lr-approximation of signals (functions) belonging to weighted w(lr,ξ(t))-class by c1⋅np...

Mishra et al. Journal of Inequalities and Applications 2013, 2013:440http://www.journalofinequalitiesandapplications.com/content/2013/1/440

RESEARCH Open Access

Lr-Approximation of signals (functions)belonging to weightedW (Lr, ξ (t))-class byC ·Np summability method of conjugateseries of its Fourier seriesVishnu Narayan Mishra1*, Vaishali Sonavane1 and Lakshmi Narayan Mishra2

*Correspondence:[email protected] of AppliedMathematics and Humanities,Sardar Vallabhbhai NationalInstitute of Technology,Ichchhanath Mahadev Road, Surat,Gujarat 395 007, IndiaFull list of author information isavailable at the end of the article

AbstractRecently, Lal (Appl. Math. Comput. 209:346-350, 2009) has determined the degree ofapproximation of a function belonging to Lipα and weightedW(Lr ,ξ (t))-classes usingproduct C1 · Np summability with non-increasing weights {pn}. In this paper, wedetermine the degree of approximation of function f̃ , conjugate to a 2π -periodicfunction f belonging to weightedW(Lr ,ξ (t))-class by dropping the monotonicity onthe generating sequence {pn} with a new (proper) set of conditions, which in turngeneralizes the results of Mishra et al. (Bull. Math. Anal. Appl., 2013) on Lip(ξ (t), r)-classand rectifies (removes) the errors of Mishra et al. (Mat. Vesn., 2013). Few examples andapplications are also highlighted in this manuscript.MSC: Primary 42B05; 42B08; 40G05; 41A10

Keywords: generalized LipschitzW(Lr ,ξ (t)) (r ≥ 1)-class of functions; conjugateFourier series; degree of approximation; C1 means; Np means; product summabilityC1 · Np transform

1 IntroductionApproximation by trigonometric polynomials is at the heart of approximation theory. Themost important trigonometric polynomials used in the approximation theory are obtainedby linear summation methods of Fourier series of π-periodic functions on the real line(i.e., Cesàro means, Nörlund means and Product Cesàro-Nörlund means, etc.). Much ofthe advance in the theory of trigonometric approximation is due to the periodicity of thefunctions. The method of summability considered here was first introduced by Woronoi[]. Summability techniques were also applied on some engineering problems like, Chenand Jeng [] implemented the Cesàro sum of order (C, ) and (C, ), in order to acceleratethe convergence rate to deal with the Gibbs phenomenon, for the dynamic response of afinite elastic body subjected to boundary traction. Chen et al. [] applied regularizationwith Cesàro sum technique for the derivative of the double layer potential. Similarly, Chenand Hong [] used Cesàro sum regularization technique for hyper singularity of dual in-tegral equation. The degree of approximation of functions belonging to Lipα, Lip(α, r),Lip(ξ (t), r) and W (Lr , ξ (t)) (r ≥ )-classes by Nörlund (Np) matrices and general summa-bility matrices has been proved by various investigators like Govil [], Khan [], Qureshi

© 2013 Mishra et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribu-tion License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in anymedium, provided the original work is properly cited.

http://www.journalofinequalitiesandapplications.com/content/2013/1/440

mailto:[email protected]

http://creativecommons.org/licenses/by/2.0

Mishra et al. Journal of Inequalities and Applications 2013, 2013:440 of 15http://www.journalofinequalitiesandapplications.com/content/2013/1/440

[], Mohapatra and Chandra [], Leindler [], Rhoades et al. [], Guven and Israfilov [],Bhardwaj and Gupta [] and Mishra et al. [–]. Here, Lal [] has assumed mono-tonicity on the generating sequence {pn} to prove their theorems. The approximation offunction f̃ , conjugate to a periodic function f ∈ W (Lr, ξ (t)) (r ≥ ) using product (C ·Np)-summability has not been studied so far. In this paper, we obtain a new theorem on thedegree of approximation of function f̃ , conjugate to a periodic function f ∈ W (Lr , ξ (t))-class using semi-monotonicity on the generating sequence {pn} and a proper set of theconditions.A bidiagonal matrix is a matrix with non-zero entries along the main diagonal and ei-

ther the diagonal above or the diagonal below. This means there are exactly two non-zerodiagonals in the matrix.When the diagonal above the main diagonal has the non-zero entries, the matrix is up-

per bidiagonal. When the diagonal below the main diagonal has the non-zero entries, thematrix is lower bidiagonal.For example, the following matrix is upper bidiagonal:

⎛⎜⎜⎜⎝

⎞⎟⎟⎟⎠

and the following matrix is lower bidiagonal:

⎛⎜⎜⎜⎝

⎞⎟⎟⎟⎠ .

