Volume 03 Issue 11-2023
56
International Journal Of Management And Economics Fundamental
(ISSN
–
2771-2257)
VOLUME
03
ISSUE
11
P
AGES
:
56-62
SJIF
I
MPACT
FACTOR
(2021:
5.
705
)
(2022:
5.
705
)
(2023:
7.
448
)
OCLC
–
1121105677
Publisher:
Oscar Publishing Services
Servi
ABSTRACT
In the work, we find sharp estimates for the root-mean-square error of the approximation of -periodic random
processes and random fields by linear positive Jackson operators.
KEYWORDS
The Jackson trigonometric polynomial (operator), -periodic random process, random field, approximation,
unimprovable inequality.
INTRODUCTION
The problem of approximation of uniformly
continuous bounded nonrandom functions has a
classical origin and has been known since the time of
Newton. There are whole mathematical areas devoted
to this theory, where the best approximations of
continuous functions of a real and complex variable by
interpolation and algebraic polynomials, trigonometric
polynomials, approximations by splines, linear positive
operators are studied, constructive characteristics of
function classes are found, the cross-sections of
function classes are estimated, and others [4]. [13],
[14].
Methods of the approximation theory of non-
random functions are also used in the study of
problems of approximation of random functions,
where well-studied, simple in construction algebraic
and trigonometric polynomials, various interpolation
formulas are chosen as the approximation apparatus.
This approach is applied in works by Azlarov T.A. [1],
Drozhzhina L.V. [5], [6], Kadyrova I.I. [7],
Research Article
SHARP ESTIMATES FOR THE APPROXIMATION OF PERIODIC RANDOM
PROCESSES AND FIELDS BY JACKSON OPERATORS
Submission Date:
November 08, 2023,
Accepted Date:
November 13, 2023,
Published Date:
November 18, 2023
Crossref doi:
https://doi.org/10.37547/ijmef/Volume03Issue11-08
Shamshiev Abdivali
Associate Professors Of The Department Of General Mathematics Of The Jizzakh State Pedagogical University,
Uzbekistan
Journal
Website:
https://theusajournals.
com/index.php/ijmef
Copyright:
Original
content from this work
may be used under the
terms of the creative
commons
attributes
4.0 licence.
Volume 03 Issue 11-2023
57
International Journal Of Management And Economics Fundamental
(ISSN
–
2771-2257)
VOLUME
03
ISSUE
11
P
AGES
:
56-62
SJIF
I
MPACT
FACTOR
(2021:
5.
705
)
(2022:
5.
705
)
(2023:
7.
448
)
OCLC
–
1121105677
Publisher:
Oscar Publishing Services
Servi
Mirzakhmedov M.A., Khudaiberganov R. [8], Nagorny
V.N., Yadrenko M.I. [9], Omarov S.O. [10], Seleznev
O.V. [11], [12], Khudoyberganov R. [15] and others.
MAIN RESULTS
Denote by
𝐶
Ω
2𝜋
(𝑅
1
)
the class of
2𝜋
-periodic,
continuous with probability one random processes
(r.p.’s),
i.e.
r.p.’s
℥(𝑡)
such
that
℥(𝑡)
is continuous and ℥(𝑡 + 2𝜋)
=
℥(𝑡)
for any
𝑡 ∊ 𝑅
1
with
probability one.
Obviously, for almost all realizarions,
℥(𝑡) ∊ 𝐿
1
(-
𝜋, 𝜋).
Therefore, we can construct the following
trigonometric Jackson polynomial [14]:
𝐷
𝑛
(℥; 𝑡) = 𝐷
𝑛
℥(𝑡) = ∫ ℥(𝑡 + 𝑥)
𝜋
−𝜋
𝐷
𝑛
(𝑥)𝑑𝑥 =
2𝜋 ∑
℥
𝑘
2𝑛−2
−(2𝑛−2)
𝜑
𝑘
(𝑛)
𝑒
𝑖𝑘𝑡
(1)
where
𝐷
𝑛
(𝑥) =
3
2𝜋(2𝑛
2
+1)𝑛
(
𝑠𝑖𝑛
𝑛𝑥
2
𝑠𝑖𝑛
𝑥
2
)
4
is the Jackson
kernel,
℥
𝑘
and 𝜑
𝑘
(𝑛)
are the Fourier coefficients of
℥(𝑡)
and
𝐷
𝑛
(𝑥)
, respectively.
