Open Access

L r -Approximation of signals (functions) belonging to weighted W ( L r , ξ ( t ) ) -class by C 1 N p summability method of conjugate series of its Fourier series

  • Vishnu Narayan Mishra1Email author,
  • Vaishali Sonavane1 and
  • Lakshmi Narayan Mishra2
Journal of Inequalities and Applications20132013:440

https://doi.org/10.1186/1029-242X-2013-440

Received: 23 January 2013

Accepted: 30 July 2013

Published: 18 September 2013

Abstract

Recently, Lal (Appl. Math. Comput. 209:346-350, 2009) has determined the degree of approximation of a function belonging to Lipα and weighted W ( L r , ξ ( t ) ) -classes using product C 1 N p summability with non-increasing weights { p n } . In this paper, we determine the degree of approximation of function f ˜ , conjugate to a 2π-periodic function f belonging to weighted W ( L r , ξ ( t ) ) -class by dropping the monotonicity on the generating sequence { p n } with a new (proper) set of conditions, which in turn generalizes the results of Mishra et al. (Bull. Math. Anal. Appl., 2013) on Lip ( ξ ( t ) , r ) -class and rectifies (removes) the errors of Mishra et al. (Mat. Vesn., 2013). Few examples and applications are also highlighted in this manuscript.

MSC: 42B05, 42B08, 40G05, 41A10.

Keywords

generalized Lipschitz W ( L r , ξ ( t ) ) ( r 1 )-class of functions conjugate Fourier series degree of approximation C 1 means N p means product summability C 1 N p transform

1 Introduction

Approximation by trigonometric polynomials is at the heart of approximation theory. The most important trigonometric polynomials used in the approximation theory are obtained by linear summation methods of Fourier series of 2π-periodic functions on the real line (i.e., Cesàro means, Nörlund means and Product Cesàro-Nörlund means, etc.). Much of the advance in the theory of trigonometric approximation is due to the periodicity of the functions. The method of summability considered here was first introduced by Woronoi [1]. Summability techniques were also applied on some engineering problems like, Chen and Jeng [2] implemented the Cesàro sum of order ( C , 1 ) and ( C , 2 ) , in order to accelerate the convergence rate to deal with the Gibbs phenomenon, for the dynamic response of a finite elastic body subjected to boundary traction. Chen et al. [3] applied regularization with Cesàro sum technique for the derivative of the double layer potential. Similarly, Chen and Hong [4] used Cesàro sum regularization technique for hyper singularity of dual integral equation. The degree of approximation of functions belonging to Lipα, Lip ( α , r ) , Lip ( ξ ( t ) , r ) and W ( L r , ξ ( t ) ) ( r 1 )-classes by Nörlund ( N p ) matrices and general summability matrices has been proved by various investigators like Govil [5], Khan [6], Qureshi [7], Mohapatra and Chandra [8], Leindler [9], Rhoades et al. [10], Guven and Israfilov [11], Bhardwaj and Gupta [12] and Mishra et al. [1320]. Here, Lal [21] has assumed monotonicity on the generating sequence { p n } to prove their theorems. The approximation of function f ˜ , conjugate to a periodic function f W ( L r , ξ ( t ) ) ( r 1 ) using product ( C 1 N p ) -summability has not been studied so far. In this paper, we obtain a new theorem on the degree of approximation of function f ˜ , conjugate to a periodic function f W ( L r , ξ ( t ) ) -class using semi-monotonicity on the generating sequence { p n } and a proper set of the conditions.

A bidiagonal matrix is a matrix with non-zero entries along the main diagonal and either the diagonal above or the diagonal below. This means there are exactly two non-zero diagonals in the matrix.

When the diagonal above the main diagonal has the non-zero entries, the matrix is upper bidiagonal. When the diagonal below the main diagonal has the non-zero entries, the matrix is lower bidiagonal.

For example, the following matrix is upper bidiagonal:
( 1 4 0 0 0 4 1 0 0 0 3 4 0 0 0 3 )
and the following matrix is lower bidiagonal:
( 1 0 0 0 2 4 0 0 0 3 3 0 0 0 4 3 ) .
Let n = 0 a n be a given infinite series with the sequence of n th partial sums { s n } . Let { p n } be a non-negative sequence of constants, real or complex, and let us write
P n = k = 0 n p k 0 n 0 , p 1 = 0 = P 1  and  P n  as  n .
The sequence to sequence transformation t n N = ν = 0 n p n ν s ν / P n defines the sequence { t n N } of Nörlund means of the sequence { s n } , generated by the sequence of coefficients { p n } . The series n = 0 a n is said to be summable N p to the sum s if lim n t n N exists and is equal to s. In the special case in which
p n = ( n + α 1 α 1 ) = ( n + α ) ( n + 1 ) ( α ) ( α > 0 ) .

The Nörlund summability N p reduces to the familiar C α summability.

