CONVERGENCE AND RATE OF APPROXIMATION

Approx. Theory & it s Appl.

　　　　　　　　　　　　　　17 : 4 , 2001 , 17 - 35

CONVERGENCE AND RATE OF APPROXIMATION IN BVφ FOR A CLASS OF INTEGRAL OPERATORS Sciamannini , S. 　and 　 Vinti , G. ( Universit àdegli Studi di Perugia , Italy )

Received 　Mar. 14 , 2001

Abstract We obtain estimates and convergence results with respect toφ2variation in spaces BVφ for a class of linear integral oper2 ators whose kernels satisfy a general homogeneity condition . Rates of approximation are also obtained . As applications , we apply our general theory to the case of Mellin convolution operators , to that one of moment operators and finally to a class of operators of f ractional order.

1 　Introduction

In [ 4 ] we have studied the problem of convergence for the family of moment type operators with respect to φ2variation. The concept of φ2variation , which represents a generalization of the classical Jordan variation , has been introduced by L. C. Young in [ 33 ] , and in [ 24 ] this concept was devel2 oped by J . Musielak and W. Orlicz in the direction of function spaces. Namely denoting by X the space of all Lebesgue measurable functions f : R →R and being φ: R0 →R0 a Φ2function , for ev2 +

+

+

+

ery f ∈X the Musielak2Orlicz φ2variation of f is defined as follows n

+

Vφ [ f ] = Vφ [ f ; R ] = sup Π

6 φ( |

f ( ti ) - f ( ti - 1 ) | ) ,

i =1

+ where the supremum is taken over all finite increasing sequences Π in R ( see [ 24 ,23 ] in case of a

bounded interval) . Then we define the space of functions with bounded φ2variation on R in the sense +

of Musielak2Orlicz , by © 1995-2004 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.

Approx . Theory & its Appl . 　　　　　　　　　　17 : 4 , 2001 1　　　　　　　　　　　 8 BVφ . = { f ∈ X : lim Vφ [λf ] = 0} λ→0

This space is connected with the theory of modular spaces , since the functional ρ: X →[ 0 , + ∞] , de2 fined by

ρ( f ) = Vφ [ f ] +| f ( a) | , for some a > 0 , f ∈X , is a convex modular on X ( see [ 23 ] ) . For references on the theory of modular spaces , see [ 23 ,17 ] . Therefore also the concept of convergence in φ2variation is connected with the modular conver2 gence ( see [ 25 ,23 ] ) and is defined in the following manner : a sequence ( f n ) n ∈N ∈ BVφ is said to be convergent in φ2variation to f ∈BVφ 　if there exists a

λ> 0 such that Vφ [λ( f n - f ) ] →0 as n →+ ∞. The sequence of integral operators taken into consideration in [ 4 ] is given by 1

Tn f ( t ) =

∫w ( s) f ( ts) d s , n

0

+

where w n : [ 0 ,1 ] →R0 is a sequence of kernel functions satisfying classical singularity assumptions n and f ∈BVφ . In the particular case of w n ( t ) = ( n + 1) t , t ∈[0 ,1 ] and n ∈N , the above operators

turn in particular into the moment operators introduced and studied by Baiada and his school ( see [ 14 , 1 ,34 ,29 ,30 ,31 ] ) ; for approximation results concerning moment type operators , see also [ 2 ,5 ,6 ,32 , 19 ,20 ] . For references about convolution operators , see [ 11 ] . It is proved that the above family of integral operators converges with respect to φ2variation and some estimates in BVφ 　are obtained. In this paper we consider a more general family of linear integral operators of the form ( Tw f ) ( s ) =

∫K ( s , t) f ( t) d t , +

w

R

+ defined for every f ∈X for which ( Tw f ) ( s ) is well2defined for every s ∈R and for every w > 0 , be2

ing Kw : R ×R →R0 a family of kernel functions satisfying a general homogeneity condition with re2 +

+

+

spect to a measurable function η. One of the main results of the paper is a convergence result of Tw f log

+ towards g , where g ∈ACφ ( R ) is of the form g ( t ) = η t ( t ) f ( t ) ( Theorem 4) . As a particular

case , for homogeneous kernels of degree - 1 , i . e. ,η( t ) = t

- 1

, we obtain a convergence result for

Tw f just towards f . Moreover , in order to study the rate of approximation for ( Tw f - g) , we introduce

a Lipschitz class which takes into account of the setting of the Vφ2variation. Such a class is connected with the modular Lipschitz class introduced to study rates of approximation in a modular frame ( see e. g. , [ 7 ,3 ] ) . In Theorem 5 a result on the order of approximation of ( Tw f - g ) is established. Finally , Section 6 contains several applications to linear operators whose kernels are homogeneous of degree - 1. This has been done in order to show some concrete examples to which the theory is ap2 ' 1995-2004 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.

