4. Degree of approximation of conjugate of signals (functions) belonging to the generalized weighted Lipschitz $W(L_r,\psi(t))$, $(r\geq 1)$-class by $(C,1)(E, q)$ means of conjugate trigonometric Fourier series

Bulletin of Mathematical Analysis and Applications ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 5 Issue 4 (2013), Pages 40-53

DEGREE OF APPROXIMATION OF CONJUGATE OF SIGNALS (FUNCTIONS) BELONGING TO THE GENERALIZED WEIGHTED LIPSCHITZ W(Lr, ξ(t)),(r≥1) - CLASS BY (C, 1) (E,

Q) MEANS OF CONJUGATE TRIGONOMETRIC FOURIER SERIES

(COMMUNICATED BY H ¨USEYIN BOR)

V. N. MISHRA, H. H. KHAN, K. KHATRI AND L. N. MISHRA

Abstract. Very recently, Sonker and Singh [21] determined the degree of ap-proximation of the conjugate of 2π-periodic signals (functions) belonging to

Lip(α, r)(0< α≤1, r≥1)-class by using Ces`aro-Euler (C,1) (E,q) means of their conjugate trigonometric Fourier series. In the present paper, we general-ize the result of Sonker and Singh [21] on the generalgeneral-ized weighted Lipschitz

W(Lr, ξ(t)),(r≥1)- class of signals (functions) by product summability (C,1) (E,q) transform of conjugate trigonometric Fourier series. Our result also generalizes the result of Lal and Singh [6]. Few applications and example of approximation of functions will also be highlighted.

1. Introduction

The degree of approximation of functions belonging toLipα, Lip(α, r), Lip(ξ(t), r) andW(Lr, ξ(t)),(r≥1) - classes by general summability matrices has been proved by various investigators like Govil [1], Khan [3-5], Mohapatra and Chandra [19], Mittal et al. [8-9], Mittal and Mishra [7], Mishra et al. [10-17] and Mishra and Mishra [18]. Recently, Sonker and Singh [21] discussed the degree of approximation of the conjugate of signals (functions) belonging to Lip(α, r) class by (C,1)(E, q) means of conjugate trigonometric Fourier series. But nothing seems to have been done so far to obtain the degree of approximation of conjugate of signals belonging to the generalized weighted Lipschitz W(Lr, ξ(t)),(r ≥ 1) class by (C,1) (E, q) product summability transforms. Weighted W(Lr, ξ(t)),(r ≥ 1) Lipschitz - class is generalization of Lipα, Lip(α, r) and Lip(ξ(t), r) - classes. In the present pa-per, an attempt to make advance study in this direction, a theorem on the degree of approximation of conjugate of signals (functions) belonging to the generalized

0_{2000 Mathematics Subject Classification: 41A10, 42B05, 42B08.}

Keywords and phrases. Degree of approximation, weighted LipschitzW(Lr, ξ(t)),(r≥1),(t >0) - class of functions, (E, q) transform, (C,1) transform, product summability (C,1) (E, q) transform, conjugate Fourier series, Lebesgue integral.

c

2013 Universiteti i Prishtin¨es, Prishtin¨e, Kosov¨e. Submitted August 9, 2013. Published October 6, 2013.

weighted Lipschitz W(Lr, ξ(t)), r ≥ 1 class by product summability (C,1) (E, q) transform of conjugate series of Fourier series has been established.

Let L2π be the space of all 2π - periodic and Lebesgue-integrable functions over

[−π, π].Then the Fourier series off ∈L2π atxis given by f(x)∼ a₂0 +

∞

X

k=1

(akcos kx+bksin kx)≡

∞

X

k=0

Ak(x), (1)

withnth partial sumssn(f;x),called trigonometric polynomial of degree (or order)

n, of the first (n+ 1) terms of the Fourier series of f , ak and bk are the Fourier coefficients off .

