ON A GENERALIZATION OF LIPSCHITZ’S CLASSES

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Generalization of Lipschitz’s Classes

Xh. Z. Krasniqi vol. 9, iss. 3, art. 73, 2008

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ON A GENERALIZATION OF LIPSCHITZ’S CLASSES

Xh. Z. KRASNIQI

Department of Mathematics and Computer Sciences, Avenue "Mother Teresa" 5

Prishtinë, 10000, Republic of Kosovo EMail:xheki00@hotmail.com

Received: 26 March, 2008

Accepted: 06 August, 2008

Communicated by: H. Bor 2000 AMS Sub. Class.: 42A20, 42A32.

Key words: Lipschitz classes, Fourier series.

Abstract: In this paper we obtain a generalization of Lipschitz’s classesΛ^m(β, p, r)defined in [1]. We give necessary conditions for even or odd functions with Fourier series to belong to the classesΛ^m(p, r, α). We also give sufficient conditions for even or odd functions with Fourier series to belong to the same classes.

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1. Definitions and Useful Statements

We consider the series

(1.1) a₀

2 +

∞

X

n=1

a_ncosnx

or (1.2)

∞

X

n=1

a_nsinnx

wherea_nare Fourier coefficients of integrable functionf.

Definition 1.1. We say that a functionf belongs toW A^p,(1< p <∞)if

∞

X

n=1

n^p−2

∞

X

k=n

|∆a_k|

!p

<+∞

where∆a_k =a_k−a_k+1(see [1]).

We say that any functionα(t)is a function of typeσ(see [4]) if it is measurable in [0,1], integrable in[δ,1]for eachδ ∈ (0,1), and there exist real numbersC_1,α >0, σandδ₀ ∈(0,1)such that

1. α(t)≥C_1,α, for allt ∈[0,1];

2. Rδ

0 α(t)t^sdt <∞for eachs > σandδ∈(0, δ₀);

3. Rδ

0 α(t)t^sdt =∞for eachs < σandδ∈(0, δ0), and Z δ

0

α(t)t^σdt≤C₂δ^σ Z 2δ

δ

α(t)dt.

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In [1], the classesΛ^m(β, p, r,)are defined in the following way Definition 1.2. f ∈Λ^m(β, p, r,), if

f

(m) β,p,r ≡

(Z 1

0

Z 2π 0

|∆_mf(x, t)|^p t^βp dx

_p^r dt

t )¹_r

<+∞,

where1< p <+∞,1≤r <+∞, β >0, m∈Nand

∆_mf(x, t) =

m

X

i=1

(−1)ⁱC_mⁱ f[x+ (m−2i)t].

Now we define classesΛ^m(p, r, α)as follows:

Definition 1.3. We sayf ∈Λ^m(p, r, α), if

f

(m) p,r,α ≡

(Z 1

0

α(t) Z 2π

0

|∆mf(x, t)|^pdx _p^r

dt )¹_r

<+∞,

whereα(t)is function of the typeσ.

For α(t) = t^−rβ−1, β > 0, we get the classes Λ^m(β, p, r), considered in [1].

Therefore classesΛ^m(p, r, α)are generalizations of classesΛ^m(β, p, r).

We need some auxiliary statements.

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Lemma 1.4 ([2]). Let a_ν, b_ν and β_n be numbers such that a_ν ≥ 0, b_ν ≥ 0 and P∞

ν=na_ν =a_nβ_n:

1. For0< p ≤1the following inequality is valid

∞

X

ν=1

a_ν

ν

X

µ=1

b_µ

!p

≥p^p

∞

X

ν=1

a_ν(b_νβ_ν)^p;

2. For1≤p < ∞we have

∞

X

ν=1

a_ν

ν

X

µ=1

b_µ

!p

≤p^p

∞

X

ν=1

a_ν(b_νβ_ν)^p.

Lemma 1.5 ([2]). Let a_ν, b_ν and γ_n be numbers such that a_ν ≥ 0, b_ν ≥ 0 and Pn

ν=1aν =bnγn:

1. For0< p ≤1, we have

∞

X

ν=1

a_ν

∞

X

µ=ν

b_µ

!p

≥p^p

∞

X

ν=1

a_ν(b_νγ_ν)^p;

2. For1≤p < ∞, we have

∞

X

ν=1

a_ν

∞

X

µ=ν

b_µ

!p

≤p^p

∞

X

ν=1

a_ν(b_νγ_ν)^p.

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Lemma 1.6 ([3]). Let µ, τ and a_ν be numbers such that 0 < µ < τ < ∞ and a_ν ≥0. Then

∞

X

ν=1

a^τ_ν

!¹_τ

≤

∞

X

ν=1

a^µ_ν

!_µ¹ .

We denote byC a constant that depends only onm, p, r and may be different in different relations.

Theorem 1.7 ([1]). Iff ∈W A^p,1< p <+∞, then

ω_p^(m)(h;f) ^p ≤Ch^mp X

n≤[h¹]

n^(m+1)p−2

∞

X

k=n

|∆ak|

!p

+C X

n>[h¹] n^p−2

∞

X

k=n

|∆ak|

!p

,

whereωp^(m)(h;f)is the integral modulus of smoothness of orderm.