Sciamannini , S . et al : Convergence and Rate of Approximation in BVφ

19

plicable in case we obtain Vφ2convergence results and rate of approximation for ( Tw f - f ) . In particu2 lar , in Section 6. 1 we consider a class of operators of Mellin convolution type. The theory of Mellin transform has been recently studied by P. L. Butzer and S. Jansche in [ 8 ,9 ,10 ] ; in these papers , the classical theory of discrete and continuous Fourier transform has been reconstructed for the discrete and continuous Mellin transform together with applications to Poisson summation formula , Kramer’s lemma and sampling theorem. Moreover , in Section 6. 2 , as particular case , the moment type operators are considered , and here we have found some previous results obtained in [ 4 ] for the Vφ2variation. In the last section ( Section 6. 3) we consider a class of fractional integral operators introduced by A. Erdé lyi and H. Kober in [ 13 ,16 ] . For results connected with the theory of fractional calculus , see e. g. ,[ 12 , 26 ,28 ,18 ,34 ,6 ,21 ,19 ] . In the applications several corollaries are established in order to prove ap2 proximation results and rates of approximation. 2 　Notations and definitions +

+

Let X be the space of all Lebesgue measurable functions f : R →R , where R = +

]0 , + ∞[ , and we set R0 = [ 0 , + ∞[ . + + Let Φ be the class of nondecreasing functions φ: R0 →R0 satisfying the following assumptions :

i ) φ( 0) = 0 , φ( u) > 0 for u > 0 ; + ii ) φ is a convex function on R0 ; - 1

+ iii ) u φ( u) →0 as u →0 .

From now on we will always suppose that φ ∈Φ. Now , for every f ∈X , we define the Musielak2Orlicz φ2variation of f as follows : n

+

Vφ [ f ] = Vφ [ f ; R ] = sup Π

6 φ( |

f ( ti ) - f ( ti - 1 ) | ) ,

i =1

+ where Π denotes an increasing finite sequence in R ( see [ 24 ,23 ] ) .

It is easy to see that the functional ρ: X →[ 0 , + ∞] , defined by

ρ( f ) = Vφ [ f ] +| f ( a) | , for some a > 0 , f ∈X , is a convex modular on X ( see [ 23 ] ) . In the following we will identify functions which differ from a constant . By means of the above modular ρ, we define the space of functions with bounded φ2variation on R in the sense of Musielak2Orlicz , as +

BVφ ( R ) = { f ∈ X :limρ(λf ) = 0} = { f ∈ X :lim Vφ [λf ] = 0}. +

λ→0

λ→0

It is possible to observe that by monotonicity and convexity of φ , it results © 1995-2004 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.

Approx . Theory & its Appl . 　　　　　　　　　　17 : 4 , 2001 2　　　　　　　　　　　 0 BVφ ( R ) = { f ∈ X : ϖλ > 0 , 　 s. t . Vφ [λf ] < + ∞} . +

+ + Moreover , if f ∈BVφ ( R ) , then f is bounded in R ( see [ 23 ] ) . + From now on , for a sake of simplicity , we will denote BVφ ( R ) simply by BVφ . + Now , we denote by ACφ ( R ) the subspace of BVφ 　consisting of all locally φ2absolutely con2 loc

+ + tinuous functions f : R →R , i . e. , the functions satisfying the following property ( see [ 24 ] ) :

there exists a constant λ > 0 such that for every bounded interval ]0 , c ] < R , with c > 0 and for +

every ε> 0 , there is aδ> 0 with the property that for every partition D = {0 < t 0 , t 1 , …, t m ≡c} of ] m

6 φ(λ|

0 , c ] with t i - t i - 1 < δfor i = 1 , …, m , the inequlity

f ( t i ) - f ( t i - 1 ) | ) < εholds .

