ETNA
Electronic Transactions on Numerical Analysis.
Volume 19, pp. 94-104, 2005.
Copyright  2005, Kent State University.
ISSN 1068-9613.
Kent State University
etna@mcs.kent.edu
CHARACTERISTICS OF BESOV-NIKOL’SKIı̆ CLASS OF FUNCTIONS ∗
SERGEY TIKHONOV†
Abstract. In this paper we consider functions from a Besov-Nikol’skiı̆ class. We give constructive characteristics of this class. We establish criteria for a function to be in this Besov-Nikol’skiı̆ class by means of conditions on
its Fourier coefficients. We also discuss embedding theorems between some classes of functions.
Key words. Moduli of smoothness, Besov-Nikol’skiı̆ class, Best approximation, Fourier coefficients, Embedding theorems.
AMS subject classifications. 42A10, 42A16, 41A50, 42A32, 42A50.
1. Introduction. If 1 ≤ p < ∞, let Lp be the space of 2π-periodic, measurable func 2π
p1
R
p
tions f (x) such that kf kp =
|f (x)| dx
< ∞. Similarly, let L∞ be the space of 2π0
periodic, continuous functions f (x) with kf k∞ = max |f (x)|. The modulus of smoothx∈[0,2π]
ness of order β (β > 0) of a function f ∈ Lp , 1 ≤ p ≤ ∞, is given by
∞
X
β(β
−
1)
·
·
·
(β
−
ν
+
1)
ωβ (f, t)p = sup (−1)ν
f (x + (β − ν)h) .
ν!
|h|≤t ν=0
p
If F (η) ≥ 0 and G(η) ≥ 0 for all η, then the notation F (η) G(η) will mean that there
exists a positive constant C not depending on η and such that F (η) ≤ C G(η) for all η. If
F (η) G(η) and G(η) F (η) hold simultaneously, then we shall write F (η) G(η).
Let α(t) be a function measurable on the interval [0, 1] and nonnegative. If there exists
R1
a real number σ such that α(t)tσ dt < ∞, then we shall say that α(t) satisfies the σ0
condition.
Denote by BH(α, β, ψ, p, θ, k) the Besov-Nikol’skiı̆ class (see, e.g., [9],[10]), that is,
the class of functions f (x) ∈ Lp , 1 ≤ p ≤ ∞, such that
 δ
 θ1
Z
Z1
θ
θ
 α(t) ωk+β
(f, t)p dt + δ βθ α(t) t−βθ ωk+β
(f, t)p dt ψ(δ),
0
δ
where β, k, θ ∈ (0, ∞), and the function α satisfies the σ-condition with σ = kθ and 0 ≤
δ ≤ 1. Here and in the sequel, we shall assume that a function ψ is a majorant, i.e., ψ is
a function, continuous and nonnegative on [0, 1], which possesses the following properties:
ψ(δ1 ) ≤ C1 ψ(δ2 ), 0 ≤ δ1 ≤ δ2 ≤ 1 ; ψ(2δ) ≤ C2 ψ(δ), 0 ≤ δ ≤ 12 , where the constants
C1 and æ2 do not depend on δ1 , δ2 and δ.
A sequence γ := {γn } of positive terms will be called quasi β-power-monotone increasing (decreasing) if there exist a natural number N := N (β, γ) and a constant K :=
K(β, γ) ≥ 1 such that
(1.1)
Knβ γn ≥ mβ γm
(nβ γn ≤ Kmβ γm )
∗ Received November 10, 2002. Accepted for publication May 10, 2003. Communicated by J. Arvesú. This work
was supported by the Russian Foundation for Fundamental Research (grant no. 03-01-00080).
† Chair of Theory of Functions and Functional Analysis, Department of Mechanics and Mathematics, Moscow
State University, Vorob’yovy gory, Moscow 119899 RUSSIA. PRESENT ADDRESS: Centre de Recerca
Matemàtica, E-08193 Barcelona, SPAIN. E-mail: tikhonov@mccme.ru.
94
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Characteristics of Besov-Nikol’skiı̆ class of functions
95
holds for any n ≥ m ≥ N .
If (1.1) holds with β = 0, then we omit the attribute ”β-power” in the inequality.
Furthermore, a sequence γ := {γn } of positive terms will be called quasi-geometrically
increasing (decreasing) if there exist natural numbers µ := µ(γ), N := N (γ) and a constant
K := K(γ) ≥ 1 such that
γn+µ ≥ 2γn and γn ≤ Kγn+1
(γn+µ ≤
1
γn and γn+1 ≤ Kγn )
2
hold for all n ≥ N .
Let
∞
(1.2)
a0 X
+
(an cos nx + bn sin nx)
2
n=1
be the Fourier series of a function f (x), and let Sn (f ) be the n-th partial sum of (1.2).
