International Journal of Analysis and Applications
ISSN 2291-8639
Volume 9, Number 1 (2015), 45-53
http://www.etamaths.com

APPLICATIONS OF SOME CLASSES OF SEQUENCES ON

APPROXIMATION OF FUNCTIONS (SIGNALS) BY ALMOST

GENERALIZED NÖRLUND MEANS OF THEIR FOURIER

SERIES

XHEVAT Z. KRASNIQI

Abstract. In this paper, using rest bounded variation sequences and head
bounded variation sequences, some new results on approximation of function-

s (signals) by almost generalized Nörlund means of their Fourier series are

obtained. To our best knowledge this the first time to use such classes of
sequences on approximations of the type treated in this paper. In addition,

several corollaries are derived from our results as well as those obtained pre-

viously by others.

1. Introduction and preliminaries

Given two sequences p := (pn) and q := (qn) the convolution (p∗ q)n is defined
by

Rn := (p∗q)n :=
n∑

m=0

pmqn−m,

and we also write Pn := (p∗ 1)n =
∑n
m=0 pm and Qn := (1 ∗ q)n =

∑n
m=0 qm =∑n

m=0 qn−m.
Let (sn) be a sequence. When Rn 6= 0 for all n, the generalized Nörlund trans-

form of the sequence (sn) is the sequence {tp,qn } obtained by putting

tp,qn =
1

Rn

n∑
m=0

pn−mqmsm.

If sn −→ s(n −→∞) induces tp,qn −→ s(n −→∞) then the method (N,pn,qn) is
called to be regular. The necessary and sufficient condition for (N,pn,qn) method
to be regular is

∑n
m=0 |pn−mqm| = O(|(p∗q)n|) and pn−m = o(|(p∗q)n|) as n −→∞

for every fixed m ≥ 0 (see Borwein [1]).
The method (N,pn,qn) reduces to Nörlund method (N,pn) if qn = 1 for all n

and to Riesz method (N,qn) if pn = 1 for all n. It is well-known that (N,pn)
mean or (N,qn) mean includes as a special case Cesàro and harmonic means or
logarithmic mean, respectively.

2010 Mathematics Subject Classification. 40C99, 40G99, 41A25, 42A16.
Key words and phrases. fourier series; almost generalized Nörlund means; degree of

approximation.

c©2015 Authors retain the copyrights of their papers, and all
open access articles are distributed under the terms of the Creative Commons Attribution License.

45



46 KRASNIQI

Let f be a 2π periodic function (signal) and Lebesgue integrable i.e. f ∈ L[0, 2π].
Then the Fourier series of the function (signal) f at the point x is given by

(1.1) f(x) ∼
a0
2

+

∞∑
m=1

(am cos mx + bm sin mx),

with its partial sums sn(f; x) being a trigonometric polynomial of order n with
n + 1 terms.

A function (signal) f ∈ Lip α if |f(x + t) −f(x)| = O(|t|α) for 0 < α ≤ 1.
A function (signal) f ∈ Lip (α,r) for a ≤ x ≤ b if

(1.2) wr(t; f) =

{∫ b
a

|f(x + t) −f(x)|r
}1/r

≤ M(|t|α)

for r ≥ 1 and 0 < α ≤ 1, where M is an absolute positive constant not necessarily
the same at each occurrence (see McFadden [5]).

It should be noted that if r −→ ∞ in Lip(p,r) class then this class reduces to
Lipα.

According to Lorentz [3] a bounded sequence (sk) of k-th sums of the Fourier
series (1.1) is said to be almost convergent to s, if

(1.3) lim
n→∞

sn,r = lim
n→∞

sr + sr+1 + · · · + sr+n
n + 1

= lim
n→∞

1

n + 1

n+r∑
k=r

sk = s.

It is said (see [9]) that the the Fourier series (1.1) is said to be almost Riesz
summable to the finite number s, if

τn,r =
1

Pn

n∑
m=0

pmsm,r −→ s as n −→∞

uniformly with respect to r, where

sm,r =
1

m + 1

m+r∑
j=r

sj.