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2. Main Results

Let us denote

A(n) :=

Z 1/n 1/(n+1)

α(t)dt,

b(n) :=b₁(n) +b₂(n) =n^mr Z 1/n

0

α(t)t^mrdt+ Z 1

1/(n+1)

α(t)dt.

We have the following first main result.

Theorem 2.1. Letmbe any natural number and

f ∈AW^p, 1< p <+∞, 1≤r <+∞.

If for the coefficients of series(1.1)or(1.2)we haveP∞

k=1|∆a_k|<+∞, then:

1. Forp≤rwe have

f

(m) p,r,α≤C

( _∞ X

n=1

n^r ¹⁻²^p ^∞

X

k=n

∆a_k

!r

b(n)

b(n) A(n)

^r_p−1)¹_r

;

2. Forp > rwe have

f

(m) p,r,α ≤C

( _∞ X

n=1

n^r ¹⁻²^p ^∞

X

k=n

∆a_k

!r

b(n) )¹_r

.

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Proof. Using the characteristics of the integral modulus of smoothness we have n

f

(m) p,r,α

or

= Z 1

0

α(t) Z 2π

0

|∆_mf(x, t)|^pdx p^r

dt

≤

∞

X

N=1

Z 1/N 1/(N+1)

α(t)

ω^(m)_p (f;t)^r dt

≤

∞

X

N=1

ω^(m)_p (f; 1/N)^r Z 1/N

1/(N+1)

α(t)dt

=

∞

X

N=1

A(N)

ω_p^(m)(f; 1/N)^r .

According to the Theorem1.7, we have n

f

(m) p,r,α

or

≤C

∞

X

N=1

A(N)N^−mr ( _N

X

n=1

n^(m+1)p−2

∞

X

k=n

|∆a_k|

!p)^r_p

+C

∞

X

N=1

A(N) ( _∞

X

n=N+1

n^p−2

∞

X

k=n

|∆ak|

!p)^r_p

=I₁+I₂.

Now we estimateI₁ andI₂. Letr/p≥1. Then according to Lemma1.4we have I₁ ≤C

∞

X

N=1

A(N)N^−mr (

N^(m+1)p−2

∞

X

k=N

|∆a_k|

!p

β_N )^r_p

.

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Now we estimate the quantityβ_N: A(N)N^−mrβ_N =

∞

X

i=N

A(i)i^−mr

=

∞

X

i=N

i+ 1 i

mr

· 1 (i+ 1)^mr

Z 1/i 1/(i+1)

α(t)dt

≤2^mr Z 1/N

0

α(t)t^mrdt, or

β_N ≤Cb1(N) A(N). Consequently

(2.1) I₁ ≤C

∞

X

N=1

A(N)N^r(¹⁻²_p)

∞

X

k=N

|∆a_k|

!r

b₁(N) A(N)

^r_p .

According to Lemma1.5, forr/p≥1we have I₂ ≤C

∞

X

N=1

A(N) (

N^p−2

∞

X

k=N

|∆a_k|

!p

γ_N )^r_p

.

We estimate the quantityγ_N: A(N)γ_N =

N

X

i=1

A(i) = Z 1

1/(N+1)

α(t)dt⇒γ_N = b₂(N) A(N).

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Consequently

(2.2) I2 ≤C

∞

X

N=1

A(N)N^r(¹⁻²p)

∞

X

k=N

|∆ak|

!r

b₂(N) A(N)

^r_p .

By (2.1) and (2.2) we get n

f

(m) p,r,α

or

≤C

∞

X

N=1

A(N)N^r(¹⁻²_p)

∞

X

k=N

|∆a_k|

!r(

b₁(N) A(N)

^r_p +

b₂(N) A(N)

^r_p) .

Finally, according to Lemma1.6, forr≥pwe have f

(m) p,r,α ≤C

( _∞ X

N=1

A(N)N^r(¹⁻²p)

∞

X

k=N

|∆a_k|

!r

b(N) A(N)

^r_p)¹_r .

Now letr/p <1. Then, according to Lemma1.6, we have I₁ ≤C

∞

X

N=1

A(N)N^−mr

N

X

n=1

n^(m+1)r−^2r^p

∞

X

k=n

|∆a_k|

!r

.

If we change the order of summation we get I1 ≤C

∞

X

n=1

n^(m+1)r−^2r^p

∞

X

k=n

|∆ak|

!r ∞

X

N=n

A(N)N^−mr

≤C

∞

X

n=1

n^r(¹⁻²_p)

∞

X

k=n

|∆a_k|

!r

b₁(n).

(2.3)

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Now we estimateI₂. Using Lemma 1.6 and changing the order of summation we have:

I₂ ≤C

∞

X

N=1

A(N)

∞

X

n=N

n^r(¹⁻²p)

∞

X

k=n

|∆a_k|

!r

=C

∞

X

n=1

n^r(¹⁻²p)

∞

X

k=n

|∆ak|

!r n

X

N=1

A(N)

=C

∞

X

n=1

n^r(¹⁻²_p)

∞

X

k=n

|∆a_k|

!r

b₂(n).