The product of C 1 summability with an N p summability defines C 1 N p summability. Thus, the C 1 N p mean is given by t n C N = 1 n + 1 k = 0 n P k 1 ν = 0 k p k ν s ν . If t n C N s as n , then the infinite series n = 0 a n or the sequence { s n } is said to be summable C 1 N p to the sum s if lim n t n C N exists and is equal to s.
s n s N p ( s n ) = t n N = P n 1 ν = 0 n p n ν s ν s , as  n , N p  method is regular , C 1 ( N p ( s n ) ) = t n C N s , as  n , C 1  method is regular , C 1 N p  method is regular .
Let f ( x ) be a 2π-periodic function and Lebesgue integrable. The Fourier series of f ( x ) is given by
f ( x ) a 0 2 + n = 1 ( a n cos n x + b n sin n x ) n = 0 A n ( x )
(1.1)

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

The conjugate series of Fourier series (1.1) is given by
n = 1 ( b n cos n x a n sin n x ) n = 1 B n ( x ) .
(1.2)
A function f ( x ) Lip α if
| f ( x + t ) f ( x ) | = O ( | t | α ) for  0 < α 1 , t > 0
and f ( x ) Lip ( α , r ) , [6] for 0 x 2 π , if
f ( x + t ) f ( x ) r = ( 0 2 π | f ( x + t ) f ( x ) | r d x ) 1 / r = O ( | t | α ) , 0 < α 1 , r 1 , t > 0 .
f ( x ) Lip ( ξ ( t ) , r ) if
f ( x + t ) f ( x ) r = ( 0 2 π | f ( x + t ) f ( x ) | r d x ) 1 / r = O ( ξ ( t ) ) , r 1 , t > 0 .
f W ( L r , ξ ( t ) ) , [18, 20] if
ω r ( t ; f ) = [ f ( x + t ) f ( x ) ] sin β ( x / 2 ) r = ( 0 2 π | f ( x + t ) f ( x ) | r sin β r ( x / 2 ) d x ) 1 / r = O ( ξ ( t ) ) , β 0 , r 1 , t > 0 ,

where ξ ( t ) is positive increasing function of t.

If β = 0 , then W ( L r , ξ ( t ) ) reduces to the class Lip ( ξ ( t ) , r ) , if ξ ( t ) = t α ( 0 < α 1 ), then Lip ( ξ ( t ) , r ) class coincides with the class Lip ( α , r ) , and if r , then Lip ( α , r ) reduces to the class Lipα.

L -norm of a function f : R R is defined by f = sup { | f ( x ) | : x R } .

L r -norm of f is defined by f r = ( 0 2 π | f ( x ) | r d x ) 1 / r , r 1 .

The degree of approximation of a function f : R R by trigonometric polynomial t n of order n under sup norm is defined by [22]
t n f = sup { | t n ( x ) f ( x ) | : x R } ,

and E n ( f ) of a function f L r is given by E n ( f ) = min n t n f r .

The conjugate function f ˜ ( x ) is defined for almost every x by
f ˜ ( x ) = 1 2 π 0 π ψ ( t ) cot t / 2 d t = lim h 0 ( 1 2 π h π ψ ( t ) cot t / 2 d t ) .

We note that t n N and t n C N are also trigonometric polynomials of degree (or order) n.

Abel’s transformation: The formula
k = m n u k v k = k = m n 1 U k ( v k v k + 1 ) U m 1 v m + U n v n ,
(1.3)

where 0 m n , U k = u 0 + u 1 + u 2 + + u k , if k 0 , U 1 = 0 , which can be verified, is known as Abel’s transformation and will be used extensively in what follows.

If v m , v m + 1 , , v n are non-negative and non-increasing, the left-hand side of (1.3) does not exceed 2 v m max m 1 k n | U k | in absolute value. In fact,
| k = m n u k v k | max | U k | { k = m n 1 ( v k v k + 1 ) + v m + v n } = 2 v m max | U k | .
(1.4)
We write throughout
ψ ( t ) = f ( x + t ) f ( x t ) , ϕ ( t ) = f ( x + t ) 2 f ( x ) + f ( x t ) , V n = 1 2 π ( n + 1 ) k = 0 n P k 1 ν = 0 k ( ν + 1 ) | p ν p ν 1 | , M ˜ n ( t ) = 1 2 π ( n + 1 ) k = 0 n P k 1 ν = 0 k p ν cos ( k v + 1 / 2 ) t sin t / 2
(1.5)

τ = [ 1 / t ] , where τ denotes the greatest integer not exceeding 1 / t . Furthermore, C denotes an absolute positive constant, not necessarily the same at each occurrence.

We note that the series, conjugate to a Fourier series, is not necessarily a Fourier series. Hence a separate study of conjugate series is desirable and attracted the attention of researchers.

2 Known theorem

Lal [21] has obtained the degree of approximation of the functions belonging to W ( L r , ξ ( t ) ) -class by C 1 N p means with monotonicity on the generating sequence { p n } . He proved the following.

Theorem 2.1 If f ( x ) is a 2π-periodic function and Lebesgue integrable on [ 0 , 2 π ] and is belonging to W ( L r , ξ ( t ) ) -class, then its degree of approximation by C 1 N p means of its Fourier series (1.1) is given by
t n C N f r = O ( ( n + 1 ) β + 1 / r ξ ( 1 n + 1 ) ) ,
(2.1)
provided ξ ( t ) satisfies the following conditions:
{ ξ ( t ) / t }  be a decreasing sequence ,
(2.2)
( 0 1 / ( n + 1 ) ( t | ϕ ( t ) | ξ ( t ) ) r sin β r t d t ) 1 / r = O ( ( n + 1 ) 1 ) ,
(2.3)
( 1 / ( n + 1 ) π ( t δ | ϕ ( t ) | ξ ( t ) ) r d t ) 1 / r = O ( ( n + 1 ) δ ) ,
(2.4)

where δ is an arbitrary number such that s ( 1 δ ) 1 > 0 , r 1 + s 1 = 1 , 1 r , conditions (2.3) and (2.4) hold uniformly in x.