Then f (x) has lacunary Fourier coefficients if an , bn = 0, n 6= 2k , and a2k = ck , b2k =
dk ; ck , dk ≥ 0 (k ∈ N ∪ {0} ). If there exists γ ≥ 0 such that nanγ ↓ 0, nbnγ ↓ 0, then f (x)
has quasimonotone Fourier coefficients.
The best trigonometrical approximation to f is given by En (f )p
=
inf kf (x) − T kp , where Tn−1 is the collection of trigonometrical polynomials of
T ∈Tn−1
total degree smaller than n.
An outline of this paper is as follows. In section 2, we consider some useful lemmas. In section 3, we give a constructive characteristic of the Besov-Nikol’skiı̆ class. In
section 4, we discuss criteria for a function to belong to a Besov-Nikol’skiı̆ class through
some conditions on its Fourier coefficients. In section 5, we consider Nikol’skiı̆ and Besov
classes and discuss embedding theorems between these classes and the Besov-Nikol’skiı̆
e
g
class. We also obtain some results concerning H(k,
p, ψ), BH(α,
β, ψ, p, θ, k), H(k, p, ψ)
and BH(α, β, ψ, p, θ, k) classes and their interrelations.
2. Auxiliary results. L EMMA 2.1 ([12]). If f (x) ∈ Lp , 1 ≤ p ≤ ∞, and if n ∈ N,
then
n+1
En (f )p ωβ (f,
X
1
)p n−β
ξ β−1 Eξ (f )p .
n
ξ=1
L EMMA 2.2 ([7]). For any positive sequence γ := {γn } the inequalities
∞
X
γk ≤ Cγn
(n = 1, 2, · · · ; C ≥ 1),
n
X
γk ≤ Cγn
(n = 1, 2, · · · ; C ≥ 1),
k=n
or
k=1
hold if and only if the sequence γ is quasi-geometrically decreasing (increasing), respectively.
L EMMA 2.3 ([8]). Let a positive sequence γ := {γn } be quasi-geometrically decreasing
(increasing). Then there exists a positive (negative) ε such that the sequence {γ n 2nε } is
quasi-geometrically decreasing (increasing).
L EMMA 2.4 ([7]). If an ≥ 0, λn > 0, and if p ≥ 1, then
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96
Sergey Tikhonov
∞
P
(a)
λn
n=1
∞
P
(b)
λn
n
P
aν
ν=1
∞
P
p
aν
ν=n
n=1
≤ pp
p
≤ pp
∞
P
n=1
∞
P
n=1
λn1−p apn
λn1−p apn
∞
P
ν=n
n
P
p
λν
p
λν
ν=1
,
.
L EMMA 2.5 ([4]). If an ≥ 0, and if 0 < p ≤ 1, then
∞
P
an
n=1
p
≤
∞
P
n=1
apn .
3. Constructive characteristics of the Besov-Nikol’skiı̆ class. T HEOREM 3.1. Let
β, k, θ > 0, and α(·) satisfies σ-condition with σ = kθ, and
2
(3.1)
n
1/2
Z
nkθ
kθ
α(t) t dt 0
n
1/2
Z
for n ∈ N ∪ {0}.
α(t)dt
1/2n+1
Suppose f (x) ∈ Lp (1 ≤ p ≤ ∞), then f (x) ∈ BH(α, β, ψ, p, θ, k) if and only if for any
n ∈ N ∪ {0}
2
(3.2)
−nβθ
n
X
2
νβθ
Λν E2θν (f )p
+
ν=0
where Λν =
ν
1/2
R
∞
X
Λν E2θν (f )p
ν=n+1
! θ1
ψ
1
2n
,
α(t) dt, ν ∈ N ∪ {0}.
1/2ν+1
Proof. Define µν =
1/ν
R
α(t) dt, ν ∈ N, i.e., µ2ν = Λν =
ν
1/2
R
α(t) dt, ν ∈
1/2ν+1
1/(2ν)
N ∪ {0}.
Using (3.1) and Lemmas
2.2 and 2.3, we get that there exists an ε > 0 such that the
sequence Λν 2−ν(kθ−ε) is quasi-geometrically decreasing. Thus,
Λl 2−l(kθ−ε) ≤ CΛm 2−m(kθ−ε)
(3.3)
holds for any m ≤ l.
1
< δ ≤ 21n . Then using the properties of modulus of
Let n be an integer such that 2n+1
1
1
≤ t ≤ 21n , see [12]), we have
smoothness (ωk+β (f, t)p ωk+β (f, 2n )p for 2n+1
Zδ
θ
α(t) ωk+β
(f, t)p dt
+δ
βθ
−n
2
Z
θ
α(t) t−βθ ωk+β
(f, t)p dt δ
2−(n+1)
2−n
Z
θ
α(t) dt ωk+β
(f,
1
1
θ
)p = Λn ωk+β
(f, n )p
2n
2
2−(n+1)
and hence,
I1 + I2 :=
(3.4)
Zδ
θ
α(t) ωk+β
(f, t)p
dt + δ
βθ
0
∞
X
2−ν
Z
ν=n −(ν+1)
2
θ
α(t) ωk+β
(f, t)p dt + 2−nβθ
Z1
θ
α(t) t−βθ ωk+β
(f, t)p dt δ
n−1
X
−ν
2
Z
ν=0 −(ν+1)
2
θ
α(t) t−βθ ωk+β
(f, t)p dt ETNA
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Characteristics of Besov-Nikol’skiı̆ class of functions
∞
X
θ
Λν ωk+β
(f,
ν=n+1
n
X
1
1
−nβθ
θ
)
+
2
2νβθ Λν ωk+β
(f, ν )p .
p
2ν
2
ν=0
Lemma 2.1 implies
∞
X
I1 + I 2 2−ν(k+β)θ Λν + 2−nβθ
n
X
2−νkθ Λν
ν=0
ν=n+1
!