i =1

Now we recall the following result about φ2variation , which we will use in the following ( see [ 24 ,4 ] ) : j ) if f 1 , f 2 , …, f n ∈X , then n

Vφ [

6

fi ] ≤

i =1

1 n

n

6

Vφ [ nf i ] .

i =1

From now on we will deal with the homotetic operator τh : X →X defined , for h ∈R , by τh f ( t ) = f +

( ht ) , for t ∈R+ . We now introduce a general homogeneity condition ( see [ 6 ,32 ,19 ,20 ,3 ] ) . Given measurable functions η: R →R and K : R ×R →R0 , we will say that K is an η2ho2 +

+

+

+

+

mogeneous function , if the following equality holds

η( t ) K ( sv , tv) = η( tv) K ( s , t ) ,

( 1)

+

for every t , s , v ∈R . Remarks . 　 α

a) As a particular case of ( 1) we may take η( t ) = t ,α ∈R and we obtain the definition of an homogeneous kernel of degree α∈R , i . e. , α

K ( sv , tv) = v K ( s , t ) +

for every s , t , v ∈R . In particular in case of α = - 1 , the previous equality is satisfied , for example , by the average kernel ( see [ 34 ,1 ,14 ,2 ,32 ,19 ,20 ,3 ]) of the form Mλ ( s , t ) = λs

- λ λ- 1

t

χ]0 , s [ ( t ) , 　　λ > 1 , 　　( s , t ) ∈ R+ ×R+ .

( 2)

　　b) We point out that equality ( 1) is not only satisfied by homogeneous kernels of some degree α ∈R. Indeed there exist kernels which aren’t homogeneous of any degree α ∈R but are η2homoge2 neous ; it is sufficient to take K ( s , t ) = H ( s , t ) η( t ) where H : R ×R →R0 is homogeneous of +

+

+

degree zero. Moreover , multiplying by η( t ) an homogeneous function of degree α ∈R , we obtain an ' 1995-2004 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.


21

α

homogeneous function with respect to t η( t ) . From now on we will denote by Kη the class of all η2homogeneous functions K. For K ∈ Kη , we define the linear integral operator

∫K( s , t) f ( t) d t ,

( Tf ) ( s ) =

( 3)

+

R

+ for every f ∈X such that ( Tf ) ( s ) is well2defined for every s ∈R .

As an example of operators ( 3) , we have , for every fixed λ> 1 , the average or moment operator Mλ ( s ) =

∫M ( s , t) f ( t) d t , +

λ

R

where Mλ is defined in ( 2) . Other examples of operators ( 3) will be discussed in Section 6 , as applications. Here , for K ∈ Kη , we put A K ∶=

∫z

-1

+

(η( z ) )

-1

K (1 , z) d z

R

and δ

A K ∶=

∫

z

| z - 1| > δ

-1

(η( z ) )

-1

K (1 , z) d z ,

for 0 < δ< 1. 3 　Estimates in BVφ

At first we establish an estimate for the operator Tf which gives an embedding theorem in BVφ . ) η( ・ ) Theorem 1. 　Let K ∈Kη and suppose that 0 < A K < + ∞. Then , if A K ( ・ ) ∈BVφ , Tf ∈BVφ and the following inequality holds , for every λ> 0 f (・ ) ] ≤ Vφ [λA K ( ・ ) η( ・ ) f (・ ) ]. Vφ [λ( Tf ) ( ・

( 4)

　　Proof . 　First we prove inequality ( 4) , since the first part of the theorem is an immediate conse2 quence , taking into account the definition of the space BVφ . Without loss of generality we can suppose λ = 1 ; then , putting t = zs , by η2homogeneity of K and setting g ( t ) = η t ( t ) f ( t ) , we have

∫K( s , t) f ( t) d t = ∫s K(1 , z) (η( z) ) = ∫z K(1 , z) (η( z) ) g ( zs) d z .

( Tf ) ( s ) =

+

+

R

-1

η( zs ) f ( zs ) d z

R

-1

-1

+

R

+

Now , we denote by D = { si } i = 0 , …, n an increasing sequence in R and we fix arbitrarily an index i ∈ © 1995-2004 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.

