INTERNATIONAL JOURNAL OF RESEARCH AND SCIENTIFIC INNOVATION (IJRSI)
ISSN No. 2321-2705 | DOI: 10.51244/IJRSI |Volume XII Issue VIII August 2025
Page 812
www.rsisinternational.org
Degree of Approximation of Function in the Generalized
Zygmund Class By (E, Q) (
,
) Means of Fourier Series
Santosh Kumar Sinha
1
and Pragya Sharma
2
1
Dept. of Mathematics, Lakhmi Chand Institute of Technology, Bilaspur (C.G.), India
2
Ghss Kudekela, Block-Dharamjaigarh Raigarh (C.G.), India
DOI: https://doi.org/10.51244/IJRSI.2025.120800068
Received: 07 Aug 2025; Accepted: 12 Aug 2025; Published: 05 September 2025
ABSTRACT
In this paper, a theorem on degree of approximation of function in the generalized Zygmund class
by (E, q) (
, Pn) means of Fourier series has been established.
Keywords : Degree of approximation , Generalized Zygmund class , (
, Pn) mean , (E, q) mean, (E, q)
(
, Pn) mean
.
MSC: 41A24, 41A25, 42B05, 42B08
INTRODUCTION
The degree of approximation of function belonging to different classes like Lip α, (Lip α, p), Lip(ξ(t),p) ,
Lip(Lp , ξ(t)) have been studied by many mathematician using different summability means. The error
estimation of function in Lipschitz and Zygmund class using different means of Fourier series and
conjugate Fourier series have been great interest among the researcher. The generalized Zygmund class of
functions has been widely studied in harmonic analysis and Fourier approximation because it captures
functions whose smoothness is characterized by a controlled modulus of continuity. This class generalizes the
well-known Zygmund class and includes functions with smoother as well as rougher behavior, making it an
appropriate setting for studying precise error bounds in trigonometric approximations.
A powerful approach for enhancing convergence involves the use of product summability methods. In
particular, the (E, q) means, when combined with the weighted Nörlund means (
, Pn) produce a generalized
summability method denoted by (E, q) (
, Pn) means. The (E, q) means accelerate convergence by
modifying partial sums, while the (
, Pn) transformation introduces flexibility through a weight sequence
{Pn}. The generalized Zygmund class was introduced by Kim [1] Leindler [2] Moricz [3], moricz and
Nemeth [4]etc. Recently Singh et. al. [7] Mishra et al. [5], Pradhan et al. [6], Sinha et al. [8] find the
results in Zygmund class by using different summability Means. In this paper we find the degree of
approximation of function in the generalized Zygmund class by (E, q) (
, Pn) means of Fourier series.
Definition
Let be a periodic function of period integrable in the sense of Lebesgue over
[π, - π]. Then the Fourier series of given by
……..(2.1)
Zygmund class z is defined as
INTERNATIONAL JOURNAL OF RESEARCH AND SCIENTIFIC INNOVATION (IJRSI)
ISSN No. 2321-2705 | DOI: 10.51244/IJRSI |Volume XII Issue VIII August 2025
Page 813
www.rsisinternational.org
.
In this paper , we introduce a generalized Zygmund
defined as
...........(2.2)
Where and is a continuous non negative and non decreasing
function. If we take , , then
class reduces to
the z class.
We write
(t) =
…….(2.3)
…….(2.4)
MAIN RESULT
In this paper we prove the following theorem.
Theorem1- Let be a periodic function ,Lebesgue integrable in
and belonging to
generalized Zygmund class
. Then the degree of approximation of function g by (E,q )(
,
Pn) product mean of Fourier series is given by
Where
denotes the Zygmund moduli of continuity such that
is positive and
increasing.
Theorem 2- Let be a periodic function, Lebesgue integrable in
and belonging to
generalized Zygmund class
. Then the degree of approximation of function g by (E,q)(
,
Pn) product mean of Fourier series is given by
where
denotes the Zygmund moduli of continuity such that
is positive and decreasing.
Lemma–To prove the theorem we need the following Lemma.