ν
X
ξ=0
θ
2ξ(k+β) E2ξ (f )p  .
Let θ ≤ 1. By Lemma 2.5 we obtain


∞
n
ν
X
X
X
−ν(k+β)θ
ξ(k+β)θ
θ
ξ(k+β)θ
θ
2
Λν 
I1 + I 2 2
E2ξ (f )p +
2
E2ξ (f )p  +
ν=n+1
ξ=0
ξ=n+1
+2
−nβθ
n
X
2
−νkθ
ν=0
Λν
ν
X
2ξ(k+β)θ E2θξ (f )p .
ξ=0
Using (3.1), we get
∞
X
(3.5)
2
−ν(k+β)θ
Λν ≤ 2
−nβθ
∞
X
2−νkθ Λν 2−n(k+β)θ Λn .
ν=n
ν=n
Thus,
n
X
I1 + I2 2−n(k+β)θ Λn
2ξ(k+β)θ E2θξ (f )p +
ξ=0
+2−nβθ
n
X
2ξ(k+β)θ E2θξ (f )p
ξ=0
∞
X
∞
X
2ξ(k+β)θ E2θξ (f )p
ξ=n
n
X
2−νkθ Λν ν=ξ
∞
X
2−ν(k+β)θ Λν +
ν=ξ
Λξ E2θξ (f )p + 2−nβθ
ξ=n
n
X
2ξβθ Λξ E2θξ (f )p .
ξ=0
Now let θ > 1. Define
I11 :=
∞
X
ν=n+1
I12 :=
∞
X
2

2−ν(k+β)θ Λν 
−ν(k+β)θ
ν=n+1
I21 := 2−nβθ
n
X
ν=0
For I12 and I21 , Lemma 2.4(a) gives
I12 ∞
X
ν=n+1
I21 2−nβθ

