Approximative properties of the Weierstrass integrals on the classes WrβHα
The work focuses on the solution of a problem of approximation theory. The task is to investigate approximative properties of the Weierstrass integrals on the classes WrβHα. We obtain asymptotic equalities for the upper borders of defluxion of functions from the classes WrβHα from the Weierstrass in...
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Інститут прикладної математики і механіки НАН України
2017
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irk-123456789-1693652020-06-11T01:27:02Z Approximative properties of the Weierstrass integrals on the classes WrβHα Grabova, U.Z. Kal'chuk, I.V. Stepaniuk, T.A. The work focuses on the solution of a problem of approximation theory. The task is to investigate approximative properties of the Weierstrass integrals on the classes WrβHα. We obtain asymptotic equalities for the upper borders of defluxion of functions from the classes WrβHα from the Weierstrass integrals. 2017 Article Approximative properties of the Weierstrass integrals on the classes WrβHα / U.Z. Grabova, I.V. Kal'chuk, T.A. Stepaniuk // Український математичний вісник. — 2017. — Т. 14, № 3. — С. 361-369. — Бібліогр.: 15 назв. — англ. 1810-3200 2010 MSC. 42A05, 41A60 http://dspace.nbuv.gov.ua/handle/123456789/169365 en Український математичний вісник Інститут прикладної математики і механіки НАН України |
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The work focuses on the solution of a problem of approximation theory. The task is to investigate approximative properties of the Weierstrass integrals on the classes WrβHα. We obtain asymptotic equalities for the upper borders of defluxion of functions from the classes WrβHα from the Weierstrass integrals. |
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Article |
| author |
Grabova, U.Z. Kal'chuk, I.V. Stepaniuk, T.A. |
| spellingShingle |
Grabova, U.Z. Kal'chuk, I.V. Stepaniuk, T.A. Approximative properties of the Weierstrass integrals on the classes WrβHα Український математичний вісник |
| author_facet |
Grabova, U.Z. Kal'chuk, I.V. Stepaniuk, T.A. |
| author_sort |
Grabova, U.Z. |
| title |
Approximative properties of the Weierstrass integrals on the classes WrβHα |
| title_short |
Approximative properties of the Weierstrass integrals on the classes WrβHα |
| title_full |
Approximative properties of the Weierstrass integrals on the classes WrβHα |
| title_fullStr |
Approximative properties of the Weierstrass integrals on the classes WrβHα |
| title_full_unstemmed |
Approximative properties of the Weierstrass integrals on the classes WrβHα |
| title_sort |
approximative properties of the weierstrass integrals on the classes wrβhα |
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Інститут прикладної математики і механіки НАН України |
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2017 |
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http://dspace.nbuv.gov.ua/handle/123456789/169365 |
| citation_txt |
Approximative properties of the Weierstrass integrals on the classes WrβHα / U.Z. Grabova, I.V. Kal'chuk, T.A. Stepaniuk // Український математичний вісник. — 2017. — Т. 14, № 3. — С. 361-369. — Бібліогр.: 15 назв. — англ. |
| series |
Український математичний вісник |
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| first_indexed |
2025-07-15T04:06:31Z |
| last_indexed |
2025-07-15T04:06:31Z |
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| fulltext |
Український математичний вiсник
Том 14 (2017), № 3, 361 – 369
Approximative properties
of the Weierstrass integrals
on the classes W r
βH
α
Uliana Z. Grabova, Inna V. Kal’chuk
and Tetiana A. Stepaniuk
(Presented by S. Ya. Mahno)
Abstract. The work focuses on the solution of the one problem of ap-
proximation theory. The problem is to investigate approximative proper-
ties of the Weierstrass integrals on the classes W r
βH
α. We obtain asymp-
totic equalities for the upper borders of defluxion of functions from the
classes W r
βH
α from the Weierstrass integrals.
2010 MSC. 42A05, 41A60.
Key words and phrases. Weierstrass integral, Weyl–Nagy class, Lip-
schitz condition, Kolmogorov–Nikolsky problem.