Let∑∞

n= an be a given infinite series with the sequence of nth partial sums {sn}. Let {pn}be a non-negative sequence of constants, real or complex, and let us write

Pn =n∑

k=pk �= ∀n≥ ,p– = = P– and Pn → ∞ as n → ∞.

The sequence to sequence transformation tNn =∑n

ν= pn–νsν/Pn defines the sequence {tNn }of Nörlundmeans of the sequence {sn}, generated by the sequence of coefficients {pn}. Theseries

∑∞n= an is said to be summableNp to the sum s if limn→∞ tNn exists and is equal to s.

In the special case in which

pn =(n + α –

α –

)=

(n + α)(n + )(α)

(α > ).

The Nörlund summability Np reduces to the familiar Cα summability.The product of C summability with an Np summability defines C · Np summability.

Thus, the C ·Np mean is given by tCNn = n+∑n

k= P–k∑k

ν= pk–νsν . If tCNn → s as n → ∞,then the infinite series

∑∞n= an or the sequence {sn} is said to be summable C ·Np to the



sum s if limn→∞ tCNn exists and is equal to s.

sn → s ⇒ Np(sn) = tNn = P–n

n∑ν=

pn–νsν → s, as n → ∞, Np method is regular,

⇒ C(Np(sn))= tCNn → s, as n→ ∞, C method is regular,

⇒ C ·Np method is regular.

Let f (x) be a π-periodic function and Lebesgue integrable. The Fourier series of f (x) isgiven by

f (x) ∼ a

+

∞∑n=

(an cosnx + bn sinnx) ≡∞∑n=

An(x) (.)

with nth partial sum sn(f ;x).The conjugate series of Fourier series (.) is given by

∞∑n=

(bn cosnx – an sinnx) ≡∞∑n=

Bn(x). (.)

A function f (x) ∈ Lipα if

∣∣f (x + t) – f (x)∣∣ = O

(|t|α) for < α ≤ , t >

and f (x) ∈ Lip(α, r), [] for ≤ x≤ π , if

∥∥f (x + t) – f (x)∥∥r =(∫ π

∣∣f (x + t) – f (x)∣∣r dx)/r

= O(|t|α), < α ≤ , r ≥ , t > .

f (x) ∈ Lip(ξ (t), r) if

∥∥f (x + t) – f (x)∥∥r =(∫ π

∣∣f (x + t) – f (x)∣∣r dx)/r

= O(ξ (t)), r ≥ , t > .

f ∈ W (Lr, ξ (t)), [, ] if

ωr(t; f ) =∥∥[f (x + t) – f (x)

]sinβ (x/)

∥∥r =(∫ π

∣∣f (x + t) – f (x)∣∣r sinβr(x/)dx

)/r

= O(ξ (t)), β ≥ , r ≥ , t > ,

where ξ (t) is positive increasing function of t.If β = , then W (Lr, ξ (t)) reduces to the class Lip(ξ (t), r), if ξ (t) = tα ( < α ≤ ), then

Lip(ξ (t), r) class coincides with the class Lip(α, r), and if r → ∞, then Lip(α, r) reduces tothe class Lipα.

L∞-norm of a function f : R → R is defined by ‖f ‖∞ = sup{|f (x)| : x ∈ R}.Lr-norm of f is defined by ‖f ‖r = (

∫ π |f (x)|r dx)/r , r ≥ .



The degree of approximation of a function f : R → R by trigonometric polynomial tn oforder n under sup norm ‖‖∞ is defined by []

‖tn – f ‖∞ = sup{∣∣tn(x) – f (x)

∣∣ : x ∈ R},

and En(f ) of a function f ∈ Lr is given by En(f ) = minn ‖tn – f ‖r .The conjugate function f̃ (x) is defined for almost every x by

f̃ (x) = –π

∫ π

ψ(t) cot t/dt = lim

h→

(–

π

∫ π

hψ(t) cot t/dt

).

We note that tNn and tCNn are also trigonometric polynomials of degree (or order) n.Abel’s transformation: The formula

n∑k=m

ukvk =n–∑k=m

Uk(vk – vk+) –Um–vm +Unvn, (.)

where ≤ m ≤ n, Uk = u + u + u + · · · + uk , if k ≥ , U– = , which can be verified, isknown as Abel’s transformation and will be used extensively in what follows.If vm, vm+, . . . , vn are non-negative and non-increasing, the left-hand side of (.) does

not exceed vm maxm–≤k≤n |Uk| in absolute value. In fact,

∣∣∣∣∣n∑

k=mukvk

∣∣∣∣∣≤ max |Uk|{ n–∑

k=m(vk – vk+) + vm + vn

}

= vm max |Uk|. (.)

