EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 13, No. 2, 2020, 351-368
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Approximation of Function in generalized Hölder Class

H. K. Nigam1, Supriya Rani1,∗

1Department of Mathematics, Central University of South Bihar, Gaya-824236 (Bihar),
India

Abstract. In the present work, we study error estimation of a function g ∈ H(η)
r (r ≥ 1) class

using Matrix-Hausdorff (T∆H) means of its Fourier series. Our Theorem 1 generalizes twelve
previously known results. Thus, the results of [4, 5, 11–16, 18, 26, 29, 30] become the particular
cases of our Theorem 1. Several useful results in the form of corollaries are also deduced from our
Theorem 1.

2020 Mathematics Subject Classifications: 41A10, 41A25, 42B05, 42A10, 40G05, 40C05

Key Words and Phrases: Error estimation, Generalized Hölder class, Fourier series, Matrix (T )
means, Hausdorff (∆H) means, Matrix-Hausdorff (T∆H) product means

1. Introduction

In the past few decades, the researchers have been greatly interested in studying the
error estimation of functions in different function spaces using summability operators
due to their variety of applications in science and engineering. In this direction, several
researchers like [2, 3, 9, 10, 19–23, 25, 28] have obtained results on error estimation of
functions in different Lipschitz classes and Hölder classes with different single summability
operators. Taking a view point that a product summability is more effective than the
individual single summability operator, researchers like [11, 13, 18, 27–29], have obtained
error estimation of functions in various Lipschitz and Hölder classes using different product
summability operators.

After reviewing the above mentioned works, we observe that these works cannot provide
the best error estimation of a function in the function spaces considered in their works.
This fact strongly motivates us to consider a more advanced class of function, which
provide the best approximation of a function using summability operator.

Therefore, in the present work, we establish a theorem on the best error approximation of

a function g in the generalized Hölder class H
(η)
r (r ≥ 1) using Matrix-Hausdorff (T∆H)

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v13i2.3667

Email addresses: hknigam@cusb.ac.in (H. K. Nigam), supriya@cusb.ac.in (Supriya Rani)

http://www.ejpam.com 351 c© 2020 EJPAM All rights reserved.



S. Rani, H. K. Nigam / Eur. J. Pure Appl. Math, 13 (2) (2020), 351-368 352

product operator of its Fourier series. Our main theorem generalizes tweleve previously
known results. Thus, the results of [4, 5, 11–16, 18, 26, 29, 30] become the particular cases
of our theorem.

2. Preliminaries

Let
∑∞

l=0 cl be an infinite series having lth partial sum sl =
∑l

ν=0 cν .

Let T ≡ (bl,j) be an infinite triangular matrix satisfying the conditions of regularity [24]
i.e. 

∑l
j=0 bl,j = 1 as l→∞;

∀ j ≥ 0, bl,j = 0 as l→∞;

∃ M > 0 ∀ l ≥ 0,
∑∞

j=0 |bl,j | < M.

(1)

The sequence-to-sequence transformation

tTl :=
l∑

j=0

bl,jsj

=
l∑

j=0

bl,l−jsl−j

defines the sequence tTl of triangular matrix means of the sequence {sl} generated by the
sequence of coefficients (bl,j).

If tTl → s as l→∞, then the infinite series
∑∞

l=0 cl or the sequence {sl} is summable to s
by triangular matrix (T ) [1].

A Hausdorff matrix H ≡ (hl,j) is an infinite lower triangular matrix [8] defined by

hl,j ≡


(
l

j

)
∆l−jµj , 0 ≤ j ≤ l;

0, j > l,

where the operator ∆ is defined ∆µj ≡ µj − µj+1 and ∆l+1µj ≡ ∆l(∆µj).

If t∆H
l =

∑l
m=0 hl,msm → s as l→∞ then the series or the sequence {sl} is summable to

the sum s by the Hausdorff method (∆H method).

A Hausdorff matrix H is regular, i.e., H preserves the limit of each convergent sequence
iff ∫ 1

0
|dξ(z)| <∞,

where the mass function ξ ∈ BV [0, 1], ξ(0+) = ξ(0) = 0, and ξ(1) = 1. In this case, µl
has the representation

µl =

∫ 1

0
zldξ(z) [17].



S. Rani, H. K. Nigam / Eur. J. Pure Appl. Math, 13 (2) (2020), 351-368 353

Superimposing T - method on ∆H method, (T∆H) is obtained. T∆H mean of the sequence
{sl} is given by

tT∆H
l :=

l∑
j=0

bl,jt
∆H
j

=
l∑

j=0

bl,j

j∑
v=0

hj,vsv.

If tT∆H
l → s as l→∞, then {sl} is summable by the T∆H means to the limit s.