Λν 
n
X
ξ=0
ν
X
θ
2ξ(k+β) E2ξ (f )p  ,
2
ξ(k+β)
ξ=n+1

2−νkθ Λν 
ν
X
ξ=0
θ
E2ξ (f )p  ,
θ
2ξ(k+β) E2ξ (f )p  .
θ

∞
∞
1−θ X
X

Λν E2θν (f )p .
2ν(k+β)θ E2θν (f )p 2−ν(k+β)θ Λν
2−ξ(k+β)θ Λξ  ν=n+1
ξ=ν
n
X
ν=0
2ν(k+β)θ E2θν (f )p 2−νkθ Λν
1−θ


n
X
ξ=ν
θ
2−ξkθ Λξ  2−nβθ
n
X
ν=0
2νβθ Λν E2θν (f )p .
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θ
and τ = θε with ε defined by (3.3). An elementary
We now estimate I11 . Let θ0 = θ−1
calculation, using the Hölder inequality, gives that
n
X
(3.6)
ξ=0

2ξ(k+β) E2ξ (f )p 
n
X
ξ=0
 θ1 
2ξ(k+β−τ )θ E2θξ (f )p  

2nτ 
n
X
ξ=0
n
X
ξ=0
 θ1
 10
θ
0
2ξτ θ 
2ξ(k+β−τ )θ E2θξ (f )p  .
Using (3.5) and then (3.6) and (3.3), we obtain

I11 2−n(k+β)θ Λn 
n
X
ξ=0
2−nβθ Λn 2−n(kθ−ε)
θ
2ξ(k+β) E2ξ (f )p  2−n(k+β−τ )θ Λn
n
X
2ξβθ 2ξ(kθ−ε) E2θξ (f )p 2−nβθ
ξ=0
n
X
2ξ(k+β−τ )θ E2θξ (f )p ξ=0
n
X
2ξβθ Λξ E2θξ (f )p .
ξ=0
Hence, we claim that (3.2) =⇒ f (x) ∈ BH(α, β, ψ, p, θ, k).
On the contrary, if f (x) ∈ BH(α, β, ψ, p, θ, k), then Lemma 2.1 and (3.4) imply (3.2).
4. Fourier coefficients of functions from Besov-Nikol’skiı̆ class. T HEOREM 4.1. Let
β, k, θ > 0, and α(·) satisfies the σ-condition with σ = kθ, and
(4.1)
Z1
α(t)dt + 2nkθ
n
1/2
Z
α(t) tkθ dt 0
1/2n
n
1/2
Z
for n ∈ N ∪ {0}.
α(t)dt
1/2n+1
If f (x) ∈ Lp (1 ≤ p ≤ ∞) has lacunary Fourier coefficients, then f (x) ∈
BH(α, β, ψ, p, θ, k) if and only if, for any n ∈ N ∪ {0} , there holds
2
(4.2)
−nβθ
n
X
θ
(cν + dν ) Λν 2
νβθ
+
ν=0
where Λν =
ν
1/2
R
∞
X
θ
(cν + dν ) Λν
ν=n+1
! θ1
ψ
1
2n
,
α(t) dt, ν ∈ N ∪ {0}.
1/2ν+1
1
Proof. Let n be an integer such that 2n+1
< δ ≤ 21n . We note that α(t) satisfies (3.1) as
well, which follows from (4.1). Thus, it is sufficient to prove
(4.2)
⇐⇒
J1 +J2 := 2
−nβθ
n
X
2
νβθ
Λν E2θν (f )p +
ν=0
ν=n+1
It is clear that condition (4.1) implies
(4.3)
n
X
ν=0
∞
X
Λν Λ n .
Λν E2θν (f )p
ψ
θ
1
2n
.
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Characteristics of Besov-Nikol’skiı̆ class of functions
Using Lemma 2.2 and 2.3, we have that there exist ε > 0 and C > 1 such that the inequality
Λl 2−lε ≤ C Λm 2−mε
(4.4)
holds for any l ≤ m.
For the case 1 ≤ p < ∞ we shall need a well-known statement for functions with lacunary
1
∞
2
P 2
2
ck + d k
. For brevity,
Fourier coefficients ([14]): E2n (f )p kf − S2n −1 (f )kp k=n
we shall suppose that the function f is even (dk = 0). Put