We write throughout

ψ(t) = f (x + t) – f (x – t), φ(t) = f (x + t) – f (x) + f (x – t),

Vn =

π (n + )

n∑k=

P–k

k∑ν=

(ν + )|pν – pν–|,

M̃n(t) =

π (n + )

n∑k=

P–k

k∑ν=

pν

cos(k – v + /)tsin t/

(.)

τ = [/t], where τ denotes the greatest integer not exceeding /t. Furthermore, C denotesan 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 se-

ries. Hence a separate study of conjugate series is desirable and attracted the attention ofresearchers.

2 Known theoremLal [] has obtained the degree of approximation of the functions belonging toW (Lr , ξ (t))-class by C ·Np means with monotonicity on the generating sequence {pn}. He proved thefollowing.



Theorem . If f (x) is a π -periodic function and Lebesgue integrable on [, π ] and isbelonging to W (Lr, ξ (t))-class, then its degree of approximation by C · Np means of itsFourier series (.) is given by

∥∥tCNn – f∥∥r = O

((n + )β+/rξ

(

n +

)), (.)

provided ξ (t) satisfies the following conditions:

{ξ (t)/t

}be a decreasing sequence, (.)(∫ /(n+)

(t|φ(t)|ξ (t)

)rsinβr t dt

)/r= O((n + )–

), (.)

(∫ π

/(n+)

(t–δ|φ(t)|

ξ (t)

)rdt)/r

= O((n + )δ

), (.)

where δ is an arbitrary number such that s(– δ) – > , r– + s– = , ≤ r ≤ ∞, conditions(.) and (.) hold uniformly in x.

Remark . The condition / sinβ (t) = O(/tβ ), /(n + ) ≤ t ≤ π used by Lal [, pp.-] in writing the proof of Theorem . is not valid since sin t → as t → π .

Remark . There is a fatal error in the proof of Theorem . of Lal [, p.], in calcu-lating

[t–βs–s+

–βs – s +

]/(n+)ε

,

note that –βs – s + < . Therefore, one has βs+s– [

εβs+s– – (n + )βs+s–], which need not

be O((n + )βs+s–), since ε might be O(/nγ ) for some γ > .

3 Main theoremIt is well known that the theory of approximations, i.e., TFA, which originated from awell-known theorem of Weierstrass, has become an exciting interdisciplinary field of study forthe last years. These approximations have assumed important new dimensions dueto their wide applications in signal analysis [], in general and in digital signal process-ing [] in particular, in view of the classical Shannon sampling theorem. Mittal et al.[, ] have obtained many interesting results on TFA using summability methods with-out monotonicity on the rows of the matrix T : a digital filter. Broadly speaking, signalsare treated as functions of one variable, and images are represented by functions of twovariables. Very recently,Mishra et al. [, ] obtained the degree of approximation of con-jugate of a function using C · Np product summability method of its conjugate series ofits Fourier series in Lipschitz and weighted spaces, respectively. Therefore, the purpose ofthe present paper is to generalize the results of Mishra et al. [] on the degree of approx-imation of a function f̃ (x), conjugate to a π-periodic function f belonging to weightedW (Lr , ξ (t))-class by C ·Np means of conjugate series of its Fourier series by dropping themonotonicity condition on the generating sequence {pn} with the help of a new (proper)set of conditions to rectify the errors of Mishra et al. []. More precisely, we state ourmain theorem as follows.



Theorem . If f̃ (x), conjugate to a π -periodic function f belonging to W (Lr , ξ (t))-class,then its degree of approximation by C ·Np means of conjugate series of Fourier series (.)is given by

∥∥t̃CNn – f̃∥∥r = O

((n)β/+r/ξ

(√n

)), (.)

provided {pn} satisfies

Vn < C, (.)

and ξ (t) satisfies the following conditions:

{ξ (t)/t

}is non-increasing in t, (.)(∫ π/

√n

( |ψ(t)|ξ (t)

)rsinβr(t/)dt

)/r= O(), (.)

(∫ π

π/√n

(t–δ|ψ(t)|

ξ (t)

)rdt)/r

= O((√n)δ), (.)

where δ is an arbitrary number such that s(β –δ)– > , r– + s– = , ≤ r ≤ ∞, conditions(.) and (.) hold uniformly in x.