The conjugate series of Fourier series (1) is given by

∞

X

k=1

(bkcos kx−aksin kx)≡

∞

X

k=1

Bk(x). (2)

LetP∞_n=0un be a given infinite series with sequence of itsnth _{partial sums{sn}.}

Hausdorff matrixH ≡(hn,k) is an infinite lower triangular matrix defined by

hn,k =

n

k

∆n−k_{µk, 0≤k≤n,}

0, k > n,

where ∆ is the forward difference operator defined by ∆µn = µn −µn+1 and ∆k+1_{µn = ∆}k_{(∆µn). If H is regular, then{µn},known as moment sequence, has}

the representation

µn= Z 1

0

undγ(u),

whereγ(u) known as mass function, is continuous atu= 0 and belongs to BV[0,1] such thatγ(0) = 0, γ(1) = 1; and for 0 < u <1, γ(u) = [γ(u+ 0) +γ(u−0)]/2.

The Hausdorff means of conjugate Fourier series off are defined by ˜

Hn(f;x) =

n

X

k=0

hn,ksk˜ , n≥0.

The conjugate Fourier series is said to be summable to s by Hausdorff means, if ˜

Hn(f;x)→s, asn→ ∞. For the mass functionγ(u) is given by

γ(u) =

0, 0≤u < a,

1, a≤u≤1,

wherea= 1/(1 +q), q >0,we can verify thatµk = 1/(1 +q)k_{, and} hn,k=

(

n k

qn₋k

(1+qn

), 0≤k≤n, 0, k > n.

Thus Hausdorff matrixH ≡(hn,k) reduces to Euler matrix (E, q) of order q >0. One more example of Hausdorff matrix [γ(u) =ufor 0≤u≤1] is the well known Ces`aro matrix of order 1 (C,1).

The (E, q) transform is defined as thenth _{partial sum of (E, q), q >0 summability}

and we denote it byEq n. If

Enq =

1 (q+ 1)n

n

X

k=0

n k

qn−k_sk

then the infinite seriesP∞_n=0un is summable (E, q) to a definite numbers[2]. If

τ

=

s

+

s

+

s

+

...

+

s

n

+ 1

=

1

n

+ 1

n

X

k=0

s

→

s, as n

→ ∞

,

(4)

then the infinite series P∞n=0un is summable to the definite number s by (C,1) method. The (C,1) transform of the (E, q) transform defines (C,1) (E, q) product transform and denote it byEq

nCn1. Thus if Cn1Enq =

1

n+ 1

n

X

k=0

(1 +q)−k k

X

v=0

k v

qk−v_sv

→s, as n→ ∞, (5) then the infinite seriesP∞_n=0un is said to be summable by (C,1) (E, q) method or summable (C,1) (E, q) to a definite number ’s’.

If C1

nEnq →s as n → ∞ then the infinite seriesP∞n=0un or the sequence {sn} is said to be summable (C,1) (E, q) to the sum s if limn→∞Cn1Enq exists and is equal

tos.

sn→s ⇒ (E, q)(sn) =Enq =

1 (1 +q)n

n

X

k=0

n k

qn−k_sk

→s, as n→ ∞,(E, q)method is regular,

⇒ (C,1)((E, q)(sn)) =Cn1Enq →s, as n→ ∞,(C,1)method is regular,

⇒ (C,1)(E, q)method is regular.

A signal (function) is said to belong to the classLipα, if

|f(x+t)−f(x)|=O(|tα|)f or 0< α≤1, t >0. (6) andf(x)∈Lip(α, r), f or 0≤x≤2π, if

kf(x+t)−f(x)kr=

Z 2π

0 |

f(x+t)−f(x)|rdx

1

r

=O

|t|α

, 0< α≤1, r≥1, t >0.

(7) For a given a positive increasing functionξ(t), f(x)∈Lip(ξ(t), r) if

kf(x+t)−f(x)kr=

Z 2π

0 |

f(x+t)−f(x)|rdx

1

r

=O

ξ(t)

, r≥1, t >0. (8) Given positive increasing function ξ(t), an integerr ≥1, a signal (function) f is said to belong to the generalized weighted LipschitzW(Lr,(ξ(t)))-class ([5]), if

[f(x+t)−f(x)]sinβ

x/2

_r=O

ξ(t)

, β ≥0, t >0. (9) Ifβ = 0, then the generalized weighted LipschitzW(Lr,(ξ(t))) class coincides with the classLip(ξ(t), r), we observe that

W(Lr,(ξ(t))−−−→β=0 Lip(ξ(t), r) ξ(t)=t

α

−−−−−→Lip(α, r)−−−→r→∞ Lip(α) f or0< α≤1, r≥1, t >0.

