Internat. J. Math. & Math. Sci.
Vol. 23, No. 9 (2000) 645–649
S0161171200002489
© Hindawi Publishing Corp.
ON THE DEGREE OF APPROXIMATION BY GAUSS
WEIERSTRASS INTEGRALS
HUZOOR H. KHAN and GOVIND RAM
(Received 1 July 1998 and in revised form 15 October 1998)
Abstract. We obtain the degree of approximation of functions belonging to class
Lip(ψ(u, v); p), p > 1 using the Gauss Weierstrass integral of the double Fourier series of
f (x, y).
Keywords and phrases. Lip(ψ(u, v); p) class, Gauss Weierstrass integral, arithmetic means,
Jackson operator, degree of approximation, Fourier series, Hölder’s inequality.
2000 Mathematics Subject Classification. Primary 41A25.
1. Introduction and results. Let the function f (x, y) be integrable in the sense of
Lebesgue over the square (ᏻ, 2π ; ᏻ, 2π ) and periodic with period 2π in each variable
outside the square. Let the double Fourier series of f (x, y) be
∞
∞ Am,n (x, y),
(1.1)
m=0 n=0
where
A0,0 (x, y) =
1
a0,0 ,
4
1
(am,0 cos mx + bm,0 sin mx),
2
1
A0,n (x, y) = (a0,n cos nx + b0,n sin nx),
2
Am,n (x, y) = am,n cos mx cos ny + bm,n cos mx sin ny
Am,0 (x, y) =
(1.2)
+ cm,n sin mx cos ny + dm,n sin mx sin ny.
Let
φ(u, v) =
1
f (x + u, y + v) + f (x − u, y + v) + f (x + u, y − v)
4
+ f (x − u, y − v) − 4f (x, y) .
(1.3)
The series
∞
∞ λm,n Am,n (x, y)
(1.4)
m=0 n=0
is called the double Fourier series associated with the function φ(u, v) such that
646
H. H. KHAN AND G. RAM

1





4
λm,n = 1



2


1
for m = 0, n = 0,
for m = 0, n > 0; m > 0, n = 0,
(1.5)
for m > 0, n > 0.
Also, the coefficients in the series (1.4) are given by
am,n =
1
π2
2π 2π
0
0
φ(u, v) cos mu cos nv du dv
(1.6)
and three other similar expressions defining bm,n , cm,n , dm,n .
We define the Gauss Weierstrass integral of f (x, y) by
∞
∞ Wm,n (x, y) = W (x, y; ξ, η) =
=
k=0 =0
π
ξ, η
π π
−π
−π
1
Ak, (x, y)
exp (k2 ξ/4) + (2 η/4)
f (x + t, y + s)
dt ds + o ξ, η ,
exp (t 2 /ξ) + (s 2 /η)
(1.7)
where o ξ, η → 0 as ξ → 0, η → 0, and ξ, η is the product of ξ and η.
A 2π periodic function f (x, y) in each variable x and y is said to belong to the
class Lip(ψ(u, v); p), p > 1, [2] if
|f (x + u, y + v) − f (x, y)| ≤ M
ψ(u, v)
,
(u, v)1/p
0 < u < π, 0 < v < π,
(1.8)
where ψ(u, v) is a positive increasing function of the variables u, v and M is a positive
number independent of x, u and v.
Yoshimitsu, [1] proved a theorem for obtaining the degree of approximation of class
of functions Lip(α, β), 0 < α < 1 and 0 < β < 1, by means of the first arithmetic means
of double Fourier series. Siddiqi and Mohammadzadeh [3] extended the result in two
directions in terms of a positive increasing function of two variables. Recently Khan,
[2] extended the result of Yoshimitsu, Siddiqi et al. for the more general operator, the
Jackson type operator, and more general class of functions, Lip(ψ(u, v); p), p > 1.
The object of the present paper is to determine the degree of approximation for the
functions belonging to the class Lip(ψ(u, v); p), p > 1, by means of Gauss Weierstrass
integral of the double Fourier series of f (x, y).
Our theorem states as follows.
Theorem 1.1. Let f (x, y) be a continuous function of period 2π with respect to
each variable x and y belonging to Lip(ψ(u, v); p), p > 1 class, then
W x, y; ξ, η − f (x, y) = ᏻ
ψ ξ, η
,
(ξ, η)(1/p)−(1/2)
(1.9)
provided
ξ η
0
0
ψ(t, s)
(t, s)1/p
1/p
p
dt ds
= ᏻ ψ ξ, η ,
(1.10)
ON THE DEGREE OF APPROXIMATION . . .
ξ π
0
η
ψ(t, s)
(t, s)2+(1/p)
1/p
p
dt ds
ψ ξ, η
=ᏻ
.
(ξ, η)2
647
(1.11)
2. Proof of the theorem. Using (1.7), we get
W x,y; ξ, η − f (x, y)
π π
π
2
2
φ(t, s)e−((t /ξ)+(s /η)) dt ds + R x, y; ξ, η
= ξ, η −π −π
π π
π
2
2
=4 φ(t, s)e−((t /ξ)+(s /η)) dt ds + ᏻ ξ, η
(2.1)
ξ, η 0 0
ξ π π η π π
ξ η
π
2
2
+
+
+
φ(t, s)e−((t /ξ)+(s /η)) dt ds
=4
ξ, η
0 0
0 η
ξ
0
ξ
η
= I1,2 (x, y) + I1,3 (x, y) + I4,2 (x, y) + I4,3 (x, y).
Applying Hölder’s inequality of two variables and the fact that φ(u, v) ∈ Lip(ψ(u,
v); p), p > 1 for I1,2 (x, y), we get
1/p
ξ η
π
p
|I1,2 (x, y)| ≤ 4 |φ(t, s)| dt ds
ξ, η
0 0
1/p
ξ η
−((t 2 /ξ)+(s 2 /η)) p
·
dt ds
,
e
0
(2.2)
0
where
p −1
1
=
p
p
1/p 1/p
ξ η
ξ η
π
ψ(t, s) p
−(t 2 /ξ)+(s 2 /η) p
dt ds
dt ds
e
≤4
(t, s)1/p
ξ, η
0 0
0 0