𝐷
𝑛
℥(𝑡)
is a linear and positive operator (l.p.o.).
Consider the approximation of a r.p.
℥(𝑡) ∊ 𝐶
Ω
2𝜋
(
𝑅
1
)
by the Jackson l.p.o.
𝐷
𝑛
(℥; 𝑡)
.
Investigate the standard deviation
𝛿
𝑛
(
℥; 𝑡
) =
{𝑀[℥(𝑡) − 𝐷
𝑛
(℥; 𝑡)]
2
}
1
2
.
Since the r.p.
℥(𝑡) − 𝐷
𝑛
(℥; 𝑡)
is
2𝜋
-periodic, the
function
𝛿
𝑛
(
℥; 𝑡
) will be the same, so it suffices to study
it on the interval [-
𝜋, 𝜋
].
Let
𝜔
℥
(𝛿) = max
|𝑡−𝑠|≤𝛿
{𝑀[℥(𝑡) − ℥(𝑠)]
2
}
1
2
be
the
modulus of continuity of the r.p.
℥(𝑡).
Theorem 1.
a
)
For any
℥(𝑡) ∊ 𝐶
Ω
2𝜋
(
𝑅
1
) and
𝑛 ∈ 𝑁
, the
inequality
max
|𝑡|≤𝜋
{𝑀[℥(𝑡) − 𝐷
𝑛
(℥; 𝑡)]
2
}
1
2
≤ (
4
3
−
45√3
76𝜋
) 𝜔
℥
(
2𝜋
𝑛
)
(2)
is valid.
b) inequality (2) is unimpovable for the class
𝐶
Ω
2𝜋
(
𝑅
1
)
in the sense that, for any
ε >
0, there exist
℥
𝜀
(t)
∈
𝐶
Ω
2𝜋
(
𝑅
1
) and
𝑛
0
∈ 𝑁
such that
max
|𝑡|≤𝜋
{𝑀[℥
𝜀
(t) − 𝐷
𝑛
0
(℥
𝜀
; 𝑡)]
2
}
1
2
> (
4
3
−
45√3
76𝜋
−
ε) 𝜔
℥
𝜀
(
2𝜋
𝑛
0
)
Proof of Theorem 1.
First of all, we note that the
Jackson kernel
𝐷
𝑛
(𝑥)
has the following property:
∫ 𝐷
𝑛
(𝑥)
𝜋
−𝜋
d
x
= 1 for any
𝑛 ∈ 𝑁
[14, p.79].
For any
℥(𝑡) ∊ 𝐶
Ω
2𝜋
(
𝑅
1
),
𝑛 ∈ 𝑁
, and t
∈
[-
𝜋, 𝜋
], uisng
the above property of
𝐷
𝑛
(𝑥)
, the Fubini theorem, and
the Cauchy-Bunyakovsky inequality, we have
𝛿
𝑛
(
℥; 𝑡
)
=
{𝑀[℥(𝑡) − 𝐷
𝑛
(℥; 𝑡)]
2
}
1
2
=
{𝑀[∫ (℥(𝑡) −
𝜋
−𝜋
℥(𝑡 + 𝑥))𝐷
𝑛
(𝑥)𝑑𝑥]
2
}
1
2
≤
≤ ∫ 𝜔
℥
(|𝑥|)
𝜋
−𝜋
𝐷
𝑛
(𝑥)𝑑𝑥
.
Using the properties of the modulus of continuity
𝜔
℥
(𝑥)
, we obtain from here that
𝛿
𝑛
(
℥; 𝑡
)
≤
2
∫ 𝜔
℥
(𝑥)
𝜋
0
𝐷
𝑛
(𝑥)𝑑𝑥
≤
2
𝜔
℥
(
2𝜋
𝑛
) ∫ (1 +]
𝑛𝑥
2𝜋
[ )
𝜋
0
𝐷
𝑛
(𝑥)𝑑𝑥
=
𝜔
℥
(
2𝜋
𝑛
)𝜆
𝑛
,
where
𝜆
𝑛
=
2
∫ (1 +]
𝑛𝑥
2𝜋
[ )
𝜋
0
𝐷
𝑛
(𝑥)𝑑𝑥
.
