Andrica-Piticari. 86KB Jun 04 2011 12:08:21 AM

AN EXTENSIONS OF THE RIEMANN-LEBESGUE

LEMMA AND SOME APLICATIONS

by Dorin Andrica,

Mihai Piticari

Abstract. We show that a similar relation as (1.2) holds for all continuous and bounded functions g:[0,∞)→R of finite Cesaro mean. A result concerning the asymptotic behavior in (2.10) when a = 0, b = T and f ∈C1[0,T] is given in Theorem 3.1. Some concrete applications and examples are presented in the last section of the paper.

Keywords: Riemann-Lebesgue Lemma, T - periodic function. Mathematics Subject Classification (2000): 26A42, 42A16

1. Introduction

The well-known Riemann-Lebesgue Lemma (see for instance [6, pp.22, Problem 1.4.11(a) and (b)] or [9, Problem9.48]) states that if f :[a,b]→R is a Riemann integrable function on [a, b] then

∫

∞ →

b a

nlim f(x)sinnxdx = →∞

∫

b a

nlim f(x)cosnxdx = 0 (1.1)

That implies a classical result in Fourier analysis, i.e. if f :R→R is a

π

2 -periodic with Fourier coefficients an, bn, then under suitable conditions

) ( )

(x f x

S_n → , where

) (x

S_n = ₀

2 1

a +

∑

= +

n

k

k

k kx b kx

a 1

) sin cos

( .

be a continuous function, where 0 ≤ a < b. Suppose the function g:[0,∞) → R to be continuous and T-periodic. Then

∞ →

n

lim

∫

=

∫

∫

b

a T

b

a f x g nx dx _T g(x)dx f(x)dx

1 ) ( ) (

0 (1.2)

(see [5] for the special case a = 0, b = T).

There are other extensions and generalization of the important result contained in (1.1). We mention here Knyazyuk A.V.[8] which gives necessary and sufficient conditions for the existence of the limit.

∞ → λlim

∫

b

ah(x,g(λ x)dx (1.3)

for a given function g : [0,∞) → R and all continuous functions h : [a,b] ×

g([0,∞)) → R. The following multidimensional generalization of Riemann-Lebesgue Lemma is given in the recent paper Canada, A., Urena, A.J. [2, Lemma 3.1] and it is used to study the asymptotic behavior of the solvability set for pendulum-type equations with linear damping and homogeneous Dirichlet conditions: Let g : R →R be a continuous and T-periodic function

with zero mean value and let u be given functions satisfying the following ₁

property are real numbers sucs that

0 0 ) ( ' :

] , 0 [

1

>    



 _∈

∑

₌

=

N

i i iu x

x

meas π ρ

then ρ₁ =...=ρ_N =0. Let B⊂C1[0,π]be such that the set

{

b':b∈B

}

is uniformly bounded in C[0,π]. Then for any given function r∈L1[0,π]we have u

0 ) ( ) ( ) ( lim

0 1 ||

||  =

 

 ₊

∫ ∑

₌

∞ →

π

ρ g ρu x b x r x dx

N

i i

i (1.4)

uniformly with respect to b∈ B. Other multidimensional version of

Some other extensions and generalizations of Riemann-Lebesgue Lemma were obtained by Bleistein, N., Handelsman, R. A., Lew, J. S. [1], Kantor, P. A. [7] and Stadije, W. [10].

In this paper we extend the relation (1.2). Our main result is contained in Theorem 2.1 and it shows that a similar relation as (1.2) holds for all continuous and bounded function g : [0,∞) → R of finite Cesaro mean. Some applications are also given.

2. The main results

Let us begin whit some auxiliary results which will help us to derive our main result.

Lemma 2.1. Let ω : [0,∞)→ R be a continuous function such that

∞ →

xlim _x

x) (

ω _{= 0. If}

1

)

(c_{n n≥} is a sequence of non-negative real numbers such that

1

≥

     

n n

n c

is bounded, then

n c_n

n

) ( limω

∞

→ = 0.