 θ2

 θ2
n
n
∞
n
X
X
X
X
J1 2−nβθ
2νβθ Λν 
2νβθ Λν 
c2ξ  2−nβθ
c2ξ  +
ν=0
+2−nβθ
n
X
ν=0
ν=0
ξ=ν

2νβθ Λν 
∞
X
ξ=n+1
ξ=ν
 θ2
c2ξ  =: J11 + J12 ,
∞
X
J2 ν=n+1

Λν 
∞
X
ξ=ν
 θ2
c2ξ  =: J21 .
To estimate J11 , J21 we now write, for θ ≤ 2, using Lemma 2.5 and (4.3), that
J11 2
−nβθ
n
X
2
νβθ
Λν
ν=0
n
X
cθξ
2
−nβθ
n
X
cθξ
∞
X
J21 cθξ
2
νβθ
Λν 2
ξ
X
n
X
cθξ 2ξβθ Λξ ;
ξ=0
Λν ν=n+1
ξ=n+1
−nβθ
ν=1
ξ=0
ξ=ν
ξ
X
∞
X
cθξ Λξ .
ξ=n+1
We note that (4.4) gives the inequality Λl ≤ CΛm (l ≤ m). Thus, for θ ≤ 2 we have
J12 Λn
∞
X
cθξ ξ=n+1
∞
X
cθξ Λξ .
ξ=n+1
Let θ > 2. Using Lemma 2.4(b) and (4.3), we get
J11 2
−nβθ
n
X
cθξ
2
ξβθ
Λξ
" ξ
1− θ2 X
2
νβθ
Λν
ν=0
ξ=0
# θ2
2
−nβθ
n
X
cθξ 2ξβθ Λξ
ξ=0
and
J21 ∞
X
cθξ
(Λξ )
1− θ2
ξ=n+1
Let θ =
θ
θ−2
(4.5)
∞
X
0
and τ =
ξ=n+1

c2ξ 
ε
θ
"
ξ
X
ν=n+1
Λν
# θ2
∞
X
cθξ Λξ .
ξ=n+1
with ε defined by (4.4). Using Hölder inequality, we have
∞
X
ξ=n+1
 θ2 
cθξ 2ξτ θ  
∞
X
ξ=n+1
 10
θ
0
2−2ξτ θ 