1. Introduction
Let L be the space of 2π–periodic summable on the period functions
having the norm ∥f∥L = ∥f∥1 =
∫ π
−π |f(t)|dt; L∞ be the space of 2π–
periodic measurable and essentially bounded functions having the norm
∥f∥∞ = ess sup
t
|f(t)|; C be the space of 2π-periodic functions having the
norm ∥f∥C = max
t
|f(t)|.
Let r > 0 and β be a fixed real number. If the series
∞∑
k=1
kr
(
ak cos
(
kx+
βπ
2
)
+ bk sin
(
kx+
βπ
2
))
(1.1)
is the Fourier series of some summable function φ, then we can introduce
the (r, β)-derivative of the function f in the Weyl–Nagy sense and denote
it by f rβ (see, e.g., [1, p. 130]). By W r
β we denote the set of all functions
satisfying this condition.
Received 10.08.2017
ISSN 1810 – 3200. c⃝ Iнститут прикладної математики i механiки НАН України
362 Approximative properties of the Weierstrass integrals
If f ∈ W r
β and, in addition, f rβ ∈ Hα, that is satisfies the Lipschitz
condition of order α:
|f rβ(x+ h)− f rβ(x)| ≤ |h|α, 0 < α ≤ 1, h ∈ R,
we say that f belongs to the class W r
βH
α. In the case where α = 0 we
set W r
βH
0 =W r
β,∞.
If β = r, r ∈ N, then the classes W r
βH
α coincide with the famous
Sobolev classes W rHα. Note that W r
∞ are the classes of functions f such
that ∥f (r)∥∞ ≤ 1.
Let f ∈ L. The quantity
W (ρ; f ;x) =
1
π
∫ π
−π
f(t+ x)
{
1
2
+
∞∑
k=1
ρk
2
cos kt
}
dt, 0 ≤ ρ < 1,
is called the Weierstrass integral of the function f . Setting ρ = e−
1
δ the
Weierstrass integral can be rewritten as follows (see, e.g., [2])
Wδ(f ;x) =
1
π
∫ π
−π
f(t+ x)
{
1
2
+
∞∑
k=1
e−
k2
δ cos kt
}
dt, δ > 0.
In the present paper we investigate asymptotic behavior of the quan-
tity
E(W r
βH
α;Wδ)C = sup
f∈W r
βH
α
∥f(·)−Wδ(f ; ·)∥C , δ → ∞. (1.2)
The problem of establishing the asymptotic equality for the quan-
tity (1.2), according to Stepanets [1, p. 198], is called the Kolmogorov–
Nikolsky problem for the Weierstrass integral Wδ on the class W r
βH
α in
the uniform metric.
In the uniform metrics the Kolmogorov–Nikolsky problem on the Zyg-
mund classes Zα, Zα :=
{
f(x) ∈ C : |f(x+h)−2f(x)+f(x−h)| ≤ 2|h|α,
0 ≤ α ≤ 2, |h| ≤ 2π
}
, Korovkin [3], Bausov [4], Falaleev [5]; on
the Sobolev classes W r
∞ — in the papers of Bausov [6], Baskakov [7].
Asymptotic equalities for approximation of the Stepanets classes [1] by
the Weierstrass integrals were obtained in the papers [2, 8, 9].
2. The approximation by Weierstrass integrals on
the classes W r
βH
α
Analogously to the paper [10] for the Weierstrass integral for r > 0
and δ > 0, we consider the following function τ(u) continuous on [0;∞):
τ(u) = τδ(r, u) =
{
(1− e−u
2
)δ
r
2 , 0 ≤ u ≤ 1√
δ
,
(1− e−u
2
)u−r, u ≥ 1√
δ
,
(2.1)
U. Z. Grabova, I. V. Kal’chuk, T. A. Stepaniuk 363
whose Fourier transform
τ̂β(t) =
1
π
∫ ∞
0
τ(u) cos
(
ut+
βπ
2
)
du (2.2)
is summable on the whole real axis (this fact is proved in [2]).
In what follows, by K, Ki, i = 1, 2, we denote constants whose values
may be different in different places.
Theorem 2.1. For r > 2, 0 < α < 1 and δ → ∞ the following asymp-
totic equality is true
E
(
W r
βH
α;Wδ
)
C
=
1
δ
sup
f∈W r
βH
α
∥∥f ′′∥∥
C
+O
(
1
δ
r+α
2
+
1
δ2
)
, (2.3)
where f ′′ be the second derivative of function f .