Since T and ∆H method are regular, then T∆H method is also regular. This can be shown
as

sl → s ⇒ t∆H
l → s, as l→∞, since the ∆H method is regular,

⇒ T (t∆H
l ) = tT∆H

l → s, as l→∞, since the T method is regular,

⇒ T∆H method is regular.

Remark 1. T∆H means reduces to

(i) (C,α)∆H or Cα∆H means when bl,j =
(l−j+α−1

α−1 )
(l+αα )

for all α ≥ −1.

(ii)
(
H, 1

l+1

)
∆H or H1/l+1∆H means if bl,j = 1

(l−j+1) log(l+1) .

(iii) (N, pl, ql)∆H or Np,q∆H means if bl,j =
pl−jqj
Rl

, Rl =
∑l

j=0 pjql−j .

(iv) (N, pl)∆H or Np∆H means if bl,j =
pl−j
Pl

where Pl =
∑l

j=0 pj , ql = 1.

(v) (Ñ , pl)∆H or Ñp∆H means if bl,j =
pj
Pl
, ql = 1 ∀ l.

(vi) (E, ql)∆H or Eq∆H means if bl,j = 1
(1+q)l

(
l
j

)
ql−j .

(vii) T (C,α) or TCα means if ξ(z) =
∏α
j=1 z

j , α ≥ 1.

(viii) T (E, ql) or TEq means if hl,j =

(
l
j

)
ql−j

(1+q)l
, 0 ≤ j ≤ l.

In above Remark 1 (iii), (iv) and (v), {pl} and {ql} are two non-negative monotonic
non-decreasing sequence of real constants.

Remark 2.

(i) (C,α)∆H or Cα∆H means further reduces to



S. Rani, H. K. Nigam / Eur. J. Pure Appl. Math, 13 (2) (2020), 351-368 354

(a) (C,α)(C,α) or CαCα means if ξ(z) =
∏α
j=1 z

j , α ≥ 1.

(b) (C,α)(E, ql) or CαEq means if hl,j =
(
l
j

) ql−j

(1+q)l
, 0 ≤ j ≤ l.

(c) (C, 1)∆H or C1∆H means if α = 1.

(ii)
(
H, 1

l+1

)
∆H or H1/l+1∆H means further reduces to

(a)
(
H, 1

l+1

)
(C,α) or H1/l+1Cα means if ξ(z) =

∏α
j=1 z

j , α ≥ 1.

(b)
(
H, 1

l+1

)
(E, ql) or H1/l+1Eq if hl,j =

(
l
j

) ql−j

(1+q)l
, 0 ≤ j ≤ l.

(iii) (N, pl, ql)∆H or Np,q∆H means further reduces to

(a) (N, pl, ql)(C,α) or Np,qCα means if ξ(z) =
∏α
j=1 z

j , α ≥ 1.

(b) (N, pl, ql)(E, ql) or Np,qEq means if hl,j =
(
l
j

) ql−j

(1+q)l
, 0 ≤ j ≤ l.

(iv) (N, pl)∆H or Np∆H means further reduces to

(a) (N, pl)(C,α) or NpCα means if ξ(z) =
∏α
j=1 z

j , α ≥ 1.

(b) (N, pl)(E, ql) or NpEq means if hl,j =
(
l
j

) ql−j

(1+q)l
, 0 ≤ j ≤ l.

(v) (Ñ , pl)∆H or Ñp∆H means further reduces to

(a) (Ñ , pl)(C,α) or ÑpCα means if ξ(z) =
∏α
j=1 z

j , α ≥ 1.

(b) (Ñ , pl)(E, ql) or ÑpEq means if hl,j =
(
l
j

) ql−j

(1+q)l
, 0 ≤ j ≤ l.

(vi) (E, ql)∆H or Eq∆H means further reduces to

(a) (E, ql)(C,α) or EqCα means if ξ(z) =
∏α
j=1 z

j , α ≥ 1.

(b) (E, ql)(E, ql) or EqEq means if hl,j =
(
l
j

) ql−j

(1+q)l
, 0 ≤ j ≤ l.

(vii) T (C,α) or TCα means further reduces to

(a) T (C, 1) or TC1 means if α = 1.

(viii) T (E, ql) or TEq means further reduces to

(a) T (E, 1) or TE1 means if ql = 1 ∀ l.

Remark 3.

(i) Above particular case (i)(b) in Remark 2 is further reduced to C1Eq, CαE1 and C1E1

means for α = 1, ql = 1 ∀ l and α = 1, ql = 1 ∀ l respectively.

(ii) Above particular cases (ii)(a) and (b) in Remark 2 are further reduced to H1/l+1C1

and H1/l+1E1 means for α = 1 and ql = 1 ∀ l respectively.