2−2nτ 
∞
X
ξ=n+1
 θ2
cθξ 2ξτ θ  .
Combining (4.3), (4.5) and (4.4), we obtain

 θ2
∞
∞
∞
∞
X
X
X
X
J12 Λn 
c2ξ  2−nτ θ Λn
cθξ 2ξτ θ 2−nε Λn
cθξ 2ξε cθξ Λξ .
ξ=n+1
ξ=n+1
ξ=n+1
ξ=n+1
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Finally, we have J1 + J2 ∞
P
ξ=n+1
cθξ 2ξθ Λξ + 2−nβθ
n
P
ξ=0
cθξ 2ξβθ Λξ and the sufficiency of
(4.2) has been proved. The necessity of (4.2) follows from evident estimates:
∞
n
P
P
cθξ Λξ .
cθξ 2ξβθ Λξ , J2 ≥
2−nβθ
J1 ≥
ξ=n+1
ξ=0
In the case of p = ∞ the proof can be developed in the same manner using the following
∞
P
inequality ([11]): E2n (f )∞ (ck + dk ).
k=n
T HEOREM 4.2. Let β, k, θ > 0, and α(·) satisfies a σ-condition with σ = kθ, and
Z1
(4.6)
α(t) dt nkθ
Z1/n
α(t) tkθ dt n
0
1/n
Z1/n
for n = 2, 3, · · ·
α(t) dt
1/(n+1)
Suppose f (x) ∈ Lp (1 < p < ∞) has quasimonotone Fourier coefficients, then f (x) ∈
BH(α, β, ψ, p, θ, k) if and only if, for any n ∈ N ,
n
−βθ
n
X
θ
(aν + bν ) λν ν
θ
βθ+θ− p
+
1/ν
R
θ
(aν + bν ) λν ν
θ
θ− p
ν=n+1
ν=1
where λν =
∞
X
! θ1
ψ
1
,
n
α(t) dt, ν ∈ N.
1/(ν+1)
Theorem 4.2 can be proved in the same way as Theorem 4.1. Another way of proof is
presented in [13].
5. Embedding theorems. Recall that the sequences {Λ} , {µ} , {λ} are defined by
ν
1/ν
1/ν
1/2
R
R
R
α(t) dt, ν ∈ N.
α(t) dt, λν =
α(t) dt, ν ∈ N ∪ {0}, µν =
Λν =
1/2ν+1
1/(ν+1)
1/(2ν)
By H(k, p, ψ) and B(α, β, p, θ) we denote Nikol’skiı̆and Besov classes, i.e., H(β, p, ψ)=
R1
{f ∈ Lp : ωβ (f, δ)p ψ(δ)} , and B(α, β, p, θ) = f ∈ Lp : α(t) ωβθ (f, t)p dt < ∞ .
0
e
The class H(β,
p, ψ) consists of all functions f ∈ Lp that satisfy ωβ (f˜, δ)p ψ(δ),
˜
where f is the conjugate function of f ([14]). In the same way we define the class
g
BH(α,
β, ψ, p, θ, k). Now we can formulate the following embedding theorems.
T HEOREM 5.1. Suppose β, k, θ > 0, 1 ≤ p ≤ ∞, and α(·) satisfies the σ-condition
with σ = kθ. If the majorant ψ(t) satisfies ([1]):
∞
P
(B)
ψ 21k = O ψ 21n ,
k=n
n
P
(Bβ )
2kβ ψ
k=0
= O 2nβ ψ
1
2n
1
2k
1
2n
,
− 1
then the classes BH(α, β, ψ, p, θ, k) and H(k + β, p, µ[1/δ] θ ψ(δ)) coincide.
1
< δ ≤ 21n . Note ([1]) that if ψ(δ) ∈ B and
Proof. Let n be an integer such that 2n+1
ψ(δ) ∈ Bβ , then for all θ > 0 we have
∞
X
k=n
ψθ
1
2k
ψθ
ψ θ (δ) ,
n
X
k=0
2kβθ ψ θ
1
2k
2nβθ ψ θ
1
2n
δ −βθ ψ θ (δ) .
ETNA
Kent State University
etna@mcs.kent.edu
101
Characteristics of Besov-Nikol’skiı̆ class of functions
Therefore,
Zδ
θ
α(t) ωk+β
(f, t)p dt
∞
X
−ν
2
Z
α(t) (µ[1/t] )−1 ψ θ (t)dt ν=n −(ν+1)
2
0
∞
X
ψθ
ν=n
1
2ν
ψ θ (δ)
and
δ
βθ
Z1
t
−βθ
θ
α(t) ωk+β
(f, t)p dt
δ
βθ
k=0
δ
In other words, if f (x) ∈ H(k + β, p, µ[1/δ]
Let f (x) ∈ BH(α, β, ψ, p, θ, k). Then
θ
ψ (δ) Zδ
n
X
θ
α(t) ωk+β
(f, t)p dt
+δ
βθ
0
Z1
− 1θ
Zδ


α(t) dt + δ βθ
1
2k
ψ θ (δ) .
ψ(δ)), then f (x) ∈ BH(α, β, ψ, p, θ, k).

−n
2
Z
 θ
θ
t−βθ α(t) dt ωk+β
(f, δ)p ,
(f, δ)p µ[1/δ] ωk+β
δ
2−(n+1)
ψ
θ
θ
t−βθ α(t) ωk+β
(f, t)p dt δ