Lemma 4(a) - For
we have
….…(4.1)
INTERNATIONAL JOURNAL OF RESEARCH AND SCIENTIFIC INNOVATION (IJRSI)
ISSN No. 2321-2705 | DOI: 10.51244/IJRSI |Volume XII Issue VIII August 2025
Page 814
www.rsisinternational.org
Proof - For
and then
Lemma 4(b) - For
we have
……….(4.2)
Proof - For
Lemma 4(c) – Let
then for
(i)
(ii)
(iii) If and are defined as in theorem then
where
Proof of Theorem 1
Let
denotes the partial sum of fourier series given in (2.1) then we have
….(5.1)
INTERNATIONAL JOURNAL OF RESEARCH AND SCIENTIFIC INNOVATION (IJRSI)
ISSN No. 2321-2705 | DOI: 10.51244/IJRSI |Volume XII Issue VIII August 2025
Page 815
www.rsisinternational.org
The (E,q) transform
of
is given by
. ……(5.2)
The (E,q)(
, Pn) transform of
is given by
……(5.3)
.
.........(5.4)
Let
then
using the generalized Minkowaski’s inequality we get
INTERNATIONAL JOURNAL OF RESEARCH AND SCIENTIFIC INNOVATION (IJRSI)
ISSN No. 2321-2705 | DOI: 10.51244/IJRSI |Volume XII Issue VIII August 2025
Page 816
www.rsisinternational.org
=
..............(5.5)
Using lemma 4(a) and 4(c) and the monotonically of
with respect to t we have
.
Using second mean value theorem of integral we have
.
..............(5.6)
For
using lemma 4(b) and 4(c) we have
..............(5.7)
from (5.5) (5.6) and (5.7) we get
..........(5.8)
Again using Lemma we have
INTERNATIONAL JOURNAL OF RESEARCH AND SCIENTIFIC INNOVATION (IJRSI)
ISSN No. 2321-2705 | DOI: 10.51244/IJRSI |Volume XII Issue VIII August 2025
Page 817
www.rsisinternational.org
..............(5.9)
from (5.8) and (5.9) we have
.
Now we write
in terms of
term of
,
in view of the monotonicity of v(t) we have
therefore we can write
Again using monotonicity of v(t)
............(5.10)
using the fact
is positive and non decreasing , we have
therefore we can write
.
So we have
.
Hence
INTERNATIONAL JOURNAL OF RESEARCH AND SCIENTIFIC INNOVATION (IJRSI)
ISSN No. 2321-2705 | DOI: 10.51244/IJRSI |Volume XII Issue VIII August 2025
Page 818
www.rsisinternational.org
This completes the proof Theorem 1.
Proof of Theorem 2 – We have from theorem 1
we assume that
is positive and decreasing in t then
.
This completes the proof Theorem 2.
REFERENCES
1. Kim, J. (2021). Degree of Approximation of Function of Class
by Cesaro Means
of Fourier series. East Asian Math. J. ,Vol. 37. No. 3 pp. 289 – 293.
2. Leinder, L. (1981). Strong approximation and generalized Zygmund class. Acta Sci. Math. 43,
no. 3-4, 301 – 309.
3. Moricz, F. (2010) . Enlarged Lipschitz and Zygmund classes of function and Fourier
transformation. East J. Approx. 16. No. 3, 259 – 271.
4. Moricz, F. & Nemeth(2007) . Generalized Zygmund classes of function and Approximation by
Fourier series . Acta Sci. Math. , no. 3 – 4, 637 – 647.
5. Mishra, A., Padhy, B. P., & Mishra, U. (2020). On approximation of signal in the
Generalized Zygmund class using (E, r) (N, qn) mean of conjugate derived Fourier series.
EJPAM , Vol 13 , No. 5, 1325 – 1336.
6. Pradhan, T., Paikray, S. K., Das, A. A. &dutta, H. (2019). On approximation of Signal in the
generalised Zygmund class via (E,1) (
) summability mean of Conjugate Fourier series.
Proyecciones (Antofagasta. Online ) , Vol 38, n. 5, 981- - 998.
7. Singh, M. V., Mittal, M. L. , & Rhoades, B. E. (2017).Approximation of functions in the
generalized Zygmund class using Hausdorff means. Journal of Inequalities and Applications.
2017:101 DOI 10.1186/s13660-017-1361-8 . pp. 1- 11.
8. Sinha, S .K. & Shrivastava, U. K. (2022). Degree of Approximation of Function of Class
by
Mean of Fourier Series. South East Asian J of Mathematics and
Mathematical Science, (18)2, 117 -124.