It is a well-known fact that a convergent sequence is almost convergent and the
limits are the same. A bounded sequence (sn) is said to be almost Riesz summable
to s if the Riesz transform of (sn) is almost convergent to s (see [2]).

The theory of approximation which is originated from a well-known theorem of
Weierstrass has been an excitatory interdisciplinary field of study till nowadays.
The approximations of the functions have a wide applications in signal analysis,
digital communications, theory of machines in mechanical engineering and in par-
ticular in digital signal processing see [7] and [8] (also the interested reader could
find several new results on these approximations and their applications into refer-
ences given in [6]).

Very recently Mishra et al [6] determined the degree of approximation of a signal
f ∈ Lip(α,r), (r ≥ 1) by almost Riesz summability means of its Fourier series.
Before we recall their results we need first some known definitions given below.

The Lr-norm of an function f : R −→ R is defined by

‖f‖r =
(∫ 2π

0

|f(x)|rdx
)1/r

, r ≥ 1.



SOME CLASSES OF SEQUENCES ON APPROXIMATION OF FUNCTIONS 47

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

A signal (function) f is approximated by trigonometric polynomial τn(f; x) of
order n and the degree of approximation En(f) of a function f ∈ Lr is given by

En(f) = min
n
‖f(x) − τn(f; x)‖r,

in terms of n.
The degree of approximation of a function f : R −→ R by a trigonometric

polynomial τn(f; x) of order n under sup norm ‖ ·‖∞ is defined by
‖f(x) − τn(f; x)‖∞ = sup{|f(x) − τn(f; x)| : x ∈ R}.

Throughout this paper we will write

ψ(t) = f(x + t) −f(x− t) − 2f(x).
Now we are able to formulate the result obtained in [6]:

Theorem 1.1. If f : R −→ R is a 2π periodic function, Lebesgue integrable and
belonging to the Lip(α,r), (r ≥ 1) class, then the degree of approximation of the
function f by almost Riesz means of its Fourier series is given by

(1.4) ‖f(t) − τn(f(t); x)‖r = O
(
P1/r−αn

)
,∀n,

and ψ(t) satisfies the following conditions

(1.5)

[∫ π/Pn
0

(
t|ψ(t)|
tα

)r
dt

]1/r
= O

(
P−1n

)
,

(1.6)

[∫ π
π/Pn

(
t−δ|ψ(t)|

tα

)r
dt

]1/r
= O

(
Pδn
)
,

where δ is a finite quantity, Riesz means are regular and r + s = rs such that
1 ≤ r ≤∞.

Note that in this theorem is not mentioned explicitly that the sequence (pn) is
a non-decreasing one but in its proof it is used this property.

We say that the the Fourier series (1.1) is said to be almost generalized Nörlund
summable to the finite number s ([1]), if

tp,qn,r =
1

Rn

n∑
m=0

pmqn−msm,r −→ s as n −→∞

uniformly with respect to r, where

sm,r =
1

m + 1

m+r∑
j=r

sj.

Now we give definitions of two classes of sequences (see [4]).
A sequence c := {cn} of nonnegative numbers tending to zero is called of Rest

Bounded Variation, or briefly c ∈ RBV S, if it has the property

(1.7)

∞∑
n=m

|cn − cn+1| ≤ K(c)cm



48 KRASNIQI

for all natural numbers m, where K(c) is a constant depending only on c.
A sequence c := {cn} of nonnegative numbers will be called of Head Bounded

Variation, or briefly c ∈ HBV S, if it has the property

(1.8)

m−1∑
n=0

|cn − cn+1| ≤ K(c)cm

for all natural numbers m, or only for all m ≤ N if the sequence c has only finite
nonzero terms, and the last nonzero term is cN .

