Trigonometric approximation of functions in generalized Lebesgue spaces with variable exponent
We investigate the approximation properties of the trigonometric system in $L_{2\pi}^{p(\cdot)}$. We consider the fractional order moduli of smoothness and obtain direct, converse approximation theorems together with a constructive characterization of a Lipschitz-type class.
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| author | Akgün, R. Акгюн, Р. |
| author_facet | Akgün, R. Акгюн, Р. |
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| description | We investigate the approximation properties of the trigonometric system in $L_{2\pi}^{p(\cdot)}$.
We consider the fractional order moduli of smoothness and obtain direct, converse approximation theorems together with a constructive characterization of a Lipschitz-type class. |
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UDC 517.938.5
R. Akgün (Balikesir Univ., Turkey)
TRIGONOMETRIC APPROXIMATION
OF FUNCTIONS IN GENERALIZED LEBESGUE SPACES
WITH VARIABLE EXPONENT
ТРИГОНОМЕТРИЧНЕ НАБЛИЖЕННЯ ФУНКЦIЙ
В УЗАГАЛЬНЕНИХ ПРОСТОРАХ ЛЕБЕГА
ЗI ЗМIННОЮ ЕКСПОНЕНТОЮ
We investigate the approximation properties of the trigonometric system in L
p(·)
2π . We consider the fractional
order moduli of smoothness and obtain direct, converse approximation theorems together with a constructive
characterization of a Lipschitz-type class.
Дослiджено властивостi наближення тригонометричної системи в L
p(·)
2π . Розглянуто модулi гладкостi
дробового порядку та отримано пряму i обернену теореми наближення разом iз конструктивною харак-
теризацiєю класу типу Лiпшиця.
1. Introduction. Generalized Lebesgue spaces Lp(x) with variable exponent and cor-
responding Sobolev-type spaces have waste applications in elasticity theory, fluid me-
chanics, differential operators [31, 10], nonlinear Dirichlet boundary-value problems
[24], nonstandard growth and variational calculus [33].
These spaces appeared first in [28] as an example of modular spaces [14, 26] and
Sharapudinov [36] has been obtained topological properties of Lp(x). Furthermore if
p∗ := ess supx∈T p(x) <∞, then Lp(x) is a particular case of Musielak – Orlicz spaces
[26]. Later various mathematicians investigated the main properties of these spaces
[36, 24, 32, 12]. In Lp(x) there is a rich theory of boundedness of integral transforms of
various type [22, 33, 9, 37].
For p(x) := p, 1 < p < ∞, Lp(x) is coincide with Lebesgue space Lp and basic
problems of trigonometric approximation in Lp are investigated by several mathemati-
cians (among others [39, 19, 30, 40, 6, 4], . . . ). Approximation by algebraic polynomials
and rational functions in Lebesgue spaces, Orlicz spaces, symmetric spaces and their
weighted versions on sufficiently smooth complex domains and curves was investigated
in [1 – 3, 15, 18, 16]. For a complete treatise of polynomial approximation we refer to
the books [5, 8, 41, 29, 35, 23].
In harmonic and Fourier analysis some of operators (for example partial sum oper-
ator of Fourier series, conjugate operator, differentiation operator, shift operator f →
→ f (·+ h) , h ∈ R) have been extensively used to prove direct and converse type
approximation inequalities. Unfortunately the space Lp(x) is not p(·)-continuous and not
translation invariant [24]. Under various assumptions (including translation invariance)
on modular space Musielak [27] obtained some approximation theorems in modular
spaces with respect to the usual moduli of smoothness. Since Lp(x) is not translation
invariant using Butzer – Wehrens type moduli of smoothness (see [7, 13]) Israfilov et all.
[17] obtained direct and converse trigonometric approximation theorems in Lp(x).
c© R. AKGÜN, 2011
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 1 3
4 R. AKGÜN
In the present paper we investigate the approximation properties of the trigonometric
system in L
p(·)
2π . We consider the fractional order moduli of smoothness and obtain
direct, converse approximation theorems together with a constructive characterization of
a Lipschitz-type class.
Let T := [−π, π] and P be the class of 2π-periodic, Lebesgue measurable functions
p = p(x) : T → (1,∞) such that p∗ < ∞. We define class Lp(·)2π := L
p(·)
2π (T ) of
2π-periodic measurable functions f defined on T satisfying∫
T
|f(x)|p(x) dx <∞.
The class Lp(·)2π is a Banach space [24] with norms
‖f(x)‖p,π := ‖f (x)‖p,π,T := inf
α > 0 :
∫
T
∣∣∣∣f(x)
α
∣∣∣∣p(x) |dx| ≤ 1
and
‖f(x)‖∗p,π := sup
∫
T
|f(x)g(x)| dx : g ∈ Lp
′(·)
2π ,
∫
T
|g(x)|p
′(x)
dx ≤ 1
having the property1
‖f‖p,π � ‖f‖
∗
p,π , (1)
where p′(x) := p(x)/ (p (x)− 1) is the conjugate exponent of p(x).
The variable exponent p(x) which is defined on T is said to be satisfy Dini –
Lipschitz property DLγ of order γ on T if
sup
x1,x2∈T
{
|p (x1)− p (x2)| : |x1 − x2| ≤ δ
}(
ln
1
δ
)γ
≤ c, 0 < δ < 1.
Let f ∈ Lp(·)2π , p ∈ P satisfy DL1, 0 < h ≤ 1 and let
σhf(x) :=
1
h
x+h/2∫
x−h/2
f(t)dt, x ∈ T ,
be Steklov’s mean operator. In this case the operator σh is bounded [37] in Lp(·)2π . Using
these facts and setting x, t ∈ T , 0 ≤ α < 1 we define
σαhf(x) := (I − σh)
α
f(x) =
=
∞∑
k=0
(−1)
k
(
α
k
)
1
hk
h/2∫
−h/2
. . .
h/2∫
−h/2
f (x+ u1 + . . .+ uk) du1 . . . duk, (2)
1X � Y means that there exist constants C, c > 0 such that cY ≤ X ≤ CY hold. Throughout this
work by c, C, c1, c2, . . . , we denote the constants which are different in different places. Xn = O (Yn) ,
n = 1, 2, . . . , means that there exists a constant C > 0 such that Xn ≤ CYn holds for n = 1, 2, . . . .
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 1
TRIGONOMETRIC APPROXIMATION IN Lp(x) 5
where f ∈ Lp(·)2π ,
(
α
k
)
:=
α (α− 1) . . . (α− k + 1)
k!
for k > 1,
(
α
1
)
:= α,
(
α
0
)
:= 1
and I is the identity operator.
Since the Binomial coefficients
(
α
k
)
satisfy [34, p. 14]
∣∣∣∣(αk
)∣∣∣∣ ≤ c(α)
kα+1
, k ∈ Z+,
we get
C(α) :=
∞∑
k=0
∣∣∣∣(αk
)∣∣∣∣ <∞
and therefore
‖σαhf‖p,π ≤ c ‖f‖p,π <∞ (3)
provided f ∈ Lp(·)2π , p ∈ P satisfy DL1 and 0 < h ≤ 1.
