EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 16, No. 2, 2023, 944-952
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Best approximation of unbounded functions by modulus
of smoothness

Alaa Adnan Auad1,∗, Mohammed A. Hilal2, Nihad Shareef Khalaf3

1 Department of Mathematics, College of Education for Pure Sciences, University of Anbar,
Ramadi, Iraq
2 Baquba Technical Institute, Middle Technical University, Baquba, Iraq
3 Department of Mathematics, College of Education for Women, Tikrit University, Tikrit,
Iraq

Abstract. In this paper, we study the approximation of unbounded functions in a weighted space
by modulus of smoothness using various linear operators. We establish direct theorems for such
approximations and analyze the properties of the modulus of smoothness within the same space.
Specifically, we investigate the behavior of the modulus of smoothness under different types of
linear operators, including the Bernstein-Durrmeyer operator, the Fejer operator, and the Jackson
operator. We also provide a detailed analysis of the convergence rate of these operators. Further-
more, we discuss the relationship between the modulus of smoothness and the Lipschitz constant
of a function. Our findings have important implications for the field of approximation theory and
may help to inform future research in this area.

2020 Mathematics Subject Classifications: 41A52, 41A44, 41A27

Key Words and Phrases: Unbounded Functions, Weighted Spaces, Approximation, Modulus
of Smoothness, Trigonometric Polynomial

1. Introduction

Let Lp ={f : f is bounded measurable function }, 1 ≤ p < ∞ be the space of all
bounded functions with the norm

∥f∥p =
(∫ π

−π
|f(x)|pdx

) 1
p

<∞.

Let W be the space of all weighted functions such that a function λ : [−π, π] → R+ is an
almost everywhere positive function which is locally integrable, that is λ ∈W .

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v16i2.4730

Email addresses: alaa.adnan.auad@uoanbar.edu.iq (A. A. Auad),
mohammed azeez hilal@mtu.edu.iq (M. A. Hilal), Nihad.shareef16@tu.edu.iq (N. S. Khalaf)

https://www.ejpam.com 944 © 2023 EJPAM All rights reserved.



A. A. Auad, M. A. Hilal, N. S. Khalaf / Eur. J. Pure Appl. Math, 16 (2) (2023), 944-952 945

Let Lp,λ[−π, π] = {f : f is unbounded function on [−π, π], 1 ≤ p <∞}, with the norm

∥f∥Lp,λ[−π,π] =

(∫ π

−π
|f(x)λ(x)|pdx

) 1
p

<∞.

Also, let N be the set of all natural numbers and for every k ∈ N ∪ {0}, we denote by
Tk the set of all trigonometric polynomials of degree less than or equal to k. For a given
function f ∈ Lp,λ[−π, π], we define

Ek(f)Lp,λ[−π,π]
= inf{∥f − g∥Lp,λ[−π,π]

; g ∈ Tk}, (1)

which is called the kth degree best approximation of f with respect to Tk. Let k ∈ N and
f ∈ Lp,λ[−π, π]. Then, we define the modulus of smoothness of f by

µk(f, δ)Lp,λ[−π,π] = sup︸︷︷︸
h≤δ

∥∥∥∆k
hf(.)

∥∥∥
Lp,λ[−π,π]

,

where δ = 1
k and ∆k

hf(x) is called the kth difference symmetric with step h at point x,
and it is given by

∆k
hf(x) =

∑k

i=1
(−1)k−1f(x+ ih). (2)

The Weierstrass approximation theorem simply states that Ek(f)Lp[a,b] converges to
zero as k → ∞ for all f ∈ Lp[a, b]. It does not say how fast Ek(f)Lp[a,b] → 0. In
1987, Prestin [20] investigated problems of estimating the deviation of functions from
their de la Vallée-Poussin sums in weighted Orlicz spaces. In 1999, Bustamante [9] stud-
ied some problems of approximation theory in the spaces Sp(1 ≤ p < ∞) and obtained
the asymptotically sharp inequalities of Jackson type that connect the best polynomial
approximations with modules of continuity of functions f ∈ Sp. In 2000, Dragomir [11]
presented some results about the development of methods for solving approximation prob-
lems using sets in normed linear spaces. Approximation of both real functions and real
data is considered by Elumalai and Vijayaragavan in 2008 and 2009 [12, 13]. In 2012, a
construction of some characterizations of best approximation in 2-normed space was stud-
ied by Dominic [10]. In 2013, Markandeya and Bharathi proved some results of b- best
approximation in uniformly 2-normed space [16]. The concept of best approximation in
2-normed space along with the concept of orthogonality in the same space were presented
and discussed in [14, 19]. The following fundamental direct estimates prerogative to Jack-
son [7, 15] assure that Ek(f)Lp[a,b] converges to zero much faster when f is smoothness.
The theory of approximation has been studied by many researchers and applied in vari-
ous fields. Auad (2019) investigated the best simultaneous approximation of unbounded
functions in weighted space using two different definitions and established the relationship
between best approximation and best simultaneous approximation [8]. In 2021, Auad et
al. discussed the algebraic polynomial’s best approximation of unbounded functions in
weighted space and obtained sharp direct inequality of algebraic approximation [6]. Ali