S. Rani, H. K. Nigam / Eur. J. Pure Appl. Math, 13 (2) (2020), 351-368 355

(iii) Above particular cases (iii)(a) and (b) in Remark 2 are further reduced to (N, pl, ql)(C, 1)
and (N, pl, ql)(E, 1) means for α = 1 and ql = 1 ∀ l respectively.

(iv) Above particular cases (iv)(a) and (b) in Remark 2 are further reduced to (N, pl)(C, 1)
and (N, pl)(E, 1) means for α = 1 and ql = 1 ∀ l respectively.

(v) Above particular cases (v)(a) and (b) in Remark 2 are further reduced to (Ñ , pl)(C, 1)
and (Ñ , pl)(E, 1) means for α = 1 and ql = 1 ∀ l respectively.

(vi) Above particular cases (vi)(a) in Remark 2 is further reduced to EqC1, E1Cα and
E1C1 means for α = 1, ql = 1 ∀ l and ql = 1 ∀ l, α = 1 respectively.

The space of the functions Lr is given by

Lr[0, 2π] =

{
g : [0, 2π] 7→ R :

∫ 2π

0
|g(x)|rdx <∞, r ≥ 1

}
.

The norm ‖ · ‖(r) by {
1

2π

∫ 2π

0
|g(x)|rdx

}1/r

, r ≥ 1.

As defined in [1], η : [0, 2π] 7→ R is an arbitrary function with η(s) > 0 for 0 < s ≤ 2π and
lims→0+ η(s) = η(0) = 0.

Now, we define

H(η)
r :=

{
g ∈ Lr[0, 2π] : sup

s 6=0

‖g(·,+s)− g(·)‖r
η(s)

<∞, r ≥ 1

}

and

‖ · ‖(η)
r = ‖g‖(η)

r = ‖g‖r + sup
s 6=0

‖g(·,+s)− g(·)‖r
η(s)

; r ≥ 1.

Clearly, ‖ · ‖(η)
r is a norm on H

(η)
r .

Note 1. η(s) and χ(s) denote moduli of continuity of order two such that η(s)
χ(s) is positive,

non-decreasing and

‖g‖(χ)
r ≤ max

(
1,
η(2π)

χ(2π)

)
‖g‖(η)

r <∞.

Thus,
H(η)
r ⊂ H(χ)

r ⊂ Lr; r ≥ 1 [1].

Remark 4.

(i) If η(s) = sα in H(η), H(η) implies H(α) class.

(ii) If η(s) = sα in H
(η)
r , H(η) implies Hα,r class.



S. Rani, H. K. Nigam / Eur. J. Pure Appl. Math, 13 (2) (2020), 351-368 356

(iii) If r →∞ in H
(η)
r , H

(η)
r implies H(η) class and Hα,r implies Hα class.

We denote the lth partial sum of the Fourier series as

sl(g;x)− g(x) =
1

2π

∫ π

0
φ(x, s)

sin
(
l + 1

2

)
s

sin s
2

ds [1].

The l-order error estimation of function g is given by

El(g) = min ‖g − tl‖r,

where tl is a trigonometric polynomial of degree l [1].

If El(g)→ 0 as l→∞, then El(g) is said to be the best approximation of g [1].

We write

φ(x, s) = g(x+ s) + g(x− s)− 2g(x);

∆bl,j = bl,j − bl,j+1;

KT∆H
l (s) =

1

2π

l∑
j=0

bl,j

j∑
a=0

∫ 1

0

(
j

a

)
za(1− z)j−a dξ(z)

sin
(
a+ 1

2

)
s

sin s
2

.

3. Main Theorem

Theorem 1. If g ∈ H(η)
r class, r ≥ 1, then the error estimation of g using T∆H product

means of its Fourier series is given by

‖tT∆H
l − g‖(χ)

r = O

(
1

l + 1

∫ π

1
l+1

η(s)

s2χ(s)
ds

)
,

where T ≡ (bl,j) is an infinite triangular matrix satisfying (1) and η, χ are as defined in
Note 1, provided

l−1∑
j=0

|∆bl,j | = O

(
1

l + 1

)
(2)

and

(l + 1)bl,l = O(1). (3)

4. Lemmas

Lemma 1. Under the conditions of regularity of matrix T ≡ (bl,j),

KT∆H
l (s) = O(l + 1) for 0 < s <

1

l + 1
.