Approx . Theory & its Appl . 　　　　　　　　　　17 : 4 , 2001 2　　　　　　　　　　　 2

{1 , …, n} ; then we may write ( Tf ) ( si ) - ( Tf ) ( si - 1 ) =

∫ +

z

-1

K ( 1 , z ) (η( z ) )

-1

[ g ( zsi ) - g ( zsi - 1 ) ]d z ,

R

and hence , by nondecreasing of φ and by Jensen inequality , we get n

6 φ(|

( Tf ) ( si ) - ( Tf ) ( si - 1 ) | )

i =1

n

　　 ≤

6 φ(∫z +

K ( 1 , z ) (η( z ) )

-1

| g ( zsi ) - g ( zsi - 1 ) | d z )

R

i =1

1

∫ 1 　　 ≤ z A∫ 　　 ≤

-1

AK

R

K

R

+

+

z

n

-1

K ( 1 , z ) (η( z ) )

-1

6 φ( A

K

| g ( zsi ) - g ( zsi - 1 ) | ) d z

i =1

-1

K ( 1 , z ) (η( z ) )

-1

Vφ [ A K g ]d z

) η( ・ ) f (・ ) ]. 　　 = Vφ [ A K g ] = Vφ [ A K ( ・ The proof follows by arbitrariety of D . Remarks . 　 - 1

a ) We remark that in case of homogeneous kernels of degree - 1 , i . e. , η( t ) = t

, inequality

( 4) takes the form Vφ [λ( Tf ) ] ≤ Vφ [λA Kf ]

with A K =

∫K(1 , z) d z . Moreover , in the particular case of the moment kernel defined in (2) , it +

R

results A K = 1 and hence inequality ( 4) becomes Vφ [λ( Tf ) ] ≤ Vφ [λf ] ,

( 5)

i . e. , the operator T maps BVφ 　in itself ; for this result see also [ 4 ] . γ

b) If K ( s , t ) is homogeneous of degree γ ∈R , i . e. , η( t ) = t , then A K =

∫z +

- (1 +γ)

K (1 ,

R

z ) d z and inequality ( 4) given in this case a result in [ 5 ] . Moreover in [ 4 ] it is shown that in this

case it is not possible to replace in ( 4) the expression ) η( ・ ) f (・ ) with f ( ・ ) ; this is possible if and only if K is homogeneous of degree - 1 , as hap2 Ak (・ pens for example for the moment or average kernel . c) We point out that condition iii) on the function φ is not used in Theorem 1 ; so in case of φ ( u) = u , u ∈R0+ , inequality ( 5) gives a result of [ 1 ] which is the property of variation non aug2 menting for Tf . Now in order to obtain an estimate for the error of approximation of ( Tf - g ) , we introduce the

φ2modules of continuity of a function f ∈BVφ 　setting ' 1995-2004 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.


ωφ ( f ,δ) =

23

sup Vφ [τs f - f ] ,

| s - 1| ≤δ

for δ∈]0 ,1[ ( see e. g. , [ 4 ] ) . So we may state the following theorem. Theorem 2. 　Let K ∈K n , and 0 < A K < + ∞. If 4 ( A K + 1) g ∈BVφ , where g ( t ) =

η t ( t ) f ( t ) , then ( Tf - g ) ∈BVφ 　and the following inequality holds Vφ [λ( Tf - g) ] ≤

δ 1 ωφ ( 2λA K g ,δ) + 1 Vφ [ 4λA K g ] A K + 1 Vφ [ 2λ( A K-1) g ] , 2 2AK 2

( 6)

forδ∈]0 ,1[ and for every λ > 0. Proof . 　We prove inequality ( 6) since the first part of the theorem is an easy consequence. For

a sake of simplicity , we may suppose λ= 1. +

Let D = { si } i = 0 ,1 , …, n be an increasing sequence in R ; for any arbitrarily fixed index i ∈{1 ,2 ,

…, n} , there results ( Tf ) ( si ) - g ( si ) - ( Tf ) ( si - 1 ) + g ( si - 1 )

　　 = ( Tf ) ( si ) - A K g ( si ) - ( Tf ) ( si - 1 ) + A K g ( si - 1 ) 　　　 + A K g ( si ) - A K g ( si - 1 ) + g ( si - 1 ) - g ( si )

∫ 　　　 ∫z 　　 =

+

z

-1

K ( 1 , z ) (η( z ) )

