ON INTEGRABILITY OF FUNCTIONS DEFINED BY TRIGONOMETRIC SERIES

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Integrability of Functions L. Leindler vol. 9, iss. 3, art. 69, 2008

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ON INTEGRABILITY OF FUNCTIONS DEFINED BY TRIGONOMETRIC SERIES

L. LEINDLER

Bolyai Institute, University of Szeged Aradi vértanúk tere 1,

H-6720 Szeged, Hungary

EMail:leindler@math.u-szeged.hu

Received: 15 January, 2008

Accepted: 19 August, 2008

Communicated by: S.S. Dragomir

2000 AMS Sub. Class.: 26D15, 26A42, 40A05, 42A32.

Key words: Sine and cosine series,L^pintegrability, equivalence of coefficient conditions, quasi power-monotone sequences.

Abstract: The goal of the present paper is to generalize two theorems of R.P. Boas Jr. pertaining toL^p(p >1)integrability of Fourier series with nonnegative coefficients and weightx^γ.In our improvement the weightx^γis replaced by a more general one, and the casep = 1is also yielded. We also generalize an equivalence statement of Boas utilizing power-monotone sequences instead of{n^γ}.

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1. Introduction

There are many classical and newer theorems pertaining to the integrability of formal sine and cosine series

g(x) :=

∞

X

n=1

λ_nsinnx, and

f(x) :=

∞

X

n=1

λ_ncosnx.

As a nice example, we recall Chen’s ([4]) theorem: Ifλ_n ↓0,thenx^−γϕ(x)∈L^p (ϕmeans eitherf org),p > 1, 1/p−1< γ < 1/p,if and only ifP

n^pγ+p−2λ^p_n <

∞.

For notions and notations, please, consult the third section.

We do not recall more theorems because a nice short survey of recent results with references can be found in a recent paper of S. Tikhonov [7], and classical results can be found in the outstanding monograph of R.P. Boas, Jr. [2].

The generalizations of the classical theorems have been obtained in two main directions: to weaken the classical monotonicity condition on the coefficientsλ_n;to replace the classical power weightx^γ by a more general one in the integrals. Lately, some authors have used both generalizations simultaneously.

J. Németh [6] studied the class ofRBV S sequences and weight functions more general than the power one in theL(0, π)space.

S. Tikhonov [8] also proved two general theorems of this type, but in theL^p-space forp=1;he also used general weights.

Recently D.S. Yu, P. Zhou and S.P. Zhou [9] answered an old problem of Boas ([2], Question 6.12.) in connection withL^p integrability considering weightx^γ, but only under the condition that the sequence {λn} belongs to the class M V BV S;

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their result is the best one among the answers given earlier for special classes of sequences. The original problem concerns nonnegative coefficients.

In the present paper we refer back to an old paper of Boas [3], which was one of the first to study theL^p-integrability with nonnegative coefficients and weightx^γ.

We also intend to prove theorems with nonnegative coefficients, but with more general weights thanx^γ.

It can be said that our theorems are the generalizations of Theorems 8 and 9 presented in Boas’ paper mentioned above. Boas names these theorems as slight improvements of results of Askey and Wainger [1]. Our theorems jointly generalize these by using more general weights thanx^γ, and broaden those to the casep = 1, as well.

Comparing our results with those of Tikhonov, as our generalization concerns the coefficients, we omit the condition {λ_n} ∈ RBV S and prove the equivalence of (2.2) and (2.3).

In proving our theorems we need to generalize an equivalence statement of Boas [3]. At this step we utilize the quasiβ-power-monotone sequences.

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2. New Results

We shall prove the following theorems.

Theorem 2.1. Let15p < ∞andλ :={λ_n}be a nonnegative null-sequence.

If the sequence γ :={γ_n}is quasiβ-power-monotone increasing with a certain β < p−1,and

(2.1) γ(x)g(x)∈L^p(0, π),

then (2.2)

∞

X

n=1

γ_nn^p−2

∞

X

k=n

k⁻¹λ_k

!p

<∞.

Ifγ is also quasiβ-power-monotone decreasing with a certainβ >−1,then condi- tion (2.2) is equivalent to

(2.3)

∞

X

n=1

γ_nn⁻²

n

X

k=1

λ_k

!p

<∞.