(10) TheLr- norm of signalf is defined by

kfkr=

Z 2π

0 |

f(x)|rdx

1

r

,(1≤r <∞), andkfk∞= sup x∈[0,2π]|

f(x)|. (11) A signal (function)f is approximated by trigonometric polynomialsτn(f) of order

En(f) = min

n kf(x)−τ(f;x)kr, (12)

in terms of n, whereτn(f;x) is a trigonometric polynomial of degree n. This method of approximation is called Trigonometric Fourier Approximation (TFA) [8]. We note thatEq

n andCn1Enq are also trigonometric polynomials of degree (or order) n.

The conjugate functionfe(x) is defined for almost everyxby

e

f(x) =−₂1_π Z π

0

ψ(t)cos t/2dt= lim

h→0

−₂1_π Z π

h

ψ(t)cos t/2dt

(see [22, p. 131]).

(13) We use the following notations throughout this paper

ψ(t) =ψx(t) =f(x+t)−f(x−t),

e

Gn(t) = 1 2π(1 +n)

   

n

X

k=0 1 (1 +q)k

k

X

v=0

k v

qk−v cos

v+ 1/2

t sin

t/2

   .

2. Known Theorem

In a recent paper, Sonker and Singh [21] have studied the degree of approximation of functions belonging toLip(α, r) - class using (C,1) (E, q) summability means of its conjugate Fourier series. They proved the following theorem:

Theorem 2.1. Let f(x)be a2π- periodic, Lebesgue integrable function and belong to theLip(α, r)- class withr≥1andαr≥1.Then the degree of approximation of

e

f(x), the conjugate of f (x) by (C,1) (E, q) means of its conjugate Fourier series is given by

kCn1Enq−fekr=O

n−α+1/r

, n= 0,1,2, ..., (14)

provided

Z π/(n+1)

0

|ψ(t)|

tα

r dt

1

r

=O((n+ 1)−1_{), (15)} Z π

π/(n+1)

t−δ|ψ(t)| tα

r dt

1

r

=O((n+ 1)δ), (16)

whereδis an arbitrary number such that(α+δ)s+1<0andr−1_+s−1_{= 1f or r >} 1.

3. Main Theorem

series. In this paper, we use the (C,1)(E, q) means of conjugate Fourier series of

f ∈W(Lr, ξ(t)) to determine the degree of approximation of the conjugate of f, which in turn generalizes the result of Sonker and Singh [21] and Lal and Singh [6]. More precisely, we prove

Theorem 3.1. If fe(x), conjugate to a 2π - periodic function f belonging to the generalized weighted LipschitzW(Lr, ξ(t)),(r≥1)- class then its degree of approx-imation by (C,1) (E, q) product summability means of conjugate series of Fourier series is given by

kC^1

nE q

n−fekr=O

n

(√n)1r+β

o

ξ

₁

√

n

, (17)

provided ξ(t) satisfies the following conditions:

Z π √n

0

t|ψ(t)|

ξ(t) r

sinβrt/2dt

1/r

=O

1 √_n

, (18)

Z π

π √n

t−δ_|ψ(t)| ξ(t)

r dt

1/r

=O

_√

n

δ

, (19)

and ξ(_tt) is non-increasing in ’t’,

whereδis an arbitrary number such thats(1−δ+β)−1>0,1

r+

1

s = 1, 1≤r≤ ∞, conditions (18) and (19) hold uniformly in x andC^1

nE q

nis(C,^1)(E, q)means of the series (2) and

f(x) =−₂1_π Z π

0

ψ(t)cos t/2dt. (20)

Note 3.2 ξ

π √ n

≤πξ

1

√ n

,for

π √ n

≥

1

√ n

.

Note 3.3.The product transform (C,1) (E, q) plays an important role in signal the-ory as a double digital filter [13] and thethe-ory of Machines in Mechanical Engineering [13].