A. A. Auad, M. A. Hilal, N. S. Khalaf / Eur. J. Pure Appl. Math, 16 (2) (2023), 944-952 946

and Pales (2022) derived an extension of the Taylor theorem related to linear differential
operators with constant coefficients and exponential polynomials, including the integral
remainder terms and mean value type theorems [5]. Approximation theory is very useful
in numerical analysis, especially when solving nonlinear equations [1]. Approximation the-
ory can be seen also in several mathematical techniques, such as finite element and finite
differences [3]. Furthermore, other applications of approximation theory in stability and
thermal science can be found in several studies [2, 4, 17, 18].

To have a basic and historical background about these direct theorems, we start by
presenting the following.

For all f ∈ Lp[a, b] and k ∈ N, the direct theorem in a bounded space can be represented
as

Ek(f, ξ)Lp[a,b] ≤ C(k)µk(f, ξ)Lp[a,b]; ξ =
1

k
,

where C is a positive constant depending on k. Also, If f ∈ Lp[a, b] has k
th derivative f (k)

for some k ∈ N, then

Ek(f, ξ)Lp[a,b] ≤ C(k)µk

(
f (k), ξ

)
Lp[a,b]

; ξ =
1

k
.

The Fourier series expansion is given as g(x) =
∑∞

i=−∞ g(i)eijx, with its Fourier co-
efficients g(i) = 1

2π

∫ π
−π g(x)e

ijxdx. If ϑ : R → R is a continuous function, then we define
the convolution function (g ∗ f)(ϑ;.) by

(g ∗ f)(ϑ;x) = 1
2π

∫ π
−π g(x)fϑ(x)f(x)dx, x ∈ [−π, π]. (3)

Clearly, (g ∗ f)(ϑ; .), (ϑ; .) ∈ Lp,λ[−π,π] with the norm

∥(g ∗ f)(ϑ; .)∥Lp,λ[−π,π]
≤ C∥g∥1∥f∥Lp,λ[−π,π],

where
C = sup︸︷︷︸

x≤π

{∥fϑ(x)∥Lp,λ[−π,π]
}.

Let l be a natural number, g ∈ Lp,λ[−π,π] and consider the following linear combination
of the convolution functions (g ∗ I)i, 1 ≤ i ≤ l as

P (g, l) = Σl
i=1(−1)l+1

(
l

i

)
(g ∗ I). (4)

Here, we consider the generalized Jackson kernel given by

Jk,r(x) = Ck,r(
sinkx

2

sinx
2

)2r, k, r ∈ N,

where the constant Ck,r > 0 is taken in such a way that

Jk,r(0) =
1

π

∫ π

0
Jk,r(x)dx = 1.



A. A. Auad, M. A. Hilal, N. S. Khalaf / Eur. J. Pure Appl. Math, 16 (2) (2023), 944-952 947

And

Jk,1(x) = kk(x) = Σk−l
i=1−k(1−

|i|
k
)eix (5)

is called the Fejer kernel such that Jk,r(x) = Ck,rFk(x) is a non-negative trigonometric
polynomial of degree r(k − 1).

The main aim of this paper is to extend this results to arbitrary weighted space
Lp,λ[−π,π] and in particular the space Lp(X), where X = [0, π] or [−1, 1], 1 ≤ p <∞.

2. Auxiliary lemmas

In this section, we recall some lemmas which we will need in our main results.

Lemma 1. Let f ∈ Lp,λ[−π,π], 1 ≤ p <∞ and k ∈ N. Then,

µk(f, δ)Lp,λ[−π,π]
≤ Ck∥f∥Lp,λ[−π,π]

,

where Ck is a positive constant depending on k.