S. Rani, H. K. Nigam / Eur. J. Pure Appl. Math, 13 (2) (2020), 351-368 357

Proof. For 0 ≤ s ≤ 1
l+1 , sin s

2 ≥
s
π , sin ls ≤ ls, we have

KT∆H
l (s) =

1

2π

l∑
j=0

bl,j

j∑
a=0

∫ 1

0

(
j
a

)
za(1− z)j−a dξ(z)

sin
(
a+ 1

2

)
s

2 sin s
2

=
1

2π

l∑
j=0

bl,j

j∑
a=0

∫ 1

0

(
j
a

)
za(1− z)j−a dξ(z)

(2a+ 1) s2
s
π

=
1

4

l∑
j=0

bl,j

{
j∑

a=0

∫ 1

0

(
j
a

)
za(1− z)j−a dξ(z)(2a+ 1)

}

=
1

4

l∑
j=0

bl,j

[
2

j∑
a=0

∫ 1

0

(
j
a

)
za(1− z)j−a a dξ(z)

]

+
1

4

l∑
j=0

bl,j

[
j∑

a=0

∫ 1

0

(
j
a

)
za(1− z)j−a dξ(z)

]
. (4)

First, we solve

2

j∑
a=0

(
j
a

)
za(1− z)j−aa = 2(1− z)j

j∑
a=0

(
j
a

)(
z

1− z

)a
a

= 2(1− z)j
j∑

a=0

(
j
a

)
daa, (5)

where
z

1− z
= d.

Now,

j∑
a=0

(
j
a

)
daa =

(
j

0

)
d00 +

(
j

1

)
d11 +

(
j

2

)
d22 + · · ·+

(
j

j

)
djj

=

(
j

1

)
d+ 2

(
j

2

)
d2 + 3

(
j

3

)
d3 · · ·+ j

(
j

j

)
dj . (6)

We observe that

(1 + d)j =

(
j

0

)
1j−0 · d0 +

(
j

1

)
1j−1 · d1 +

(
j

2

)
1j−2 · d2 + · · ·+

(
j

j

)
1j−j · dj

(1 + d)j =

(
j

0

)
+

(
j

1

)
d+

(
j

2

)
d2 + · · ·+

(
j

j

)
dj

j(1 + d)j−1 = 0 +

(
j

1

)
+ 2

(
j

2

)
d+ 3

(
j

3

)
d2 + · · ·+ j

(
j

j

)
dj−1

(by differentiating w.r.t d)



S. Rani, H. K. Nigam / Eur. J. Pure Appl. Math, 13 (2) (2020), 351-368 358

jd(1 + d)j−1 =

(
j

1

)
d+ 2

(
j

2

)
d2 + 3

(
j

3

)
d3 + · · ·+ j

(
j

j

)
dj (7)

(multiplying both side by d).

Now, from (6) and (7), we get

j∑
a=0

(
j

a

)
daa = jd(1 + d)j−1

= j

(
z

1− z

)(
1

(1− z)j−1

)
=

jz

(1− z)j
. (8)

Thus, from (5) and (8), we get

2

j∑
a=0

(
j
a

)
za(1− z)j−aa = 2(1− z)j

j∑
a=0

(
j

a

)
daa

= 2(1− z)j jz

(1− z)j
= 2jz. (9)

Now,

j∑
a=0

(
j

a

)
za(1− z)j−a =

(
j

0

)
z0(1− z)j +

(
j

1

)
z1(1− z)j−1 + · · ·+

(
j

j

)
zj(1− z)j−j

= (1− z + z)j

= 1. (10)

Thus, from (4), (9) and (10), we get

KT∆H
l (s) =

1

4

l∑
j=0

bl,j

[
j∑

a=0

∫ 1

0

(
j

a

)
za(1− z)j−a(2a+ 1) dξ(z)

]

=
1

4

l∑
j=0

bl,j

∫ 1

0
(2jz + 1) dz

=
1

4

l∑
j=0

bl,j(j + 1).

= O(l + 1)

l∑
j=0

bl,j

= O(l + 1).



S. Rani, H. K. Nigam / Eur. J. Pure Appl. Math, 13 (2) (2020), 351-368 359

Lemma 2. Under the conditions of regularity of matrix T ≡ (bl,j),

KT∆H
l (s) = O

(
1

s2(l + 1)

)
for

1

l + 1
≤ s ≤ π.