In [10], it is shown that
sup
𝑛≥1
𝜆
𝑛
=
𝜆
3
=
4
3
−
45√3
76𝜋
=
1,00688858…
Volume 03 Issue 11-2023
58
International Journal Of Management And Economics Fundamental
(ISSN
–
2771-2257)
VOLUME
03
ISSUE
11
P
AGES
:
56-62
SJIF
I
MPACT
FACTOR
(2021:
5.
705
)
(2022:
5.
705
)
(2023:
7.
448
)
OCLC
–
1121105677
Publisher:
Oscar Publishing Services
Servi
It follows from here that
𝛿
𝑛
(
℥; 𝑡
)
≤
(
4
3
−
45√3
76𝜋
) 𝜔
℥
(
2𝜋
𝑛
)
for any
℥(𝑡) ∊ 𝐶
Ω
2𝜋
(
𝑅
1
),
𝑛 ∈ 𝑁
, and t
∈
[-
𝜋, 𝜋
], hence,
max
|𝑡|≤𝜋
𝛿
𝑛
(℥; 𝑡)
≤
(
4
3
−
45√3
76𝜋
) 𝜔
℥
(
2𝜋
𝑛
)
.
Part a) of Theorem 1 is proved.
To prove part b) of Theorem 1, consider an even,
2𝜋
-
periodic, non-random function defined on the interval
[0,
𝜋
] in a following way:
𝑓
ε
(
x
)=
{
𝑥
ε
if 0 ≤ 𝑥 ≤ ε,
1 if ε ≤ 𝑥 ≤
2𝜋
3
,
1 +
1
ε
(𝑥 −
2𝜋
3
) if
2𝜋
3
≤ 𝑥 ≤
2𝜋
3
+ ε,
2 if
2𝜋
3
+ ε ≤ 𝑥 ≤ 𝜋
}
where
ε
is a sufficiently small number.
Let a r.v.
℥
0
be such that M
℥
0
2
= 1.
Obviously, the r.p.
℥
ε
(
t
) =
℥
0
𝑓
ε
(
x
)
∈
𝐶
Ω
2𝜋
(
𝑅
1
) and
𝜔
℥
ε
(
2𝜋
3
) = 1.
For
℥
ε
(
t
), we have
max
|𝑡|≤𝜋
𝛿
3
(℥
ε
; 𝑡)
≥
𝛿
3
(℥
ε
; 0)
=
{𝑀[∫ ℥
ε
(𝑥)𝐷
3
(𝑥)
𝜋
−𝜋
𝑑𝑥]
2
}
1
2
=
=
∫ 𝑓
ε
(𝑥)𝐷
3
(𝑥)
𝜋
−𝜋
𝑑𝑥
= 2{
∫
𝑥
ε
𝐷
3
(𝑥)
ε
0
𝑑𝑥
+
∫ 𝐷
3
(𝑥)
2𝜋
3
ε
𝑑𝑥
+
+
∫
[1 +
1
ε
(𝑥 −
2𝜋
3
)] 𝐷
3
(𝑥)
2𝜋
3
+ε
2𝜋
3
𝑑𝑥
+
2 ∫
𝐷
3
(𝑥)
𝜋
2𝜋
3
+ε
} =
= 2[
∫ 𝐷
3
(𝑥)
2𝜋
3
0
𝑑𝑥 + 2 ∫ 𝐷
3
(𝑥)
𝜋
2𝜋
3
𝑑𝑥
]
–
2[
∫ 𝐷
3
(𝑥)
ε
0
𝑑𝑥 −
∫
𝑥
ε
𝐷
3
(𝑥)
ε
0
𝑑𝑥
]
–
–
2[
2 ∫
𝐷
3
(𝑥)
2𝜋
3
+ε
2𝜋
3
𝑑𝑥 − ∫
[1 +
1
ε
(𝑥 −
2𝜋
3
+ε
2𝜋
3
2𝜋
3
)] 𝐷
3
(𝑥) 𝑑𝑥
] =
= 2
∫ (1+]
2𝜋
3
[ )𝐷
3
(𝑥)
𝜋
0
𝑑𝑥
- 2
∫ (1 −
𝑥
ε
) 𝐷
3
(𝑥)
ε
0
𝑑𝑥
–
–
2
∫
[1 +
1
ε
(𝑥 −
2𝜋
3
)] 𝐷
3
(𝑥)
2𝜋
3
+ε
2𝜋
3
𝑑𝑥
=
𝜆
3
−
𝐼
ε
(1)
−
𝐼
ε
(2)
(
𝜆
3
−
𝐼
ε
(1)
−
𝐼
ε
(2)
) 𝜔
℥
ε
(
2𝜋
3
).