4. Lemmas

For the proof of our theorem, the following lemmas are required:

Lemma 4.1. |Gne (t)|=O

1

t

+ ((n+ 1)t) f or0≤t≤ _√π n ≤

Proof. For 0≤t≤√π n,sin

t

2 ≥

t

π and|cos nt| ≤1.

|Gne (t)| ≤ _2π(n1_{+ 1)} n X k=0 1 (1 +q)k

k X v=0 k v

qk−v cos

v+ 1/2 t sin t/2 = 1

2π(n+ 1) n X k=0 1 (1 +q)k

k X v=0 k v qk−v cos

v+ 1−1/2 t sin t/2 ≤ 1

(n+ 1)

n

X

k=0 1 (1 +q)k

k X v=0 k v qk−v cos

v+ 1 t cos t/2 +sin

v+ 1 t sin t/2 sin t/2 = 1

(n+ 1)

n

X

k=0 1 (1 +q)k

k X v=0 k v qk−v cos

v+ 1 tcos t/2 +sin

v+ 1 t sin t/2 sin t/2 = 1

(n+ 1)

n

X

k=0 1 (1 +q)k

k X v=0 k v

qk−v

O 1 t +O

sin(v+ 1)t

= O

" 1 (n+ 1)t

n

X

k=0 1 (1 +q)k

k X v=0 k v

qk−v

# +O

" 1 (n+ 1)

n

X

k=0 1 (1 +q)k

k X v=0 k v

qk−v_{(v+ 1)t}

#

= O

1

(n+ 1)t(n+ 1)

+O

1

(n+ 1)(n+ 1)(n+ 1)t = O 1 t

+O[(n+ 1)t],

in view ofsin(v+ 1)t≤(v+ 1)tfor 0≤t≤_√π n ≤

π √_v.

This completes the proof of Lemma (4.1)

Lemma 4.2. |Gne (t)|=O

1

t

+O(1) f or_√π n ≤

π

Proof. For 0≤ √π n ≤

π √

v ≤t≤π,sin t

2 ≥

t

π and|cos nt| ≤1.

|Gne (t)| ≤ _2π(n1_{+ 1)} n X k=0 1 (1 +q)k

k X v=0 k v

qk−v cos

v+ 1/2 t sin t/2 = 1

2π(n+ 1) n X k=0 1 (1 +q)k

k X v=0 k v

qk−v cos

v+ 1−1/2 t sin t/2

≤ _{(n+ 1)}1

n

X

k=0 1 (1 +q)k

k X v=0 k v

qk−v

cos

v+ 1 tcos t/2 +sin

v+ 1 t sin t/2 sin t/2 = 1

(n+ 1)

n

X

k=0 1 (1 +q)k

k X v=0 k v

qk−v

cos

v+ 1 tcos t/2 +sin

v+ 1 t sin t/2 sin t/2 = 1

(n+ 1)

n

X

k=0 1 (1 +q)k

k X v=0 k v qk−v O ₁ t +O 1 = O " 1 (n+ 1)t

n

X

k=0 1 (1 +q)k

k X v=0 k v qk−v # +O " 1 (n+ 1)

n

X

k=0 1 (1 +q)k

k X v=0 k v qk−v # = O ₁

(n+ 1)t(n+ 1)

+O

₁

(n+ 1)(n+ 1)

= O

₁

t

+O[1],

in view of|sin(v+ 1)t| ≤1 for 0≤t≤ _√π n ≤

π √

v ≤t≤π.

This completes the proof of Lemma (4.2).

5. Proof of Theorem 3.1

Let ˜sn(x) denotes the partial sum of series (2), we have

e

sn(x)−fe(x) = 1 2π

Z π

0

ψ(t)

cos

n+ 1/2 t sin t/2 dt.

Therefore, using (2) the (C,1)transformC1

n ofesn is given by

e

Cn1−fe(x) =

1 2π(n+ 1)

Z π

0

ψ(t)

n

X

k=0

cos

Now denoting(C,^1)(E, q) transform ofesn as (C^1

nE q

n), we write

^

(C1

nE q

n)−fe(x) = 1

2π(n+ 1) _Xn

k=0 1 (1 +q)k

Z π

0

ψ(t)

sintt 2 k X v=0 k v

qk−v_cos

v+1 2 tdt = Z π 0

ψ(t)Gne (t)dt

=

Z π/√n

0 +

Z π

π/√n

ψ(t)Gne (t)dt

= I1+I2,(say). (21)

We consider,

|I1| ≤ Z π/√n

0 |

ψ(t)| Gne (t)

dt.