Proof. For 1
l+1 ≤ s ≤ π, sin s

2 ≥
s
π , sin2 ls ≤ 1 and sup0≤z≤1 |ξ′(z)| = N , we have

KT∆H
l (s) =

1

π

l∑
j=0

bl,j

j∑
a=0

∫ 1

0

(
j

a

)
za(1− z)j−a dξ(z)

sin
(
a+ 1

2

)
s

2 sin s
2

=
1

2π

l∑
j=0

bl,j

j∑
a=0

∫ 1

0

(
j

a

)
za(1− z)j−a dξ(z)

sin
(
a+ 1

2

)
s

s
π

=
1

2s

n∑
j=0

bl,j

j∑
a=0

∫ 1

0

(
j

a

)
za(1− z)j−a dξ(z) sin

(
a+

1

2

)
s

≤ N

2s

∣∣∣∣ l∑
j=0

bl,jIm

j∑
a=0

∫ 1

0

(
j

a

)
za(1− z)j−aei(a+ 1

2)s dξ(z)

∣∣∣∣. (11)

Now, first we solve

j∑
a=0

∫ 1

0

(
j

a

)
za(1− z)j−a sin

(
a+

1

2

)
s dξ(z) = (1− z)j

j∑
a=0

∫ 1

0

(
j

a

)(
z

1− z

)a
Im
{
ei(a+ 1

2)s
}
dξ(z)

= (1− z)j
j∑

a=0

∫ 1

0

(
j

a

)(
z

1− z

)a
Im
{
eias · e

is
2

}
dξ(z)

= (1− z)jIm

[
e
is
2

j∑
a=0

∫ 1

0

(
j

a

)(
zeis

1− z

)a
dξ(z)

]

= Im

[
e
is
2

∫ 1

0
(1− z + zeis)jdz

]
= Im

[
e
is
2

∫ 1

0

{
1 + z(eis − 1)

}j
dz

]
= Im

[
ei(j+1)s − 1

(1 + j)(e
is
2 − e

−is
2 )

]

= Im

[
ei(j+1)s − 1

(j + 1)2i sin s
2

]

= Im

[
cos(j + 1)s+ i sin(j + 1)s− 1

2i(j + 1) sin s
2

]
=

sin2(j + 1) s2
(j + 1) sin s

2

. (12)



S. Rani, H. K. Nigam / Eur. J. Pure Appl. Math, 13 (2) (2020), 351-368 360

Now, from (11) and (12), we get

KT∆H
l (s) ≤ N

2s

∣∣∣∣ l∑
j=0

bl,j
sin2(j + 1) s2
(j + 1) sin s

2

∣∣∣∣
≤ N

2s

∣∣∣∣ l∑
j=0

bl,j
1

(j + 1) sπ

∣∣∣∣
=
Nπ

2s2

∣∣∣∣ l∑
j=0

bl,j
1

j + 1

∣∣∣∣
Using Abel’s Lemma, we have

KT∆H
l (s) =

Nπ

2s2

∣∣∣∣ l−1∑
j=0

(bl,j − bl,j+1)

j∑
k=0

1

k + 1
+ bl,l

l∑
j=0

1

j + 1

∣∣∣∣
≤ Nπ

2s2

∣∣∣∣ l−1∑
j=0

∆bl,j

j∑
k=0

1

k + 1

∣∣∣∣+ bl,l

∣∣∣∣ l∑
j=0

1

j + 1

∣∣∣∣
≤ Nπ

2s2

 l−1∑
j=0

|∆bl,j |+ bl,l

 max
0≤j≤p

∣∣∣∣ p∑
j=0

1

j + 1

∣∣∣∣
=
Nπ

2s2

[
O

(
1

l + 1

)
+ O

(
1

l + 1

)]
= O

(
1

s2(l + 1)

)
.

Lemma 3. [28] Let g ∈ H(η)
r , then for 0 < s ≤ π :

(i) ‖φ(·, s)‖r = O(η(s));

(ii) ‖φ(·+ z, s)− φ(·, s)‖r =

{
O(η(s))

O(η(z)).

(iii) If η(s) and χ(s) are as defined in Note 1, then ‖φ(·+z, s)−φ(·, s)‖r = O
(
χ(|z|)

(
η(s)
χ(s)

))
.

5. Proof of the main theorem

5.1. Proof of Theorem 1

Proof. Following [7], sl(g;x) of Fourier series

sl(g;x)− g(x) =
1

2π

∫ π

0
φ(x, s)

sin
(
l + 1

2

)
s

sin s
2

ds.



S. Rani, H. K. Nigam / Eur. J. Pure Appl. Math, 13 (2) (2020), 351-368 361

The Hausdorff matrix mean of sl(x), denoted by t∆H
l (x), we get

t∆H
l (x)− g(x) =

l∑
j=0

hl,j(sj(x)− g(x))

=
l∑

j=0

(
l

j

)
∆l−jµj

{
1

2π

∫ π

0
φ(x, s)

sin
(
j + 1

2

)
s

sin s
2

ds

}

=
1

2π

∫ π

0
φ(x, s)

l∑
j=0

(
l

j

)
∆l−j

(∫ 1

0
zj dξ(z)

)
sin
(
j + 1

2

)
s

sin s
2

ds

=
1

2π

∫ π

0
φ(x, s)

l∑
j=0

∫ 1

0

(
l

j

)
zj(1− z)l−j dξ(z)

sin
(
j + 1

2

)
s

sin s
2

ds.