It is obvious that
𝐼
ε
(1)
≤
2
∫ 𝐷
3
(𝑥)
ε
0
𝑑𝑥
⟶
0,
and
𝐼
ε
(2)
≤
2 ∫
𝐷
3
(𝑥)
2𝜋
3
+ε
2𝜋
3
0
𝑑𝑥 ⟶ 0
as
ε
⟶
0,
i.e.
max
|𝑡|≤𝜋
𝛿
𝑛
(℥
ε
; 𝑡)
≥ [
𝜆
3
−
𝛼(ε)
]
𝜔
℥
ε
(
2𝜋
3
) , 𝛼(ε) ⟶ 0
as
ε
⟶
0
where
𝛼(ε) = 𝐼
ε
(1)
+
𝐼
ε
(2)
.
This leads to the statement of Part b) of Theorem 1.
Theorem 1 is proved.
Theorem 2.
For the class of r.p.’s
𝐶
Ω
2𝜋
(
𝑅
1
)
, the relation
s u p
𝑛 ∊𝑁,℥∊ 𝐶
Ω
2𝜋
(𝑅
1
)
max
|𝑡|≤𝜋
{ M|℥(𝑡)−𝐷𝑛(℥,𝑡)|
2
}
1
2
𝜔
℥
(
2𝜋
𝑛
)
=
4
3
−
45√3
76𝜋
takes place.
Proof of Theorem 2
follows from Theorem 1.
Let us proceed to finding the sharp estimate for the
approximation of random fields by the Jackson l.p.o.
’s.
Denote by
𝐶
Ω
2𝜋
(
𝑅
2
) the class of
2𝜋
-periodic in each
argument and continuous with probability one r.p.
’s
℥(𝑡, 𝑠).
The function
𝜔
℥
(1)
(
𝑥
1
, 𝑥
2
)
=
sup
|𝑡 − 𝑡
′
| ≤ 𝑥
1
|𝑠 − 𝑠
′
| ≤ 𝑥
2
{M
|℥(𝑡, 𝑠) − ℥(𝑡
′
, 𝑠
′
)|
2
}
1
2
,
𝑥
1
, 𝑥
2
≥ 0
Volume 03 Issue 11-2023
59
International Journal Of Management And Economics Fundamental
(ISSN
–
2771-2257)
VOLUME
03
ISSUE
11
P
AGES
:
56-62
SJIF
I
MPACT
FACTOR
(2021:
5.
705
)
(2022:
5.
705
)
(2023:
7.
448
)
OCLC
–
1121105677
Publisher:
Oscar Publishing Services
Servi
is said to be the modulus of continuity of the first type
of a r.p.
℥(𝑡, 𝑠) ∈ 𝐶
Ω
2𝜋
(
𝑅
2
) [1], [5].
The function
𝜔
℥
(2)
(𝑥) =
𝑠𝑢𝑝
(𝑡 − 𝑡
′
)
2
+ (𝑠 − 𝑠
′
)
2
≤ 𝑥
2
{M|℥(𝑡, 𝑠)
− ℥(𝑡
′
, 𝑠
′
)|
2
}
1
2
, 𝑥 ≥ 0
is said to be the modulus of continuity of the second
type of a r.p.
℥(𝑡, 𝑠) ∈ 𝐶
Ω
2𝜋
(
𝑅
2
).
The modules of continuity of a r.p.
℥(𝑡, 𝑠) ∈
𝐶
Ω
2𝜋
(
𝑅
2
) have the following properties:
1
0
. For any
0 ≤ 𝑥
1
≤ 𝑥
1
′
,
0 ≤ 𝑥
2
≤ 𝑥
2
′
, the inequalities
𝜔
℥
(1)
( 𝑥
1
, 𝑥
2
) ≤ 𝜔
℥
(1)
(𝑥
1
′
, 𝑥
2
) ≤ 𝜔
℥
(1)
(𝑥
1
′
, 𝑥
2
′
)
take place.
2
0
.