Proof. Similarly, as in the paper [11], let us rewrite the function τ(u)
defined by the relation (2.1) in the form τ(u) = φ(u) + µ(u), where
φ(u) =
{
u2δ
r
2 , 0 ≤ u ≤ 1√
δ
,
u2−r, u ≥ 1√
δ
,
(2.4)
µ(u) =
{ (
1− e−u
2 − u2
)
δ
r
2 , 0 ≤ u ≤ 1√
δ
,(
1− e−u
2 − u2
)
u−r, u ≥ 1√
δ
.
(2.5)
It is known that Fourier transforms φ̂(t) and µ̂(t) of kind (2.2) of func-
tions φ(u) and µ(u) are summable on whole real axis (see, e.g., [2]).
According to the theorem 3 of Bausov [6], if the integrals
A(α, φ) =
1
π
∫ ∞
−∞
|t|α
∣∣∣∣∫ ∞
0
φ(u) cos
(
ut+
βπ
2
)
du
∣∣∣∣ dt, (2.6)
A(α, µ) =
1
π
∫ ∞
−∞
|t|α
∣∣∣∣∫ ∞
0
µ(u) cos
(
ut+
βπ
2
)
du
∣∣∣∣ dt, (2.7)
are convergent and A(α, µ) = o
(
A(α, φ)
)
, then the following asymptotic
equality holds
E(W r
βH
α;Wδ)C =
1
δ
r
2
sup
f∈W r
βH
α
∥fφ∥C +O
(
1
δ
r+α
2
A(α, µ)
)
, (2.8)
where fφ(x) :=
∫∞
−∞
(
f rβ
(
x+ t√
δ
)
− f rβ(x)
)
φ̂(t)dt.
364 Approximative properties of the Weierstrass integrals
To prove the convergence of the integral A(α, φ), according to theo-
rem 1 of the paper of Bausov [6], let us consider the integrals∫ 1
2
0
u1−α
∣∣dφ′(u)
∣∣,∫ 3
2
1
2
|u− 1|1−α
∣∣dφ′(u)
∣∣,∫ ∞
3
2
(u− 1)
∣∣dφ′(u)
∣∣, (2.9)
∣∣∣ sin βπ
2
∣∣∣ ∫ ∞
0
∣∣φ(u)∣∣
u1+α
du,
∫ 1
0
|φ(1− u)− φ(1 + u)|
uα+1
du, (2.10)
and obtain the upper bounds.
For the first integral from (2.9) we get∫ 1
2
0
u1−α
∣∣dφ′(u)
∣∣ = ∫ 1√
δ
0
u1−α
∣∣dφ′(u)
∣∣+ ∫ 1
2
1√
δ
u1−α
∣∣dφ′(u)
∣∣
= 2δ
r
2
∫ 1√
δ
0
u1−αdu+(2−r)(1−r)
∫ 1
2
1√
δ
u1−r−αdu = O
(
1
δ1−
r+α
2
)
, r > 2.
(2.11)
In view of the fact that the function |u− 1|1−α|dφ′(u)| is continuous
on the segment [12 ,
3
2 ] it is bounded on this segment. Thus∫ 3
2
1
2
|u− 1|1−α|dφ′(u)| = O(1). (2.12)
Now we estimate the third integral from (2.9):∫ ∞
3
2
(u− 1)|dφ′(u)| ≤
∫ ∞
3
2
u|dφ′(u)|
= (2− r)(1− r)
∫ ∞
3
2
u1−rdu ≤ K, r > 2. (2.13)
For first integral from (2.10) we get∫ ∞
0
|φ(u)|
u1+α
du = δ
r
2
∫ 1√
δ
0
u1−αdu+
∫ ∞
1√
δ
u1−r−αdu
= O
(
1
δ1−
r+α
2
)
, r > 2. (2.14)
Let us estimate the second integral from (2.10). Similarly to formula
(30) of [12], it can be shown that for the function φ, given by the relation
(2.4), we have the equality∫ 1
0
|φ(1− u)− φ(1 + u)|
u1+α
du =
∫ 1
0
|λ(1− u)− λ(1 + u)|
u1+α
du+O(H(α, φ)),
(2.15)
U. Z. Grabova, I. V. Kal’chuk, T. A. Stepaniuk 365
where λ(u) = 1− u2 and
H(α, φ) =
∣∣φ(0)∣∣+ ∣∣φ(1)∣∣+ ∫ 1
2
0
u1−α
∣∣dφ′(u)
∣∣+ ∫ ∞
3
2
(u− 1)
∣∣dφ′(u)
∣∣.