Note . ξ ( π√n ) ≤ πξ ( √

n ), for (π√n ) ≥ ( √

n ).

Note . Condition Vn < C ⇒ npn < CPn, [].

Note . The product transform C · Np plays an important role in signal theory as adouble digital filter [] and theory of machines in mechanical engineering [].

4 LemmasWe need the following lemmas for the proof of our theorem.

Lemma . |M̃n(t)| = O[/t] for < t ≤ π/√n.

Proof For < t ≤ π/√n, sin(t/) ≥ (t/π ) and | cosnt| ≤ .

∣∣M̃n(t)∣∣ =∣∣∣∣∣ π (n + )

n∑k=

P–k

k∑v=

pν


∣∣∣∣∣≤

π (n + )

n∑k=

P–k

k∑v=

pν

| cos(k – v + /)t|| sin t/|

≤ t(n + )

n∑k=

P–k

k∑v=

pν =

t(n + )

n∑k=

P–k Pk

= O[τ ].

This completes the proof of Lemma .. �



Lemma . Let {pn} be a non-negative sequence and satisfy (.), then

∣∣M̃n(t)∣∣ = O(τ ) + O

(τ

n

)( n∑k=τ

P–k

k–∑ν=

|�pν |)

uniformly in < t ≤ π . (.)

Proof We have

M̃n(t) =

π (n + )

n∑k=

P–k

k∑v=

pν


=

π (n + )

(τ–∑k=

+n∑

k=τ

)P–k

k∑v=

pν


=(M̃n(t)

) +(M̃n(t)

), say, (.)

where

∣∣(M̃n(t))

∣∣ =∣∣∣∣∣ π (n + )

τ–∑k=

P–k

k∑v=

pν


∣∣∣∣∣≤

π (n + )

τ–∑k=

P–k

k∑v=

pν

| cos(k – v + /)t|| sin t/|

≤ t(n + )

τ–∑k=

P–k

k∑v=

pν

= O(

τ

(n + )

), (.)

and using Abel’s transformation and sin(t/) ≥ (t/π ), for < t ≤ π , we get

∣∣(M̃n(t))

∣∣ =∣∣∣∣∣ π (n + )

n∑k=τ

P–k

k∑v=

pν


∣∣∣∣∣≤

t(n + )

n∑k=τ

P–k

{ k–∑v=

|�pν |∣∣∣∣∣(

ν∑γ=

cos(k – γ + /)t)∣∣∣∣∣

+

∣∣∣∣∣( k∑

γ=cos(k – γ + /)t

)∣∣∣∣∣pk}

=O(t–)

t(n + )

( n∑k=τ

P–k

k–∑v=

|�pν | +n∑

k=τ

P–k pk

)

by virtue of the fact that∑μ

k=λ exp(–ikt) = O(t–), ≤ λ ≤ k ≤ μ.

∣∣(M̃n(t))

∣∣ = O(

τ

(n + )

)( n∑k=τ

P–k

k–∑v=

|�pν | +n∑

k=τ

P–k pk

kk

)

= O(

τ

(n + )

)( n∑k=τ

P–k

k–∑v=

|�pν | + (n + )τ

)



= O(τ ) + O(

τ

(n + )

) n∑k=τ

P–k

k–∑v=

|�pν |,

∣∣(M̃n(t))

∣∣ = O(τ ) + O(

τ

n

) n∑k=τ

P–k

k–∑v=

|�pν | (.)

in view of Note .. Combining (.), (.) and (.) yields (.).This completes the proof of Lemma .. �

5 Proof of theoremLet s̃n(f ;x) denote the partial sum of series (.), we have

s̃n(f ;x) – f̃ (x) = π

∫ π

ψ(t)cos(n + /)t

sin t/dt.

Denoting C ·Np means of s̃n(f ;x) by t̃CNn , we write

t̃CNn (x) – f̃ (x) =∫ π

ψ(t)

π (n + )

n∑k=

P–k

k∑ν=

pν


dt

=∫ π

ψ(t)M̃n(t)dt

=[∫ π/

√n

+∫ π

π/√n

]ψ(t)M̃n(t)dt

= I + I (say). (.)

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,

{∫ π

∣∣(ψ(x + t) – ψ(t))

sinβ (x/)∣∣r dx}/r

≤{∫ π

∣∣(f (u + x + t) – f (u + x))


+{∫ π

∣∣(f (u – x – t) – f (u – x))


= O(ξ (t)).

