y | ) ) + O ( ξ ( | y | ) ) = O ( ξ ( | y | ) ) .
(4.1)
Also, we can write
(4.2)
For t | y | , we obtain
ψ ( + y , t ) ψ ( , t ) r = O ( ξ ( t ) ) = O ( u ( | y | ) ξ ( t ) u ( t ) ) .
Since ξ ( t ) u ( t ) is positive, non-decreasing, if t | y | , then ξ ( t ) u ( t ) ξ ( | y | ) u ( | y | ) , so that
ψ ( + y , t ) ψ ( , t ) r = O ( ξ ( | y | ) ) = O ( u ( | y | ) ξ ( t ) u ( t ) ) .
(4.3)

 □

Lemma 4.6 The inequality
{ 1 2 π 0 2 π | a b g ( x , t ) d t | r d x } 1 / r a b { 1 2 π 0 2 π | g ( t , x ) | r d x } 1 / r d t

is known as generalized Minkowski’s inequality where the generalization is simply replacing a finite sum by a definite (Lebesgue) integral (see Chui [16]).

5 Proof of Theorem 3.1

The n th partial sum of conjugate series (2.2) is given by
s ˜ n ( f ; x ) f ˜ ( x ) = 1 2 π 0 π ψ ( x , t ) cos ( n + 1 2 ) t sin t 2 d t .
Denoting the Hausdorff matrix summability transform of s ˜ n ( f ; x ) by t ˜ n H ( x ) , we get
t ˜ n H ( x ) f ˜ ( x ) = k = 0 n h n , k { s ˜ k ( f ; x ) f ˜ ( x ) } = k = 0 n ( n k ) Δ n k μ k { 1 2 π 0 π ψ ( x , t ) cos ( k + 1 2 ) t sin t 2 d t } = 1 2 π 0 π ψ ( x , t ) k = 0 n ( n k ) Δ n k ( 0 1 z k d α ( z ) ) cos ( k + 1 2 ) t sin t 2 d t = 1 2 π 0 π ψ ( x , t ) k = 0 n 0 1 ( n k ) z k ( 1 z ) n k d α ( z ) cos ( k + 1 2 ) t sin t 2 d t .
The ( E , q ) transform of t ˜ n H ( x ) , i.e., the E ( q ) Δ H transform of s ˜ n ( f ; x ) denoted by t ˜ n E H ( x ) , is given by
(5.1)
Thus,
t ˜ n E C ( x ) f ˜ ( x ) 0 π | ψ ( x , t ) | | ( E H ˜ ) n ( t ) | d t = O ( 0 π t α ( ( E H ˜ ) n ( t ) ) d t ) , by Lemma 4.3 = O ( 0 1 ( n + 1 ) t α ( E H ˜ ) n ( t ) d t ) + O ( 1 ( n + 1 ) π t α ( E H ˜ ) n ( t ) d t ) = I 1 + I 2 say .
(5.2)
Applying Lemma 4.1 and Lemma 4.3, we have
I 1 = O ( 0 1 ( n + 1 ) t α ( E H ˜ ) n ( t ) d t ) = O ( 0 1 ( n + 1 ) t α 1 d t ) = O ( ( n + 1 ) α ) .
(5.3)
Now, by Lemma 4.2 and Lemma 4.3, we get
I 2 = O ( 1 ( n + 1 ) π t α ( n + 1 ) t 2 d t ) = O [ 1 ( n + 1 ) 1 ( n + 1 ) π t α 2 d t ] = { O [ ( 1 ( n + 1 ) ) ( 1 1 α ) ( 1 ( n + 1 ) α 1 π α 1 ) ] , 0 < α < 1 , O ( log ( n + 1 ) π ( n + 1 ) ) , α = 1 = { O ( ( n + 1 ) α ) , 0 < α < 1 , O ( log ( n + 1 ) π ( n + 1 ) ) , α = 1 .
(5.4)
Collecting equations (5.2) to (5.4), the result is
t ˜ n E H f ˜ = ess sup 0 x 2 π | t ˜ n E H ( x ) f ˜ ( x ) | = { O ( ( n + 1 ) α ) , 0 < α < 1 , O ( log ( n + 1 ) π ( n + 1 ) ) , α = 1 .