-1

[ g ( zsi ) - g ( si ) ]d z

R

-1

+

K ( 1 , z ) (η( z ) )

-1

[ g ( zsi - 1 ) - g ( si - 1 ) ]d z

R

　　　 + A K [ g ( si ) - g ( si - 1 ) ] + [ g ( si - 1 ) - g ( si ) ] 　　 =

∫z

-1

+

K ( 1 , z ) (η( z ) )

-1

[ (τz g - g ) ( si ) - (τz g - g ) ( si - 1 ) ]d z

R

　　　 + [ A K - 1 ] [ g ( si ) - g ( si - 1 ) ] . Now we evaluate the expression Vφ [ Tf - g ] ; by convexity and nondecreasing of φ , we have n

6 φ( |

( Tf ) ( si ) - g ( si ) - ( Tf ) ( si - 1 ) + g ( si - 1 ) | )

　　 ≤

1 2

i =1

　　　 +

n

6 φ(2∫z +

-1

K ( 1 , z ) (η( z ) )

-1

| (τz g - g) ( si ) - (τz g - g ) ( si - 1 ) | d z )

R

i =1

n

6 φ(2 |

1 2

A K-1 | | g ( si ) - g ( si - 1 ) | ) = I1 + I2 .

i =1

We evaluate I1 . Applying Jensen inequality , we obtain I1 ≤

1

∫

2AK

+

R

z

n

-1

K ( 1 , z ) (η( z ) )

-1

6 φ(2 A

K

| (τz g - g ) ( si ) - (τz g - g ) ( si - 1 ) | ) d z

i =1

© 1995-2004 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.


1 2AK

∫z K(1 , z) (η( z) ) 1 = { + + }z 2A ∫ ∫ ∫ ≤

-1

-1

+

Vφ [2 A K (τz g - g) ]d z

R

K

1

1- δ

1 +δ

+∞

0

1- δ

1 +δ

2

-1

-1

K ( 1 , z ) (η( z ) )

Vφ [ 2 A K (τz g - g ) ]d z

3

= I1 + I1 + I1 . Now , taking into account the definition of ωφ ( ・,δ) , we evaluate I1 as follows. 2

1 +δ

1

2

I1 ≤

∫z

2AK

1- δ

-1

-1

K ( 1 , z ) (η( z ) ) ωφ (2 A K g ,δ) d z

1 ≤ ωφ (2 A K g ,δ) . 2 1

From condition j) in Section 2 , we have , for I1 : 1

1

I1 ≤

1- δ

∫z

4AK

-1

K (1 , z ) (η( z ) )

-1

{ Vφ [ 4 A Kτz g ] + Vφ [ 4 A K g ]}d z .

0

Since Vφ [4 A Kτz g ] ≤Vφ [4 A K g ] , we have 1- δ

1

1

I1 ≤

2AK

∫z

Vφ [4 A K g ]

-1

K ( 1 , z ) (η( z ) )

-1

-1

K ( 1 , z ) (η( z ) )

-1

dz ,

0

and , analogously , we obtain 3

+∞

1

I1 ≤

2AK

∫

Vφ [4 A K g ]

z

1 +δ

dz.

Hence I1 ≤

=

1 ωφ ( 2 A K g ,δ) + 1 Vφ [ 4 A K g ] 2 2AK

∫

z

-1

K ( 1 , z ) (η( z ) )

| z - 1| > δ

-1

dz

1 ωφ ( 2 A K g ,δ) + 1 Vφ [ 4 A K g ] AδK . 2 2AK

Finally , it remains to evaluate I2 . We have I2 =

1 2

n

6 φ(2 |

A K - 1 | | g ( s i ) - g ( si - 1 ) | ) ≤

i =1

1 Vφ [ 2 ( A K - 1) g ] . 2

Therefore ( 6) follows from the arbitrariety of D . Remark . 　We remark that Theorems 1 and 2 can be stated replacing the convexity assumption on

φ ∈Φ with the slighter assumption of quasi2convexity of φ , as defined in [ 19 ] ; see also [ 15 ] . 4 　An approximation theorem +

+

+

Let K = { Kw } w > 0 be a family of kernels Kw : R ×R →R0 , wiht Kw ∈ Kη for every w > 0 , and we put ' 1995-2004 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.