Proof. We have

∆k
hf(x) =

∑k

i=1
(−1)k−i

(
k

i

)
f(x+ ih)),

∥∆k
hf(.)∥Lp,λ[−π,π]

= ∥
∑k

i=1
(−1)k−i

(
k

i

)
f(x+ ih))∥Lp,λ[−π,π]

,

sup∥∆k
hf(.)∥Lp,λ[−π,π]

≤ sup{
k∑

i=1

(−1)k−i

(
k

i

)
∥f(.)∥Lp,λ[−π,π]

},

thus,

µk(f, δ)Lp,λ[−π,π]
≤ max{sup{

k∑
i=1

(−1)k−i

(
k

i

)
∥f(.)∥Lp,λ[−π,π]

}.

Now, we can take that

max{sup{
k∑

i=1

(−1)k−i

(
k

i

)
}} ≤ Ck,

which implies,
µk(f, δ)Lp,λ[−π,π]

≤ Ck∥f∥Lp,λ[−π,π]
.

Lemma 2. Let f ∈ Lp,λ[−π,π], 1 ≤ p <∞, h > 0 and r ∈ N. Then,

µr(f,
1

k
)Lp,λ[−π,π]

≤ Ck µr−k(f,
1

k
)Lp,λ[−π,π]

.



A. A. Auad, M. A. Hilal, N. S. Khalaf / Eur. J. Pure Appl. Math, 16 (2) (2023), 944-952 948

Proof. We have

∆k
hf(x) = ∆r−1

h (∆1
hf(x)) = ∆r−1

h (f(x+ h)− f(x− h)).

So,
∥∆k

hf(.)∥Lp,λ[−π,π]
≤ ∥∆r−1

h (f(+h)− f(−h))∥Lp,λ[−π,π]
, C > 0.

Take k = 1, we obtain

µr(f, 1)Lp,λ[−π,π]
≤ {maxC} µr−k(f, 1)Lp,λ[−π,π]

,

which completes the proof.

Lemma 3. If f, f ′ ∈ Lp,λ[−π,π], 1 ≤ p <∞, f ′ is the derivative of f and r, k ∈ N . Then,

µr(f,
1

k
)Lp,λ[−π,π]

≤ Ck µr(f ′,
1

k
)Lp,λ[−π,π],

where Ck is a positive constant.

Proof. The proof of this lemma goes in the same way as the proof of lemma 2.

Lemma 4. If f ∈ Lp,λ[−π,π], 1 ≤ p <∞ and r, k ∈ N. Then,

µr(f,
α

k
)Lp,λ[−π,π]

≤ Ck µr(f,
1

k
)Lp,λ[−π,π]

,

where Ck is a positive constant depending on k and α > 0.

Proof. We have

µr(f,
α

k
)Lp,λ[−π,π]

= sup︸︷︷︸
|h|≤α

k

∥∆r
hf(.)∥Lp,λ[−π,π]

≤ sup︸︷︷︸
|h|≤α

k

∥∆r
α
k
f(.)∥Lp,λ[−π,π]

≤ sup︸︷︷︸
|h|≤α

k

∥(α
k
)rDrf(.)∥Lp,λ[−π,π]

≤ max|α|r{sup∥∆r
α
k
f(.)∥Lp,λ[−π,π]

} ≤ max(αk)
rµr(f,

1

k
)Lp,λ[−π,π]

.

Substituting max|α|r = Ck, we obtain

µr(f,
α

k
)Lp,λ[−π,π]

≤ Ck µr(f,
1

k
)Lp,λ[−π,π]

.



A. A. Auad, M. A. Hilal, N. S. Khalaf / Eur. J. Pure Appl. Math, 16 (2) (2023), 944-952 949

Lemma 5. If f ∈ Lp,λ[−π,π], 1 ≤ p <∞, l ∈ N and g(0) = 1. Then,

∥f − p(f)∥Lp,λ[−π,π]
≤ Ck µl(f,

1

l
)Lp,λ[−π,π]

∑l

i=0

(
l

i

)
lG(g, i),

where

G(g, i) =
1

2π

∫ π

−π
|x|ig(x)dx

that belongs to the subspace of Lp,λ[−π,π] is a positive constant.

Proof.

p(f)− f =
(−1)l+1

2π

∫ π

−π
g(x)∆xl(f)dx,

and

∥p(f)− f∥Lp,λ[−π,π]
=

(−1)l+1

2π

∫ π

−π
∥g(.)∆xl(f)∥Lp,λ[−π,π]

dx

≤ (−1)l+1

2π
sup∥∆xl(f)∥Lp,λ[−π,π]

∫ π

−π
|g(x)|dx.