Using Lemma 4.1, H¨older’s inequality, Minkowski’s inequality and condition 18, we have

|I1| ≤

Z π/√n

0

t|ψ(t)|

ξ(t) r

sinβrt/2dt

1/r Z π/√n

0

ξ(t)|Gne (t)|

tsinβ_t/2

s dt 1/s = O 1 √ n

Z π/√n

0

ξ(t)|Gne (t)|

tsinβ_t/2

s dt 1/s = O 1 √ n

Z π/√n

0

ξ(t)

t2_sinβ_t/2 +

ξ(t)(n+ 1)t tsinβ_t/2

s dt 1/s = O 1 √ n

Z π/√n

h

ξ(t)

t2+β

s dt

1/s

+O(√n)

Z π/√n

h

ξ(t)

tβ

s dt

1/s

, as h→0.

Since ξ(t) is a positive increasing function and using second mean value theorem for integrals, we have

|I1| = O

₁ √ nξ _π √ n

Z π/√n

h

₁

t2+β

s dt 1/s +O _√ nξ _π √ n

Z π/√n

h ₁ tβ s dt 1/s

, as h→0

= O 1 √ nπξ 1 √ n

Z π/√n

h

t−βs−2s_dt

1/s +O _√ nπξ 1 √ n

Z π/√n

h

t−βs_dt

1/s

, as h→0.

Note that ξ _π √ n ≤πξ ₁ √ n ,

|I1| = O ₁ √ nξ ₁ √ n

(√n)β+2−1/s +O ξ ₁ √ n

(√n)β+1−1/s = O ξ ₁ √ n

(√n)β+1−1/s

= O ξ ₁ √ n

(√n)β+1/r

∴ 1

r+

1

s = 1, 1≤r≤ ∞. (22)

Now, we consider,

|I2| ≤Rπ/π√n|ψ(t)|

Gne (t)

Using Lemma 4.2, H¨older’s inequality, Minkowski’s inequality and condition 19, we have

|I2| ≤ Z π

π/√n

t−δ_sinβ_t/2.|ψ(t)| ξ(t)

r dt

1/r Z π

π/√n

ξ(t)|Gne (t)|

t−δ_sinβ_t/2

s dt

1/s

= O

(√n)δ

Z π

π/√n

ξ(t)|Gne (t)|

t−δ_sinβ_t/2

s dt

1/s

= O

(√n)δ

Z π

π/√n

ξ(t)

t−δ+1_sinβ_t/2+

ξ(t)

t−δ_sinβ_t/2

s dt

1/s

= O

(√n)δ

Z π

π/√n

ξ(t)

t−δ+β+1 s

dt

1/s

+ Z π

π/√n

ξ(t)

t−δ+β

s dt

1/s .

Now puttingt= 1/y,

|I2| ≤ O

(√n)δ

Z (√n)/π

1/π

ξ(1/y)

yδ−β−1 s

dy y2

1/s

+

Z (√n/π)

1/π

ξ(1/y)

yδ−β

s dy y2

1/s .

Since ξ(t) is a positive increasing function so ξ(1_1/y/y) is also a positive increasing function and using second mean value theorem for integrals, we have

|I2| = O

(√n)δ

ξ

π/√n

π/√n

" Z (√n)/π

1/π

dy y−βs+δs+2

1/s

+

Z (√n)/π

1/π

dy y−βs+δs+s+2

1/s# .

= O

(√n)δ+1ξ

1 √

n

"

(y−δs−1+βs₎(√n)/π

1/π

1/s

+

(y−δs−1+βs−s₎(√n)/π

1/π

1/s# .

= O

(√n)δ+1ξ

₁

√

n (

√

n)−δ−1/s+β_{) + (}√_n)−δ−1/s+β−1_.

= O

ξ

₁

√_n h(√n)−δ−1/s+β+δ+1_{+ (}√_n)−δ−1/s+β−1+δ+1i_.