2
kβθ
ψ(δ)).
i.e., f (x) ∈ H(k + β, p, µ[1/δ]
Using Theorems 4.2 and 5.1, one can prove
C OROLLARY 5.2. Under the conditions of Theorem 5.1, if f (x) ∈ L p (1 < p < ∞) has
quasimonotone Fourier coefficients and α(·) satisfies (4.6), then
− θ1
1
1
− θ1
f (x) ∈ H(k + β, p, µ[1/δ]
ψ(δ)) ⇐⇒ an , bn µn ψ
n−(1− p ) .
n
− θ1
T HEOREM 5.3. Suppose β, k, θ > 0, 1 ≤ p ≤ ∞, and α(·) satisfies the σ-condition
with σ = kθ. If the majorant ψ(t) satisfies the following condition: ψ(t) ≥ C for 0 ≤ t ≤ 1,
then the classes BH(α, β, ψ, p, θ, k) and B(α, k + β, p, θ) coincide.
Proof. Let f (x) ∈ B(α, k + β, p, θ). We see that
 θ1
 δ
Z
Z1
θ
θ
 α(t) ωk+β
(f, t)p dt + δ βθ t−βθ α(t) ωk+β
(f, t)p dt 0
δ
 1
 1θ
Z
θ
 α(t) ωk+β
(f, t)p dt C ≤ ψ (δ) .
0
Let f (x) ∈ BH(α, β, ψ, p, θ, k). Then
R1
0
θ
α(t) ωk+β
(f, t)p dt ψ θ (1) < ∞.
Using Theorems 4.2 and 5.3, one can prove
C OROLLARY 5.4. Under the conditions of Theorem 5.3, if f (x) ∈ L p (1 < p < ∞) has
quasimonotone Fourier coefficients and α(·) satisfies (4.6), then
f (x) ∈ B(α, k + β, p, θ) ⇐⇒
∞
X
ν=1
θ
(aν + bν )θ λν ν θ− p < ∞.
ETNA
Kent State University
etna@mcs.kent.edu
102
Sergey Tikhonov
T HEOREM 5.5. Let β, k > 0, 0 < θ ≤ 1, p = 1, ∞, and α(t) = t−1 . Then
e
e + β, p, ψ).
BH(α, β, ψ, p, θ, k) ⊂ H(β,
p, ψ) ⊂ H(k
Proof. Let f (x) ∈ BH(α, β, ψ, p, θ, k) and α(t) = t−1 . It is easy to prove that Λν ∞
P
1 (ν ∈ N ∪ {0}). We need here the Stechkin inequality ([1]): E2n (f˜)p E2ν (f )p . If
ν=n
we combine this with Lemma 2.1, 2.5 and Theorem 3.1, we get
!θ
n
X
1
1
2νβ E2ν (f˜)p
ωk+β (f˜, n )p ωβθ (f˜, n )p 2−nβ
2
2
ν=0

θ

θ
n
∞
n
n
X
X
X
X
2−nβθ
2νβθ 
E2ξ (f )p  2−nβθ
2νβθ 
E2ξ (f )p  +
ν=0
+2−nβθ
n
X
ν=0
ν=0
ξ=ν

2νβθ 
∞
X
ξ=n+1
θ
E2ξ (f )p  2−nβθ
2
(5.1)
−nβθ
n
X
2
ξβθ
n
X
ξ=ν
E2θξ (f )p
ξ
X
2νβθ +
ν=0
ξ=0
E2θξ (f )p
∞
X
+
ξ=0
∞
X
E2θξ (f )p ξ=n+1
E2θξ (f )p
ψ
θ
ξ=n+1
1
2n
.
R EMARK 5.6. Let β1 , β2 > 0, 0 < θ ≤ 1. If f (x) ∈ Lp , p = 1, ∞, then