For 0 ≤ α < 1 and r = 1, 2, 3, . . . we define the fractional modulus of smoothness
of index r + α for f ∈ Lp(·)2π , p ∈ P, satisfy DL1 and 0 < h ≤ 1 as
Ωr+α (f, δ)p(·) := sup
0≤hi,h≤δ
∥∥∥∥∥
r∏
i=1
(I − σhi)σαhf
∥∥∥∥∥
p,π
and
Ωα (f, δ)p(·) := sup
0≤h≤δ
‖σαhf‖p,π .
We have by (3) that
Ωr+α (f, δ)p(·) ≤ c ‖f‖p,π
where f ∈ Lp(·)2π , p ∈ P satisfy DL1, 0 < h ≤ 1 and the constant c > 0 dependent only
on α, r and p.
Remark 1. The modulus of smoothness Ωα(f, δ)p(·), α ∈ R+, has the follow-
ing properties for p ∈ P satisfying DL1: (i) Ωα (f, δ)p(·) is non-negative and non-
decreasing function of δ ≥ 0, (ii) Ωα (f1 + f2, ·)p(·) ≤ Ωα (f1, ·)p(·) + Ωα (f2, ·)p(·) ,
(iii) lim
δ→0
Ωα(f, δ)p(·) = 0.
Let
En(f)p(·) := inf
T∈Tn
‖f − T‖p,π , n = 0, 1, 2, . . . ,
be the approximation error of function f ∈ Lp(·)2π where Tn is the class of trigonometric
polynomials of degree not greater than n.
For a given f ∈ L1, assuming ∫
T
f(x)dx = 0, (4)
we define α-th fractional (α ∈ R+) integral of f as [42, v. 2, p. 134]
Iα(x, f) :=
∑
k∈Z∗
ck (ik)
−α
eikx,
where ck :=
∫
T
f(x)e−ikxdx for k ∈ Z∗ := {±1,±2,±3, . . .} and
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 1
6 R. AKGÜN
(ik)
−α
:= |k|−α e(−1/2)πiα sign k
as principal value.
Let α ∈ R+ be given. We define fractional derivative of a function f ∈ L1, satisfy-
ing (4), as
f (α)(x) :=
d[α]+1
dx[α]+1
I1+[α]−α(x, f)
provided the right-hand side exists, where [x] denotes the integer part of a real number x.
Let Wα
p(·), p ∈ P, α > 0, be the class of functions f ∈ Lp(·)2π such that f (α) ∈ Lp(·)2π .
Wα
p(·) becomes a Banach space with the norm
‖f‖Wα
p(·)
:= ‖f‖p,π +
∥∥f (α)∥∥
p,π
.
Main results of this work are following.
Theorem 1. Let f ∈ Wα
p(·), α ∈ R+, and p ∈ P satisfy DLγ with γ ≥ 1, then
for every natural n there exists a constant c > 0 independent of n such that
En(f)p(·) ≤
c
(n+ 1)α
En(f (α))p(·)
holds.
Corollary 1. Under the conditions of Theorem 1
En(f)p(·) ≤
c
(n+ 1)
α
∥∥f (α)∥∥
p,π
with a constant c > 0 independent of n = 0, 1, 2, 3, . . . .
Theorem 2. If α ∈ R+, p ∈ P satisfy DLγ with γ ≥ 1 and f ∈ Lp(·)2π , then there
exists a constant c > 0 dependent only on α and p such that for n = 0, 1, 2, 3, . . .
En(f)p(·) ≤ cΩα
(
f,
2π
n+ 1
)
p(·)
holds.
The following converse theorem of trigonometric approximation holds.
Theorem 3. If α ∈ R+, p ∈ P satisfy DLγ with γ ≥ 1 and f ∈ Lp(·)2π , then for
n = 0, 1, 2, 3, . . .
Ωα
(
f,
π
n+ 1
)
p(·)
≤ c
(n+ 1)α
n∑
ν=0
(ν + 1)
α−1
Eν(f)p(·)
hold where the constant c > 0 dependent only on α and p.
Corollary 2. Let α ∈ R+, p ∈ P satisfy DLγ with γ ≥ 1 and f ∈ Lp(·)2π . If
En(f)p(·) = O
(
n−σ
)
, σ > 0, n = 1, 2, . . . ,
then
Ωα (f, δ)p(·) =
O(δσ), α > σ,
O (δσ |log (1/δ)|) , α = σ,
O(δα), α < σ,
hold.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 1
TRIGONOMETRIC APPROXIMATION IN Lp(x) 7
Definition 1. For 0 < σ < α we set
Lipσ (α, p(·)) :=
{
f ∈ Lp(·)2π : Ωα (f, δ)p(·) = O (δσ), δ > 0
}
.
Corollary 3. Let 0 < σ < α, p ∈ P satisfy DLγ with γ ≥ 1 and f ∈ Lp(·)2π be
fulfilled. Then the following conditions are equivalent:
(a) f ∈ Lipσ (α, p(·)) ,
(b) En(f)p(·) = O (n−σ), n = 1, 2, . . . .
Theorem 4. Let p ∈ P satisfy DLγ with γ ≥ 1 and f ∈ Lp(·)2π . If β ∈ (0,∞) and
∞∑
ν=1
νβ−1Eν(f)p,π <∞
then f ∈W β
p(·) and
En(f
(β))p(·) ≤ c
(
(n+ 1)βEn(f)p(·) +
∞∑
ν=n+1
νβ−1Eν (f)p(·)
)
hold where the constant c > 0 dependent only on β and p.
Corollary 4. Let p ∈ P satisfy DLγ with γ ≥ 1, f ∈ Lp(·)2π , β ∈ (0,∞) and
∞∑
ν=1
να−1Eν(f)p(·) <∞
for some α > 0. In this case for n = 0, 1, 2, . . . there exists a constant c > 0 dependent
only on α, β and p such that
Ωβ
(
f (α),
π
n+ 1
)
p(·)
≤ c
(n+ 1)β
n∑
ν=0
(ν + 1)
α+β−1
Eν(f)p(·)+c
∞∑
ν=n+1
να−1Eν(f)p(·)
hold.
The following simultaneous approximation theorem holds.
Theorem 5. Let β ∈ [0,∞), p ∈ P satisfy DLγ with γ ≥ 1 and f ∈ Lp(·)2π . Then
there exist a T ∈ Tn and a constant c > 0 depending only on α and p such that∥∥f (β) − T (β)
∥∥
p,π
≤ cEn
(
f (β)
)
p(·)
holds.