(2.4)

From (2.3) and (2.4) we deduce f

(m) p,r,α ≤C

( _∞ X

n=1

n^r ¹⁻²^p ^∞

X

k=n

∆a_k

!r

b(n) )¹_r

,

which fully demonstrates Theorem2.1.

Theorem 2.2. Letmbe any natural number and

1< p ≤2, 1≤r <+∞, 1/p+ 1/q= 1.

Ifa_nare the coefficients of series(1.1)or(1.2), then:

1. Forr≤qwe have

f

(m) p,r,α ≥C

( _∞ X

n=1

n^−mr|a_n|^rb₁(n)

b₁(n) A(n)

^r_q⁻¹)¹_r

;

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2. Forr > qwe have

f

(m) p,r,α ≥C

( _∞ X

n=1

n^−mr|a_n|^rb₁(n) )¹_r

.

Proof. Letf be an even function. Iff is an odd function then the proof of the theorem is analogous to the even case. It is not difficult to see that the Fourier series of

∆_mf(x, t)is

∆mf(x, t)∼











(−1)^m²2^m

∞

P

n=1

ancosnxsin^mnt, formeven

(−1)^m−1² ⁻¹2^m

∞

P

n=1

ansinnxsin^mnt, formodd.

According to the well-known Hausdorf-Young’s theorem we find C

Z 2π 0

|∆mf(x, t)|^pdx ^r_p

≥

∞

X

n=1

|an|^q|sinnt|^mq

!^r_q ,

and then n

f

(m) p,r,α

or

≥C

∞

X

ν=1

Z 1/ν 1/(ν+1)

α(t)

ν

X

n=1

|a_n|^q|sinnt|^mq

!^r_q dt.

Using the well-known inequalitysinB ≥ _π²B for0≤B ≤ ^π₂, we get n

f

(m) p,r,α

or

≥C

∞

X

ν=1

A(ν)

ν

X

n=1

n^−mq|a_n|^q

!^r_q .

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Letr ≤q, then according to Lemma1.4we have n

f

(m) p,r,α

or

≥C

∞

X

ν=1

A(ν)

ν^−mq|a_ν|^qβ_ν^r_q .

It is easy to prove thatβ_ν ≥ ^b_A(ν)¹^(ν), from which we get

(2.5) n

f

(m) p,r,α

or

≥C

∞

X

ν=1

ν^−mr|a_ν|^rb₁(ν)

b1(ν) A(ν)

^r_q−1

.

Letq < r, then according to Lemma1.6 and with the change of the order of summation we have

n f

(m) p,r,α

or

≥C

∞

X

ν=1

A(ν)

ν

X

n=1

n^−mr|a_n|^r

=C

∞

X

n=1

n^−mr|a_n|^r

∞

X

ν=n

A(ν)

≥C

∞

X

n=1

n^−mr|an|^rb1(n).

(2.6)

Relations (2.5) and (2.6) prove Theorem2.2.

We can deduce three corollaries from Theorem2.1and Theorem2.2.

Corollary 2.3. Under the conditions of Theorem2.1 and with b(n) ≤ CA(n), we have

f

(m) p,r,α ≤C

( _∞ X

n=1

n^r ¹⁻²^p ^∞

X

k=n

∆a_k

!r

b(n) )¹_r

.

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Corollary 2.4. Under the conditions of Theorem2.2 and withb₁(n) ≤ CA(n), we have

f

(m) p,r,α≥C

( _∞ X

n=1

n^−mr|a_n|^rb₁(n) )¹_r

.

As a special case, forα(t) =t^−βr−1, it is easy to prove the estimates:

A(n)≤Cn^βr−1 and b(n)≤Cn^βr.

From Theorem 2.1 and the last estimates we can deduce the following result proved in [1].

Corollary 2.5 ([1]). Letmbe any natural number and

0< β ≤m, 1< p <+∞, 1≤r <+∞, 1/p+ 1/q= 1.

If the coefficients of series (1.1) or (1.2) satisfyP∞

k=1|∆a_k|<+∞, then f

(m) β,p,r ≤C

( _∞ X

n=1

n^r(^β+¹q)⁻¹

∞

X

k=n

∆a_k

!r)¹_r .

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References

[1] T.Sh. TEVZADZE, Some classes of functions and trigonometric Fourier series, Some Questions of Function Theory, v. II, 31–92, Tbilisi University Press, 1981 (in Russian).

[2] M.K. POTAPOVANDM. BERISHA, Moduli of smoothnes and Fourier coeffi- cients of functions of one variable, Publ. Inst. Math. (Beograd) (N.S.), 26(40) (1979), 215–228 (in Russian).

[3] B. HARDY, E. LITLEWOOD AND G. POLYA, Inequalities, GIIL Moscow, 1948, 1–456 (in Russian).

[4] M.K. POTAPOV, A certain imbedding theorem, Mathematica (Cluj), 14(37) (1972), 123–146.