Then f ∈W (Lr , ξ (t)) ⇒ ψ ∈ W (Lr, ξ (t)).Using Hölder’s inequality, ψ(t) ∈ W (Lr , ξ (t)), condition (.), sin(t/) ≥ (t/π ), for <

t ≤ π , Lemma ., Note . and second mean value theorem for integrals, we have

|I| ≤[∫ π/

√n

( |ψ(t) sinβ (t/)|ξ (t)

)rdt]/r[∫ π/

√n

(ξ (t)|M̃n(t)|sinβ (t/)

)sdt]/s

= O()[∫ π/

√n

(ξ (t)t+β

)sdt]/s

= O{ξ

(π√n

)}[∫ π/√n

h

(

t+β

)sdt]/s

, as h→

= O((n)β/+r/ξ

(√n

)), r– + s– = . (.)



Using Lemma ., we have

|I| = O[∫ π

π/√n

|ψ(t)|t

dt]+O[∫ π

π/√n

|ψ(t)|tn

(τ

n∑k=τ

P–k

k–∑ν=

|pν |)dt]= O(I) +O(I).

Using Hölder’s inequality, conditions (.) and (.), | sin t| ≤ , sin(t/) ≥ (t/π ), for <t ≤ π , Note . and second mean value theorem for integrals, we have

|I| ≤[∫ π

π/√n

(t–δ|ψ(t)| sinβ (t/)

ξ (t)

)rdt]/r[∫ π

π/√n

(ξ (t)

t–δ+ sinβ (t/)

)sdt]/s

= O((√n)δ)[∫ π

π/√n

(ξ (t)

t–δ++β

)sdt]/s

= O{(√n)δ}[∫ √

n/π

/π

(ξ (/y)yδ––β

)s dyy

]/s

= O((√n)δ ξ (π/

√n)

π/√n

)[∫ √n/π

/π

(dy

y(δ–β)s+

)]/s

= O((√n)δ+ξ

(√n

))((√n)(β–δ)s– – (π )(–β+δ)s+

(β – δ)s –

)/s

= O((n)β/+r/ξ

(√n

)). (.)

Similarly, as conditions (.), (.) and (.) above, | sin t| ≤ , sin(t/) ≥ (t/π ), for <t ≤ π , Note . and second mean value theorem for integrals, we have

|I| ≤[∫ π

π/√n

(t–δ|ψ(t)| sinβ (t/)

ξ (t)

)rdt]/r

×[∫ π

π/√n

(ξ (t)

t–δ+ sinβ (t/)n

(τ

n∑k=τ

P–k

k–∑ν=

|pν |))s

dt]/s

= O((√n)δ

n

)[∫ π

π/√n

(ξ (t)

t–δ++β

(τ

n∑k=τ

P–k

k–∑ν=

|pν |))s

dt]/s

= O((√n)δ

n

)[∫ π

π/√n

(ξ (t)

t–δ++β

( n∑k=τ

P–k

k–∑ν=

(ν + )|pν |))s

dt]/s

= O((√n)δ

n

)[∫ π

π/√n

(ξ (t)

t–δ++β

( n∑k=

P–k

k∑ν=

(ν + )|pν |))s

dt]/s

= O((√n)δ

n

)[∫ π

π/√n

(ξ (t)

t–δ++βVnπ (n)

)sdt]/s

= O((√n)δ)[∫ π

π/√n

(ξ (t)

t–δ++β

)sdt]/s

= O{(√n)δ}[∫ √

n/π

/π

(ξ (/y)yδ––β

)s dyy

]/s

= O((n)β/+r/ξ

(√n

)). (.)



Collecting (.)-(.), we have

∣∣t̃CNn – f̃∣∣ = O

((n)β/+r/ξ

(√n

)). (.)

Now, using the Lr-norm of a function, we get

∥∥t̃CNn – f̃∥∥r ={∫ π

∣∣t̃CNn (x) – f̃ (x)∣∣r dx}/r

= O(∫ π

((n)β/+r/ξ

(√n

))rdx)/r

= O((n)β/+r/ξ

(√n

)(∫ π

dx)/r)

= O((n)β/+r/ξ

(√n

)).

This completes the proof of Theorem ..

6 ApplicationsThe theory of approximation is a very extensive field, which has various applications. Asmentioned in [], the Lp-space in general and L and L∞ in particular play an impor-tant role in the theory of signals and filters. From the point of view of the applications, theSharper estimates of infinite matrices are useful to get bounds for the lattice norms (whichoccur in solid state physics) ofmatrix-valued functions and enable to investigate perturba-tions of matrix-valued functions and compare them. Some interesting applications of theCesàro summability can be seen in [–]. The following corollaries can be derived fromTheorem ..