Definition 2 (Hardy space of variable exponent Hp(·) on the unit disc D with the
boundary T := ∂D) [21]. Let p(z) : T→(1,∞), be measurable function. We say that a
complex valued analytic function Φ in D belongs to the Hardy space Hp(·) if
sup
0<r<1
2π∫
0
∣∣Φ (reiϑ)∣∣p(ϑ) dϑ < +∞
where p(ϑ) := p
(
eiϑ
)
, ϑ ∈ [0, 2π] (and therefore p (ϑ) is 2π-periodic function). Let
p := infz∈T p(z) and p := supz∈T p(z). If p > 0, then it is obvious that Hp ⊂ Hp(·) ⊂
⊂ Hp. Therefore if f ∈ Hp(·) and p > 0, then there exist nontangential boundary-values
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 1
8 R. AKGÜN
f
(
eiθ
)
a.e. on T and f
(
eiθ
)
∈ Lp(·)2π (T) . Under the conditions 1 < p and p <∞, Hp(·)
becomes a Banach space with the norm
‖f‖Hp(·) :=
∥∥f (eiθ)∥∥
p,π,T = inf
λ > 0:
∫
T
∣∣∣∣∣f
(
eiθ
)
λ
∣∣∣∣∣
p(θ)
dθ ≤ 1
.
Theorem 6. If p ∈ P satisfy DLγ with γ ≥ 1, f belongs to Hardy space Hp(·)
on D and r ∈ R+, then there exists a constant c > 0 independent of n such that∥∥∥∥∥f(z)−
n∑
k=0
ak(f)zk
∥∥∥∥∥
Hp(·)
≤ cΩr
(
f
(
eiθ
)
,
1
n+ 1
)
p(·)
, n = 0, 1, 2, . . . ,
where ak(f), k = 0, 1, 2, 3, . . . , are the Taylor coefficients of f at the origin.
2. Some auxiliary results. We begin with the following lemma.
Lemma A [20]. For r ∈ R+ we suppose that
(i) a1 + a2 + . . .+ an + . . . ,
(ii) a1 + 2ra2 + . . .+ nran + . . .
be two series in a Banach space (B, ‖·‖). Let
R〈r〉n :=
n∑
k=1
(
1−
(
k
n+ 1
)r )
ak
and
R〈r〉∗n :=
n∑
k=1
(
1−
(
k
n+ 1
)r )
krak
for n = 1, 2, . . . . Then ∥∥∥R〈r〉∗n
∥∥∥ ≤ c, n = 1, 2, . . . ,
for some c > 0 if and only if there exists a R ∈ B such that∥∥∥R〈r〉n −R∥∥∥ ≤ C
nr
,
where c and C are constants depending only on one another.
Lemma B [38]. If p ∈ P satisfy DLγ with γ ≥ 1 and f ∈ L
p(·)
2π then there are
constants c, C > 0 such that ∥∥f̃∥∥
p,π
≤ c ‖f‖p,π (5)
and ∥∥Sn(·, f)
∥∥
p,π
≤ C ‖f‖p,π (6)
hold for n = 1, 2, . . . .
Remark 2. Under the conditions of Lemma B
(i) It can be easily seen from (5) and (6) that there exists constant c > 0 such that∥∥f − Sn(·, f)
∥∥
p,π
≤ cEn(f)p(·) � En
(
f̃
)
p(·).
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 1
TRIGONOMETRIC APPROXIMATION IN Lp(x) 9
(ii) From generalized Hölder inequality [24] (Theorem 2.1) we have
L
p(·)
2π ⊂ L1.
For a given f ∈ L1 let
f(x) v
a0
2
+
∞∑
k=1
(ak cos kx+ bk sin kx) =
∞∑
k=−∞
cke
ikx (7)
and
f̃(x) v
∞∑
k=1
(ak sin kx− bk cos kx)
be the Fourier and the conjugate Fourier series of f, respectively. Putting Ak(x) :=
:= cke
ikx in (7) we define
Sn(f) := Sn(x, f) :=
n∑
k=0
(Ak(x) +A−k(x)) =
=
a0
2
+
n∑
k=1
(ak cos kx+ bk sin kx), n = 0, 1, 2, . . . ,
R〈α〉n (f, x) :=
n∑
k=0
(
1−
(
k
n+ 1
)α)
(Ak(x) +A−k(x))
and
Θ〈r〉m :=
1
1−
(
m+ 1
2m+ 1
)rR〈r〉2m −
1(
2m+ 1
m+ 1
)r
− 1
R〈r〉m , for m = 1, 2, 3, . . . . (8)
Under the conditions of Lemma B using (6) and Abel’s transformation we get∥∥R〈α〉n (f, x)
∥∥
p,π
≤ c ‖f‖p,π , n = 1, 2, 3, . . . , x ∈ T , f ∈ Lp(·)2π , (9)
and therefore from (8) and (9)∥∥Θ〈r〉m (f, x)
∥∥
p,π
≤ c ‖f‖p,π , m = 1, 2, 3, . . . , x ∈ T , f ∈ Lp(·)2π .
From the property [25] ((16))
Θ〈r〉m (f)(x) =
=
1∑2m
k=m+1
[
(k + 1)r − kr
] 2m∑
k=m+1
[(k + 1)r − kr]Sk(x, f), x ∈ T , f ∈ L1,
it is known [25] ((18)) that
Θ〈r〉m (Tm) = Tm (10)
for Tm ∈ Tm, m = 1, 2, 3, . . . .
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 1
10 R. AKGÜN
Lemma 1. Let Tn ∈ Tn, p ∈ P satisfy DLγ with γ ≥ 1 and r ∈ R+. Then there
exists a constant c > 0 independent of n such that∥∥T (r)
n
∥∥
p,π
≤ cnr ‖Tn‖p,π
holds.
Proof. Without loss of generality one can assume that ‖Tn‖p,π = 1. Since
Tn =
n∑
k=0
(Ak(x) +A−k (x))
we get
T̃n
nr
=
n∑
k=1
[
(Ak(x)−A−k(x)) /nr
]
and
T
(r)
n
nr
= ir
n∑
k=1
kr
[
(Ak(x)−A−k(x)) /nr
]
.
In this case we have by (9) and (5) that∥∥∥∥∥R〈r〉n
(
T̃n
nr
)∥∥∥∥∥
p,π
≤ c
nr
∥∥T̃n∥∥p,π ≤ c
nr
‖Tn‖p,π =
c
nr
and hence applying Lemma A (with R = 0) to the series
n∑
k=1
[
(Ak(x)−A−k (x)) /nr
]
+ 0 + 0 + . . .+ 0 + . . . ,
n∑
k=1
kr
[
(Ak(x)−A−k (x)) /nr
]
+ 0 + 0 + . . .+ 0 + . . . ,
we find ∥∥∥∥∥
n∑
k=1
(
1−
(
k
n+ 1
)r )
kr
[
(Ak(x)−A−k(x)) /nr
]∥∥∥∥∥
p,π
≤ c,
namely,∥∥∥∥∥R〈r〉n
(
T
(r)
n
nr
)∥∥∥∥∥
p,π
=
∥∥∥∥∥ir
n∑
k=1
(
1−
(
k
n+ 1
)r )
kr
[
(Ak(x)−A−k(x)) /nr
]∥∥∥∥∥
p,π
=
=
∥∥∥∥∥
n∑
k=1
(
1−
(
k
n+ 1
)r )
kr
[
(Ak(x)−A−k(x)) /nr
]∥∥∥∥∥
p,π
≤ c∗.