If the sequenceγis quasiβ-power-monotone decreasing with a certainβ >−1−p, and

(2.4)

∞

X

n=1

γ_nn^p−2

∞

X

k=n

|∆λ_k|

!p

<∞, then (2.1) holds.

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Theorem 2.2. Letpandλbe defined as in Theorem2.1.

If the sequenceγis quasiβ-power-monotone increasing with a certainβ < p−1, and

(2.5) γ(x)f(x)∈L^p(0, π),

then (2.2) holds.

If the sequenceγ is quasiβ-power-monotone decreasing with a certainβ >−1, then (2.4) implies (2.5).

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3. Notions and Notations

We shall say that a sequenceγ :={γ_n}of positive terms is quasiβ-power-monotone increasing (decreasing) if there exist a natural numberN :=N(β, γ)and a constant K :=K(β, γ)=1such that

(3.1) Kn^βγn =m^βγm (n^βγn 5Km^βγm) holds for anyn=m=N.

If (3.1) holds with β = 0, then we omit the attribute "β-power" and use the symbols↑(↓).

We shall also use the notations L R at inequalities if there exists a positive constantK such thatL5KR.

A null-sequencec:={c_n}(c_n→0)of positive numbers satisfying the inequali-

ties ∞

X

n=m

|∆c_n|5K(c)c_m, (∆c_n :=c_n−c_n+1), m∈N,

with a constant K(c) > 0 is said to be a sequence of rest bounded variation, in symbols,c∈RBV S.

A nonnegative sequencecis said to be a mean value bounded variation sequence, in symbols,c∈M V BV S,if there exist a constantK(c)>0and aλ=2such that

2n

X

k=n

|∆c_k|5K(c)n⁻¹

[λn]

X

k=[λ⁻¹n]

c_k, n∈N,

where[α]denotes the integral part ofα.

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In this paper a sequenceγ := {γ_n}and a real numberp = 1are associated to a functionγ(x) (=γ_p(x)),being defined in the following way:

γπ n

:=γ_n^1/p, n∈N; and K₁(γ)γ_n 5γ(x)5K₂(γ)γ_n holds for allx∈ _n+1^π ,^π_n

.

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4. Lemmas

To prove our theorems we recall one known result and generalize one of Boas’ lemmas ([2, Lemma 6.18]).

Lemma 4.1 ([5]). Letp=1, α_n =0andβ_n >0.Then (4.1)

∞

X

n=1

β_n

n

X

k=1

α_k

!p

5p^p

∞

X

n=1

β_n^1−p

∞

X

k=n

β_k

!p

α^p_n,

and (4.2)

∞

X

n=1

β_n

∞

X

k=n

α_k

!p

5p^p

∞

X

n=1

β_n^1−p

n

X

k=1

β_k

!p

α^p_n. Lemma 4.2. Ifb_n =0, p =1, s > 0,then

(4.3) X

1 :=

∞

X

n=1

β_n

∞

X

k=n

b_k

!p

<∞ implies

(4.4) X

2 :=

∞

X

n=1

βnn^−sp

n

X

k=1

k^sbk

!p

<∞

ifn^δβ_n ↓with a certainδ > 1−sp;and ifn^δβ_n ↑with a certainδ < 1,then (4.4) implies (4.3).

Thus, if both monotonicity conditions for{β_n}hold, then the conditions (4.3) and (4.4) are equivalent.

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Proof of Lemma4.2. First, suppose (4.3) holds. Write Tn :=

∞

X

k=n

bk; then

X

2 =

∞

X

n=1

β_nn^−sp

n

X

k=1

k^s(T_k−T_k+1)

!p

.

By partial summation we obtain X

2 s^p

∞

X

n=1

β_nn^−sp

n

X

k=1

k^s−1T_k

!p

=:X

3. Sincen^δβ_n ↓withδ >1−sp,Lemma4.1with (4.1) shows that

X

3

∞

X

n=1

(n^s−1T_n)^p(β_nn^−sp)^1−p

∞

X

k=n

β_kk^−sp

!p

∞

X

n=1

β_nT_n^p =X

1, this proves that (4.3)⇒(4.4).