Remark 2.2 The condition 1 / sin β ( t ) = O ( 1 / t β ) , 1 / ( n + 1 ) t π used by Lal [[21], pp.349-350] in writing the proof of Theorem 2.1 is not valid since sin t 0 as t π .

Remark 2.3 There is a fatal error in the proof of Theorem 2.1 of Lal [[21], p.349], in calculating
[ t β s s + 1 β s s + 1 ] ϵ 1 / ( n + 1 ) ,

note that β s s + 1 < 0 . Therefore, one has 1 β s + s 1 [ 1 ϵ β s + s 1 ( n + 1 ) β s + s 1 ] , which need not be O ( ( n + 1 ) β s + s 1 ) , since ϵ might be O ( 1 / n γ ) for some γ > 1 .

3 Main theorem

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 approximations have assumed important new dimensions due to their wide applications in signal analysis [23], in general and in digital signal processing [24] in particular, in view of the classical Shannon sampling theorem. Mittal et al. [25, 26] have obtained many interesting results on TFA using summability methods without monotonicity on the rows of the matrix T: a digital filter. Broadly speaking, signals are treated as functions of one variable, and images are represented by functions of two variables. Very recently, Mishra et al. [13, 18] obtained the degree of approximation of conjugate of a function using C 1 N p product summability method of its conjugate series of its Fourier series in Lipschitz and weighted spaces, respectively. Therefore, the purpose of the present paper is to generalize the results of Mishra et al. [13] on the degree of approximation of a function f ˜ ( x ) , conjugate to a 2π-periodic function f belonging to weighted W ( L r , ξ ( t ) ) -class by C 1 N p means of conjugate series of its Fourier series by dropping the monotonicity condition on the generating sequence { p n } with the help of a new (proper) set of conditions to rectify the errors of Mishra et al. [18]. More precisely, we state our main theorem as follows.

Theorem 3.1 If f ˜ ( x ) , conjugate to a 2π-periodic function f belonging to W ( L r , ξ ( t ) ) -class, then its degree of approximation by C 1 N p means of conjugate series of Fourier series (1.2) is given by
t ˜ n C N f ˜ r = O ( ( n ) β / 2 + r / 2 ξ ( 1 n ) ) ,
(3.1)
provided { p n } satisfies
V n < C ,
(3.2)
and ξ ( t ) satisfies the following conditions:
{ ξ ( t ) / t }  is non-increasing in  t ,
(3.3)
( 0 π / n ( | ψ ( t ) | ξ ( t ) ) r sin β r ( t / 2 ) d t ) 1 / r = O ( 1 ) ,
(3.4)
( π / n π ( t δ | ψ ( t ) | ξ ( t ) ) r d t ) 1 / r = O ( ( n ) δ ) ,
(3.5)

where δ is an arbitrary number such that s ( β δ ) 1 > 0 , r 1 + s 1 = 1 , 1 r , conditions (3.4) and (3.5) hold uniformly in x.

Note 3.2 ξ ( π n ) π ξ ( 1 n ) , for ( π n ) ( 1 n ) .

Note 3.3 Condition V n < C n p n < C P n , [27].

Note 3.4 The product transform C 1 N p plays an important role in signal theory as a double digital filter [14] and theory of machines in mechanical engineering [14].

4 Lemmas

We need the following lemmas for the proof of our theorem.

Lemma 4.1 | M ˜ n ( t ) | = O [ 1 / t ] for 0 < t π / n .

Proof For 0 < t π / n , sin ( t / 2 ) ( t / π ) and | cos n t | 1 .
| M ˜ n ( t ) | = | 1 2 π ( n + 1 ) k = 0 n P k 1 v = 0 k p ν cos ( k v + 1 / 2 ) t sin t / 2 | 1 2 π ( n + 1 ) k = 0 n P k 1 v = 0 k p ν | cos ( k v + 1 / 2 ) t | | sin t / 2 | 1 2 t ( n + 1 ) k = 0 n P k 1 v = 0 k p ν = 1 2 t ( n + 1 ) k = 0 n P k 1 P k = O [ τ ] .