The purpose of this paper is to determine the degree of approximation of a
function (signal) f ∈ Lip(α,r), (r ≥ 1) by almost generalized Nörlund means of
its Fourier series under conditions that (pn) ∈ HBV S and (qn) ∈ RBV S. As is
pointed out in Figure 2 constructed in [10] the class of sequences RBV S is a wider
one than that of monotone sequences. This fact shows that in some way our results
are very extensive results.

2. Main Results

We prove the following main result.

Theorem 2.1. Let (pn) ∈ HBV S and (qn) ∈ RBV S. If f : R −→ R is a 2π
periodic function, Lebesgue integrable and belonging to the Lip(α,r), (r ≥ 1) class,
then the degree of approximation of the function f by almost generalized Nörlund
means of its Fourier series tp,qn,r(f(t); x) is given by

(2.1) ‖f(t) − tp,qn,r(f(t); x)‖r = O
(
R1/r−αn

)
, ∀n,

and ψ(t) satisfies the following conditions

(2.2)

[∫ π/Rn
0

(
t|ψ(t)|
tα

)r
dt

]1/r
= O

(
R−1n

)
,

(2.3)

[∫ π
π/Rn

(
t−δ|ψ(t)|

tα

)r
dt

]1/r
= O

(
Rδn
)
,

where δ is a finite quantity, generalized Nörlund means are regular and r + s = rs
such that 1 ≤ r ≤∞.

Proof. It is almost a routine that for partial sums sk(f(t); x) of the Fourier series
(1.1) the equality

sk,r(f(t); x) −f(t) =
1

2π(k + 1)

∫ π
0

ψ(t)
cos(rt) − cos(k + r + 1)t

2 sin2 t
2

dt

holds true.



SOME CLASSES OF SEQUENCES ON APPROXIMATION OF FUNCTIONS 49

Whence, for almost generalized Nörlund means of sk,r(f(t); x) we have

tp,qn,r(f(t); x) −f(t)

=
1

Rn

n∑
m=0

pmqn−m{sm,r(f(t); x) −f(t)}

=
1

2πRn

∫ π
0

ψ(t)

n∑
m=0

pmqn−m
m + 1

·
cos(rt) − cos(m + r + 1)t

2 sin2 t
2

dt

=
1

2πRn

(∫ π/Rn
0

+

∫ π
π/Rn

)
ψ(t)

n∑
m=0

pmqn−m
m + 1

·
sin(m + 2r + 1) t

2
· sin(m + 1) t

2

2 sin2 t
2

dt

:= L1 + L2.(2.4)

Applying Hölder’s inequality, f(t) ∈ Lip(α,s) =⇒ ψ(t) ∈ Lip(α,s) on [0,π] (see
[5]), condition (2.2), the well-known inequalities

(2.5) sin u ≥
2

π
u, for u ∈ (0,π/2],

(2.6) |sin(mu)| ≤ m|sin u| for all u ∈ R,m ∈ N,

r + s = rs such that 1 ≤ r ≤∞, we obtain

|L1| ≤
1

2πRn

[∫ π/Rn
0

(
t|ψ(t)|
tα

)r
dt

]1/r
×

×

[∫ π/Rn
0

(
1

t1−α

∣∣∣∣∣
n∑

m=0

pmqn−m
m + 1

·
sin(m + 2r + 1) t

2
· sin(m + 1) t

2

2 sin2 t
2

∣∣∣∣∣
)s

dt

]1/s

= O
(
R−2n

)[∫ π/Rn
0

(
1

t1−α

∣∣∣∣∣
n∑

m=0

pmqn−m ·
1

t

∣∣∣∣∣
)s

dt

]1/s

= O
(
R−2n

)[∫ π/Rn
0

Rsn · t
(α−2)s dt

]1/s

= O
(
R−1n

)
·O

(
1

R
α−2+ 1

s
n

)
= O

(
1

R
α−1

r
n

)
.(2.7)

To estimate |L2| from above we again apply Hölder’s inequality to obtain

|L2| ≤
1

2πRn

[∫ π
π/Rn

(
t−δ|ψ(t)|

tα

)r
dt

]1/r
×

×

[∫ π
π/Rn

(
tδ+α

∣∣∣∣∣
n∑

m=0

pmqn−m
m + 1

·
sin(m + 2r + 1) t

2
· sin(m + 1) t

2

2 sin2 t
2

∣∣∣∣∣
)s

dt

]1/s
.