Proof. We consider the following cases. Case 1. There exists _n

n→∞c

lim and it is +∞. In that case we have

n c_n) (

ω ₌

n n n

n n

c c M n c c

c ) ( )

( ω

ω _{⋅ ≤ , (2.1)}

where M = sup

   



 _:n≥1

n c_n

. Since

∞ →

xlim _x

x) (

ω _{= 0, from (2.1) it follows}

0 ) (

lim =

∞ → n

c_n

n

ω _{hence the desired conclusion.}

Case 2. The sequence (c_n)_n≥1 is bounded. Consider A > 0 such that

A

c_n ≤ for all positive integers n≥1, and define K= sup ( )

] , 0 [

x

A x∈ ω

. It is clear that

n K n

c_n

≤

) (

ω _{for all n ≥1, i.e.}

n c_n

n

) ( limω

∞

Case 3. The sequence

( )

c_{n n≥1} is unbounded and _n n→∞c

lim does not exist. For ε >0 there exists δ >0 such that

M x

x ε

ω( ) _<

for any x >δ , where

M = sup

   



 _:n≥1

n c_n

. Consider the sets

A_δ = {n∈N*:cn≤δ } and Bδ = {n∈N* : cn>δ }

If one of these sets is finite, then we immediately derive the desired result. Assume that A_δ and B_δ are both infinite. Since

n c_n

A n

) ( limω

δ

∈ = 0, it

follows that there exists N₁(ε) with the property ω <ε

n c_n) (

for all n∈ A_δ

with n ≥ N₁(ε). For n∈B_δ, we have

n c_n) (

ω ₌ ω _{⋅ <} ε _{M =ε} M

n c c

c _n

n n)

(

, (2.2)

i.e

n c_n) (

ω _<

ε for any n ≥N₁(ε). Finally

n c_n

n

) ( limω

∞

→ =0.

Lemma 2.2 Let f :R→R be a continuous non-constant periodic function. If F is an antiderivative of f, then

F(x) = 1 f(t)dt x g(x)

T

T

o  +

 





∫

_{, (2.2)}

where T > 0 is the period of f and g : R →R is a T-periodic function.

Proof. Using the relation f(t + T) = f(t) for any t∈R it follows F(x + t)

-F(x) =

∫

T

o f(t)dt for all x∈R. Considering the function h(x) = _T f t dt x T

o 

 



we have h(x + T) – h(x) =

∫

T

o f(t)dt, i.e. F(x + T) - h(x + T)=F(x) - h(x). That is

the function defined by g(x) = F(x) - h(x), x∈R, is periodic of period T. Moreover the formula (2.2) holds.

Lemma 2.3 Consider 0≤ a < b to be real numbers and let f:[a,b]→R be a function of class C1. Let g : [0,∞) →R be a continuous function such that

∫

=

∞ →

x

x

L dt t g x 0 ( )

1

lim (finite). Then the following relation holds:

∫

∫

=

∞ →

b a b

a

nlim f(x)g(nx)dx L f(x)dx

(2.3)

Proof. If G(x)=

∫

xg t dt

0 ( ) , then define the function ω(x)=G(x)−Lx,

0 ≥

x . It is clear that ω is differentiable and it satisfies lim ( ) =0 ∞

→ _x

x x

ω _.

We have

∫

=

a

b n

dx nx g x

f( ) ( ) 1 − −

∫

b

a

dx nx G x f n na G a f b f nb

G( ) ( ) ( ) ( )) 1 '( ) ( )

( =

= − −

∫

+

b

a

dx nx Lnx

x f n na G a f b f nb G

n '( )( ( ))

1 )) ( ) ( ) ( ( ( ( 1

ω =

= − −

∫

−

∫

=

b

a

a

b

dx x f nx n

xdx x f L na G a f b f nb G

n ( ) '( )

1 )

( ' ))

( ) ( ) ( ) ( ( 1

ω

+ −

−

− ( ) ( )) ( ( ) ( )) )

( ) ( ( 1

a af b bf L na G a f b f nb G n

+ nx f x dx

n dx x f L

b

a b

a

) ( ' ) ( 1 )

(

∫

∫

− ω

0 ) ( ' ) ( 1

lim

∫

=

∞

→ n nx f x dx

b a

n ω (2.4)

Indeed, from the relations

dx x f nx n

b

a ( ) '( )

1

∫

ω ≤ nx f x dx

n b

a| ( )|| '( )| 1

∫

ω =

=

∫

b

a n

dx x f n

c

, | ) ( ' | | ) ( |ω

where na<c_n <nb, by applying Lemma 2.1 it follows lim ( ) =0 ∞

→ _n

c_n

n

ω

, i.e. the equality (2.4) holds.