Since R〈r〉n (cf) = cR
〈r〉
n (f) for every real c we obtain from (10) and the last inequality
that ∥∥T (r)
n
∥∥
p,π
=
∥∥∥Θ〈r〉n
(
T (r)
n
)∥∥∥
p,π
= nr
∥∥∥∥ 1
nr
Θ〈r〉n
(
T (r)
n
)∥∥∥∥
p,π
=
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 1
TRIGONOMETRIC APPROXIMATION IN Lp(x) 11
= nr
∥∥∥∥∥Θ〈r〉n
(
T
(r)
n
nr
)∥∥∥∥∥
p,π
≤ c∗nr = c∗n
r ‖Tn‖p,π .
General case follows immediately from this.
Lemma 2. If p ∈ P satisfy DLγ with γ ≥ 1, f ∈W 2
p(·) and r = 1, 2, 3, . . . , then
Ωr (f, δ)p(·) ≤ cδ
2Ωr−1 (f ′′, δ)p(·) , δ ≥ 0,
with some constant c > 0.
Proof. Putting
g(x) :=
r∏
i=2
(I − σhi) f(x)
we have
(I − σh1
) g(x) =
r∏
i=1
(I − σhi) f(x)
and
r∏
i=1
(I − σhi) f(x) =
1
h1
h1/2∫
−h1/2
(g(x)− g (x+ t)) dt =
= − 1
2h1
h1/2∫
0
2t∫
0
u/2∫
−u/2
g′′ (x+ s) dsdudt.
Therefore from (1) ∥∥∥∥∥
r∏
i=1
(I − σhi) f (x)
∥∥∥∥∥
p,π
≤
≤ c
2h1
sup
∫
T
∣∣∣∣∣∣∣
h1/2∫
0
2t∫
0
u/2∫
−u/2
g′′ (x+ s) dsdudt
∣∣∣∣∣∣∣ |g0 (x)| dx :
g0 ∈ Lp
′(·)
2π and
∫
T
|g0(x)|p
′(x)
dx ≤ 1
≤
≤ c
2h1
h1/2∫
0
2t∫
0
u
∥∥∥∥∥∥∥
1
u
u/2∫
−u/2
g′′ (x+ s) ds
∥∥∥∥∥∥∥
p,π
dudt ≤
≤ c
2h1
h1/2∫
0
2t∫
0
u ‖g′′‖p,π dudt = ch21 ‖g′′‖p,π .
Since
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 1
12 R. AKGÜN
g′′(x) =
r∏
i=2
(I − σhi) f ′′(x),
we obtain that
Ωr (f, δ)p(·) ≤ sup
0<hi≤δ
i=1,2,...,r
ch21 ‖g′′‖p,π = cδ2 sup
0<hi≤δ
i=2,...,r
∥∥∥∥∥
r∏
i=2
(I − σhi) f ′′(x)
∥∥∥∥∥
p,π
=
= cδ2 sup
0<hj≤δ
j=2,...,r−1
∥∥∥∥∥∥
r−1∏
j=1
(
I − σhj
)
f ′′(x)
∥∥∥∥∥∥
p,π
= cδ2Ωr−1 (f ′′, δ)p(·) .
Lemma 2 is proved.
Corollary 5. If r = 1, 2, 3, . . . , p ∈ P satisfy DLγ with γ ≥ 1 and f ∈ W 2r
p(·),
then
Ωr (f, δ)p(·) ≤ cδ
2r
∥∥∥f (2r)∥∥∥
p,π
, δ ≥ 0,
with some constant c > 0.
Lemma 3. Let α ∈ R+, p ∈ P satisfy DLγ with γ ≥ 1, n = 0, 1, 2, . . . and
Tn ∈ Tn. Then
Ωα
(
Tn,
π
n+ 1
)
p(·)
≤ c
(n+ 1)α
∥∥∥T (α)
n
∥∥∥
p,π
hold where the constant c > 0 dependent only on α and p.
Proof. Firstly we prove that if 0 < α < β, α, β ∈ R+ then
Ωβ (f, ·)p(·) ≤ cΩα (f, ·)p(·) . (11)
It is easily seen that if α ≤ β, α, β ∈ Z+, then
Ωβ (f, ·)p(·) ≤ c (α, β, p) Ωα (f, ·)p(·) . (12)
Now, we assume that 0 < α < β < 1. In this case putting Φ(x) := σαhf(x) we have
σβ−αh Φ(x) =
∞∑
j=0
(−1)
j
(
β − α
j
)
1
hj
h/2∫
−h/2
. . .
h/2∫
−h/2
Φ (x+ u1 + . . . uj) du1 . . . duj =
=
∞∑
j=0
(−1)
j
(
β − α
j
)
1
hj
h/2∫
−h/2
. . .
h/2∫
−h/2
∞∑
k=0
(−1)
k
(
α
k
)
1
hk
h/2∫
−h/2
. . .
. . .
h/2∫
−h/2
f (x+ u1 + . . . uj + uj+1 + . . . uj+k) du1 . . . dujduj+1 . . . duj+k
=
=
∞∑
j=0
∞∑
k=0
(−1)
j+k
(
β − α
j
)(
α
k
)
×
×
1
hj+k
h/2∫
−h/2
. . .
h/2∫
−h/2
f(x+ u1 + . . . uj+k)du1 . . . duj+k
=
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 1
TRIGONOMETRIC APPROXIMATION IN Lp(x) 13
=
∞∑
υ=0
(−1)
υ
(
β
υ
)
1
hυ
h/2∫
−h/2
. . .
h/2∫
−h/2
f (x+ u1 + . . . uυ) du1 . . . duυ = σβhf (x) a.e.
Then ∥∥∥σβhf(x)
∥∥∥
p,π
=
∥∥∥σβ−αh Φ(x)
∥∥∥
p,π
≤ c ‖σαhf(x)‖p,π
and
Ωβ (f, ·)p(·) ≤ cΩα (f, ·)p(·) . (13)
We note that if r1, r2 ∈ Z+, α1, β1 ∈ (0, 1) taking α := r1 + α1, β := r2 + β1 for the
remaining cases r1 = r2, α1 < β1 or r1 < r2, α1 = β1 or r1 < r2, α1 < β1 it can
easily be obtained from (12) and (13) that the required inequality (11) holds.
Using (11), Corollary 5 and Lemma 1 we get
Ωα
(
Tn,
π
n+ 1
)
p(·)
≤ cΩ[α]
(
Tn,
π
n+ 1
)
p(·)
≤ c
(
π
n+ 1
)2[α] ∥∥∥T (2[α])
n
∥∥∥
p,π
≤
≤ c
(n+ 1)2[α]
(n+ 1)[α]−(α−[α])
∥∥∥T (α)
n
∥∥∥
p,π
=
c
(n+ 1)α
∥∥∥T (α)
n
∥∥∥
p,π
the required result.