This completes the proof of Lemma 4.1. □

Lemma 4.2 Let { p n } be a non-negative sequence and satisfy (3.2), then
| M ˜ n ( t ) | = O ( τ ) + O ( τ 2 n ) ( k = τ n P k 1 ν = 0 k 1 | Δ p ν | ) uniformly in  0 < t π .
(4.1)
Proof We have
M ˜ n ( t ) = 1 2 π ( n + 1 ) k = 0 n P k 1 v = 0 k p ν cos ( k v + 1 / 2 ) t sin t / 2 = 1 2 π ( n + 1 ) ( k = 0 τ 1 + k = τ n ) P k 1 v = 0 k p ν cos ( k v + 1 / 2 ) t sin t / 2 = ( M ˜ n ( t ) ) 1 + ( M ˜ n ( t ) ) 2 , say ,
(4.2)
where
| ( M ˜ n ( t ) ) 1 | = | 1 2 π ( n + 1 ) k = 0 τ 1 P k 1 v = 0 k p ν cos ( k v + 1 / 2 ) t sin t / 2 | 1 2 π ( n + 1 ) k = 0 τ 1 P k 1 v = 0 k p ν | cos ( k v + 1 / 2 ) t | | sin t / 2 | 1 2 t ( n + 1 ) k = 0 τ 1 P k 1 v = 0 k p ν = O ( τ 2 ( n + 1 ) ) ,
(4.3)
and using Abel’s transformation and sin ( t / 2 ) ( t / π ) , for 0 < t π , we get
| ( M ˜ n ( t ) ) 2 | = | 1 2 π ( n + 1 ) k = τ n P k 1 v = 0 k p ν cos ( k v + 1 / 2 ) t sin t / 2 | 1 2 t ( n + 1 ) k = τ n P k 1 { v = 0 k 1 | Δ p ν | | ( γ = 0 ν cos ( k γ + 1 / 2 ) t ) | + | ( γ = 0 k cos ( k γ + 1 / 2 ) t ) | p k } = O ( t 1 ) 2 t ( n + 1 ) ( k = τ n P k 1 v = 0 k 1 | Δ p ν | + k = τ n P k 1 p k )
by virtue of the fact that k = λ μ exp ( i k t ) = O ( t 1 ) , 0 λ k μ .
| ( M ˜ n ( t ) ) 2 | = O ( τ 2 ( n + 1 ) ) ( k = τ n P k 1 v = 0 k 1 | Δ p ν | + k = τ n P k 1 p k k k ) = O ( τ 2 ( n + 1 ) ) ( k = τ n P k 1 v = 0 k 1 | Δ p ν | + ( n + 1 ) τ ) = O ( τ ) + O ( τ 2 ( n + 1 ) ) k = τ n P k 1 v = 0 k 1 | Δ p ν | , | ( M ˜ n ( t ) ) 2 | = O ( τ ) + O ( τ 2 n ) k = τ n P k 1 v = 0 k 1 | Δ p ν |
(4.4)

in view of Note 3.3. Combining (4.2), (4.3) and (4.4) yields (4.1).

This completes the proof of Lemma 4.2. □

5 Proof of theorem

Let s ˜ n ( f ; x ) denote the partial sum of series (1.2), we have
s ˜ n ( f ; x ) f ˜ ( x ) = 1 2 π 0 π ψ ( t ) cos ( n + 1 / 2 ) t sin t / 2 d t .
Denoting C 1 N p means of s ˜ n ( f ; x ) by t ˜ n C N , we write
t ˜ n C N ( x ) f ˜ ( x ) = 0 π ψ ( t ) 1 2 π ( n + 1 ) k = 0 n P k 1 ν = 0 k p ν cos ( k v + 1 / 2 ) t sin t / 2 d t = 0 π ψ ( t ) M ˜ n ( t ) d t = [ 0 π / n + π / n π ] ψ ( t ) M ˜ n ( t ) d t = I 1 + I 2 ( say ) .
(5.1)

Clearly, | ψ ( x + t ) ψ ( t ) | | f ( u + x + t ) f ( u + x ) | + | f ( u x t ) f ( u x ) | .

Hence, by Minkowski’s inequality,
{ 0 2 π | ( ψ ( x + t ) ψ ( t ) ) sin β ( x / 2 ) | r d x } 1 / r { 0 2 π | ( f ( u + x + t ) f ( u + x ) ) sin β ( x / 2 ) | r d x } 1 / r + { 0 2 π | ( f ( u x t ) f ( u x ) ) sin β ( x / 2 ) | r d x } 1 / r = O ( ξ ( t ) ) .

Then f W ( L r , ξ ( t ) ) ψ W ( L r , ξ ( t ) ) .