50 KRASNIQI

Next, using again the fact that f(t) ∈ Lip(α,s) =⇒ ψ(t) ∈ Lip(α,s) on [0,π] (see
[5]), conditions (2.3), (2.5), (2.6), and r + s = rs such that 1 ≤ r ≤∞, we get

|L2| = O
(
R−1n

)
×

×

[∫ π
π/Rn

(
tδ+α

n∑
m=0

∣∣pmqn−m sin(m + 2r + 1) t2∣∣ · (m + 1) ∣∣sin t2∣∣
2(m + 1) sin2 t

2

)s
dt

]1/s

= O
(
R−1n

)
O
(
Rδn
)[∫ π

π/Rn

(
tδ+α

sin t
2

n∑
m=0

∣∣∣∣pmqn−m sin(m + 2r + 1) t2
∣∣∣∣
)s

dt

]1/s
.

Since (pk) ∈ HBV S, then by (1.8) we have

pm −pn ≤ |pm −pn| ≤
n−1∑
k=m

|pk −pk+1| ≤
n−1∑
k=0

|pk −pk+1| ≤ K(p)pn

which implies

(2.8) pm ≤ (K(p) + 1)pn,∀m ∈ [0,n].

Also, since (qk) ∈ RBV S, then by (1.7) we have

qn−m ≤
∞∑
k=m

|qn−k −qn−k−1| ≤
∞∑
k=0

|qn−k −qn−k−1| ≤ K(q)qn

which implies

(2.9) qn−m ≤ K(q)qn,∀m ∈ [0,n].

Using the well-known fact

d∑
`=j

e−i`t = O
(
t−1
)
, 0 ≤ j ≤ d,

(2.8) and (2.9) we find that

n∑
m=0

∣∣∣∣pmqn−m sin(m + 2r + 1) t2
∣∣∣∣

≤ (K(p) + 1)K(q)pnqn max
0≤j≤n

j∑
m=0

sin(m + 2r + 1)
t

2
= O

(
Rnt

−1) .
Subsequently, we obtain

|L2| = O
(
Rδ−1n

)[∫ π
π/Rn

(
Rnt

δ+α−2)s dt
]1/s

= O

(
1

R
α−1

r
n

)
.(2.10)

Inserting (2.7) and (2.10) into (2.4) we immediately obtain

|f(t) − tp,qn,r(f(t); x)| = O
(
R1/r−αn

)
.



SOME CLASSES OF SEQUENCES ON APPROXIMATION OF FUNCTIONS 51

Finaly, using Lr-norm and the lastest estimate we find that

‖f(t) − tp,qn,r(f(t); x)‖r =
[∫ 2π

0

|f(t) − tp,qn,r(f(t); x)|
r dt

]1/r
=

[∫ 2π
0

O
(
R1/r−αn

)r
dt

]1/r
= O

(
R1/r−αn

)
.

The proof of the theorem is completed. �

If we take qn = 1 for all n ≥ 0 then we obtain:

Corollary 2.1. Let (pn) ∈ HBV S. If f : R −→ R is a 2π periodic function,
Lebesgue integrable and belonging to the Lip(α,r), (r ≥ 1) class, then the degree of
approximation of the function f by almost Riesz means tpn,r(f(t); x) of its Fourier
series is given by

‖f(t) − tpn,r(f(t); x)‖r = O
(
P1/r−αn

)
, ∀n,

and ψ(t) satisfies the following conditions

(2.11)

[∫ π/Pn
0

(
t|ψ(t)|
tα

)r
dt

]1/r
= O

(
P−1n

)
,

(2.12)

[∫ π
π/Pn

(
t−δ|ψ(t)|

tα

)r
dt

]1/r
= O

(
Pδn
)
,

where δ is a finite quantity, Riesz means are regular and r + s = rs such that
1 ≤ r ≤∞.