˜ δ)p 
ωβ1 (f,
Zδ
t−1 ωβθ 2 (f, t)p dt + δ β1 θ
0
Z1
δ
 θ1
t−β1 θ−1 ωβθ 2 (f, t)p dt .
Proof. Applying Lemma 2.1 to (5.1), we obtain
n
∞
X
X
1
ωβθ 1 (f˜, n )p 2−nβ1 θ
2ξβ1 θ ωβθ 2 (f, 2−ξ )p +
ωβθ 2 (f, 2−ξ )p 2
ξ=0
2
−nβ1 θ
n−1
X
ξ=n+1
ωβθ 2 (f, 2−ξ )p
ξ=0
+
∞
X
ξ=n+1
ωβθ 2 ( f , 2−ξ )p
1/2
Z
t−β1 θ−1 dt + ωβθ 2 (f, 2−n )p +
1/2ξ+1
ξ
t−1 dt 2−nβ1 θ
1/2ξ+1
+ 2−nβ1 θ
ξ
1/2
Z
1/2
Z
n−1
X
ξ=0
ξ
1/2
Z
t−β1 θ−1 ωβθ 2 (f, t)p dt +
1/2ξ+1
n
1/2n+1
t−β1 θ−1 ωβθ 2 (f, t)p dt +
∞
X
ξ
1/2
Z
ξ=n+1
1/2ξ+1
t−1 ωβθ 2 (f, t)p dt.
ETNA
Kent State University
etna@mcs.kent.edu
103
Characteristics of Besov-Nikol’skiı̆ class of functions
Let n be an integer such that
ωβθ 1 (f˜, δ)p 2−nβ1 θ
n−1
X
ξ=0
+
∞
X
1/2
Z
1
2n+1
ξ
1/2
Z
<δ≤
t
Hence,
t−β1 θ−1 ωβθ 2 (f, t)p dt + δ β1 θ
ωβθ 2 (f, t)p
n
1/2
Z
t−β1 θ−1 ωβθ 2 (f, t)p dt +
1/2n+1
1/2ξ+1
ξ
−1
1
2n .
dt ξ=n+1
1/2ξ+1
Zδ
t
−1
ωβθ 2 (f, t)p dt
0
+δ
β1 θ
Z1
t−β1 θ−1 ωβθ 2 (f, t)p dt.
δ
C OROLLARY 5.7. Suppose the sequence {µn } is quasi-monotone increasing and
quasi ε-power-monotone decreasing with ε = −kθ. Under the conditions of Theorem
g
5.1, if p = 1, ∞, then the classes BH(α, β, ψ, p, θ, k), BH(α,
β, ψ, p, θ, k), H(k +
− θ1
− 1θ
e
β, p, µ[1/δ]
ψ(δ)) and H(k + β, p, µ[1/δ]
ψ(δ)) coincide.
Proof.
By Theorem 5.1, it is sufficient to prove that the classes H(k +
1
− 1
e + β, p, µ[1/δ] − θ ψ(δ)) coincide. This easily follows from
β, p, µ[1/δ] θ ψ(δ)) and H(k
the conditions on {µn } and the inequality of Remark 5.6 (θ = 1, β1 = β2 = k + β).
T HEOREM 5.8. Suppose β, k, θ > 0, p = 1, ∞, and α(·) satisfies the σ-condition with
σ = kθ, and
Z1
1/2n
α(t)dt n
1/2
Z
α(t)dt
for n ∈ N.
1/2n+1
g
Then the classes BH(α, β, ψ, p, θ, k) and BH(α,
β, ψ, p, θ, k) coincide.
Proof. According to Theorem 3.1,
f (x) ∈ BH(α, β, ψ, p, θ, k) ⇐⇒ 2−nβθ
∞
X
1
2νβθ Λν Eνθ (f )p +
Λν Eνθ (f )p ψ θ
.
n
2
ν=0
ν=n+1
n
X
∞
˜ p P E2ν (f )p and following the line of proof of TheUsing the inequality ([1]) E2n (f)
ν=n
n
P
orem 3.1 and 3.2 it easy to show that 2−nβθ
2νβθ Λν Eνθ (f˜)p +
ν=0
g
ψ θ 21n , i.e., f (x) ∈ BH(α,
β, ψ, p, θ, k).
∞
P
ν=n+1
˜p Λν Eνθ (f)
Note. Let k, β, θ > 0, k + β ∈ N and α(t) = t−rθ−1 , 0 < r < k. Then Theorems
3.1, 5.1, 5.3 are proved in [6]. If additionally f (x) has monotonic Fourier coefficients, then
Theorem 4.2 is established in [3], in Corollary 5.2 in [5], and Corollary 5.4 in [2].
Acknowledgments. The author would like to thank Professor Mikhail K. Potapov for
useful comments on this work.
ETNA
Kent State University
etna@mcs.kent.edu
104
Sergey Tikhonov
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