Since
∫ 1
0
|λ(1−u)−λ(1+u)|
u1+α
du = O(1), according to (2.11) and (2.12) from
(2.15) we get∫ 1
0
|φ(1− u)− φ(1 + u)|
u1+α
du = O
( 1
δ
1−r−α
2
)
, r > 2. (2.16)
Applying theorem 1 of [6] and taking into account relations (2.11)–
(2.16) we show that the Fourier transform of the function φ of the form
(2.2) is summable on the real axis, and the following estimate holds
A(α,φ) = O
( 1
δ
1−r−α
2
)
, r > 2. (2.17)
We now show the convergence of the integral A(α, µ). Let us consider
the following integrals∫ 1
2
0
u1−α
∣∣dµ′(u)∣∣,∫ 3
2
1
2
|u− 1|1−α
∣∣dµ′(u)∣∣,∫ ∞
3
2
(u− 1)
∣∣dµ′(u)∣∣, (2.18)
∣∣∣ sin βπ
2
∣∣∣ ∫ ∞
0
∣∣µ(u)∣∣
u1+α
du,
∫ 1
0
|µ(1− u)− µ(1 + u)|
uα+1
du, (2.19)
and get the corresponding upper bounds.
Let us investigate the first integral from (2.18). Similarly, as in the
proof of Lemma 1 from [13], we divide the segment
[
0, 12
]
into two parts:[
0, 1√
δ
]
,
[
1√
δ
, 12
]
. According to the inequality
2u2e−u
2 − e−u
2
+ 1 ≤ 3u2, u ∈ R, (2.20)
and the fact that if u ∈
[
0, 1√
δ
]
then µ′′(u) ≤ 0, we can write
∫ 1√
δ
0
u1−α|dµ′(u)| = 2δ
r
2
∫ 1√
δ
0
u1−α(2u2e−u
2 − e−u
2
+ 1)du
≤ 6δ
r
2
∫ 1√
δ
0
u3−αdu ≤ K
δ2−
r+α
2
. (2.21)
366 Approximative properties of the Weierstrass integrals
Let us estimate the integral on the segment
[
1√
δ
, 12
]
. From (2.5) we
get
µ′′(u) = (1− e−u
2 − u2)r(r + 1)u−r−2 + 4(−r)u(e−u2 − 1)u−r−1
+2u−r(e−u
2 − 2u2e−u
2 − 1). (2.22)
One can verify that µ′′(u) ≤ 0. Further, according to (2.20) and, in
addition, to the inequalities
e−u
2
+ u2 − 1 ≤ u4
2
, 1− e−u
2 ≤ u2, u ∈ R, (2.23)
we have∫ 1
2
1√
δ
u1−α|dµ′(u)| ≤ r(r + 1)
∫ 1
2
1√
δ
(u2 + e−u
2 − 1)u−1−r−αdu
+4r
∫ 1
2
1√
δ
(1− e−u
2
)u1−r−αdu+ 2
∫ 1
2
1√
δ
(2u2e−u
2 − e−u
2
+ 1)u1−r−αdu
≤ K
∫ 1
2
1√
δ
u3−r−αdu ≤ K1 +
K2
δ2−
r+α
2
. (2.24)
In view of (2.21) and (2.24) we find∫ 1
2
0
u1−α|dµ′(u)| = O
(
1 +
1
δ2−
r+α
2
)
. (2.25)
By the same way as to estimate the second integral from (2.18), we
get ∫ 3
2
1
2
|u− 1|1−α|dµ′(u)| = O(1). (2.26)
To estimate the last integral from (2.18) we use (2.22) and the fol-
lowing inequalities
e−u
2 ≤ 1, 1− e−u
2 ≤ 1, u2e−u
2 ≤ 1, u ∈ R. (2.27)
We obtain∫ ∞
3
2
(u− 1)|dµ′(u)| ≤
∫ ∞
3
2
u|dµ′(u)| ≤ K
∫ ∞
3
2
u−r+1du ≤ K1, r > 2.