 1/p
ξ η
ψ ξ, η
2
2
= ᏻ √ 
e−(t p
/ξ) e−(s p
/η) dt ds
using condition (1.10)
0 0
ξ, η
 1/p
 √ξp
√ηp
√
ξ
η
ψ
ξ,
η
2
2
= ᏻ √ 
e−u √ du
e−v √ dv
p
p
0
0
ξ, η

 1/p
ψ ξ, η
ψ ξ, η
= ᏻ √ ᏻ ξ, η
=ᏻ (1/p)−(1/2) ,
ξ, η
ξ, η
(2.3)
where u = t p
/ξ and v = s p
/η.
Applying Hölder’s inequality of two variables to I1,3 (x, y) and using the fact that
φ(u, v) ∈ Lip(ψ(u, v); p), p > 1, we get
648
H. H. KHAN AND G. RAM
√
4 π
|I1,3 (x, y)| ≤
√ ξ, η
p
1/p ξ π −(t 2 /ξ)+(s 2 /η)p
1/p
ξ π
e
φ(t, s) dt ds
·
dt ds
2
(t, s)−2p
0 η (t, s) 0 η
1/p
p
√
ξ π
ψ(t, s)
4 π
dt
ds
≤ √ (t, s)2+(1/p)
0 η
ξ, η
·
ξ
0
e−t
2 p
/ξ
· t 2p
dt
π
η
e−s
2 p
/η 2p
s
1/p
ds

1/p
2p  



√ξp

ξ
ξ
ψ ξ, η
2


e−u  √ u  √  du
=ᏻ 5/2 
p
p

ξ, η
 0
1/p
2p
√ √

η
η

v
dv
e
using condition (1.11)
√
√
√

p
p
ηp
2+(1/p
)
ψ ξ, η
ψ ξ, η
=ᏻ =ᏻ 5/2 ᏻ ξ, η
(1/p)−(1/2) .
ξ, η
ξ, η
(2.4)

(π /√ηp
)
·
−v 2
Similarly, we can prove that
ψ ξ, η
(1/p)−(1/2) ,
ξ, η
|I4,2 (x, y)| = ᏻ ψ ξ, η
(1/p)−(1/2) .
ξ, η
|I4,3 (x, y)| = ᏻ (2.5)
Finally, we get
W x, y; ξ, η − f (x, y) = ᏻ ψ ξ, η
(1/p)−(1/2) .
ξ, η
(2.6)
Remark 2.1. It may also be remarked that by giving different values to ψ(u, v),
we get some interesting results:
(i) If ψ(u, v) = uα ∗ v β , [1], 0 < α < 1, 0 < β < 1, then we have
W (x, y; ξ, η) − f (x, y) = ᏻ ξ α+(1/2)−(1/p) ∗ ηβ+(1/2)−(1/p) .
(2.7)
(ii) If ψ(u, v) = J(u, v)(u, v)1/p , [3], where J(u, v) is a positive increasing function
of variables u and v, then
W (x, y; ξ, η) − f (x, y) = ᏻ J ξ,
η
(2.8)
−1/2 .
ξ, η
References
[1]
[2]
Yoshimitsu H., On summabilities of double Fourier series, Kōdai Math. Sem. Rep. 15 (1963),
226–238. MR 28#5299. Zbl 127.29302.
Huzoor H. Khan, On the degree of approximation by Jackson type operators, Preprint of International Centre for Theoretical Physics, Trieste, Itlay (October 1994). IC/94/344.
ON THE DEGREE OF APPROXIMATION . . .
[3]
649
A. H. Siddiqui and M. Mohammadzadeh, Approximation by Cesáro and B means of double
Fourier series, Math. Japon. 21 (1976), no. 4, 343–349. MR 56#977. Zbl 367.40003.
Khan and Ram: Department of Mathematics, Aligarh Muslim University, Aligarh202 002, India