From the properties of the modulus of smoothness, we obtain

∥p(f)− f∥Lp,λ[−π,π]
≤ Cµl (f,

1

l
)Lp,λ[−π,π]

1

2π

∫ π

−π
|l|i|g(x)|dx

≤ Cµl (f,
1

l
)Lp,λ[−π,π]

l∑
i=0

(
l

i

)
l
1

2π

∫ π

−π
|l|i|g(x)|dx.

3. Main Results

In this section, we introduce direct theorems of unbounded functions in weighted space
by using some linear operators.

Theorem 1. Let f ∈ Lp,λ[−π,π], 1 ≤ p <∞ , l ∈ N and r ∈ N ∪ 0. Then,

Er(f, ξ)Lp,λ[−π,π]
≤ inf∥p(f)− f∥Lp,λ[−π,π]

≤ Cr inf{µl(f, ξ)Lp,λ[−π,π]

∑l

i=0

(
l

i

)
lG(g, i)},

where Cr is a positive constant and ξ > 0 .

Proof. Taking Equation (1) and Equation (4), and applying Lemma 5 using the fact
that P (f) ∈ Tl, the proof of this theorem is completed.



A. A. Auad, M. A. Hilal, N. S. Khalaf / Eur. J. Pure Appl. Math, 16 (2) (2023), 944-952 950

Theorem 2. Let f ∈ Lp,λ[−π,π], 1 ≤ p <∞ , ξ > 0 and l, k ∈ N . Then,

Ek(f, ξ)Lp,λ[−π,π]
≤ ∥jk,l(.)− f∥Lp,λ[−π,π]

≤ Ckµl(f,
1

l
)Lp,λ[−π,π]

,

where Ck is a positive constant and Ji,j(x) is the Jackson operator with x ∈ [−π, π] that
takes i = [(k + 3)/2] and j = [l/i] + 1.

Proof. We have the operator Ji,j(x) belongs to the space Tk. Therefore, by Theorem
1, we obtain

Ek(f, ξ)Lp,λ[−π,π]
≤ ∥jk,l(.)− f∥Lp,λ[−π,π]

≤ Ckµl(f, ξ)Lp,λ[−π,π]

∑l

i=0

(
l

i

)
lG(g, i)

and this completes the proof.

Theorem 3. Let {ψk}k=0,1,2,. . . be a sequence of operators in the space Tk satisfying
ψk(p) = p, for each p that belongs to the subspace Sk of Lp,λ[−π,π] and l ∈ N. Then, for
all f ∈ Lp,λ[−π,π], we have

∥f − ψk(f)∥Lp,λ[−π,π]
≤ (∥ψk∥Lp,λ[−π,π]

+ 1)Ek(f,
1

k
)Lp,λ[−π,π]

≤ Ck(∥ψk∥Lp,λ[−π,π]
+ 1)µk(f,

1

k
)Lp,λ[−π,π]

.

Proof. Let p be a function in the space ψk .Then

∥f − ψk(f)∥Lp,λ[−π,π]
≤ ∥f − p∥Lp,λ[−π,π]

+ ∥f − ψk∥Lp,λ[−π,π]

≤ (∥ψk∥Lp,λ[−π,π]
+ 1))∥f − p∥Lp,λ[−π,π]

.

From Equation (1), we have

∥f − ψk(f)∥Lp,λ[−π,π]
≤ (∥ψk∥Lp,λ[−π,π]

+ 1)Ek(f,
i

k
)Lp,λ[−π,π]

.

Also, by using Theorem 2, we obtain

∥f − ψk(f)∥Lp,λ[−π,π]
≤ Ck(∥ψk∥Lp,λ[−π,π]

+ 1)µk(f,
i

k
)Lp,λ[−π,π]

,

and consequently the proof follows.



REFERENCES 951

4. Conclusion

In this study, we have demonstrated the direct trigonometric approximation theorems
of unbounded functions in a weighted space defined on the interval [−π, π]. Our results
are based on the use of various linear operators and provide insights into the properties
of the modulus of smoothness within the same space. While we did not provide a specific
example in this paper, our findings are applicable to a wide range of functions and have
important implications for the field of approximation theory. We believe that our results
will inspire further research in this area.

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