𝜔
℥
(1)
( 𝑛𝑥
1
, 𝑛𝑥
2
) ≤ 𝑛𝜔
℥
(1)
( 𝑥
1
, 𝑥
2
)
for any
n
∈ 𝑁,
0 ≤
𝑥
1
≤
𝑥
2
.
3
0
𝜔
℥
(2)
( 𝑥
1
) ≤ 𝜔
℥
(2)
( 𝑥
2
)
for any
0 ≤ 𝑥
1
≤ 𝑥
2
.
4
0
.
𝜔
℥
(2)
(𝑛𝑥) ≤ 𝑛𝜔
℥
(2)
(𝑥)
for any
n
∈ 𝑁
,
0 ≤ 𝑥
1
≤ 𝑥
2
.
5
0
. 𝜔
℥
(1)
(
√2
2
𝑥,
√2
2
𝑥) ≤
𝜔
℥
(2)
(𝑥)
≤ 𝜔
℥
(1)
(𝑥, 𝑥) ≤
𝜔
℥
(2)
(𝑥√2)
,
x
≥
0.
Consider the approximation of a r.p.
℥(𝑡, 𝑠) ∈ 𝐶
Ω
2𝜋
(
𝑅
2
)
by the Jackson l.p.o.
𝐷
𝑛,𝑛
(
℥; t, s)
=
∫ ∫ ℥(𝑡 + 𝑥, 𝑠 +
𝜋
−𝜋
𝜋
−𝜋
𝑦) 𝐷
𝑛
(𝑥)𝐷
𝑛
(𝑦)
dxdy
.
(3)
Theorem 3.
a) For any
℥(𝑡, 𝑠) ∈ 𝐶
Ω
2𝜋
(
𝑅
2
) and
𝑛 ∈ 𝑁
,
the inequality
max
|𝑡|≤𝜋
|𝑠|≤𝜋
{𝑀|℥(𝑡, 𝑠) − 𝐷
𝑛,𝑛
(℥; t, s)|
2
}
1
2
≤
[2
-
(
2
3
−
45√3
76𝜋
)
2
]
𝜔
℥
(1)
(
2𝜋
𝑛
,
2𝜋
𝑛
)
(4)
holds.
b) inequality (4) is unimprovable in the following sense:
for any
ε >
0, there exist
℥
𝜀
(t,s)
∈ 𝐶
Ω
2𝜋
(
𝑅
2
) and
𝑛
0
∈ 𝑁
such that
max
|𝑡|≤𝜋
|𝑠|≤𝜋
{𝑀[℥
𝜀
(t, s) − 𝐷
𝑛
0
,𝑛
0
(℥
𝜀
; 𝑡, 𝑠)]
2
}
1
2
> [2
–
(
2
3
−
45√3
76𝜋
)
2
–
ε
]
𝜔
℥
𝜀
(1)
(
2𝜋
𝑛
0
,
2𝜋
𝑛
0
)
.
Proof of Theorem 3.
For any
℥(𝑡, 𝑠) ∊ 𝐶
Ω
2𝜋
(
𝑅
2
),
𝑛 ∈
𝑁
, and (t,s)
∈
[-
𝜋, 𝜋
]
2
, using the propertt of the Jackson
kernels, the Fubini theorem and the Cauchy-
Bunyakovsky inequality, we have
𝛿
𝑛,𝑛
(
℥; 𝑡, 𝑠)
≡
{𝑀[℥(𝑡, 𝑠) − 𝐷
𝑛,𝑛
(℥; 𝑡, 𝑠)]
2
}
1
2
=
=
{M[∫ ∫ [℥(𝑡, 𝑠) − ℥(𝑡 + 𝑥, 𝑠 +
𝜋
−𝜋
𝜋
−𝜋
𝑦)] 𝐷
𝑛
(𝑥)𝐷
𝑛
(𝑦)𝑑𝑥𝑑𝑦]
2
}
1
2
≤
≤
∫ ∫ 𝜔
℥
(1)
(|𝑥|, |𝑦|)
𝜋
−𝜋
𝜋
−𝜋
𝐷
𝑛
(𝑥)𝐷
𝑛
(𝑦)𝑑𝑥𝑑𝑦
≤
≤
𝜔
℥
(1)
(
2𝜋
𝑛
,
2𝜋
𝑛
)
∫ ∫ (1 +
𝜋
−𝜋
𝜋
−𝜋
max {]
𝑛|𝑥|
2𝜋
[, ]
𝑛|𝑦|
2𝜋
[}) 𝐷
𝑛
(𝑥)𝐷
𝑛
(𝑦)𝑑𝑥𝑑𝑦
=
=
4
𝜔
℥
(1)
(
2𝜋
𝑛
,
2𝜋
𝑛
) ∫ ∫ (1 +
𝜋
0
𝜋
0
max {]
𝑛|𝑥|
2𝜋
[, ]
𝑛|𝑦|
2𝜋
[}) 𝐷
𝑛
(𝑥)𝐷
𝑛
(𝑦)𝑑𝑥𝑑𝑦
=
K
n
𝜔
℥
(1)
(
2𝜋
𝑛
,
2𝜋
𝑛
)
where
K
n
=
4
∫ ∫ (1 +
𝜋
0
𝜋
0
max {]
𝑛|𝑥|
2𝜋
[, ]
𝑛|𝑦|
2𝜋
[}) 𝐷
𝑛
(𝑥)𝐷
𝑛
(𝑦)𝑑𝑥𝑑𝑦
.