Definition 3. For p ∈ P, f ∈ L
p(·)
2π , δ > 0 and r = 1, 2, 3, . . . the Peetre K-
functional is defined as
K
(
δ, f ;L
p(·)
2π ,W
r
p(·)
)
:= inf
g∈W r
p(·)
{
‖f − g‖p,π + δ
∥∥∥g(r)∥∥∥
p,π
}
. (14)
Theorem 7. If p ∈ P satisfy DLγ with γ ≥ 1 and f ∈ L
p(·)
2π , then the K-
functional K
(
δ2r, f ;L
p(·)
2π ,W
2r
p(·)
)
in (14) and the modulus Ωr (f, δ)p(·) , r = 1, 2, 3, . . .
are equivalent.
Proof. If h ∈W 2r
p(·), then we have by Corollary 5 and (14) that
Ωr (f, δ)p(·) ≤ c ‖f − h‖p,π + cδ2r
∥∥∥h(2r)∥∥∥
p,π
≤ cK
(
δ2r, f ;L
p(·)
2π ,W
2r
p(·)
)
.
We estimate the reverse of the last inequality. The operator Lδ defined by
(Lδf) (x) := 3δ−3
δ/2∫
0
2t∫
0
u/2∫
−u/2
f (x+ s) ds du dt, x ∈ T ,
is bounded in Lp(·)2π because
‖Lδf‖p,π ≤ 3δ−3
δ/2∫
0
2t∫
0
u ‖σuf‖p,π du dt ≤ c ‖f‖p,π .
We prove
d2
dx2
Lδf =
c
δ2
(I − σδ) f
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 1
14 R. AKGÜN
with a real constant c. Since
(Lδf) (x) = 3δ−3
δ/2∫
0
2t∫
0
u/2∫
−u/2
f (x+ s) ds du dt =
= 3δ−3
δ/2∫
0
2t∫
0
x+u/2∫
0
f (s) ds−
x−u/2∫
0
f (s) ds
du dt
using Lebesgue Differentiation Theorem
d
dx
(Lδf) (x) = 3δ−3
δ/2∫
0
2t∫
0
d
dx
x+u/2∫
0
f (s) ds− d
dx
x−u/2∫
0
f (s) ds
du dt =
= 3δ−3
δ/2∫
0
2t∫
0
[
f (x+ u/2)− f (x− u/2)
]
du dt =
= 6δ−3
δ/2∫
0
x+t∫
x
f(u)du+
x−t∫
x
f(u)du
dt a.e.
Using Lebesgue Differentiation Theorem once more
d2
dx2
(Lδf) (x) = 6δ−3
δ/2∫
0
d
dx
x+t∫
x
f(u)du+
d
dx
x−t∫
0
f(u)du
dt =
= 6δ−3
δ/2∫
0
[f (x+ t)− f (x) + f (x− t)− f(x)] dt =
=
6
δ3
δ/2∫
0
f (x+ t) dt+
δ/2∫
0
f (x− t) dt− δf(x)
=
=
6
δ2
1
δ
δ/2∫
0
f (x+ t) dt+
1
δ
0∫
−δ/2
f (x+ t) dt− f(x)
=
=
6
δ2
1
δ
δ/2∫
−δ/2
f (x+ t) dt− f(x)
=
=
−6
δ2
f(x)− 1
δ
δ/2∫
−δ/2
f (x+ t) dt
=
−6
δ2
(I − σδ) f(x) a.e.
The last equality implies by induction on r that
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 1
TRIGONOMETRIC APPROXIMATION IN Lp(x) 15
d2r
dx2r
Lrδf =
c
δ2r
(I − σδ)r f, r = 1, 2, 3, . . . a.e.
Indeed, for r = 2
d4
dx4
L2
δf =
d2
dx2
(
d2
dx2
L2
δf
)
=
d2
dx2
(
d2
dx2
Lδ (Lδf =: u)
)
=
=
d2
dx2
(
d2
dx2
Lδu
)
=
d2
dx2
(
−6
δ2
(I − σδ)u
)
=
=
−6
δ2
(
d2
dx2
(I − σδ)u
)
=
−6
δ2
(
d2
dx2
(I − σδ)Lδf
)
a.e.
Since
d2
dx2
σδ (Lδf) = σδ
(
d2
dx2
Lδf
)
we get
d2
dx2
(I − σδ)Lδf =
d2
dx2
Lδf −
d2
dx2
σδ (Lδf) =
=
d2
dx2
Lδf − σδ
(
d2
dx2
Lδf
)
= (I − σδ)
[
d2
dx2
Lδf
]
a.e.
and therefore
d4
dx4
L2
δf =
−6
δ2
(
d2
dx2
(I − σδ)Lδf
)
=
−6
δ2
(I − σδ)
[
d2
dx2
Lδf
]
=
=
−6
δ2
(I − σδ)
[
−6
δ2
(I − σδ) f
]
=
c
δ4
(I − σδ)2 f a.e.
Now let be
d2(r−1)
dx2(r−1)
L
(r−1)
δ f =
c
δ2(r−1)
(I − σδ)(r−1) f a.e. Then
d2r
dx2r
Lrδf =
d2
dx2
[
d2(r−1)
dx2(r−1)
L
(r−1)
δ (Lδf := u)
]
=
d2
dx2
[
d2(r−1)
dx2(r−1)
L
(r−1)
δ u
]
=
=
d2
dx2
[ c
δ2(r−1)
(I − σδ)(r−1) u
]
=
d2
dx2
[ c
δ2(r−1)
(I − σδ)(r−1) Lδf
]
=
=
c
δ2(r−1)
(I − σδ)(r−1)
[
d2
dx2
Lδf
]
=
c
δ2r
(I − σδ)r f a.e.
Letting Arδ := I − (I − Lrδ)
r we prove that
∥∥∥∥ d2rdx2r
Arδf
∥∥∥∥
p,π
≤ c
∥∥∥∥ d2rdx2r
Lrδf
∥∥∥∥
p,π
and
Arδf ∈W 2r
p(·). For r = 1 we have A1
δf := I −
(
I − L1
δf
)1
= L1
δf and
∥∥∥∥ d2dx2A1
δf
∥∥∥∥
p,π
=
=
∥∥∥∥ d2dx2L1
δf
∥∥∥∥
p,π
. Since
d2
dx2
Lδf =
c
δ2
(I − σδ) f we get A1
δf ∈ W 2
p(·). For r =
= 2, 3, . . . using
Arδ := I − (I − Lrδ)
r
=
r−1∑
j=0
(−1)
r−j+1
(
r
j
)
L
r(r−j)
δ
we obtain
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 1
16 R. AKGÜN
∥∥∥∥ d2rdx2r
Arδf
∥∥∥∥
p,π
≤
r−1∑
j=0
(
r
j
)∥∥∥∥ d2rdx2r
L
r(r−j)
δ f
∥∥∥∥
p,π
.