Moreover we have

   



 ₋

=

− _→∞

∞

→ ( )

) ( ) ( ) ( lim )) ( ) ( ) ( ) ( ( 1

lim f a

na na G a b f nb

nb G b a

f na G b f nb G

n n

n

=

= L(bf(b)−af(a)) Using (2.4) and (2.5) the relation (2.3) follows.

Theorem 2.1 Let 0 ≤ a < b, be real numbers and let f : [a,b] →R be a continuous function. Consider the function g : [0,∞) →R to be continuous and bounded such that

∫

=

∞ →

x

x _x 0g(t)dt L

1

lim (finite). Then

∫

∞ →

b

a n

lim f x g nx dx L b f x dx a ( )

) ( )

( =

∫

(2.6)

) 2 )( (b−a L+

ε _{. Then we can find a positive integer M(ε) such that} ε

< − ( )| )

(

| f_m x f x ’ for any m ≥ M

( )

ε and for all x∈[a,b]. If m≥I(ε), then we have

dx nx xg f dx nx g x

f b

a m

b

a ( ) ( )

∫

( ( )

∫

− ≤ b f_m(x) f(x)

a −

∫

( ) '( )

a b dx nx

g ≤ε − ,

for all positive n. Therefore, for m ≥M

( )

ε and n∈N*

≤ ≤

−

−

∫

∫

f x g nx dx b a b f x g nx dx a

m b

a ( ) ( ) ( ) ( ) ( )

' ε

b a b f_m x g nx dx

a ( ) ( )

) (

'

∫

+ −

≤ε (2.7)

from Lemma 2.3. it follows

∞ → n

lim f (x)g(nx)dx L b f_m(x)dx. a

m b

a

∫

∫

=

Hence, for ε > 0 one can find a positive integer N

( )

ε,m such that for all ≥

n N

( )

ε,m we have

).L f (x) '(b a) f (x)g(nx)dx L b f_m(x)dx '(b a a

m b a m

b

a − − <

∫

<

∫

+ −

∫

ε ε

Using the last inequalities and (2.7) we get

< <

−

−

∫

∫

f x dx b a f x g nx dx

L b

a m

b

a ( ) 2 ( ) ( ) ( )

'

ε

L f (x)dx 2 '(b a).

b

a

m + −

<

∫

ε (2.8)

But for all m≥M(ε) we have f(x)-ε ' < fm(x) < f(x)+ε

'

for any

x∈[a,b]. Thus

) ( )

( )

( )

( )

(x dx L ' b a f x dx L f x dx L ' b a f

L

b

a b

a m b

a

− +

< <

−

−

∫

∫

From (2.8) and (2.9) it follows

' '

) 2 )( ( ) ( )

( ) ( )

2 )( ( )

( − − + ε <

∫

<

∫

+ − + ε

∫

f x dx b a L f x g nx dx L f x dx b a L L

b

a b

a b

a

for all positive integers n≥N(ε,m), i.e.

ε ε = + − < −

∫

∫

'

) 2 )( ( ) ( )

( )

(x g nx dx L f x dx b a L f

b

a b

a

and the desired conclusion is obtained.

Corollary 2.1 Let 0≤a< b be real numbers and let f : [a,b]→R be a continuous function. Consider the function g : [0,∞) → R to be continuous such that g x L

x→∞ =

) (

lim (finite). Then

∞ →

xlim

∫

=

∫

b

a b

a

dx x f L dx nx g x

f( ) ( ) ( ) .

Proof. Because g is continuous and limg(x)

x→∞ = L we obtain that g is

bounded. Moreover, applying L’Hospital rule it follows

∞ →

x

lim

∫

x

dt t g

0

) ( =

∞ →  x

lim g(x) = L

and the conclusion follows from Theorem 2.1.

Corollary 2.2 (Riemann-Lebesgue Lemma) Let f : [0,b] → R be a continuous function, where 0 ≤ a ≤ b. Consider the function g : [0,∞) → R

∫

∫

∫

=

∞ →

b

a T

b

a

n f x g nx dx _T g(x)dx f(x)dx

1 ) ( ) ( lim

0

. (2.10)

Proof. Using Lemma 2.2 it follows that the function G(x) = 

  

 

∫

x

dt t g T 0 ( )

1

has the representation

G(x) = 

  

 

∫

T

dt t g T 0 ( )

1

x + h(x), x∈[0,∞)

where h is continuous and periodic. We have

x x G( )

=

∫

Tg t dt T 0 ( )

1

+

x x h( )

,

hence

∞ →

xlim _x

1

∫

x dt t g

0 ( ) =

∫

T

dt t g T 0 ( )

1

, and the relation (2.10) follows from Theorem 2.1.