The T transform of t∆H
l (x) denoted by tT∆H

l (x), is given by

tT∆H
l (x)− g(x) =

l∑
j=0

bl,j

(
1

2π

∫ π

0
φ(x, s)

j∑
a=0

∫ 1

0

(
j

a

)
za(1− z)j−a dξ(z)

sin
(
a+ 1

2

)
s

sin s
2

ds

)

=
1

2π

∫ π

0
φ(x, s)

l∑
j=0

bl,j

j∑
a=0

∫ 1

0

(
j

a

)
za(1− z)j−a dξ(z)

sin
(
a+ 1

2

)
s

sin s
2

ds

=

∫ π

0
φ(x, s)KT∆H

l (s) ds.

Let

Tl(x) = tT∆H
l (x)− g(x) =

∫ π

0
φ(x, s)KT∆H

l (s) ds.

Then

Tl(x+ z)− Tl(x) =

∫ π

0
(φ(x+ z, s)− φ(x, s))KT∆H

l (s) ds.

Using generalized Minkowski’s inequality [6], we obtain

‖Tl(·,+z)− Tl(·)‖r ≤
∫ π

0
‖φ(·+ z, s)− φ(·, s)‖rKT∆H

l (s) ds

=

∫ 1
l+1

0
‖φ(·+ z, s)− φ(·, s)‖rKT∆H

l (s) ds

+

∫ π

1
l+1

‖φ(·+ z, s)− φ(·, s)‖rKT∆H
l (s) ds

= I1 + I2. (13)

Using Lemmas 1 and 3 (iii), we get

I1 =

∫ 1
l+1

0
‖φ(·+ z, s)− φ(·, s)‖rKT∆H

l (s) ds



S. Rani, H. K. Nigam / Eur. J. Pure Appl. Math, 13 (2) (2020), 351-368 362

= O(l + 1)

(
χ(|z|)

∫ 1
l+1

0

η(s)

χ(s)
ds

)

=

(
χ(|z|)

η( 1
l+1)

χ( 1
l+1)

)
. (14)

Also, using Lemmas 2 and 3 (iii), we get

I2 =

∫ π

1
l+1

‖φ(·+ z, s)− φ(·, s)‖rKT∆H
l (s) ds

= O

(
1

l + 1

∫ π

1
l+1

χ(|z|) η(s)

s2χ(s)
ds

)
. (15)

From (13), (14) and (15), we have

sup
z 6=0

‖Tl(·,+z)− Tl(·)‖r
χ(|z|)

= O

 η
(

1
l+1

)
χ
(

1
l+1

)
+ O

(
1

l + 1

∫ π

1
l+1

η(s)

s2χ(s)
ds

)
. (16)

Again applying Minkowski’s inequality and using Lemmas 1, 2 and 3 (i), we obtain

‖Tl(·)‖r = ‖tT∆H
l − g‖r

≤

(∫ 1
l+1

0
+

∫ π

1
l+1

)
‖φ(·, s)‖rKT∆H

l (s) ds

= O

(
(l + 1)

∫ 1
l+1

0
η(s) ds

)
+ O

(
1

l + 1

∫ π

1
l+1

η(s)

s2
ds

)

= O

(
η

(
1

l + 1

))
+ O

(
1

l + 1

∫ π

1
l+1

η(s)

s2
ds

)
. (17)

We know that

‖Tl(·)‖(χ)
r = ‖Tl(·)‖r + sup

z 6=0

‖Tl(·,+z)− Tl(·)‖r
χ(|z|)

. (18)

Now, using (16), (17) and (18), we get

‖Tl(·)‖(χ)
r = O

(
η

(
1

l + 1

))
+ O

(
1

l + 1

∫ π

1
l+1

η(s)

s2
ds

)

+ O

 η
(

1
l+1

)
χ
(

1
l+1

)
+ O

(
1

l + 1

∫ π

1
l+1

η(s)

s2χ(s)
ds

)
. (19)



S. Rani, H. K. Nigam / Eur. J. Pure Appl. Math, 13 (2) (2020), 351-368 363

By the monotonicity of χ(s), η(s) = η(s)
χ(s)χ(s) ≤ χ(π) η(s)

χ(s) for 0 < s ≤ π, we get

‖Tl(·)‖(χ)
r = O

 η
(

1
l+1

)
χ
(

1
l+1

)
+ O

(
1

l + 1

∫ π

1
l+1

η(s)

s2χ(s)
ds

)
. (20)