We estimate
∥∥∥∥ d2rdx2r
L
r(r−j)
δ f
∥∥∥∥
p,π
as the following
∥∥∥∥ d2rdx2r
L
r(r−j)
δ f
∥∥∥∥
p,π
=
∥∥∥∥ d2rdx2r
Lrδ
(
L
(r−j)
δ f =: u
)∥∥∥∥
p,π
=
=
∥∥∥∥ d2rdx2r
Lrδu
∥∥∥∥
p,π
=
∥∥∥ c
δ2r
(I − σδ)r u
∥∥∥
p,π
=
=
∥∥∥ c
δ2r
(I − σδ)r
[
L
(r−j)
δ f
]∥∥∥
p,π
=
c
δ2r
∥∥∥(I − σδ)r
[
L
(r−j)
δ f
]∥∥∥
p,π
≤
≤ c
δ2r
∥∥∥∥∥
r∑
i=0
(−1)
i
(
r
i
)
σiδ
[
L
(r−j)
δ f
]∥∥∥∥∥
p,π
.
Since σδ (Lδf) = Lδ (σδf) we have σiδ
[
L
(r−j)
δ f
]
= L
(r−j)
δ
(
σiδf
)
and hence
∥∥∥∥ d2rdx2r
L
r(r−j)
δ f
∥∥∥∥
p,π
≤ c
δ2r
∥∥∥∥∥
r∑
i=0
(−1)i
(
r
i
)
σiδ
[
L
(r−j)
δ f
]∥∥∥∥∥
p,π
≤
≤ c
δ2r
∥∥∥∥∥
r∑
i=0
(−1)
i
(
r
i
)
L
(r−j)
δ
(
σiδf
)∥∥∥∥∥
p,π
=
=
c
δ2r
∥∥∥∥∥L(r−j)
δ
[
r∑
i=0
(−1)
i
(
r
i
)
σiδf
]∥∥∥∥∥
p,π
≤ C
δ2r
∥∥∥∥∥
r∑
i=0
(−1)
i
(
r
i
)
σiδf
∥∥∥∥∥
p,π
=
=
C
δ2r
‖(I − σδ)r f‖p,π =
∥∥∥∥ Cδ2r (I − σδ)r f
∥∥∥∥
p,π
= c1
∥∥∥∥ d2rdx2r
Lrδf
∥∥∥∥
p,π
.
From the last inequality∥∥∥∥ d2rdx2r
Arδf
∥∥∥∥
p,π
≤ c
∥∥∥∥ d2rdx2r
Lrδf
∥∥∥∥
p,π
and Arδf ∈W 2r
p(·).
Therefore we find∥∥∥∥ d2rdx2r
Arδf
∥∥∥∥
p,π
≤ c
∥∥∥∥ d2rdx2r
Lrδf
∥∥∥∥
p,π
=
c
δ2r
‖(I − σδ)r‖p,π ≤
c
δ2r
Ωr(f, δ)p(·).
Since
I − Lrδ = (I − Lδ)
r−1∑
j=0
Ljδ
we get
‖(I − Lrδ) g‖p,π ≤ c ‖(I − Lδ) g‖p,π ≤
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 1
TRIGONOMETRIC APPROXIMATION IN Lp(x) 17
≤ 3cδ−3
δ/2∫
0
2t∫
0
u ‖(I − σu) g‖p,π dudt ≤ c sup
0<u≤δ
‖(I − σu) g‖p,π .
Taking into account
‖f −Arδf‖p,π = ‖(I − Lrδ)
r
f‖p,π
by a recursive procedure we obtain
‖f −Arδf‖p,π ≤ c sup
0<t1≤δ
∥∥∥(I − σt1) (I − Lrδ)
r−1
f
∥∥∥
p,π
≤
≤ c sup
0<t1≤δ
sup
0<t2≤δ
∥∥∥(I − σt1) (I − σt2) (I − Lrδ)
r−2
f
∥∥∥
p,π
≤ . . .
. . . ≤ c sup
0<ti≤δ
i=1,2,...,r
∥∥∥∥∥
r∏
i=1
(I − σti) f(x)
∥∥∥∥∥
p,π
= cΩr (f, δ)p(·) .
Theorem 7 is proved.
3. Proofs of the main results. Proof of Theorem 1. We set Ak(x, f) := ak cos kx+
+ bk sin kx. Since the set of trigonometric polynomials is dense [22] in Lp(·)2π for given
f ∈ Lp(·)2π we have En(f)p(·) → 0 as n → ∞. From the first inequality in Remark 2,
we have f(x) =
∑∞
k=0
Ak(x, f) in ‖·‖p,π norm. For k = 1, 2, 3, . . . we can find
Ak(x, f) = ak cos k
(
x+
απ
2k
− απ
2k
)
+ bk sin k
(
x+
απ
2k
− απ
2k
)
=
= Ak
(
x+
απ
2k
, f
)
cos
απ
2
+Ak
(
x+
απ
2k
, f̃
)
sin
απ
2
and
Ak
(
x, f (α)
)
= kαAk
(
x+
απ
2k
, f
)
.
Therefore
∞∑
k=0
Ak(x, f) =
= A0(x, f) + cos
απ
2
∞∑
k=1
Ak
(
x+
απ
2k
, f
)
+ sin
απ
2
∞∑
k=1
Ak
(
x+
απ
2k
, f̃
)
=
= A0(x, f) + cos
απ
2
∞∑
k=1
k−αAk
(
x, f (α)
)
+ sin
απ
2
∞∑
k=1
k−αAk
(
x, f̃ (α)
)
and hence
f(x)− Sn(x, f) = cos
απ
2
∞∑
k=n+1
1
kα
Ak
(
x, f (α)
)
+ sin
απ
2
∞∑
k=n+1
1
kα
Ak
(
x, f̃ (α)
)
.
Since
∞∑
k=n+1
k−αAk
(
x, f (α)
)
=
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 1
18 R. AKGÜN
=
∞∑
k=n+1
k−α
[(
Sk
(
·, f (α)
)
− f (α)(·)
)
−
(
Sk−1
(
·, f (α)
)
− f (α)(·)
)]
=
=
∞∑
k=n+1
(
k−α − (k + 1)−α
)(
Sk
(
·, f (α)
)
− f (α)(·)
)
−
−(n+ 1)−α
(
Sn
(
·, f (α)
)
− f (α)(·)
)
and
∞∑
k=n+1
k−αAk
(
x, f̃ (α)
)
=
∞∑
k=n+1
(
k−α − (k + 1)−α
) (
Sk
(
·, f̃ (α)
)
− f̃ (α)(·)
)
−
−(n+ 1)−α
(
Sn
(
·, f̃ (α)
)
− f̃ (α)(·)
)
we obtain
‖f(·)− Sn (·, f)‖p,π ≤
∞∑
k=n+1
(
k−α − (k + 1)
−α
)∥∥∥Sk (·, f (α))− f (α)(·)∥∥∥
p,π
+
+(n+ 1)−α
∥∥∥Sn (·, f (α))− f (α)(·)∥∥∥
p,π
+
+
∞∑
k=n+1
(
k−α − (k + 1)−α
) ∥∥∥Sk (·, f̃ (α))− f̃ (α)(·)∥∥∥
p,π
+
+(n+ 1)−α
∥∥∥Sn (·, f̃ (α))− f̃ (α)(·)∥∥∥
p,π
≤
≤ c
[ ∞∑
k=n+1
(
k−α − (k + 1)
−α
)
Ek
(
f (α)
)
p(·)
+ (n+ 1)−αEn
(
f (α)
)
p(·)
]
+
+c
[ ∞∑
k=n+1
(
k−α − (k + 1)
−α
)
Ek
(
f̃ (α)
)
p(·)
+ (n+ 1)−αEn
(
f̃ (α)
)
p(·)
]
.