Sciamannini , S . et al : Convergence and Rate of Approximation in BVφ Aw ∶= A K

w

=

∫z

-1

(η( z ) )

+

-1

25

Kw ( 1 , z ) d z

R

and δ

δ

Aw ∶= A Kw =

∫

z

-1

| z - 1| > δ

(η( z ) )

-1

Kw ( 1 , z ) d z ,

for 0 < δ< 1. We will say that the family K is singular if Kw . 1) sup A w = A < + ∞ and lim A w = 1 ; w →+ ∞

w >0

δ

( Kw . 2) for every δ∈]0 ,1[ , there results : lim A w = 0. w →+ ∞

Now , we consider the following family of integral operators ( Tw f ) ( s ) =

∫K ( s , t) f ( t) d t , +

w

R

+ defined for every f ∈X such that ( Tw f ) ( s ) is well2defined for evry s ∈R and for every w > 0.

Now we introduce the concept of convergence with respect to φ2variation ( see [ 24 ,25 ,23 ,4 ] ) . A sequence ( f n ) n ∈BVφ is said to be convergent in φ2variation to f ∈BVφ 　if there exists a λ > 0 , such that Vφ [λ( f n - f ) ] →0 as n →+ ∞. In order to formulate the following Lemma 1 , Theorem 3 and Corollary 1 , without loss of generali2 + ty , we will consider functions belonging to BVφ ( R0 ) , i . e. , we extend with continuity our functions + + in t = 0. in this way , if f ∈BVφ ( R ) , then f ∈BVφ ( R0 ) , and

Vφ [λf ; R ] = Vφ [λf ; R0 ] , for every λ> 0. +

+

loc + Lemma 1. 　If f ∈ACφ ( R0 ) , there exists a λ > 0 such that for everyε> 0 there are constants

c , δ> 0 such that if D = { t 0 ≡0 , t 1 , …, t m ≡c} is a partition of [0 , c ] with t i - t i - 1 < δ, we have

a) Vφ [λf ,[ c , + ∞) ] < ε; m

b)

6

Vφ [λf ,[ t i - 1 , t i ] ] < ε;

i =1

c) the f unctionν: R0 →R defined by +

ν( t ) =

f ( ti - 1 ) ,

ti - 1 ≤ t < ti ,

f ( c) ,

t ≥c ,

　　　i = 1 , …, m

satisfies the property : Vφ [λ( f - ν) ; R0 ] 0 such that Vφ [λ gf ; R0 ] < + ∞ and we have +

+

Vφ [λ gf ; R0 ] = lim Vφ [λ gf ; [0 , cn ] ] , +

n →+ ∞

+ where ( cn ) n ∈N is an arbitrary increasing sequence in R .

Now , by a well2known property of the φ2variation ( see property 1. 17 of [ 24 ] ) , we have © 1995-2004 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.

Approx . Theory & its Appl . 　　　　　　　　　　17 : 4 , 2001 2　　　　　　　　　　　 6 Vφ [λ gf ; [0 , cn ] ] + Vφ [λ gf ; [ cn , + ∞) ] ≤ Vφ [λ gf ; R0 ] , +

and so Vφ [λ gf ; [ cn , + ∞) ] ≤ Vφ [λ gf ; R0 ] - Vφ [λ gf ; [ 0 , cn ] ] . +

From the last inequality , it immediately follows lim Vφ [λ gf ; [ cn , + ∞) ] = 0.

n →+ ∞

Therefore we have proved that there exists a λ g > 0 such that for every ε> 0 there is a constant c > 0 such that Vφ [λ gf ; [ c , + ∞) ] 0 such that if D = { t0 ≡0 , t1 , …, t m ≡ c} is a partition of [ 0 , c ] with the property that t i - t i - 1 < δ, then , m

6

Vφ [λf ,[ t i - 1 , t i ] ] < ε,

i =1

i . e. , b) holds.