Since η and χ are as defined in Note 1, therefore

1

l + 1

∫ π

1
l+1

η(s)

s2χ(s)
ds ≥

η
(

1
l+1

)
χ
(

1
l+1

) ( 1

l + 1

)∫ π

1
l+1

1

s2
ds ≥

η
(

1
l+1

)
2χ
(

1
l+1

) .
Then,

η
(

1
l+1

)
χ
(

1
l+1

) = O

(
1

l + 1

∫ π

1
l+1

η(s)

s2χ(s)
ds

)
. (21)

From (20) and (21), we get

‖Tl(·)‖(χ)
r = O

(
1

l + 1

∫ π

1
l+1

η(s)

s2χ(s)
ds

)
,

‖tT∆H
l − g‖(χ)

r = O

(
1

l + 1

∫ π

1
l+1

η(s)

s2χ(s)
ds

)
. (22)

6. Corollaries

Corollary 1. Let g ∈ H(α),r; r ≥ 1 and 0 ≤ β < α ≤ 1, then

‖tT∆H
l − g‖(β),r =

{
O((l + 1)β−α) if 0 ≤ β < α < 1

O
(

log π(l+1)
l+1

)
if β = 0, α = 1.

Proof. The proof is obtained by putting η(s) = sα, χ(s) = sβ, 0 ≤ β < α ≤ 1 in
Theorem 1.

Corollary 2. Following the Remark 1(i), we obtain

‖tCα∆H
l − g‖(χ)

r = O

(
1

l + 1

∫ π

1
l+1

η(s)

s2χ(s)
ds

)
.

Corollary 3. Following the Remark 1(ii), we obtain

‖tH1/l+1∆H

l − g‖(χ)
r = O

(
1

l + 1

∫ π

1
l+1

η(s)

s2χ(s)
ds

)
.



S. Rani, H. K. Nigam / Eur. J. Pure Appl. Math, 13 (2) (2020), 351-368 364

Corollary 4. Following the Remark 1(iii), we obtain

‖tNp,q∆H

l − g‖(χ)
r = O

(
1

l + 1

∫ π

1
l+1

η(s)

s2χ(s)
ds

)
.

Corollary 5. Following the Remark 1(iv), we obtain

‖tNp∆H

l − g‖(χ)
r = O

(
1

l + 1

∫ π

1
l+1

η(s)

s2χ(s)
ds

)
.

Corollary 6. Following the Remark 1(v), we obtain

‖tÑp∆H

l − g‖(χ)
r = O

(
1

l + 1

∫ π

1
l+1

η(s)

s2χ(s)
ds

)
.

Corollary 7. Following the Remark 1(vi), we obtain

‖tEq∆H

l − g‖(χ)
r = O

(
1

l + 1

∫ π

1
l+1

η(s)

s2χ(s)
ds

)
.

Corollary 8. Following the Remark 1(vii), we obtain

‖tTCαl − g‖(χ)
r = O

(
1

l + 1

∫ π

1
l+1

η(s)

s2χ(s)
ds

)
.

Corollary 9. Following the Remark 1(viii), we obtain

‖tTEql − g‖(χ)
r = O

(
1

l + 1

∫ π

1
l+1

η(s)

s2χ(s)
ds

)
.

Remark 5.

(i) Corollary 2 can be further reduced for CαEq and C1∆H means in view of Remark 2
(i)(b) and (c) respectively.

(ii) Corollary 3 can be further reduced for H1/l+1Cα and H1/l+1Eq means in view of
Remark 2 (ii)(a) and (b) respectively.

(iii) Corollary 4 can be further reduced for Np,qCα and Np,qEq in view of Remark 2 (iii)(a)
and (b) respectively.

(iv) Corollary 5 can be further reduced for NpCα and NpEq means in view of Remark 2
(iv)(a) and (b) respectively.

(v) Corollary 6 can be further reduced for ÑpCα and ÑpEq means in view of Remark 2
(v)(a) and (b) respectively.



S. Rani, H. K. Nigam / Eur. J. Pure Appl. Math, 13 (2) (2020), 351-368 365

(vi) Corollary 7 can be further reduced for EqCα means in view of Remark 2 (vi)(a).

(vii) Corollaries 8 can be further reduced for TC1 means in view of Remark 2 (vii)(a).

(viii) Corollaries 9 can be further reduced for TE1 means in view of Remark 2 (viii)(a).

Remark 6.

(i) In our Theorem 1, if r →∞ in H
(η)
r class, then this turns down to H(η) class. Also

putting η(s) = sα and χ(s) = sβ in our Theorem 1, H(η) class then this turns down
to Hα class. Then for β = 0 in Hα class, this turns down to Lipα class.