It is obvious that K
1
= K
2
=1.
In [2], it is shownm that
sup
𝑛≥1
K
𝑛
= K
3
= 2- (
2
3
−
45√3
76𝜋
)
2
= 1,0137297…
.
It follows from here the proof of part a) of Theorem 2.
Volume 03 Issue 11-2023
60
International Journal Of Management And Economics Fundamental
(ISSN
–
2771-2257)
VOLUME
03
ISSUE
11
P
AGES
:
56-62
SJIF
I
MPACT
FACTOR
(2021:
5.
705
)
(2022:
5.
705
)
(2023:
7.
448
)
OCLC
–
1121105677
Publisher:
Oscar Publishing Services
Servi
To prove part b) of Theorem 2, consider an even
2𝜋
-
periodic in each argument non-random function
defined on [0,
𝜋
]
2
as follows:
𝑓
ε
(𝑡, 𝑠) =
{
𝑥
ε
if 0 ≤ 𝑥 ≤ ε , 0 ≤ 𝑦 ≤ 𝑥
𝑦
ε
if 0 ≤ 𝑦 ≤ ε , 0 ≤ 𝑥 ≤ 𝑦
1 if {
ε ≤ 𝑥 ≤
2𝜋
3
, 0 ≤ 𝑦 ≤ 𝑥
ε ≤ 𝑦 ≤
2𝜋
3
, 0 ≤ 𝑥 ≤ 𝑦
}
1 +
1
ε
(𝑥 −
2𝜋
3
) if
2𝜋
3
≤ 𝑥 ≤
2𝜋
3
+ ε, 0 ≤ 𝑦 ≤ 𝑥
1 +
1
ε
(𝑦 −
2𝜋
3
) if
2𝜋
3
≤ 𝑦 ≤
2𝜋
3
+ ε, 0 ≤ 𝑥 ≤ 𝑦
2 if {
2𝜋
3
+ ε ≤ 𝑥 ≤ 𝜋, 0 ≤ 𝑦 ≤ 𝑥
2𝜋
3
+ ε ≤ 𝑦 ≤ 𝜋, 0 ≤ 𝑥 ≤ 𝑦
}
}
where
ε >
0 is a sufficiently small number.
Let a r.v.
℥
0
be such that M
℥
0
2
= 1. Then the r.p.
℥
ε
(
t,s
)
=
℥
0
𝑓
ε
(
t,s
)
∈
∈ 𝐶
Ω
2𝜋
(
𝑅
2
) and
𝜔
℥
ε
(
2𝜋
3
,
2𝜋
3
) = 1.