λ and Vφ [ 2λ( f - ν) ; [ 0 , c ] ] < ε ( the last inequality Now , we take λ> 0 so small that 2λ ≤g follows from Theorem 2. 21 of [ 24 ]) . From property 1. 17 of [ 24 ] , being φ convex , we have Vφ [λ( f - ν) ; R0 ] ≤ +

1 { Vφ [ 2λ( f - ν) ; [ 0 , c ] ] + Vφ [ 2λ( f - ν) ; [ c , + ∞) ]}. 2

Now , being ν( t ) = f ( c) for t ≥c , we have from a) Vφ [2λ( f - ν) ; [ c , + ∞) ] = Vφ [2λf ; [ c , + ∞) ] < ε. + Therefre , being Vφ [ 2λ( f - ν) ; [ 0 , c ] ] < ε, we get Vφ [λ( f - ν) ; R0 ] < ε. So the Lemma is

proved. The above Lemma 1 is a fundamental tool in order to prove the following theorem. loc

+ Theorem 3. 　If f ∈ACφ ( R0 ) , then there exists a constant λ > 0 such that

lim Vφ [λ(τz f - f ) ] = 0. z →1

　　Proof . 　By Theorem 1 of [ 4 ] , it suffices to prove that lim+ Vφ [λ(τz f - f ) ] = 0.

z →1

The proof is analogous to that given in Theorem 1 of [ 4 ] , taking η = min{ applying Lemma 1 and taking into account that Vφ [

ti ti - 1

} , with 1 < z < η < c ,

λ (τν- ν) ; [ c , + ∞) ] = 0. 4 z

As an easy conseqence of Theorem 3 , we have the following corollary. loc

+ Corollary 1. 　If f ∈ACφ ( R0 ) , then there exists a constant λ> 0 such that

limωφ (λf ,δ) = 0.

δ→0

' 1995-2004 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.


27

　　Now we are ready to formulate the main approximation theorem with respect to φ2variation. Theorem 4. 　Let K < Kη be a f amily of singular kernels . If g ( t ) = η t ( t) f ( t) ∈ loc

ACφ ( R ) , there exists a constant λ > 0 sucht that +

lim Vφ [λ( Tw f - g) ] = 0.

w →+ ∞

+ 　　Proof . 　Since g ∈ACφ ( R ) , by Corollary 1 we may take a λ g > 0 such that wφ (λg ,δ) 0 ,0 < λ≤g gg ] < + ∞. Moreover , we may take a λ > 0 so small that 4λ( A + 1) 0. Remarks . 　

a) In the particular case of a family of kernels homogeneous of degree - 1 , as happens for example in the case of the moment kernel , the previous theorem gives the existence of a constant λ> 0 such that lim Vφ [λ( Tw f - f ) ] = 0 ,

w →+ ∞

that is we have in this case the convergence in φ2variation of Tw f just towards f and we extend a result given in [ 4 ] for the moment type operator. 5 　Rate of approximation Here we will introduce a concept of order of approximation with respect to φ2variation , i . e. , with respect to the modular taken into consideration. For the concept of order of approximation in a modular sense , see e. g. , [ 7 ,3 ] . © 1995-2004 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.

Approx . Theory & its Appl . 　　　　　　　　　　17 : 4 , 2001 2　　　　　　　　　　　 8 + + Let Γ be the class of all measurable functions γ: R →R0 such that γ( 1 ) = 0 , and γ( s ) ≠0

for s ≠1. For a fixed γ ∈Γ , we define the class + Lipγ ( Vφ) = { f ∈ BVφ ( R ) : ϖν > 0 , s. t . Vφ [ν(τs f - f ) = O (γ( s ) ) , as s →1} ,

where for any two functions f , g ∈X , f ( s ) = O ( g ( s ) ) as s →1 means that there are constants C > 0 ,δ> 0 , such that | f ( s ) | ≤C| g ( s ) | , for s ∈[ 1 - δ,1 + δ] . Let now X be the class of all f unctions ξ: R0 →R0 such that ξ is continuous at u = 0 , +

+

ξ( 0) = 0 and ξ( u) > 0 for u > 0. Let ξ∈X be fixed. Then we will say that the family K < Kη is ξ2singular if i ) there exist constants A , B > 0 , such that B ≤Aw ≤A , for every w > 0 ; - 1 ii ) Ωw = O (ξ( w ) ) as w →+ ∞, being Ωw = A w - 1 , for w > 0 ;

δ

iii ) for every δ∈]0 ,1[ , Aw = O (ξ( w

- 1

) ) , as w →+ ∞.

Now , we formulate the following theorem.