Corollary If β = , then the generalized weighted Lipschitz W (Lr, ξ (t)) (r ≥ )-class re-duces to the class Lip(ξ (t), r), and the degree of approximation of a function f̃ (x), conjugateto a π -periodic function f belonging to the class Lip(ξ (t), r), is given by

∣∣t̃CNn (f ;x) – f̃ (x)∣∣ = O

(nr/ξ (/

√n)). (.)

Proof The result follows by setting β = in Theorem ., we have

∥∥t̃CNn (f ;x) – f̃ (x)∥∥r ={∫ π

∣∣t̃CNn (f ;x) – f̃ (x)∣∣r dx}/r

= O(nr/ξ (/

√n)), r ≥ .

Thus, we get

∣∣t̃CNn (f ;x) – f̃ (x)∣∣≤ {∫ π

∣∣t̃CNn (x) – f̃ (x)∣∣r dx}/r

= O(nr/ξ (/

√n)), r ≥ .

This completes the proof of Corollary . �



Corollary If β = , ξ (t) = tα , < α ≤ , then the generalized weighted LipschitzW (Lr , ξ (t)) (r ≥ )-class reduces to the class Lip(α, r), (/r) < α < and the degree of ap-proximation of a function f̃ (x), conjugate to a π -periodic function f belonging to the classLip(α, r), is given by

∣∣t̃CNn (f ;x) – f̃ (x)∣∣ = O

((n)–α/+r/).

Proof Putting β = , ξ (t) = tα , < α ≤ in Theorem ., we have

∥∥t̃CNn (f ;x) – f̃ (x)∥∥r ={∫ π

∣∣t̃CNn (f ;x) – f̃ (x)∣∣r dx}/r

or,

O(nβ/+r/ξ

(√n

))={∫ π

∣∣t̃CNn (f ;x) – f̃ (x)∣∣r dx}/r

or,

O() ={∫ π

∣∣t̃CNn (f ;x) – f̃ (x)∣∣r dx}/r

O(

nβ/+r/ξ ( √

n )

),

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

∣∣t̃CNn (f ;x) – f̃ (x)∣∣ = O

(n–α/nr/

)= O(n–α/+r/).


Corollary If β = , ξ (t) = tα for < α < and r → ∞ in (.), then f ∈ Lipα. In thiscase, the degree of approximation of a function f̃ (x), conjugate to a π -periodic function fbelonging to the class Lipα ( < α < ) is given by

∣∣t̃CNn (f ;x) – f̃ (x)∣∣ = O

((n)–α/).

Proof For r → ∞ in Corollary , we get

∥∥t̃CNn (f ;x) – f̃ (x)∥∥∞ = sup

≤x≤π

∣∣t̃CNn (f ;x) – f̃ (x)∣∣ = O

((n)–α/).

Thus, we have

∣∣t̃CNn (f ;x) – f̃ (x)∣∣≤ ∥∥t̃CNn (f ;x) – f̃ (x)

∥∥∞

= sup≤x≤π

∣∣t̃CNn (f ;x) – f̃ (x)∣∣ = O

((n)–α/).




Examples (i) From Theorem of Hardy’s ‘Divergent Series,’ if a Nörlund method (N ,p)has increasing weights {pn}, then it is stronger than (C, ).Therefore, there is an easy way to find a sequence that is (C, ) summable, but not con-

vergent. One such sequence is

sn :=n∑

k=(–)k .

Here s = , s = , s = , s = , . . . .For (C, ) summability,

σ n =

s + s + · · · + snn

.

Case (i) If n is even, i.e., n = m, then

σ n =

s + s + · · · + smm

=( + + + · · · + + )

m

=mm

=.

Case (ii) If n is odd, i.e., n = m + , then

σ n =

s + s + · · · + sm+

m +

=( + + + · · · + + )

m +

=.

Thus, number is assigned to the infinite series

∑(–)n.

This sequence is summable (C, ) to /, but is clearly not convergent.Now, take (N ,p) to be the Nörlund matrix generated by pn = n + . Then this sequence

is summable by (N ,p) method.For

Nn(f ;x) =

(n + )(n + )

n∑k=

(n – k + )sk(f ;x) =

⎧⎨⎩, k even,, k odd.