= O

ξ

1

√_n h(√n)β+1/r+ (√n)−1+β+1/ri_∴ 1 r +

1

s = 1, 1≤r≤ ∞

= O

ξ

1 √

n

(√n)β+1/r

1 + (√n)−1

= O

ξ

1 √

n

(√n)β+1/r

. (23)

CombiningI1andI2 yields

|C^1

nEnq−fe|=O

(√n)β+1/r_ξ

₁

√

n

Now, using theLr -norm of a function, we get kC^1

nE q

n−fekr =

Z 2π

0 |

^ C1

nE q

n−fe|rdx

1/r

= O

Z 2π

0

(√n)β+1/rξ

₁

√

n

r1/r

= O

(√n)β+1/r_ξ

₁

√

n

Z 2π

0

dx

1/r

= O

(√n)β+1/rξ

1 √_n

.

This completes the proof of Theorem 3.1.

6. Applications

The theory of approximation is a very extensive field, which has various applica-tions. As mentioned in [20], theLp-space in general, and L2and L∞ in particular

play an important role in the theory of signals and filters. From the point of view of the applications, 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 enables to investigate perturbations of matrix valued functions and compare them. The following corollaries may be derived from Theorem 3.1.

Corollary 6.1. Ifξ(t) =tα_{,0< α≤1, then the weighted class W(Lr, ξ(t)), r≥1} reduces to the class Lip(α, r),(1/r) < α < 1 and the degree of approximation of a function fe(x), conjugate to a 2π - periodic function f belonging to the class Lip(α, r), is given by

|C^1

nE q

n−fekr=O

1 (√n)α−1/r

. (25)

Proof. Puttingβ= 0, ξ(t) =tα_{,0< α≤1 in Theorem 3.1, we have}

kC^1

nE q

n−fekr=

Z 2π

0 |

^ C1

nE q

n(f;x)−fe(x)|rdx

1/r

or,

O

(√n)1/r_ξ

₁

√

n

=

Z 2π

0 |

^ C1

nEnq(f;x)−fe(x)|rdx

1/r

or,

O(1) = Z 2π

0 |

^ C1

nEnq(f;x)−fe(x)|rdx

1/r O

₁

(√n)1/r_ξ(√1

n) ,

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

C^n1E

q

n(f;x)−fe(x)

=O

₁

√

n

α

(√n)1/r

=O

₁

(√nα−1/r

.

Corollary 6.2. Ifξ(t) =tα_{,0< α≤1andr→ ∞in corollary 6.1, thenf ∈Lipα} and

C^n1E

q n−fe

=O

₁

√

n

α

. (26)

Proof. Forr→ ∞in corollary 6.1, we get kC^1

nE q

n−fek∞= sup

0≤x≤2π

C^n1E

q

n(f;x)−fe(x)

=O

1 √_n

α

.

Thus, we have C^n1E

q

n(f;x)−fe(x)

≤

C^n1E

q

n(f;x)−fe(x)

∞

= sup

x∈[0,2π] C^n1E

q

n(f;x)−fe(x)

= O

_√

n

−α

This completes the proof of Corollary 6.2.

7. An Example

Since Hausdorff matrices commute, therefore it is very easy to find an example of a positive non-convergent sequence which is (C,1) summable. Let us consider an infinite series

∞

X

n=0

(−1)n(2q+ 1)ncos nx (27)

Thenth _{partial sumssn of series (27) at x =0 is given by} sn =

n

X

r=0

(−1)r(2q+ 1)rcos rx

≤ 1−(−1)

n+1_{(2q+ 1)}n+1 2(1 +q)

Since limn→∞sn does not exist. Therefore the series (27) is non-convergent.

Now, the (E, q) transform of (27) is given by

Eq

n =

1 (1 +q)n

n

X

k=0

n q

qn−k_{sk, q >0}

≤ _{(1 +}1_q)n n

X

k=0

n k

qn−k

₁

−(−1)k+1_{(2q+ 1)}k+1 2(1 +q)

= 1

2(1 +q)+

(1 + 2q)(q−1−2q)n

2(1 +q)(q+ 1)n

= 1

2(1 +q)+

(1 + 2q)(−1)n

Here, limn→∞Enq not exist. Hence the series (27) is not summable.