If we take pn = 1 for all n ≥ 0 then we obtain:

Corollary 2.2. Let (qn) ∈ RBV S. If f : R −→ R is a 2π periodic function,
Lebesgue integrable and belonging to the Lip(α,r), (r ≥ 1) class, then the degree
of approximation of the function f by almost Nörlund means tqn,r(f(t); x) of its
Fourier series is given by

‖f(t) − tqn,r(f(t); x)‖r = O
(
Q1/r−αn

)
, ∀n,

and ψ(t) satisfies the following conditions

(2.13)

[∫ π/Qn
0

(
t|ψ(t)|
tα

)r
dt

]1/r
= O

(
Q−1n

)
,

(2.14)

[∫ π
π/Qn

(
t−δ|ψ(t)|

tα

)r
dt

]1/r
= O

(
Qδn
)
,

where δ is a finite quantity, Nörlund means are regular and r + s = rs such that
1 ≤ r ≤∞.

If we take r →∞ then Lip(α,r) ≡ Lipα and we derive the following.



52 KRASNIQI

Corollary 2.3. Let (pn) ∈ HBV S and (qn) ∈ RBV S. If f : R −→ R is a
2π periodic function, Lebesgue integrable and belonging to the Lipα class, then the
degree of approximation of the function f by almost generalized Nörlund means of
its Fourier series is given by

|f(t) − tp,qn,r(f(t); x)| = O
(
R−αn

)
, ∀n,

and ψ(t) satisfies the following conditions (2.2) and (2.3), where δ is a finite quan-
tity, generalized Nörlund means are regular and r + s = rs such that 1 ≤ r ≤∞.

Proof. For r →∞ and Theorem 2.1 we have

|f(t) − tp,qn,r(f(t); x)|∞ = sup
0≤x≤2π

|f(t) − tp,qn,r(f(t); x)| = O
(
R−αn

)
.

Subsequently, we find that

|f(t) − tp,qn,r(f(t); x)| ≤ |f(t) − t
p,q
n,r(f(t); x)|∞ = O

(
R−αn

)
,

which completes the proof. �

Finally, if for all n ≥ 0 we take qn = 1 or pn = 1 in Corollary 2.3 respectively,
then we obtain the following two corollaries.

Corollary 2.4. Let (pn) ∈ HBV S. If f : R −→ R is a 2π periodic function,
Lebesgue integrable and belonging to the Lipα class, then the degree of approximation
of the function f by almost Riesz means of its Fourier series is given by

|f(t) − tpn,r(f(t); x)| = O
(
P−αn

)
, ∀n,

and ψ(t) satisfies the following conditions (2.11) and (2.12), where δ is a finite
quantity, Riesz means are regular and r + s = rs such that 1 ≤ r ≤∞.

Corollary 2.5. Let (qn) ∈ RBV S. If f : R −→ R is a 2π periodic function,
Lebesgue integrable and belonging to the Lipα class, then the degree of approximation
of the function f by almost Nörlund means of its Fourier series is given by

|f(t) − tqn,r(f(t); x)| = O
(
Q−αn

)
, ∀n,

and ψ(t) satisfies the following conditions (2.13) and (2.14), where δ is a finite
quantity, Nörlund means are regular and r + s = rs such that 1 ≤ r ≤∞.

Remark 2.1. If we had assumed in Theorem 2.1 that (pn) is a non-decreasing
sequence and (qn) is a non-increasing one, then it would also hold true. Thus,
taking qn = 1 for all n ≥ 0, then all results obtained in [6] are immediate results of
ours.

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Mathematics and Informatics, Avenue ”Mother Theresa” 5, 10000 Prishtina, Kosovo