This completes the proof of Theorem 3.1.

6 Proof of Theorem 3.2

Following the proof of Theorem 3.1, using equation (5.1), we have
t ˜ n E H f ˜ r = [ 0 2 π | 0 π ψ ( x , t ) ( ( E H ˜ ) n ( t ) ) d t | r d x ] 1 r 0 π { 0 2 π | ψ ( x , t ) | r d x } 1 / r | ( E H ˜ ) n ( t ) | d t = O ( 0 π ξ ( t ) ( ( E H ˜ ) n ( t ) ) d t ) , by Lemma 4.6 = O ( 0 1 ( n + 1 ) ξ ( t ) ( E H ˜ ) n ( t ) d t ) + O ( 1 ( n + 1 ) π ξ ( t ) ( E H ˜ ) n ( t ) d t ) = I 1 + I 2 say .
(6.1)
Applying Lemmas 4.1 and 4.4 and condition (3.2), we obtain
I 1 = O ( 0 1 ( n + 1 ) ξ ( t ) ( E H ˜ ) n ( t ) d t ) = O ( 0 1 ( n + 1 ) ξ ( t ) t d t ) = O ( ξ ( 1 n + 1 ) ) .
(6.2)
By Lemmas 4.2 and 4.4, we have
I 2 = O ( 1 ( n + 1 ) π ξ ( t ) ( n + 1 ) t 2 d t ) = O ( 1 ( n + 1 ) 1 ( n + 1 ) π ξ ( t ) t 2 d t ) .
(6.3)
Next,
1 ( n + 1 ) 1 ( n + 1 ) π ξ ( t ) t 2 d t 1 ( n + 1 ) ξ ( 1 ( n + 1 ) ) 1 ( n + 1 ) π 1 t 2 d t = ξ ( 1 ( n + 1 ) ) { 1 1 ( n + 1 ) π } 1 2 ξ ( 1 ( n + 1 ) ) .
Then
ξ ( 1 ( n + 1 ) ) = O ( 1 ( n + 1 ) 1 ( n + 1 ) π ξ ( t ) t 2 d t ) .
(6.4)
Combining equations (6.1) to (6.4), we get
t ˜ n E H f ˜ r = O ( 1 ( n + 1 ) 1 ( n + 1 ) π ξ ( t ) t 2 d t ) .

This completes the proof of Theorem 3.2.

7 Proof of Theorem 3.3

By equation (5.2) of the proof of Theorem 3.1 and Lemma 4.6, we get
(7.1)
(7.2)
(7.3)
Using Lemma 4.1 and Lemma 4.5 and (3.5), we obtain
I 1 = 0 1 n + 1 ψ ( + y , t ) ψ ( , t ) r ( E H ) n ( t ) d t = O ( 0 1 n + 1 u ( | y | ) ξ ( t ) t u ( t ) d t ) = O ( u ( | y | ) ξ ( 1 n + 1 ) u ( 1 n + 1 ) ) .
(7.4)
Also, using Lemma 4.2 and Lemma 4.5, we have
I 2 = 1 n + 1 π ψ ( + y , t ) ψ ( , t ) r ( E H ) n ( t ) d t = O ( 1 n + 1 π u ( | y | ) ξ ( t ) u ( t ) 1 ( n + 1 ) t 2 d t ) , by Lemma 4.2 , = O ( 1 n + 1 1 n + 1 π u ( | y | ) ξ ( t ) t 2 u ( t ) d t ) .
(7.5)
By (7.3), (7.4) and (7.5), we have
l ˜ n ( + y ) l ˜ n ( ) r = O ( u ( | y | ) ξ ( 1 n + 1 ) u ( 1 n + 1 ) ) + O ( 1 n + 1 1 n + 1 π u ( | y | ) ξ ( t ) t 2 u ( t ) d t ) .
Thus,
sup y 0 l ˜ n ( + y ) l ˜ n ( ) r u ( | y | ) = O ( ξ ( 1 n + 1 ) u ( 1 n + 1 ) ) + O ( 1 n + 1 1 n + 1 π ξ ( t ) t 2 u ( t ) d t ) .
(7.6)
Since ξ and u are the modulus of continuity such that ξ ( t ) u ( t ) is positive, non-decreasing, therefore
1 n + 1 1 n + 1 π ξ ( t ) t 2 u ( t ) d t ξ ( 1 n + 1 ) u ( 1 n + 1 ) ( 1 n + 1 ) 1 n + 1 π d t t 2 ξ ( 1 n + 1 ) 2 u ( 1 n + 1 ) .
Then,
ξ ( 1 n + 1 ) u ( 1 n + 1 ) = O ( 1 n + 1 1 n + 1 π ξ ( t ) t 2 u ( t ) ) d t .
(7.7)
Thus,
l ˜ n ( ) r ( u ) = sup y 0 l ˜ n ( + y ) l ˜ n ( ) r u ( | y | ) = O ( 1 n + 1 1 n + 1 π ξ ( t ) t 2 u ( t ) d t ) .
(7.8)