(ii) In our Theorem 1, by putting η(s) = sα, χ(s) = sβ in H
(η)
r class, H

(η)
r class then

this turns down to Hα,r class. Then for β = 0 in Hα,r class, this turns down to
Lip(α, r) class.

Remark 7.

(i) If ζ(s) = sα and r →∞ then Lip(ζ(s), r) class turns down to Lipα class. Thus, the
results of [12], [15], [16] and [30] reduces to Lipα class.

(ii) If β = 0, ζ(s) = sα and r → ∞ then W (Lr, ζ(s)) class turns down to Lipα class.
Thus, the results of [11], [13] and [14] reduces to Lipα class.

7. Particular cases

(i) Using Remark 6(i) and putting hl,j = 1
l+1 , 0 ≤ j ≤ l in our Theorem 1, the result of

Dhakal [4] follows.

(ii) Using Remark 6(i) , putting bl,j =
pl−jqj
Rl

, Rl =
∑l

j=0 pjql−j and hl,j = 1
l+1 , 0 ≤ j ≤ l

in our Theorem 1, the result of Dhakal [5] follows.

(iii) Using Remark 6(i), putting bl,j = 1
2l

(
l
j

)
and hl,j = 1

l+1 , 0 ≤ j ≤ l in our Theorem

1, then in view of Remark 7(ii), the result of Nigam [11] follows.

(iv) Using Remark 6(i), putting ξ(z) =
∏α
j=1 z

j , α ≥ 1 and hl,j = 1
l+1 , 0 ≤ j ≤ l in our

Theorem 1, then in view of Remark 7(i), the result of Nigam [12] follows.

(v) Using Remark 6(i), putting bl,j = 1
l+1 and hl,j = 1

(1+q)l

(
l
j

)
ql−j in our Theorem 1,

in view of Remark 7(ii), the result of Nigam [13] follows.

(vi) Using Remark 6(i) and 6(ii), putting bl,j =
pl−j
Pj
,
∑l

j=0 pj 6= 0, ql = 1 ∀ l and

hl,j = 1
l+1 , 0 ≤ j ≤ l in our Theorem 1 then in view of Remark 7(ii), the result of

Nigam and Sharma [14] follows.



REFERENCES 366

(vii) Using Remark 6(i), putting bl,j = 1
l+1 and hl,j = 1

(1+q)l

(
l
j

)
ql−j in our Theorem 1,

then in view of Remark 7(i), the result of Nigam and Sharma [15] follows.

(viii) Using Remark 6(i), putting bl,j = 1
2l

(
l
j

)
and hl,j = 1

l+1 , 0 ≤ j ≤ l in our Theorem

1, then in view of Remark 7(i), the result of Nigam and Sharma [16] follows.

(ix) Using Remark 6(ii), putting bl,j =
pl−jqj
Rl

, Rl =
∑l

j=0 pjql−j and hl,j = 1
l+1 , 0 ≤ j ≤ l

in our Theorem 1, the result of Kushwaha and Dhakal [18] follows.

(x) Using Remark 6(i), putting ξ(z) =
∏α
j=1 z

j , α ≥ 1 and hl,j = 1
l+1 , 0 ≤ j ≤ l in our

Theorem 1, the result of Tiwari and Bariwal [26] follows.

(xi) Using Remark 6(i), putting bl,j = 1
l+1 and hl,j = 1

(1+q)l

(
l
j

)
ql−j in our Theorem 1,

the result of Lal [29] follows.

(xii) Using Remark 6(i), putting hl,j = 1
l+1 , 0 ≤ j ≤ l in our Theorem 1, then in view of

Remark 7(i), the result of Shrivastava, Rathore and Shukla [30] follows.

8. Conclusion

In this paper, we obtain the error estimation of the function g in the Hölder space H
(η)
r

(r ≥ 1) by Matrix-Hausdorff (T∆H) product means of its Fourier series. Since, in view
of Remark 1, the product summability means Cα∆H , H1/l+1∆H , Np,q∆H , Np∆H , Ñp∆H ,
Eq∆H , TCα and TEq are the particular cases of T∆H product means. Some useful results
are also deduced in the form of corollaries from our theorem.

Some other studies regarding modulus of continuity (smoothness) of functions using
more generalized functional spaces may be the future interest of a few investigators in the
direction of this work.

Acknowledgements

The first author expresses his gratitude towards his mother for her blessings. The first
author also expresses his gratitude towards his father in heaven, whose soul is always guid-
ing and encouraging him. The first author is also thankful to Council of Scientific and In-
dustrial Research, Government of India for support under the scheme 25/(0225)/13/EMR-
II. The second author also expresses her gratitude towards her parents for their blessings.

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