Now suppose that (4.4) holds. First we show that (4.5)

∞

X

n=1

b_n<∞.

Denote

H_n :=

n

X

k=1

k^sb_k.

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Then (4.6)

N

X

k=n

b_k =

N

X

k=n

k^−s(H_k−H_k−1)5s

N−1

X

k=n

k^−s−1H_k+H_NN^−s.

Ifp > 1then by Hölder’s inequality, we obtain (4.7)

N−1

X

k=n

k^−s−1H_kβ

1 p−¹

p

k 5

N−1

X

k=n

H_k^pβ_kk^−sp

!¹_p _N₋₁ X

k=n

(k⁻¹β_k^−1/p)^p/(p−1)

!^p−1_p .

Since,n^δβ_n↑withδ <1,thus

∞

X

k=1

(k^−pβ_k⁻¹k^−δ+δ)^1/(p−1)

∞

X

k=1

k^δ−p^p−1 <∞.

This, (4.4) and (4.7) imply that (4.8)

∞

X

k=1

k^−s−1Hk <∞,

thusH_NN^−stends to zero, herewith, by (4.6), (4.5) is verified, furthermore, (4.9)

∞

X

k=n

b_k

∞

X

k=n

k^−s−1H_k.

Ifp = 1, then without Hölder’s inequality, the assumptionn^δβ_n ↑with a certain δ <1and (4.4) clearly imply (4.8).

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Thus we can apply (4.9) and Lemma4.1 with (4.2) for anyp = 1, whence, by n^δβ_n↑withδ <1,we obtain that

∞

X

n=1

β_n

∞

X

k=n

b_k

!p

∞

X

n=1

β_n

∞

X

k=n

k^−s−1H_k

!p

∞

X

n=1

(n^−s−1H_n)^pβ_n^1−p

n

X

k=1

β_k

!p

∞

X

n=1

β_nn^−spH_n^p; herewith (4.4)⇒(4.3) is also proved.

The proof of Lemma4.2is complete.

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5. Proof of the Theorems

Proof of Theorem2.1. First we prove that (2.1) impliesg(x)∈L(0, π)and (2.2). If p >1,then, by Hölder’s inequality, we get withp⁰ :=p/(p−1)

Z π

0

|g(x)|dx5 Z π

0

|g(x)γ(x)|^pdx

_p¹ Z π

0

γ(x)^−p⁰dx _p¹0

.

Denotex_n:= ^π_n, n∈N.Sinceγ_nn^β ↑ (β < p−1) Z π

0

γ(x)^−p⁰dx

∞

X

n=1

γ_n^1/(1−p) Z xn

xn+1

dx

=

∞

X

n=1

n⁻²(γ_nn^β)^1/(1−p)n^β/(p−1) 1, that is,g(x)∈L.

Ifp= 1, thenγ_nn^β ↑with someβ <0,thusγ_n↑, whence Z π

0

|g(x)|dx

∞

X

n=1

1 γ_n

Z xn

xn+1

|g(x)|γ(x)dx 1 γ₁

Z π

0

|g(x)|γ(x)dx1.

Integratingg(x),we obtain G(x) :=

Z x

0

g(t)dt =

∞

X

n=1

λ_n

n (1−cosnx) = 2

∞

X

n=1

λ_n

n sin² nx 2 . Hence

G(x_2k)

2k

X

n=k

λ_n n .

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Denote

g_n:=

Z xn

xn+1

|g(x)|dx, n∈N. Then

∞

X

k=n

k⁻¹λ_k=

∞

X

ν=0 2^ν+1n

X

k=2^νn

k⁻¹λ_k

∞

X

ν=0

G(2^ν+1n)

∞

X

ν=0

∞

X

k=2^ν+1n

g_k

∞

X

ν=0

1 2^ν+1n

2^ν+1n

X

i=2^νn

∞

X

k=2^ν+1n

g_k

∞

X

ν=0 2^ν+1n

X

i=2^νn

1 i

∞

X

k=i

g_k

∞

X

i=n

1 i

∞

X

k=i

g_k. (5.1)

Now we have X

1 :=

∞

X

n=1

n^p−2γ_n

∞

X

k=n

k⁻¹λ_k

!p

∞

X

n=1

n^p−2γ_n

∞

X

k=n

k⁻¹

∞

X

i=k

g_i

!p

.