Consequently from equivalence in Remark 2 (i) we have
‖f(x)− Sn(x, f)‖p,π ≤
≤ c
[ ∞∑
k=n+1
(
k−α − (k + 1)
−α
)
+ (n+ 1)−α
]{
Ek
(
f (α)
)
p(·)
+ En
(
f̃ (α)
)
p(·)
}
≤
≤ cEn
(
f (α)
)
p(·)
[ ∞∑
k=n+1
(
k−α − (k + 1)−α
)
+ (n+ 1)−α
]
≤ c
(n+ 1)α
En
(
f (α)
)
p(·)
.
Theorem 1 is proved.
Proof of Theorem 2. We put r − 1 < α < r, r ∈ Z+. For g ∈ W 2r
p(·) we have by
Corollary 1, (14) and Theorem 7 that
En(f)p(·) ≤ En (f − g)p(·) + En (g)p(·) ≤ c
[
‖f − g‖p,π + (n+ 1)−2r
∥∥∥g(2r)∥∥∥
p,π
]
≤
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 1
TRIGONOMETRIC APPROXIMATION IN Lp(x) 19
≤ cK
(
(n+ 1)−2r, f ;L
p(·)
2π ,W
2r
p(·)
)
≤ cΩr
(
f,
1
n+ 1
)
p(·)
as required for r ∈ Z+. Therefore by the last inequality
En(f)p(·) ≤ cΩr (f, 1/(n+ 1))p(·) ≤ cΩr (f, 2π/(n+ 1))p(·) , n = 0, 1, 2, 3, . . . ,
and by (11) we get
En(f)p(·) ≤ cΩr (f, 2π/(n+ 1))p(·) ≤ cΩα (f, 2π/(n+ 1))p(·)
and the assertion follows.
Proof of Theorem 3. Let Tn ∈ Tn be the best approximating polynomial of f ∈
∈ Lp(·)2π and let m ∈ Z+. Then by Remark 1 (ii)
Ωα (f, π/n+ 1)p(·) ≤ Ωα (f − T2m , π/(n+ 1))p(·) + Ωα (T2m , π/(n+ 1))p(·) ≤
≤ cE2m(f)p(·) + Ωα (T2m , π/(n+ 1))p(·) .
Since
T
(α)
2m (x) = T
(α)
1 (x) +
m−1∑
ν=0
{
T
(α)
2ν+1(x)− T (α)
2ν (x)
}
we get by Lemma 3 that
Ωα (T2m , π/(n+ 1))p(·) ≤
c
(n+ 1)α
{∥∥∥T (α)
1
∥∥∥
p,π
+
m−1∑
ν=0
∥∥∥T (α)
2ν+1 − T (α)
2ν
∥∥∥
p,π
}
.
Lemma 1 gives∥∥∥T (α)
2ν+1 − T (α)
2ν
∥∥∥
p,π
≤ c2να ‖T2ν+1 − T2ν‖p,π ≤ c2
να+1E2ν (f)p(·)
and ∥∥∥T (α)
1
∥∥∥
p,π
=
∥∥∥T (α)
1 − T (α)
0
∥∥∥
p,π
≤ cE0(f)p(·).
Hence
Ωα (T2m , π/(n+ 1))p(·) ≤
c
(n+ 1)α
{
E0(f)p(·) +
m−1∑
ν=0
2(ν+1)αE2ν (f)p(·)
}
.
Using
2(ν+1)αE2ν (f)p(·) ≤ c∗
2ν∑
µ=2ν−1+1
µα−1Eµ(f)p(·), ν = 1, 2, 3, . . . ,
we obtain
Ωα (T2m , π/(n+ 1))p(·) ≤
≤ c
(n+ 1)α
E0(f)p(·) + 2αE1(f)p(·) + c
m∑
ν=1
2ν∑
µ=2ν−1+1
µα−1Eµ(f)p(·)
≤
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 1
20 R. AKGÜN
≤ c
(n+ 1)α
{
E0(f)p(·) +
2m∑
µ=1
µα−1Eµ(f)p(·)
}
≤ c
(n+ 1)
α
2m−1∑
ν=0
(ν + 1)
α−1
Eν(f)p(·).
If we choose 2m ≤ n+ 1 ≤ 2m+1, then
Ωα (T2m , π/(n+ 1))p(·) ≤
c
(n+ 1)α
n∑
ν=0
(ν + 1)
α−1
Eν(f)p(·),
E2m(f)p(·) ≤ E2m−1 (f)p(·) ≤
c
(n+ 1)α
n∑
ν=0
(ν + 1)
α−1
Eν (f)p(·) .
Last two inequalities complete the proof.
Proof of Theorem 4. For the polynomial Tn of the best approximation to f we have
by Lemma 1 that∥∥∥T (β)
2i+1 − T (β)
2i
∥∥∥
p,π
≤ C(β)2(i+1)β ‖T2i+1 − T2i‖p,π ≤ 2C(β)2(i+1)βE2i(f)p(·).
Hence
∞∑
i=1
‖T2i+1 − T2i‖Wβ
p(·)
=
∞∑
i=1
∥∥∥T (β)
2i+1 − T (β)
2i
∥∥∥
p,π
+
∞∑
i=1
‖T2i+1 − T2i‖p,π ≤
≤ c
∞∑
m=2
mβ−1Em(f)p(·) <∞.
Therefore
‖T2i+1 − T2i‖Wβ
p(·)
→ 0 as i→∞.
This means that {T2i} is a Cauchy sequence in Lp(·)2π . Since T2i → f in Lp(·)2π and W β
p(·)
is a Banach space we obtain f ∈W β
p(·).
On the other hand since ∥∥∥f (β) − Sn(f (β))
∥∥∥
p,π
≤
≤
∥∥∥S2m+2(f (β))− Sn(f (β))
∥∥∥
p,π
+
∞∑
k=m+2
∥∥∥S2k+1(f (β))− S2k(f (β))
∥∥∥
p,π
we have for 2m < n < 2m+1∥∥∥S2m+2(f (β))− Sn(f (β))
∥∥∥
p,π
≤ c2(m+2)βEn(f)p(·) ≤ c (n+ 1)
β
En(f)p(·).