Using Hölder’s inequality, ψ ( t ) W ( L r , ξ ( t ) ) , condition (3.4), sin ( t / 2 ) ( t / π ) , for 0 < t π , Lemma 4.1, Note 3.2 and second mean value theorem for integrals, we have
| I 1 | [ 0 π / n ( | ψ ( t ) sin β ( t / 2 ) | ξ ( t ) ) r d t ] 1 / r [ 0 π / n ( ξ ( t ) | M ˜ n ( t ) | sin β ( t / 2 ) ) s d t ] 1 / s = O ( 1 ) [ 0 π / n ( ξ ( t ) t 1 + β ) s d t ] 1 / s = O { ξ ( π n ) } [ h π / n ( 1 t 1 + β ) s d t ] 1 / s , as  h 0 = O ( ( n ) β / 2 + r / 2 ξ ( 1 n ) ) , r 1 + s 1 = 1 .
(5.2)
Using Lemma 4.2, we have
| I 2 | = O [ π / n π | ψ ( t ) | t d t ] + O [ π / n π | ψ ( t ) | t n ( τ k = τ n P k 1 ν = 0 k 1 | p ν | ) d t ] = O ( I 21 ) + O ( I 22 ) .
Using Hölder’s inequality, conditions (3.3) and (3.5), | sin t | 1 , sin ( t / 2 ) ( t / π ) , for 0 < t π , Note 3.2 and second mean value theorem for integrals, we have
| I 21 | [ π / n π ( t δ | ψ ( t ) | sin β ( t / 2 ) ξ ( t ) ) r d t ] 1 / r [ π / n π ( ξ ( t ) t δ + 1 sin β ( t / 2 ) ) s d t ] 1 / s = O ( ( n ) δ ) [ π / n π ( ξ ( t ) t δ + 1 + β ) s d t ] 1 / s = O { ( n ) δ } [ 1 / π n / π ( ξ ( 1 / y ) y δ 1 β ) s d y y 2 ] 1 / s = O ( ( n ) δ ξ ( π / n ) π / n ) [ 1 / π n / π ( d y y ( δ β ) s + 2 ) ] 1 / s = O ( ( n ) δ + 1 ξ ( 1 n ) ) ( ( n ) ( β δ ) s 1 ( π ) ( β + δ ) s + 1 ( β δ ) s 1 ) 1 / s = O ( ( n ) β / 2 + r / 2 ξ ( 1 n ) ) .
(5.3)
Similarly, as conditions (3.2), (3.3) and (3.5) above, | sin t | 1 , sin ( t / 2 ) ( t / π ) , for 0 < t π , Note 3.2 and second mean value theorem for integrals, we have
| I 22 | [ π / n π ( t δ | ψ ( t ) | sin β ( t / 2 ) ξ ( t ) ) r d t ] 1 / r × [ π / n π ( ξ ( t ) t δ + 1 sin β ( t / 2 ) 1 n ( τ k = τ n P k 1 ν = 0 k 1 | p ν | ) ) s d t ] 1 / s = O ( ( n ) δ n ) [ π / n π ( ξ ( t ) t δ + 1 + β ( τ k = τ n P k 1 ν = 0 k 1 | p ν | ) ) s d t ] 1 / s = O ( ( n ) δ n ) [ π / n π ( ξ ( t ) t δ + 1 + β ( k = τ n P k 1 ν = 0 k 1 ( ν + 1 ) | p ν | ) ) s d t ] 1 / s = O ( ( n ) δ n ) [ π / n π ( ξ ( t ) t δ + 1 + β ( k = 0 n P k 1 ν = 0 k ( ν + 1 ) | p ν | ) ) s d t ] 1 / s = O ( ( n ) δ n ) [ π / n π ( ξ ( t ) t δ + 1 + β V n 2 π ( n ) ) s d t ] 1 / s = O ( ( n ) δ ) [ π / n π ( ξ ( t ) t δ + 1 + β ) s d t ] 1 / s = O { ( n ) δ } [ 1 / π n / π ( ξ ( 1 / y ) y δ 1 β ) s d y y 2 ] 1 / s = O ( ( n ) β / 2 + r / 2 ξ ( 1 n ) ) .
(5.4)
Collecting (5.1)-(5.4), we have
| t ˜ n C N f ˜ | = O ( ( n ) β / 2 + r / 2 ξ ( 1 n ) ) .
(5.5)
Now, using the L r -norm of a function, we get
t ˜ n C N f ˜ r = { 0 2 π | t ˜ n C N ( x ) f ˜ ( x ) | r d x } 1 / r = O ( 0 2 π ( ( n ) β / 2 + r / 2 ξ ( 1 n ) ) r d x ) 1 / r = O ( ( n ) β / 2 + r / 2 ξ ( 1 n ) ( 0 2 π d x ) 1 / r ) = O ( ( n ) β / 2 + r / 2 ξ ( 1 n ) ) .

This completes the proof of Theorem 3.1.

6 Applications

The theory of approximation is a very extensive field, which has various applications. As mentioned in [24], the Lp-space in general and L 2 and L in particular play an important role in the theory of signals and filters. From the point of view of the applications, the Sharper estimates of infinite matrices are useful to get bounds for the lattice norms (which occur in solid state physics) of matrix-valued functions and enable to investigate perturbations of matrix-valued functions and compare them. Some interesting applications of the Cesàro summability can be seen in [24]. The following corollaries can be derived from Theorem 3.1.

Corollary 1 If β = 0 , then the generalized weighted Lipschitz W ( L r , ξ ( t ) ) ( r 1 )-class reduces to the class Lip ( ξ ( t ) , r ) , and the degree of approximation of a function f ˜ ( x ) , conjugate to a 2π-periodic function f belonging to the class Lip ( ξ ( t ) , r ) , is given by
| t ˜ n C N ( f ; x ) f ˜ ( x ) | = O ( n r / 2 ξ ( 1 / n ) ) .
(6.1)
Proof The result follows by setting β = 0 in Theorem 3.1, we have
t ˜ n C N ( f ; x ) f ˜ ( x ) r = { 0 2 π | t ˜ n C N ( f ; x ) f ˜ ( x ) | r d x } 1 / r = O ( n r / 2 ξ ( 1 / n ) ) , r 1 .
Thus, we get
| t ˜ n C N ( f ; x ) f ˜ ( x ) | { 0 2 π | t ˜ n C N ( x ) f ˜ ( x ) | r d x } 1 / r = O ( n r / 2 ξ ( 1 / n ) ) , r 1 .