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Applying Lemma4.1with (4.2) we obtain that X

1

∞

X

n=1

n⁻¹

∞

X

i=n

gi

!p

(n^p−2γn)^1−p

n

X

k=1

k^p−2γk

!p

.

Sinceγ_nn^β ↑withβ < p−1,we have (5.2)

n

X

k=1

γ_kk^βk^p−2−β γ_nn^β

n

X

k=1

k^p−2−β γ_nn^p−1, and thus

(n^p−2γ_n)^1−p

n

X

k=1

k^p−2γ_k

!p

γ_nn^2p−2, whence we get

X

1

∞

X

n=1

γ_nn^2p−2 n⁻¹

∞

X

i=n

g_i

!p

.

Using again Lemma4.1with (4.2) we have X

1

∞

X

n=1

n^−pg^p_n(n^2p−2γn)^1−p

n

X

k=1

k^2p−2γk

!p

. A similar calculation and consideration as in (5.2) give that

n

X

k=1

k^2p−2γ_k γ_nn^2p−1, and

(n^2p−2γ_n)^1−p

n

X

k=1

k^2p−2γ_k

!p

γ_nn^3p−2,

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thus

(5.3) X

1

∞

X

n=1

γ_nn^2p−2g^p_n. Since

∞

X

n=1

γ_nn^2p−2g_n^p =

∞

X

n=1

γ_nn^2p−2 Z xn

xn+1

|g(x)|dx p

∞

X

n=1

γ_nn^2p−2 Z xn

xn+1

|g(x)|^pdx Z xn

xn+1

dx ^p−1

∞

X

n=1

Z xn

xn+1

|γ(x)g(x)|^pdx

= Z π

0

|γ(x)g(x)|^pdx.

This and (5.3) prove the implication (2.1)⇒(2.2).

Next we verify that (2.4) implies (2.1). Letx ∈(x_n+1, x_n].Then, using the Abel transformation and the well-known estimation

De_n(x) :=

k

X

n=1

sinnx

x⁻¹,

we obtain

(5.4) |g(x)| x

n

X

k=1

kλ_k+

∞

X

k=n+1

λ_ksinkx

x

n

X

k=1

kλ_k+n

∞

X

k=n

|∆λ_k|.

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Denote

∆_n:=

∞

X

k=n

|∆λ_k|.

It is easy to see that

n∆_nn⁻¹

n

X

k=1

k∆_k and, byλn →0,

λn 5∆n. Thus, by (5.4), we have

|g(x)| n⁻¹

n

X

k=1

k∆_k.

Hence Z π

0

|γ(x)g(x)|^pdx=

∞

X

n=1

Z xn

xn+1

|γ(x)g(x)|^pdx

∞

X

n=1

γnn^−2−p

n

X

k=1

k∆k

!p

.

Applying Lemma4.1with (4.1), we obtain Z π

0

|γ(x)g(x)|^pdx

∞

X

n=1

(n∆_n)^p(γ_nn^−2−p)^1−p

∞

X

k=n

γ_kk^−2−p

!p

.

Sinceγ_nn^β ↓withβ >−1−p,we have

∞

X

k=n

γ_kk^βk^{−2−p−β} γ_nn^β

∞

X

k=n

k^{−2−p−β} γ_nn^−1−p,

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and thus

(γ_nn^−2−p)^1−p

∞

X

k=n

γ_kk^−2−p

!p

γ_nn⁻². Collecting these estimations we obtain

Z π

0

|γ(x)g(x)|^pdx

∞

X

n=1

γ_nn^p−2∆^p_n=

∞

X

n=1

γ_nn^p−2

∞

X

k=n

|∆λ_k|

!p

, herewith the implication (2.4)⇒(2.1) is also proved.

In order to prove the equivalence of the conditions (2.2) and (2.3), we apply Lemma4.2with

s= 1, β_n=γ_nn^p−2 and b_k=k⁻¹b_k.

Then the assumptionsn^δβ_n ↑withδ < 1andn^δβ_n ↓withδ > 1−p,determine the following conditions pertaining toγ_n;

(5.5) n^βγ_n ↑ with β < p−1 and n^βγ_n ↓ with β >−1.