(2.28)
Analogously to the estimation (36) of [14], to estimate the first inte-
gral with (2.19) we divide the interval [0,∞) into three parts:
[
0, 1√
δ
]
,
U. Z. Grabova, I. V. Kal’chuk, T. A. Stepaniuk 367
[
1√
δ
, 1
]
,
[
1,∞
]
. According to the first inequalities from (2.23) and (2.27)
we get ∫ ∞
0
|µ(u)|
u1+α
du = δ
r
2
∫ 1√
δ
0
e−u
2
+ u2 − 1
u1+α
du
+
(∫ 1
1√
δ
+
∫ ∞
1
)
(e−u
2
+ u2 − 1)u−1−r−αdu ≤ 1
2
δ
r
2
∫ 1√
δ
0
u3−αdu
+
1
2
∫ 1
1√
δ
u3−r−αdu+
∫ ∞
1
u1−r−αdu ≤ K1 +
K2
δ2−
r+α
2
. (2.29)
Let us estimate the second integral from (2.19). Similarly, as for function
φ, the following relation holds∫ 1
0
|µ(1− u)− µ(1 + u)|
u1+α
du =
∫ 1
0
|λ(1− u)− λ(1 + u)|
u1+α
du+O(H(α, µ)),
(2.30)
where λ(u) = e−u
2
+ u2 and
H(α, µ) =
∣∣µ(0)∣∣+ ∣∣µ(1)∣∣+ ∫ 1
2
0
u1−α
∣∣dµ′(u)∣∣+ ∫ ∞
3
2
(u− 1)
∣∣dµ′(u)∣∣.
Since
∫ 1
0
|λ(1−u)−λ(1+u)|
u1+α
du = O(1), according to (2.25) and (2.28) from
(2.30) we get∫ 1
0
|µ(1− u)− µ(1 + u)|
u1+α
du = O
(
1 +
1
δ2−
r+α
2
)
, r > 2. (2.31)
Applying theorem 1 from the paper of Bausov [6] and taking into
account estimates (2.29) and (2.31), we have
A(α, µ) = O
(
1 +
1
δ2−
r+α
2
)
, r > 2. (2.32)
Thus, we showed the convergence of integrals A(α,φ), A(α, µ) and
validity of the relation A(α, µ) = o (A(α, φ)). So equality (2.8) holds.
Whereas an estimation (2.32) from (2.8) we receive
E(W r
βH
α;Wδ)C =
1
δ
r
2
sup
f∈W r
βH
α
∥fφ∥C +O
(
1
δ
r+α
2
+
1
δ2
)
. (2.33)
It is well known that Fourier series of the function fφ(x) (see, e.g.,
[15]) is of the form:
S[fφ(x)] =
∞∑
k=1
φ
(
k√
δ
)
kr(ak cos kx+ bk sin kx).
368 Approximative properties of the Weierstrass integrals
From here, considering formulas (2.4) and (1.1) we have that
S[fφ(x)] =
∞∑
k=1
k2
δ1−
r
2
(ak cos kx+ bk sin kx)
=
1
δ1−
r
2
f
(2)
0 (x) = − 1
δ1−
r
2
f ′′(x), (2.34)
where f ′′ be the second derivative of function f .
Substituting (2.34) into (2.33), we obtain (2.3). Theorem is proved.
References
[1] A. I. Stepanets, Methods of Approximation Theory. Part 1, Institute of Mathe-
matics, Ukrainian Academy of Sciences, Kiev, 2002.
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Contact information
Uliana Z.
Grabova
Lesya Ukrainka Eastern European National
University, Lutsk, Ukraine
E-Mail: grabova_u@ukr.net
Inna V.
Kalchuk
Lesya Ukrainka Eastern European National
University, Lutsk, Ukraine
E-Mail: k.inna80@gmail.com
Tetiana A.
Stepaniuk
Graz University of Technology,
Graz, Austria
E-Mail: tania_stepaniuk@ukr.net
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