This completes the proof of Corollary 1. □

Corollary 2 If β = 0 , ξ ( t ) = t α , 0 < α 1 , then the generalized weighted Lipschitz W ( L r , ξ ( t ) ) ( r 1 )-class reduces to the class Lip ( α , r ) , ( 1 / r ) < α < 1 and the degree of approximation of a function f ˜ ( x ) , conjugate to a 2π-periodic function f belonging to the class Lip ( α , r ) , is given by
| t ˜ n C N ( f ; x ) f ˜ ( x ) | = O ( ( n ) α / 2 + r / 2 ) .
Proof Putting β = 0 , ξ ( t ) = t α , 0 < α 1 in Theorem 3.1, we have
t ˜ n C N ( f ; x ) f ˜ ( x ) r = { 0 2 π | t ˜ n C N ( f ; x ) f ˜ ( x ) | r d x } 1 / r
or,
O ( n β / 2 + r / 2 ξ ( 1 n ) ) = { 0 2 π | t ˜ n C N ( f ; x ) f ˜ ( x ) | r d x } 1 / r
or,
O ( 1 ) = { 0 2 π | t ˜ n C N ( f ; x ) f ˜ ( x ) | r d x } 1 / r O ( 1 n β / 2 + r / 2 ξ ( 1 n ) ) ,

since otherwise, the right-hand side of the equation above will not be O ( 1 ) .

Hence
| t ˜ n C N ( f ; x ) f ˜ ( x ) | = O ( n α / 2 n r / 2 ) = O ( n α / 2 + r / 2 ) .

This completes the proof of Corollary 2. □

Corollary 3 If β = 0 , ξ ( t ) = t α for 0 < α < 1 and r in (3.1), then f Lip α . In this case, the degree of approximation of a function f ˜ ( x ) , conjugate to a 2π-periodic function f belonging to the class Lipα ( 0 < α < 1 ) is given by
| t ˜ n C N ( f ; x ) f ˜ ( x ) | = O ( ( n ) α / 2 ) .
Proof For r in Corollary 2, we get
t ˜ n C N ( f ; x ) f ˜ ( x ) = sup 0 x 2 π | t ˜ n C N ( f ; x ) f ˜ ( x ) | = O ( ( n ) α / 2 ) .
Thus, we have
| t ˜ n C N ( f ; x ) f ˜ ( x ) | t ˜ n C N ( f ; x ) f ˜ ( x ) = sup 0 x 2 π | t ˜ n C N ( f ; x ) f ˜ ( x ) | = O ( ( n ) α / 2 ) .

This completes the proof of Corollary 3. □

Examples (i) From Theorem 20 of Hardy’s ‘Divergent Series,’ if a Nörlund method ( N , p ) has increasing weights { p n } , then it is stronger than ( C , 1 ) .

Therefore, there is an easy way to find a sequence that is ( C , 1 ) summable, but not convergent. One such sequence is
s n : = k = 0 n ( 1 ) k .

Here s 1 = 1 , s 2 = 0 , s 3 = 1 , s 4 = 0 , … .

For ( C , 1 ) summability,
σ n 1 = s 1 + s 2 + + s n n .
Case (i) If n is even, i.e., n = 2 m , then
σ n 1 = s 1 + s 2 + + s 2 m 2 m = ( 1 + 0 + 1 + + 1 + 0 ) 2 m = m 2 m = 1 2 .
Case (ii) If n is odd, i.e., n = 2 m + 1 , then
σ n 1 = s 1 + s 2 + + s 2 m + 1 2 m + 1 = ( 1 + 0 + 1 + + 0 + 1 ) 2 m + 1 = 1 2 .

Thus, number 1 2 is assigned to the infinite series ( 1 ) n .

This sequence is summable ( C , 1 ) to 1 / 2 , but is clearly not convergent.

Now, take ( N , p ) to be the Nörlund matrix generated by p n = n + 1 . Then this sequence is summable by ( N , p ) method.

For
N n ( f ; x ) = 2 ( n + 1 ) ( n + 2 ) k = 0 n ( n k + 1 ) s k ( f ; x ) = { 0 , k  even , 1 , k  odd .
Since it is summable ( C , 1 ) , also, and ( N , p ) method is stronger than ( C , 1 ) method. Therefore, it is then summable by the product ( C , 1 ) ( N , p ) method.
  1. (ii)

    In the first example, we discuss the case in which the Nörlund method is stronger than ( C , 1 ) . The product C 1 N sums a divergent sequence, where N is the Nörlund method.

     

We know already that ( C , 1 ) sums the sequence s n = k = 0 n ( 1 ) k . We will now show that CN is stronger than C, for any regular Nörlund matrix (with non-negative generating sequence).

The statement CN is stronger than C is equivalent to showing that the matrix C N C 1 is regular.