We
obtain from here that
max
|𝑡|≤𝜋
|𝑠|≤𝜋
𝛿
𝑛,𝑛
(℥
ε
; 𝑡, 𝑠) ≥ 𝛿
3,3
(℥
ε
; 0,0)
= ∫ ∫ 𝑓
ε
(𝑡, 𝑠)
𝜋
−𝜋
𝜋
−𝜋
𝐷
𝑛
(𝑥)𝐷
𝑛
(𝑦)𝑑𝑥𝑑𝑦 =
=
4
∫ ∫ 𝑓
ε
(𝑡, 𝑠)
𝜋
0
𝜋
0
𝐷
𝑛
(𝑥)𝐷
𝑛
(𝑦)𝑑𝑥𝑑𝑦
=
4
∑
∬ 𝑓
ε
(𝑡, 𝑠)
𝐴
𝑘
4
𝑘=1
𝐷
3
(𝑥)𝐷
3
(𝑦)𝑑𝑥𝑑𝑦
(5)
where
𝐴
1
=
{
(
x,y
) :
0 ≤ 𝑥 ≤ ε ,
0 ≤ 𝑦 ≤ ε }
,
𝐴
2
=
{
(
x,y
):
ε ≤ 𝑥 ≤
2𝜋
3
,
0 ≤ 𝑦 ≤ 𝑥
}
U
{
(
x,y
):
ε ≤ 𝑥 ≤
2𝜋
3
,
0 ≤ 𝑥 ≤ 𝑦
},
𝐴
3
=
{
(
x,y
):
2𝜋
3
≤ 𝑥 ≤
2𝜋
3
+
ε
,
0 ≤ 𝑦 ≤ 𝑥
}
U
{
(
x,y
):
2𝜋
3
≤
𝑦 ≤
2𝜋
3
+ ε
,
0 ≤ 𝑥 ≤ 𝑦
},
𝐴
4
=
{
(
x,y
):
2𝜋
3
+ ε ≤ 𝑥 ≤ 𝜋
,
0 ≤ 𝑦 ≤ 𝑥
}
U
{
(
x,y
):
2𝜋
3
+
ε ≤ 𝑦 ≤ 𝜋
,
0 ≤ 𝑥 ≤ 𝑦
}.
Taking into account definition of the function
𝑓
ε
(𝑡, 𝑠)
,
we have
∑
∬ 𝑓
ε
(𝑡, 𝑠)
𝐴
𝑘
4
𝑘=1
𝐷
3
(𝑥)𝐷
3
(𝑦)𝑑𝑥𝑑𝑦 =
∬ 𝑚𝑎𝑥 {
𝑥
ε
𝐴
1
,
𝑦
ε
}𝐷
3
(𝑥)𝐷
3
(𝑦)𝑑𝑥𝑑𝑦
+
+
∬ 𝐷
3
(𝑥)𝐷
3
(𝑦)
𝐴
2
𝑑𝑥𝑑𝑦
+
+ ∬ 𝑚𝑎𝑥{1 +
1
ε
(𝑥 −
2𝜋
3
) , 1 +
1
ε
(𝑦 −
𝐴
3
2𝜋
3
)} 𝐷
3
(𝑥)𝐷
3
(𝑦)𝑑𝑥𝑑𝑦
+
+2 ∬ 𝐷
3
(𝑥)𝐷
3
(𝑦)
𝐴
4
𝑑𝑥𝑑𝑦
=
∬
𝐷
3
(𝑥)𝐷
3
(𝑦)
𝐴
1
U𝐴
2
𝑑𝑥𝑑𝑦 +
+2 ∬
𝐷
3
(𝑥)𝐷
3
(𝑦)
𝐴
3
U𝐴
4
𝑑𝑥𝑑𝑦
–
− ∬ (1 −
𝐴
1
𝑚𝑎𝑥 {
𝑥
ε
,
𝑦
ε
})𝐷
3
(𝑥)𝐷
3
(𝑦)𝑑𝑥𝑑𝑦
–
–
∬ (2 − 𝑚𝑎𝑥 {1 +
1
ε
(𝑥 −
2𝜋
3
) , 1 +
1
ε
(𝑦 −
𝐴
3
2𝜋
3
)}) 𝐷
3
(𝑥)𝐷
3
(𝑦)𝑑𝑥𝑑𝑦
=
∫ ∫ (1 +
𝜋
0
𝜋
0
max {]
3𝑥
2𝜋
[, ]
3𝑦
2𝜋
[}
)
𝐷
3
(𝑥)𝐷
3
(𝑦)𝑑𝑥𝑑𝑦
–
∬ (1 −
𝐴
1
𝑚𝑎𝑥 {
𝑥
ε
,
𝑦
ε
}𝐷
3
(𝑥)𝐷
3
(𝑦)𝑑𝑥𝑑𝑦
–
− ∬ (1 +
2𝜋
3ε
− 𝑚𝑎𝑥 {
𝑥
ε
,
𝑦
ε
})
𝐴
3
𝐷
3
(𝑥)𝐷
3
(𝑦)𝑑𝑥𝑑𝑦
≡
1
4
K
3
-
𝐼
ε
(3)
− 𝐼
ε
(4)
≥
1
4
K
3
-
|𝐼
ε
(3)
| − |𝐼
ε
(4)
|
.