Thus, Theorem 3.3 is completely established.

8 Applications

The following corollaries can be derived from our theorem.

Corollary 8.1 Let ξ ( t ) = t α β , 0 β < α 1 , and f Lip ( ξ , r ) , r 1 . Then Theorem  3.2 becomes
t ˜ n E H f ˜ r = { O ( ( n + 1 ) β α ) , 0 β < α < 1 , O ( log ( n + 1 ) π ( n + 1 ) ) , β = 0 , α = 1 .
Proof Putting ξ ( t ) = t α β , 0 β < α 1 , in (3.3), we have
t ˜ n E H f ˜ r = O ( 1 ( n + 1 ) 1 ( n + 1 ) π t α β 2 d t ) = O { ( ( n + 1 ) β α ) , 0 β < α < 1 , O ( log ( n + 1 ) π ( n + 1 ) ) , β = 0 , α = 1 .

 □

Corollary 8.2 Let 0 β < α 1 and f L ( α ) r r 1 . Then
t ˜ n E H f ˜ r ( β ) = { O ( ( n + 1 ) β α ) , 0 β < α < 1 , O ( log ( n + 1 ) π n + 1 ) , β = 0 , α = 1 .
Proof Putting ξ ( t ) = t α , u ( t ) = t β , 0 β < α 1 , in (3.4), we have
t ˜ n E H f ˜ r ( β ) = O ( 1 ( n + 1 ) 1 n + 1 π t α β 2 d t ) = { O ( 1 ( n + 1 ) ( ( n + 1 ) β α + 1 ( 1 π ) β α + 1 β α + 1 ) ) , 0 β < α 1 , O ( log ( n + 1 ) π n + 1 ) , β = 0 , α = 1 = { O ( ( n + 1 ) β α ) , 0 β < α 1 , O ( log ( n + 1 ) π n + 1 ) , β = 0 , α = 1 .

 □

Corollary 8.3 If f : [ 0 , 2 π ] R is 2π-periodic, Lebesgue integrable on [ 0 , 2 π ] and belongs to the class Lip ( ξ , r ) ( r 1 ) and condition (3.2) holds, then the degree of approximation of a function f ˜ , the conjugate of f, by ( E , 1 ) ( C , 1 ) means
t ˜ n E 1 C 1 = 1 2 n ν = 0 n ( n ν ) 1 ν + 1 k = 0 ν s ˜ k
of conjugate series (2.2) is given by
t ˜ n E 1 C 1 f ˜ r = O ( 1 n + 1 1 n + 1 π ξ ( t ) t 2 d t ) .
Corollary 8.4 If f : [ 0 , 2 π ] R is 2π-periodic, Lebesgue integrable on [ 0 , 2 π ] and belongs to the class Lip ( α , r ) , r 1 , 0 < α 1 , then the degree of approximation of its conjugate function f ˜ by t n E H means of conjugate series (2.2) satisfies
t ˜ n E H f ˜ r = { O ( ( n + 1 ) α ) , 0 < α < 1 , O ( log ( n + 1 ) π ( n + 1 ) ) , α = 1 .
Corollary 8.5 In addition to the conditions of Theorem  3.2, if also ( ξ ( t ) t ) is non-increasing, then
t ˜ n E H f ˜ r = O ( ξ ( 1 n + 1 ) log ( n + 1 ) π ) .
Corollary 8.6 In addition to the conditions of Theorem  3.3, if also ( ξ ( t ) t u ( t ) ) is non-increasing, then
t ˜ n E H f ˜ r ( u ) = O ( ξ ( 1 n + 1 ) u ( 1 n + 1 ) log ( n + 1 ) π ) .