The equivalence of (2.2) and (2.3) clearly holds if both monotonicity conditions required in (5.5) hold.

This completes the proof of Theorem2.1.

Proof of Theorem2.2. As in the proof of Theorem2.1, first we prove that (2.5) implies (2.2) andf(x)∈L.The proof off(x)∈Lruns as that ofg(x)∈Lin Theorem 2.1.

Integratingf(x), we obtain F(x) :=

Z x

0

f(t)dt =

∞

X

n=1

λn

n sinnx,

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and integratingF(x)we get F₁(x) :=

Z x

0

F(t)dt = 2

∞

X

n=1

λn

n² sin² nx 2 . Thus we obtain

F₁ π 2k

2k

X

n=k

λ_n n². Denote

f_n:=

Z xn

xn+1

|f(x)|dx, n ∈N,

x_n= π n

.

Then

F1(x2n) = Z x2n

0

F(t)dt

∞

X

k=2n

Z xk

x_k+1

Z xk

0

|f(t)|dt

du

∞

X

k=2n

1 k²

∞

X

`=k

Z x`

x`+1

|f(t)|dt=

∞

X

k=2n

1 k²

∞

X

`=k

f_`,

thus 2n

X

k=n

λ_k k n

∞

X

k=2n

1 k²

∞

X

`=k

f`.

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Using the estimation obtained above we have

∞

X

k=n

k⁻¹λ_k =

∞

X

ν=0 2^ν+1n

X

k=2^νn

k⁻¹λ_k

∞

X

ν=0

2^νn

∞

X

k=2^ν+1n

k⁻²

∞

X

`=k

f_`

∞

X

ν=0

2^νn

∞

X

i=ν 2ⁱ⁺²n

X

k=2ⁱ⁺¹n

k⁻²

∞

X

`=2ⁱ⁺¹n

f_`

∞

X

ν=0

2^νn

∞

X

i=ν

(2ⁱ_n)⁻¹

∞

X

`=2ⁱ⁺¹n

f_`

∞

X

i=0

(2ⁱn)⁻¹

∞

X

`=2ⁱ⁺¹n

f_`

i

X

ν=0

2^νn

!

∞

X

i=0

∞

X

`=2ⁱ⁺¹n

f_`.

Hereafter, as in (5.1), we get that

∞

X

k=n

k⁻¹λ_k

∞

X

i=n

1 i

∞

X

`=i

f_`,

and following the method used in the proof of Theorem2.1 withf_n in place ofg_n, the implication (2.5)⇒(2.2) can be proved.

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The proof of the statement (2.4)⇒(2.5) is easier. Namely

|f(x)|5

n

X

k=1

λ_k+

∞

X

k=n+1

λ_kcoskx

n

X

k=1

λ_k+ 1 x

∞

X

k=n

|∆λ_k|.

Using the notations of Theorem2.1and assumingx∈(x_n+1, x_n],we obtain Z xn

xn+1

|γ(x)f(x)|^pdxγ_nn⁻²

n

X

k=1

λ_k

!p

+γ_nn⁻² n

∞

X

k=n

|∆λ_k|

!p

and thus, byλ_n→0, (5.6)

Z π

0

|γ(x)f(x)|^pdx

∞

X

n=1

γ_nn⁻²

n

X

k=1

∞

X

m=k

|∆λ_m|

!p

+

∞

X

n=1

γ_nn^p−2

∞

X

k=n

∆λ_k

!p

. To estimate the first sum, we again use Lemma4.1with (4.1), thus, byγ_nn^β ↓with someβ >−1,

∞

X

n=1

γ_nn⁻²

n

X

k=1

∆_k

!p

∞

X

n=1

∆^p_n(γ_nn⁻²)^1−p

∞

X

k=n

γ_kk⁻²

!p

∞

X

n=1

γ_nn^p−2∆^p_n ≡

∞

X

n=1

γ_nn^p−2

∞

X

k=n

|∆λ_k|

!p

. This and (5.6) imply the second assertion of Theorem2.2, that is, (2.4)⇒(2.5).

We have completed our proof.

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References

[1] R. ASKEYANDS. WAINGER, Integrability theorems for Fourier series. Duke Math. J., 33 (1966), 223–228.

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