Obviously,
|𝐼
ε
(3)
| ≤ ∬ 𝐷
3
(𝑥)𝐷
3
(𝑦)
𝐴
1
𝑑𝑥𝑑𝑦 →
0 as
ε →
0,
|𝐼
ε
(4)
| ≤ ∬ 𝐷
3
(𝑥)𝐷
3
(𝑦)
𝐴
3
𝑑𝑥𝑑𝑦 →
0 as
ε →
0.
Thus,
Volume 03 Issue 11-2023
61
International Journal Of Management And Economics Fundamental
(ISSN
–
2771-2257)
VOLUME
03
ISSUE
11
P
AGES
:
56-62
SJIF
I
MPACT
FACTOR
(2021:
5.
705
)
(2022:
5.
705
)
(2023:
7.
448
)
OCLC
–
1121105677
Publisher:
Oscar Publishing Services
Servi
∑
∬ 𝑓
ε
(𝑡, 𝑠)
𝐴
𝑘
4
𝑘=1
𝐷
3
(𝑥)𝐷
3
(𝑦)𝑑𝑥𝑑𝑦
≥
K
3
–
β
(
ε
),
β
(
ε
)
→
0
as
ε →
0.
Taking into account relation (5), we obtain from here
that
max
|𝑡|≤𝜋
|𝑠|≤𝜋
𝛿
𝑛,𝑛
(℥
ε
; 𝑡, 𝑠) ≥ 𝛿
3,3
(℥
ε
; 0,0) ≥ 𝐾
3
–
β
(
ε
) = [
𝐾
3
–
β
(
ε
)]
𝜔
℥
𝜀
(1)
(
2𝜋
3
,
2𝜋
3
)
Let
ε
1
> 0 be an arbitrary number. Choosng
β
(
ε
) such
that
β
(
ε
) <
ε
1
, we come to the assertion of the second
part of Theorem 3.
Theorem 3 is proved.
Theorem 4.
For the class of r.p.’s
𝐶
Ω
2𝜋
(
𝑅
2
)
, the relation
𝑠 𝑢 𝑝
𝑛 ∊𝑁 ,℥(𝑡,𝑠)∊ 𝐶
Ω
2𝜋
(𝑅
1
)
max
(𝑡,𝑠)∊ 𝑅2
{ M|℥(𝑡,𝑠)−𝐷𝑛,𝑛(℥;𝑡,𝑠) |
2
}
1
2
𝜔
℥
(
2𝜋
𝑛
)
=
4
3
−
45√3
76𝜋
holds.
The proof of Theorem 4
follows from Theorem 3.
Note that Theorems 1
–
4 in the case when
℥
(t) and
℥(𝑡, 𝑠)
are nonrandom functions coincide with the
results of [2] obtained there by another method.
REFERENCES
1.
Azlarov T.A., One remark on the interpolation of
random fields. In:
“
Limit Theorems for Random
Processes and Statistical Inference
”
. FAN,
Tashkent, 1981, p. 3-6 (in Russian).
2.
Bugaets V.P., Martynyuk V.T., Exact constant for
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26
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3.
Gihman I.I., Skorokhod A.V., The Theory of
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Dzyadyk V.K., Introduction to Theory of Uniform
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А
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Kadyrova I.I., On approximation of periodic mean-
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Mirzakhmedov M.A., Khudaiberganov R., On the
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Nagorny V. N., Yadrenko M. I., Polynomial
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Volume 03 Issue 11-2023
62
International Journal Of Management And Economics Fundamental
(ISSN
–
2771-2257)
VOLUME
03
ISSUE
11
P
AGES
:
56-62
SJIF
I
MPACT
FACTOR
(2021:
5.
705
)
(2022:
5.
705
)
(2023:
7.
448
)
OCLC
–
1121105677
Publisher:
Oscar Publishing Services
Servi
11.
Seleznev O.V., Approximation of periodic Gaussian
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Akad. Nauk SSSR,
250
:1 (1980), p. 35
–
38.
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Seleznev O.V., On the approximation of continuous
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Gaussian
processes
by
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Timan A.F., Theory of Approximation of Functions
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