Now,

σn1=

1

n+ 1

n

X

k=0

sk ≤ _{n+ 1}1

n

X

k=0 ₁

−(−1)k+1_{(2q+ 1)}k+1 2(1 +q)

= 1

2(1 +q)+

(1 + 2q) 2(1 +q)(n+ 1)

∞

X

n=0

(−1)k(2q+ 1)k

= 1

2(1 +q)+

(1 + 2q)

4(1 +q)2_{(n+ 1)}{1 + (2q+ 1)

n+1 }

= 1

2(1 +q)+

(1 + 2q) 4(1 +q)2_{(n+ 1)} +

(1 + 2q)n+2 4(1 +q)2_{(n+ 1)} Here, limn→∞σn1 does not exist, the series (27) is not (C,1) summable.

In

n

X

k=0

n k

qn−k

Change the variable of summation byj= (n−k) to get 0

X

j=n

n n−j

qj ₌ n

X

j=0

n j

qj _{= (1 +q)}n_.

Now integrate both sides from 0 toxto get

n

X

j=0

n j

xj+1

j+ 1 =

(1 +q)n+1

n+ 1 x

0

=(1 +x)

n+1₋₁

n+ 1 . Thus

n

X

j=0

n j

_xj j+ 1 =

(1 +x)n+1₋₁

x(n+ 1) . Finally, the (C, 1) (E, q) transform of (27) is given by

EnqCn1 =

1 (1 +q)n

n

X

k=0

n k

qn−k_σ1

k

≤ 1

(1 +q)n n

X

k=0

n k

qn−k

₁

2(1 +q)+

(1 + 2q) 4(1 +q)2_{(k+ 1)} +

(1 + 2q)k+2 4(1 +q)2_{(k+ 1)}

= 1

2(1 +q)+

(1 + 2q) 4(1 +q)n+2 −

n

X

k=0

n k

_qn−k

(k+ 1) +

(1 + 2q)2 4(1 +q)n+2

n

X

k=0

n k

_qn−k

(k+ 1)

= 1

2(1 +q)+

(1 + 2q) 4(1 +q)n+2

_{(1 +q)}n+1₋₁ (n+ 1)q

+ (1 + 2q) 2

4(1 +q)n+2

_{(1 +q)}n+1₋₁ (n+ 1)q

= 1

2(1 +q)+

(1 + 2q) 4(1 +q)(n+ 1)q

1−_{(1 +}1_q)n+1

+ (1 + 2q) 2

4(1 +q)(n+ 1)q

1−_{(1 +}1_q)n+1

. Eq

nCn1→ 2(1+1q) asn→ ∞, the series (27) is (E, q)(C,1) summable.

Therefore the series (27) is neither (E, q) summable nor (C,1) summable but it is (E, q)(C,1) summable to 1

approximation than the individual methods (E, q) and (C,1).

Acknowledgement: Authors are grateful to the anonymous learned referees for their valuable suggestions and comments, leading to an overall improvement in the paper. Authors mention their sense of gratitude to Professor H¨useyin Bor, editorial board member of BMAA and Professor Faton Z. Merovci, secretary of BMAA for for their efforts in sending the reports of the manuscript timely. LNM computed an example of (E, q)(C,1) in its detail in the present manuscript.

Kejal Khatri and Lakshmi Narayan Mishra, one of the authors, are thankful to the financial support of Ministry of Human Resource Development (MHRD), New Delhi, Govt. of India, India to carry out their research article.

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Vishnu Narayan Mishra and Kejal Khatri

Applied Mathematics and Humanities Department, Sardar Vallabhbhai National Insti-tute of Technology, Ichchhanath Mahadev Road, Surat,

Surat-395 007 (Gujarat), India.

E-mail address_{: [email protected]; vishnu [email protected] and}

[email protected]

Huzoor H. Khan

Department of Mathematics, Aligarh Muslim University Aligarh, India.

E-mail address_{: [email protected]}

Lakshmi Narayan Mishra

L. 1627 Awadh Puri Colony Beniganj, Phase -III, Opp. I.T.I., Ayodhya Main Road Faizabad-224 001, (Uttar Pradesh).

Department of Mathematics, National Institute of Technology, Silchar - 788 010, District - Cachar (Assam), India.