Since C 1 is a bidiagonal matrix c n n 1 = n + 1 and c n + 1 1 = ( n 1 ) , for k < n ,
( C N C 1 ) n , k = j = k n ( C N ) n j ( C 1 ) j k = ( C N ) n k ( C 1 ) k k + ( C N ) n , k + 1 ( C 1 ) k + 1 , k = ( C N ) n k ( k + 1 ) + ( C N ) n , k + 1 ( ( k + 1 ) ) = ( k + 1 ) ( ( C N ) n k ( C N ) n , k + 1 ) = ( k + 1 ) [ j = k n 1 ( n + 1 ) p j k P j j = k + 1 n 1 ( n + 1 ) p j k P j ] = ( k + 1 n + 1 ) p 0 P n > 0 .
Also,
( C N C 1 ) n n = ( 1 n + 1 ) p 0 P n ( n + 1 ) = p 0 P n > 0 .
Therefore,
C N C 1 = C N C 1 ( e ) ,

where e denotes the sequence of all ones. C 1 ( e ) = e , so that C N ( e ) = C ( N ( e ) ) = C e = e . Thus, not only does C N C 1 have a finite norm, but it also has row sums equal to one.

Let { e k } denote the column vector with one in position k and zeros elsewhere. Then
b n = C 1 e k = { 0 , n < k , ( k + 1 ) , n = k , ( k + 1 ) , n = k + 1 , 0 , n > k + 1 .

Therefore, { b n } is a null sequence. Since N is a regular Nörlund matrix, with b : = { b n } , t = N ( b ) is a null sequence. Since C is also regular. Ct is a null sequence, and C N C 1 has null columns. Therefore, it is regular, and CN is stronger than C.

7 Conclusion

Various results concerning the degree of approximation of periodic signals (functions) belonging to the generalized weighted class by matrix operator have been reviewed, and the condition of monotonicity on the weights { p n } has been relaxed (i.e., weakening the conditions on the filter, we improve the quality of digital filter). Further, a proper (correct) set of conditions has been discussed to rectify the errors pointed out in Remarks 2.2 and 2.3. Some interesting application of the operator ( C 1 N p ) used in this paper is pointed out in Note 3.4. The theorem of this paper is an attempt to formulate the problem of approximation of the function f ˜ , conjugate to a periodic function f W ( L r , ξ ( t ) ) ( r 1 )-class through trigonometric polynomials generated by the product summability ( C 1 N p ) -transform of conjugate series of Fourier series of f in a simpler manner by dropping monotonicity on the generating sequence { p n } with the help of proper (new) set of conditions. Few applications and examples on product summability ( C 1 N p ) -transform of signals (functions) are also discussed in this manuscript.

Authors’ information

Dr. VNM is currently working as an Assistant Professor of Mathematics at SVNIT, Surat, Gujarat, India, and he is a very active researcher in various fields of mathematics. A Ph.D. in Mathematics, he is a double gold medalist, ranking first in the order of merit in both B.Sc. and M.Sc. Examinations from the Dr. Ram Manohar Lohia Avadh University, Faizabad.

Dr. VNM has undergone rigorous training from IIT, Roorkee, Mumbai in computer oriented mathematical methods and has experience of teaching post graduate, graduate and engineering students. The second author VS is a research scholar (R/S) in Applied Mathematics and Humanities Department at the Sardar Vallabhbhai National Institute of Technology, Ichchhanath Mahadev Road, Surat (Gujarat), India under the supervision of Dr. VNM. Recently, the third author LNM has joined as a full time research scholar in the Department of Mathematics, National Institute of Technology, Silchar-788010, District-Cachar, Assam, and he is also a very good active researcher in approximation theory and operator analysis.

Declarations

Acknowledgements

The authors would like to thank the anonymous learned referees for several useful interesting comments and suggestions about the paper. Special thanks are due to our great master and friend academician Prof. Ravi P. Agarwal, Texas A&M University, Kingsville, TX, USA, for kind cooperation, smooth behavior during communication and for his efforts to send the reports of the manuscript timely.

Authors’ Affiliations

(1)
Department of Applied Mathematics and Humanities, Sardar Vallabhbhai National Institute of Technology
(2)