Since it is summable (C, ), also, and (N ,p) method is stronger than (C, ) method. There-fore, it is then summable by the product (C, )(N ,p) method.(ii) In the first example, we discuss the case in which the Nörlund method is stronger

than (C, ). The productCN sums a divergent sequence, whereN is the Nörlundmethod.We know already that (C, ) sums the sequence sn =

∑nk=(–)k . We will now show that

CN is stronger than C, for any regular Nörlund matrix (with non-negative generating se-quence).The statement CN is stronger than C is equivalent to showing that the matrix CNC– is

regular.



Since C– is a bidiagonal matrix c–nn = n + and c–n+ = –(n – ), for k < n,

(CNC–)

n,k =n∑j=k

(CN)nj(C–)

jk

= (CN)nk(C–)

kk + (CN)n,k+(C–)

k+,k

= (CN)nk(k + ) + (CN)n,k+(–(k + )

)= (k + )

((CN)nk – (CN)n,k+

)= (k + )

[ n∑j=k

(n + )

pj–kPj

–n∑

j=k+

(n + )

pj–kPj

]

=(k + n +

)p

Pn> .

Also,

(CNC–)

nn =(

n +

)p

Pn(n + ) =

p

Pn> .

Therefore,

∥∥CNC–∥∥ = CNC–(e),

where e denotes the sequence of all ones. C–(e) = e, so that CN(e) = C(N(e)) = Ce = e.Thus, not only does CNC– have a finite norm, but it also has row sums equal to one.Let {ek} denote the column vector with one in position k and zeros elsewhere. Then

bn = C–ek =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

, n < k,–(k + ), n = k,(k + ), n = k + ,, n > k + .

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

7 ConclusionVarious results concerning the degree of approximation of periodic signals (functions) be-longing to the generalized weighted class by matrix operator have been reviewed, and thecondition of monotonicity on the weights {pn} has been relaxed (i.e., weakening the con-ditions on the filter, we improve the quality of digital filter). Further, a proper (correct) setof conditions has been discussed to rectify the errors pointed out in Remarks . and ..Some interesting application of the operator (C ·Np) used in this paper is pointed out inNote .. The theorem of this paper is an attempt to formulate the problem of approxima-tion of the function f̃ , conjugate to a periodic function f ∈W (Lr , ξ (t)) (r ≥ )-class throughtrigonometric polynomials generated by the product summability (C ·Np)-transform of



conjugate series of Fourier series of f in a simpler manner by dropping monotonicity onthe generating sequence {pn} with the help of proper (new) set of conditions. Few appli-cations and examples on product summability (C · Np)-transform of signals (functions)are also discussed in this manuscript.

Competing interestsThe authors declare that they have no competing interests.

Authors’ contributionsLNM computed lemmas and established the main theorem with a proper (a new) set of conditions to remove the errorsin this direction. LNM studied examples (i) and (ii) in its depth. LNM and VNM conceived of the study and participated inits design and coordination. LNM, VNM contributed equally and significantly in writing this paper. All the authors, i.e.,VNM, VS and LNM drafted the manuscript, read and approved the final manuscript.

Author details1Department of Applied Mathematics and Humanities, Sardar Vallabhbhai National Institute of Technology, IchchhanathMahadev Road, Surat, Gujarat 395 007, India. 2L. 1627 Awadh Puri Colony Beniganj, Opp. I.T.I., Ayodhya Main Road,Faizabad, Uttar Pradesh 224 001, India.

Authors’ informationDr. VNM is currently working as an Assistant Professor of Mathematics at SVNIT, Surat, Gujarat, India, and he is a very activeresearcher in various fields of mathematics. A Ph.D. in Mathematics, he is a double gold medalist, ranking first in the orderof 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 andhas 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 hasjoined 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.

AcknowledgementsThe authors would like to thank the anonymous learned referees for several useful interesting comments andsuggestions 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 hisefforts to send the reports of the manuscript timely.

Received: 23 January 2013 Accepted: 30 July 2013 Published: 18 September 2013

References1. Woronoi, GF: Extension of the notion of the limit of the sum of terms of an infinite series. Ann. Math. 33, 422-428

(1932)2. Chen, JT, Jeng, YS: Dual series representation and its applications to a string subjected to support motions. Adv. Eng.