On the other hand we find
∞∑
k=m+2
∥∥∥S2k+1(f (β))− S2k(f (β))
∥∥∥
p,π
≤ c
∞∑
k=m+2
2(k+1)βE2k(f)p(·) ≤
≤ c
∞∑
k=m+2
2k∑
µ=2k−1+1
µβ−1Eµ(f)p(·) =
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 1
TRIGONOMETRIC APPROXIMATION IN Lp(x) 21
= c
∞∑
ν=2m+1+1
νβ−1Eν (f)p(·) ≤ c
∞∑
ν=n+1
νβ−1Eν(f)p(·).
Theorem 4 is proved.
Proof of Theorem 5. We set Wn(f) := Wn(x, f) :=
1
n+ 1
∑2n
ν=n
Sν(x, f), n =
= 0, 1, 2, . . . . Since
Wn(·, f (α)) = W (α)
n (·, f)
we have ∥∥∥f (α)(·)− T (α)
n (·, f)
∥∥∥
p,π
≤
∥∥∥f (α)(·)−Wn(·, f (α))
∥∥∥
p,π
+
+
∥∥∥T (α)
n (·,Wn(f))− T (α)
n (·, f)
∥∥∥
p,π
+
∥∥∥W (α)
n (·, f)− T (α)
n (·,Wn(f))
∥∥∥
p,π
:=
:= I1 + I2 + I3.
We denote by T ∗n(x, f) the best approximating polynomial of degree at most n to f
in Lp(·)2π . In this case, from the boundedness of the operator Sn in Lp(·)2π we obtain the
boundedness of operator Wn in Lp(·)2π and there holds
I1 ≤
∥∥∥f (α)(·)− T ∗n(·, f (α))
∥∥∥
p,π
+
∥∥∥T ∗n(·, f (α))−Wn(·, f (α))
∥∥∥
p,π
≤
≤ cEn(f (α))p(·) +
∥∥∥Wn(·, T ∗n(f (α))− f (α))
∥∥∥
p,π
≤ cEn(f (α))p(·).
From Lemma 1 we get
I2 ≤ cnα ‖Tn(·,Wn(f))− Tn(·, f)‖p,π
and
I3 ≤ c (2n)
α ‖Wn(·, f)− Tn(·,Wn(f))‖p,π ≤ c (2n)
α
En (Wn(f))p(·) .
Now we have
‖Tn(·,Wn(f))− Tn(·, f)‖p,π ≤
≤ ‖Tn(·,Wn(f))−Wn(·, f)‖p,π + ‖Wn(·, f)− f(·)‖p,π + ‖f(·)− Tn(·, f)‖p,π ≤
≤ cEn (Wn(f))p(·) + cEn(f)p(·) + cEn(f)p(·).
Since
En (Wn(f))p(·) ≤ cEn(f)p(·)
we get ∥∥∥f (α)(·)− T (α)
n (·, f)
∥∥∥
p,π
≤ cEn(f (α))p(·) + cnαEn (Wn(f))p(·) +
+cnαEn(f)p(·) + c (2n)
α
En (Wn(f))p(·) ≤ cEn(f (α))p(·) + cnαEn(f)p(·).
Since by Theorem 1
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 1
22 R. AKGÜN
En(f)p(·) ≤
c
(n+ 1)α
En(f (α))p(·)
we obtain ∥∥∥f (α)(·)− T (α)
n (·, f)
∥∥∥
p,π
≤ cEn(f (α))p(·).
Theorem 5 is proved.
Proof of Theorem 6. Let f ∈ Hp(·)(D). First of all if p(x), defined on T , satisfy
Dini – Lipschitz property DLγ for γ ≥ 1 on T , then p
(
eix
)
, x ∈ T , defined on T,
satisfy Dini – Lipschitz property DLγ for γ ≥ 1 on T. Since Hp(·) ⊂ H1 (D) for 1 < p,
let
∑∞
k=−∞
βke
ikθ be the Fourier series of the function f
(
eiθ
)
, and Sn(f, θ) :=
:=
∑n
k=−n
βke
ikθ be its nth partial sum. From f
(
eiθ
)
∈ H1 (D) , we have [11, p. 38]
βk =
0, for k < 0;
ak(f), for k ≥ 0.
Therefore ∥∥∥∥∥f(z)−
n∑
k=0
ak(f)zk
∥∥∥∥∥
Hp(·)
= ‖f − Sn (f, ·)‖p,π . (15)
If t∗n is the best approximating trigonometric polynomial for f(eiθ) in Lp(·)2π , then from
(6), (15) and Theorem 2 we get∥∥∥∥∥f(z)−
n∑
k=0
ak(f)zk
∥∥∥∥∥
Hp(·)
≤
∥∥f (eiθ)− t∗n(θ)
∥∥
p,π
+ ‖Sn (f − t∗n, θ)‖p,π ≤
≤ cEn
(
f
(
eiθ
))
p(·) ≤ cΩr
(
f
(
eiθ
)
,
1
n+ 1
)
p(·)
.
Theorem 6 is proved.
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after revision — 22.10.10
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 1
|
| id | umjimathkievua-article-2695 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:28:29Z |
| publishDate | 2011 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/28/492b1904c94f245a2959c00efcc07f28.pdf |
| spelling | umjimathkievua-article-26952020-03-18T19:34:07Z Trigonometric approximation of functions in generalized Lebesgue spaces with variable exponent Тригонометричне наближення функцiй в узагальнених просторах Лебега зi змiнною експонентою Akgün, R. Акгюн, Р. We investigate the approximation properties of the trigonometric system in $L_{2\pi}^{p(\cdot)}$. We consider the fractional order moduli of smoothness and obtain direct, converse approximation theorems together with a constructive characterization of a Lipschitz-type class. Дослiджено властивостi наближення тригонометричної системи в $L_{2\pi}^{p(\cdot)}$. Розглянуто модулi гладкостi дробового порядку та отримано пряму i обернену теореми наближення разом iз конструктивною характеризацiєю класу типу Лiпшиця. Institute of Mathematics, NAS of Ukraine 2011-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2695 Ukrains’kyi Matematychnyi Zhurnal; Vol. 63 No. 1 (2011); 3-23 Український математичний журнал; Том 63 № 1 (2011); 3-23 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2695/2146 https://umj.imath.kiev.ua/index.php/umj/article/view/2695/2147 Copyright (c) 2011 Akgün R. |
| spellingShingle | Akgün, R. Акгюн, Р. Trigonometric approximation of functions in generalized Lebesgue spaces with variable exponent |
| title | Trigonometric approximation of functions in generalized Lebesgue spaces with variable exponent |
| title_alt | Тригонометричне наближення функцiй в узагальнених просторах Лебега зi змiнною експонентою |
| title_full | Trigonometric approximation of functions in generalized Lebesgue spaces with variable exponent |
| title_fullStr | Trigonometric approximation of functions in generalized Lebesgue spaces with variable exponent |
| title_full_unstemmed | Trigonometric approximation of functions in generalized Lebesgue spaces with variable exponent |
| title_short | Trigonometric approximation of functions in generalized Lebesgue spaces with variable exponent |
| title_sort | trigonometric approximation of functions in generalized lebesgue spaces with variable exponent |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2695 |
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