References

  1. Woronoi GF: Extension of the notion of the limit of the sum of terms of an infinite series. Ann. Math. 1932, 33: 422–428. 10.2307/1968525MathSciNetView ArticleGoogle Scholar
  2. Chen JT, Jeng YS: Dual series representation and its applications to a string subjected to support motions. Adv. Eng. Softw. 1996, 27: 227–238. 10.1016/S0965-9978(96)00027-0MathSciNetView ArticleGoogle Scholar
  3. Chen JT, Hong H-K, Yeh CS, Chyuan SW: Integral representations and regularizations for a divergent series solution of a beam subjected to support motions. Earthq. Eng. Struct. Dyn. 1996, 25: 909–925. 10.1002/(SICI)1096-9845(199609)25:9<909::AID-EQE591>3.0.CO;2-MView ArticleGoogle Scholar
  4. Chen JT, Hong H-K: Review of dual boundary element methods with emphasis on hypersingular integrals and divergent series. Appl. Mech. Rev. 1999, 52: 17–33. 10.1115/1.3098922View ArticleGoogle Scholar
  5. Govil NK: On the summability of a class of the derived conjugate series of a Fourier series. Can. Math. Bull. 1965, 8: 637–645. 10.4153/CMB-1965-047-0MathSciNetView ArticleGoogle Scholar
  6. Khan HH:On the degree of approximation of a functions belonging to the class Lip ( α , p ) . Indian J. Pure Appl. Math. 1974, 5: 132–136.MathSciNetGoogle Scholar
  7. Qureshi K: On the degree of approximation of functions belonging to the Lipschitz class by means of a conjugate series. Indian J. Pure Appl. Math. 1981, 12(9):1120–1123.MathSciNetGoogle Scholar
  8. Mohapatra RN, Chandra P: Degree of approximation of functions in the Holder metric. Acta Math. Hung. 1983, 41: 67–76. 10.1007/BF01994063MathSciNetView ArticleGoogle Scholar
  9. Leindler L:Trigonometric approximation in L p -norm. J. Math. Anal. Appl. 2005, 302(1):129–136. 10.1016/j.jmaa.2004.07.049MathSciNetView ArticleGoogle Scholar
  10. Rhoades BE, Ozkoklu K, Albayrak I: On degree of approximation to a functions belonging to the class Lipschitz class by Hausdorff means of its Fourier series. Appl. Math. Comput. 2011, 217: 6868–6871. 10.1016/j.amc.2011.01.034MathSciNetView ArticleGoogle Scholar
  11. Guven A, Israfilov DM:Trigonometric approximation in generalized Lebesgue space L p ( x ) . J. Math. Inequal. 2010, 4(2):285–299.MathSciNetView ArticleGoogle Scholar
  12. Bhardwaj VK, Gupta S: Cesàro summable difference sequence space. J. Inequal. Appl. 2013., 2013: Article ID 315 10.1186/1029-242X-2013-315Google Scholar
  13. Mishra, VN, Khan, HH, Khan, IA, Mishra, LN: Approximation of signals (functions) belonging to Lip ( ξ ( t ) , r ) -class by C 1 N p summability method of conjugate series of its Fourier series. Bull. Math. Anal. Appl. (2013)Google Scholar
  14. Mishra VN, Khatri K, Mishra LN: Product summability transform of conjugate series of Fourier series. Int. J. Math. Math. Sci. 2012., 2012: Article ID 298923 10.1155/2012/298923Google Scholar
  15. Mishra VN, Khatri K, Mishra LN:Product ( N , p n ) ( C , 1 ) summability of a sequence of Fourier coefficients. Math. Sci. 2012., 6: Article ID 38 10.1186/2251-7456-6-38Google Scholar
  16. Mishra VN, Khatri K, Mishra LN:Using linear operators to approximate signals of Lip ( α , p ) , ( p 1 )-class. Filomat 2013, 27(2):355–365. 10.2298/FIL1302353MMathSciNetView ArticleGoogle Scholar
  17. Mishra, VN, Khatri, K, Mishra, LN: Strong Cesàro summability of triple Fourier integrals. Fasc. Math. 53 (2014)Google Scholar
  18. Mishra, VN, Khatri, K, Mishra, LN: Approximation of functions belonging to the generalized Lipschitz class by C 1 N p summability method of conjugate series of Fourier series. Mat. Vesn. (2013)Google Scholar
  19. Mishra LN, Mishra VN, Sonavane V:Trigonometric approximation of functions belonging to Lipschitz class by matrix ( C 1 N p ) operator of conjugate series of Fourier series. Adv. Differ. Equ. 2013., 2013: Article ID 127 10.1186/1687-1847-2013-127Google Scholar
  20. Mishra VN, Sonavane V, Mishra LN:On trigonometric approximation of W ( L p , ξ ( t ) ) ( p 1 ) function by product ( C , 1 ) ( E , 1 ) means of its Fourier series. J. Inequal. Appl. 2013., 2013: Article ID 300 10.1186/1029-242X-2013-300Google Scholar
  21. Lal S:Approximation of functions belonging to the generalized Lipschitz class by C 1 N p summability method of Fourier series. Appl. Math. Comput. 2009, 209: 346–350. 10.1016/j.amc.2008.12.051MathSciNetView ArticleGoogle Scholar
  22. Zygmund A 1. In Trigonometric Series. 2nd edition. Cambridge University Press, Cambridge; 1959:114–115.Google Scholar
  23. Proakis JG: Digital Communications. McGraw-Hill, New York; 1985.Google Scholar
  24. Psarakis EZ, Moustakides GV:An L 2 -based method for the design of 1-D zero phase FIR digital filters. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 1997, 44: 591–601. 10.1109/81.596940MathSciNetView ArticleGoogle Scholar
  25. Mittal ML, Rhoades BE, Mishra VN, Singh U:Using infinite matrices to approximate functions of class Lip ( α , p ) using trigonometric polynomials. J. Math. Anal. Appl. 2007, 326: 667–676. 10.1016/j.jmaa.2006.03.053MathSciNetView ArticleGoogle Scholar
  26. Mittal ML, Rhoades BE, Mishra VN:Approximation of signals (functions) belonging to the weighted W ( L P , ξ ( t ) ) , ( p 1 )-class by linear operators. Int. J. Math. Math. Sci. 2006., 2006: Article ID 53538Google Scholar
  27. Hille E, Tamarkin JD: On the summability of Fourier series. I. Trans. Am. Math. Soc. 1932, 34(4):757–783. 10.1090/S0002-9947-1932-1501662-XMathSciNetView ArticleGoogle Scholar

Copyright

© Mishra et al.; 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.