9 Remarks

  1. 1.

    If we take ξ ( t ) = t α in Theorem 3.2, then it reduces to Corollary 8.4.

     
  2. 2.

    The independent proofs of Corollaries 8.3 to 8.6 can be developed along the same lines as the theorems.

     
  3. 3.

    Some interesting estimates parallel to the main theorems and Corollary 8.4, Corollary 8.5 and Corollary 8.6 can also be obtained for ( E , q ) ( C , δ ) , ( E , 1 ) ( C , δ ) , ( E , q ) ( C , 1 ) , ( E , 1 ) ( C , 1 ) summability operators.

     
  4. 4.

    The degree of approximation of functions of Lip ( ξ ; r ) class has been determined without using conditions like (1.1) and (1.2) of Theorem 1.2.

     

Declarations

Acknowledgements

Authors are thankful to DST-CIMS, Banaras Hindu University, Varanasi for encouragement to this work.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, Banaras Hindu University, Varanasi, 221005, India

References

  1. Alexits G: Über die Annagerung einer stetigen Fundtions durch die Cesàrochen Mittel ihrer Fourieroeihe. Math. Ann. 1928, 100: 264–277. 10.1007/BF01448845MathSciNetView ArticleGoogle Scholar
  2. Sahney BN, Goel DS: On the degree of approximation of continuous functions. Ranchi Univ. Math. J. 1973, 4: 50–53.MathSciNetGoogle Scholar
  3. Chandra P: On degree of approximation of functions belonging to the Lipschitz class. Nanta Math. 1975, 8(1):88–91.MathSciNetGoogle Scholar
  4. Qureshi K: On degree of approximation of a periodic function f by almost Nörlund means. Tamkang J. Math. 1981, 12(1):35–38.MathSciNetGoogle Scholar
  5. Qureshi K: On degree of approximation of functions belonging to the class Lip α . Indian J. Pure Appl. Math. 1982, 13(8):898–903.MathSciNetGoogle Scholar
  6. Qureshi K, Nema HK: A class of functions and their degree of approximation. Ganita 1990, 41(1):37–42.MathSciNetGoogle Scholar
  7. Lal S, Singh PN:Degree of approximation of conjugate of Lip ( α , p ) function by ( C , 1 ) ( E , 1 ) means of conjugate series of a Fourier series. Tamkang J. Math. 2002, 33(3):269–274.MathSciNetGoogle Scholar
  8. Nigam HK, Sharma K: A study on degree of approximation by Karamata summability method. J. Inequal. Appl. 2011., 2011: Article ID 85Google Scholar
  9. Nigam HK, Sharma K:Degree of approximation of a class of functions by ( C , 1 ) ( E , 1 ) means of Fourier series. IAENG. Int. J. Appl. Math. 2011, 41(2):128–132.MathSciNetGoogle Scholar
  10. Boos J, Cass P: Classical and Modern Methods in Summability. Oxford University Press, New York; 2000.Google Scholar
  11. Hardy GH: Divergent Series. 1st edition. Oxford University Press, London; 1949.Google Scholar
  12. Zygmund A I. In Trigonometric Series. 2nd edition. Cambridge University Press, Cambridge; 1968.Google Scholar
  13. Titchmarsh EC: The Theory of Functions. 2nd edition. Oxford University Press, London; 1939.Google Scholar
  14. McFaddin L: Absolute Nörlund summability. Duke Math. J. 1942, 9: 168–207. 10.1215/S0012-7094-42-00913-XMathSciNetView ArticleGoogle Scholar
  15. Khan, HH: Approximation by class of functions. Thesis, Aligarh Muslim University, Aligarh, India (1974)Google Scholar
  16. Chui CK 1. In An Introduction to Wavelets (Wavelet Analysis and Its Applications). Academic Press, San Diego; 1992.Google Scholar

Copyright

© Lal and Mishra; licensee Springer 2013

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Advertisement