Remark. If g : [0,∞) → R is continuous and almost-periodic, then its Cesaro mean

∞ →

xlim _x

1

∫

x dt t g

0 ( ) exists and it is finite (see for instance our paper

[1] or any book on almost-periodic functions). Therefore the class of continuous functions g in Theorem 2.1 is more general than the class of functions g in Corollary 2.2.

3. Some applications

First of all we derive a result concerning the asymptotic behavior in (2.10) when a = 0, b = T and f∈C1[0,T].

Theorem 3.1. Let g : [0,∞) → R be continuous and T-periodic and let C1 [0,T]. Then

∫

∫

∫

=

∞ →

b

a T

b

a

n f x g nx dx _T g(x)dx f(x)dx

1 ) ( ) ( lim

    ₋ −

= f T f G t T

∫

G x dx T ( ( ) (0)) ( ) ₀ ( )

1

(3.1)

where =

∫

x dt t g x G 0 . ) ( ) (

Proof. Denote by =

∫

T

g g t dt

T M 0 , ) ( 1

the Cesaro’s mean of g on the interval [0,T]. According to Lemma 2.2 we have G(x)=M_g ⋅x+h(h),

) , 0 [ ∞ ∈

x , where h is continuous and T-periodic. We can write

∫

∫

∫

∫

∫

∫

∫

∫

∫

∫

′ − = = ′ − + − − = ′ − ′ − − + = ′ − = = ′ − = ′ = T T g T T g g g T T g T g

T T T

T dx nx h x f dx x f nM dx nx h x f dx x f nM T Tf nM T Tf nM dx nx h x f dx x f x nM nT h nT M T f dx nx G x f nT G T f dx nx G x f nx G x f dx nx G x f dx nx g x f n 0 0 0 0 0 0 0

0 0 0

0 . ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )) ( )( ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (

That is equivalent to

∫

∫

∫

∫

_=− ′



T ₋ T T T

dx nx h x f dx x f dx x g T dx nx g x f n 0

0 0 0

. ) ( ) ( ) ( ) ( 1 ) ( ) ( (3.2)

  

 ₋

− =

= ′

− =

= ′

= ′

∫

∫

∫

∫

∫

∫

∞ →

T

T T

T T

T

n

T G dx x G f

T f T

dx x f dx x h f

T f T

dx x f dx x h T dx nx h x f

0

0 0

0 0

0

) ( )

( )) 0 ( ) ( ( 1

) ( )

( )) 0 ( ) ( ( 1

) ( )

( 1 ) ( ) ( lim

and the desired formula is obtained from (3.2).

Remark. In the similar way we get the following result when a = kT, b = (k+1)T, where k is a fixed positive integer:

∞ → n

lim 

∫

+ −

∫

T

∫

k+ T

kT T

k

kT f x g nx dx _T 0 g x dx f x dx )

1 ( )

1 (

) ( )

( 1 )

( )

( =

= 1(f((k 1)T) f(xT))

T + − 

 

 ₋

∫

+

dx x G T

G k T

kT

) 1 (

) ( )

( . (3.3)

The asymptotic behavior of the Fourier coefficients is given in the following result.

Corolary 3.2. Let f : [0,∞) → R be a function of class C1. Then

∞ → n

limn

∫

+ π π

) 1 ( 2

2 ( )sin

k

k f x nxdx = f(2kπ ) - f(2(k+1)π) (3.4)

∞ → n

limn 2( 1) ( )cos 0

2 =

∫

k+ f x nxdx

k

π

π . (3.5)

Proof. In formula (3.3) take g(x) = sinx, respectively g(x) = cos and x

obtain the desired results.

In what follows we shall present some concrete applications.

∞ → n

lim

∫

1 =

0 f(x)f(nx)dx

2 1

0 ( ) 

 





∫

_{f x dx . (3.6)}

2) If f is of class C1 on [0,∞) then

∞ → n limn

  



  

 −

∫

∫

1 2

0 1

0 f(x)f(nx)dx f(x)dx = 0. (3.7)

Indeed, from Corollary 2.2 we have

∞ → n

lim

∫

1 =

0 f(x)f(nx)dx ₁

1

∫

∫

∫

1 = 

0

2 1

0 1

0 f(x)dx f(x)dx f(x)dx ,

that is (3.6). In order to obtain (3.7) we apply Theorem 3.1 and the relation ).