Softw. 27, 227-238 (1996)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. 25, 909-925 (1996)4. Chen, JT, Hong, H-K: Review of dual boundary element methods with emphasis on hypersingular integrals and

divergent series. Appl. Mech. Rev. 52, 17-33 (1999)5. Govil, NK: On the summability of a class of the derived conjugate series of a Fourier series. Can. Math. Bull. 8, 637-645

(1965)6. Khan, HH: On the degree of approximation of a functions belonging to the class Lip(α,p). Indian J. Pure Appl. Math. 5,

132-136 (1974)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. 12(9), 1120-1123 (1981)8. Mohapatra, RN, Chandra, P: Degree of approximation of functions in the Holder metric. Acta Math. Hung. 41, 67-76

(1983)9. Leindler, L: Trigonometric approximation in Lp-norm. J. Math. Anal. Appl. 302(1), 129-136 (2005)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. 217, 6868-6871 (2011)11. Guven, A, Israfilov, DM: Trigonometric approximation in generalized Lebesgue space Lp(x) . J. Math. Inequal. 4(2),

285-299 (2010)12. Bhardwaj, VK, Gupta, S: Cesàro summable difference sequence space. J. Inequal. Appl. 2013, Article ID 315 (2013).

doi:10.1186/1029-242X-2013-31513. Mishra, VN, Khan, HH, Khan, IA, Mishra, LN: Approximation of signals (functions) belonging to Lip(ξ (t), r)-class by

C1 · Np summability method of conjugate series of its Fourier series. Bull. Math. Anal. Appl. (2013)14. Mishra, VN, Khatri, K, Mishra, LN: Product summability transform of conjugate series of Fourier series. Int. J. Math.

Math. Sci. 2012, Article ID 298923 (2012). doi:10.1155/2012/29892315. Mishra, VN, Khatri, K, Mishra, LN: Product (N,pn)(C, 1) summability of a sequence of Fourier coefficients. Math. Sci. 6, 38

(2012). doi:10.1186/2251-7456-6-38


http://dx.doi.org/10.1186/1029-242X-2013-315

http://dx.doi.org/10.1155/2012/298923

http://dx.doi.org/10.1186/2251-7456-6-38


16. Mishra, VN, Khatri, K, Mishra, LN: Using linear operators to approximate signals of Lip(α,p), (p≥ 1)-class. Filomat 27(2),355-365 (2013). doi:10.2298/FIL1302353M

17. Mishra, VN, Khatri, K, Mishra, LN: Strong Cesàro summability of triple Fourier integrals. Fasc. Math. 53 (2014)18. Mishra, VN, Khatri, K, Mishra, LN: Approximation of functions belonging to the generalized Lipschitz class by C1 · Np

summability method of conjugate series of Fourier series. Mat. Vesn. (2013)19. Mishra, LN, Mishra, VN, Sonavane, V: Trigonometric approximation of functions belonging to Lipschitz class by matrix

(C1 · Np) operator of conjugate series of Fourier series. Adv. Differ. Equ. 2013, Article ID 127 (2013).doi:10.1186/1687-1847-2013-127

20. Mishra, VN, Sonavane, V, Mishra, LN: On trigonometric approximation ofW(Lp ,ξ (t)) (p≥ 1) function by product(C, 1)(E, 1) means of its Fourier series. J. Inequal. Appl. 2013, Article ID 300 (2013). doi:10.1186/1029-242X-2013-300

21. Lal, S: Approximation of functions belonging to the generalized Lipschitz class by C1 · Np summability method ofFourier series. Appl. Math. Comput. 209, 346-350 (2009)

22. Zygmund, A: Trigonometric Series, 2nd edn., vol. 1, pp. 114-115 Cambridge University Press, Cambridge (1959)23. Proakis, JG: Digital Communications. McGraw-Hill, New York (1985)24. Psarakis, EZ, Moustakides, GV: An L2-based method for the design of 1-D zero phase FIR digital filters. IEEE Trans.

Circuits Syst. I, Fundam. Theory Appl. 44, 591-601 (1997)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. 326, 667-676 (2007)26. Mittal, ML, Rhoades, BE, Mishra, VN: Approximation of signals (functions) belonging to the weightedW(LP ,ξ (t)),

(p≥ 1)-class by linear operators. Int. J. Math. Math. Sci. 2006, Article ID 53538 (2006)27. Hille, E, Tamarkin, JD: On the summability of Fourier series. I. Trans. Am. Math. Soc. 34(4), 757-783 (1932)

doi:10.1186/1029-242X-2013-440Cite this article as:Mishra et al.: Lr -Approximation of signals (functions) belonging to weightedW(Lr ,ξ (t))-class byC1 · Np summability method of conjugate series of its Fourier series. Journal of Inequalities and Applications 20132013:440.


http://dx.doi.org/10.2298/FIL1302353M

http://dx.doi.org/10.1186/1687-1847-2013-127

http://dx.doi.org/10.1186/1029-242X-2013-300

lr-approximation of signals (functions) belonging to weighted w(lr,ξ(t))-class by c1⋅np...

Documents