1 ( ) 0

( f

f =

Application 2. The following relation holds:

=

∫

∞ →

π π

2 sin

lim dx

x nx

n ln2.

2

π (3.8)

Let us apply Corollary 2.2 to functions f :[π,2π]→R,

x x

f( )= 1 and →

∞] , 0 [ :

g R, g(x) = sinx . Taking into account that g is periodic of period

π, it follows

. 2 ln 2 1 sin

1 sin

lim 2

0 2

π

π π π

π

π =

∫

∫

=

∫

∞

→ _x dx _x xdx _xdx

x

n

Application 3. The following relation holds

=

∫

∞

→ _x dx

nx n

n

π π 4 2 2

sin

lim ₂

16 3

In formula (3.3) take g(x)=sinx, ( ) 1₂,

x x

f = T = 2π and k = 1. We

have G(x)=−cosx+1 and

∫

π =

π π

4

2 G(x)dx 2 . Then

. 16

3 ) 2 ( 4

1 16

1 2

1 sin

lim 4 _{2 2 2}

2 2 π π π π π

π

π _ − =

 



 ₋

=

∫

∞

→ _x dx

nx n

n

For a fixed positive integer k we have

2 2 )

1 ( 2

2 2 _{4( ( 1)}

1 2 sin

lim

π

π

π +

+ =

∫

+ ∞

→ _{k k}

k x

nx

n k

k

n (3.10)

(see also formula (3.4) in Corollary 3.2).

Application 4. ([9, Problem 9.2]) The following relation holds:

2 cos

1 sin lim

0 + 2 =

∫

∞

→ _nxdx

x n

π

. (3.11)

We apply Corollary 2.2 to functions f :[π,2π]→R, f(x)=sinx and →

∞] , 0 [ :

g R,

x x

g ₂

cos 1

1 )

(

+

= . The function g g is periodic of period π, hence

= +

=

+

∫

∫

∫

∞

→ _nxdx _xdx xdx

x

n

π π

π

π 0 2 0

0 2 _{1 cos} sin

1 1

cos 1

sin lim

=

∫

∫

∫

∞

+ =

+ =

+ 0 2

2

0 2

0 2 ₁

4 cos

1 4 cos

1 2

u du x

dx x

dx

π π

π

π π

= 2

2 2

4

0=

∞

u arctg

References:

[1] Andrica, D., On a large class of derivatives (Romanian), GM-A, No. 4 (1986), 169-177.

[2] Bleinstein, N., Handelsman, R.A., Lew, J. S., Functions whose Fourier

transforms decay at infinity; An extension of Riemann- Lebesque Lemma,

SIAM j. Math Anal. 3(1972), 485-495.

[3] Canada, A., Urena, A. J., Asymptotic behavior of the solvability set for

pendulum type equations damping and homogeneous Dirichlet conditions,

USA-Chile Workshop on Nonlinear Analysis, Electron. J. Diff. Eqns., Conf. 06, 2001, pp. 55-64.

[4] Canada, A., Riuz D., Periodic perturbations of a class of resonant problems, Preprint 2001.

[5] Dumitrel, F., Problems in Mathematical Analysis (Romanian), Editura SCRIBUL, 2002.

[6] Kaczor, W. J., Nowak, M. T., Problems in Mathematical Analysis III, Student Mathematical Library Volume 21, American Mathematical Society, 2003.

[7] Kantor, P. A., Extension of the Riemann - Lebesque lemma,J. Mathematical Phys. 11 (1970), 3099 -3103

[8] Knyazyuk, A. V., On a generalization of Riemann lemma (Russian), Dok 1 Akad. Nauk Ukrain SSR Ser A (1982), nr. 1, 19-22

[9] Siretki, Gh, Mathematical Analysis II, Advanced problems in Differential

and Integral Calculus (Romanian), University of Bucharest, 1982.

[10] Stadje, W., Eine Erweiterung des Riemann-Lebesque Lemmas, Monatsh. Math. 89(1980), no. 4, 315-322.

Authors:

Dorin Andrica - “Babes-Bolyai” University, Cluj Napoca, Romania, E-mail address: dandrica@math.ubbcluj.ro