Γξ Theorem 5. 　Let γ ∈ , ∈X be fixed and let K < Kη be a f amily of ξ2singular kernels . As2 sume that there exists a δ> 0 such that

∫

z

| z - 1| < δ

-1

Kw ( 1 , z ) (η( z ) ) γ( z ) d z = O (ξ( w ) ) , as w →+ ∞. -1

-1

( 7)

Then , if f ∈X is such that g ( t ) = η t ( t ) f ( t ) ∈Lipγ ( Vφ) , then for sufficiently smallλ > 0 , we have Vφ [λ( Tw f - g ) ] = O (ξ( w ) ) , 　　 as w →+ ∞. -1

　　Proof . 　Since g ∈BVφ we may take λ > 0 so small that Vφ [ 4λDg ] < + ∞, being D = max { A , C}. From Theorem 2 applied to the family of integral operators ( Tw ) w > 0 , we have , for such a λ > 0 and for a fixed δ∈]0 ,1[ Vφ [λ( Tw f - g ) ] ≤

1

∫z

2 Aw +

1 +δ 1- δ

-1

Kw ( 1 , z ) (η( z ) )

-1

Vφ [ 2λAw (τz g - g) ]d z

δ 1 1 Vφ [ 4λA w g ] Aw + Vφ [ 2λ( A w - 1) g ] = I1 + I2 + I3 . 2 Aw 2

Since g ∈Lipγ ( Vφ) , there is a δ g > 0 and a ν> 0 such that Vφ [ν(τz g - g) ] ≤ Cγ( z )

for s ∈[1 - δ g ,1 +δ g ] and for a suitable constant C > 0. Moreover , let δ> 0 be such that ( 7) is satisfied ; so for δ ~ = min{δ,δ g} , we have , by i) of ξ2 singularity of K and 2λA ≤ν, I1 ≤

C 2 Aw

1+δ

∫z 1- δ

-1

Kw ( 1 , z ) (η( z ) ) γ( z ) d z -1



≤

29

1+δ

C 2B

∫z

-1

1- δ

Kw ( 1 , z ) (η( z ) ) γ( z ) d z -1

-1 = O (ξ( w ) ) , 　　 as w →+ ∞.

Moreover , by i) and iii ) of ξ2singularity of K , there results I2 =

δ δ 1 1 -1 Vφ [ 4λAw g ] Aw ≤ Vφ [ 4λA g ] Aw = O (ξ( w ) ) , 　　as w →+ ∞, 2 Aw 2B

since Vφ [ 4λAg ] < + ∞. Finally , since I3 =

1 Ωw g ] , by ii ) of ξ2singularity , by continuity of ξ at zero and by Vφ [ 2λ 2

convexity of φ ∈Φ , we have 1 2

I3 ≤ ξ( w ) Vφ [ 2λCg ] , -1

for a suitable constant C > 0 ; so being Vφ [ 2λCg ] < + ∞, by ii ) of ξ2singularity , we obtain I3 = O (ξ( w ) ) , 　　as w →+ ∞. -1

So the assertion follows. 6 　Applications In this section we will show some concrete examples of operators to which the previous theory is applicable. At first we will discuss the case of the Mellin convolution operators and , as a particular case , the moment type operators ; then we will consider a class of operators of fractional order. For the above classes of operators we will show convergence results and we will study the degree of approxima2 tion. Moreover , we remark that , since the kernels of the family taken into consideration are homoge2 neous of degree - 1 , we get , as corollaries , convergence and order of approximation results just for ( Tw f - f ) , being in this case g ( t ) = f ( t ) , for t ∈R+ . 611 　Mellin type convolution operators In this section we consider a family of kernels which are homogeneous of degree - 1 , i . e. , sati2 sfying the η2homogeneity condition with η( t ) = t +

+

- 1

+ , for every t ∈R . Let K = ( Kw ) w > 0 be a family

+

of kernels with Kw : R ×R →R0 of the form -1

Kw ( s , t ) = Hw ( st ) t +

-1

+

, 　　　s , t ∈ R ,

+

for every w > 0 , where Hw : R →R0 is a family of kernels satisfying the following assumptions ~ K w . 1)

∫H ( t

sup w >0

+

w

-1

) t- 1 d t = : A < + ∞

R



and lim

∫H ( t

w →+ ∞ R+

-1

) t- 1d t = 1 ;

w

　　K ~ w . 2) 　　for every δ∈]0 ,1[ , there results lim

∫

-1

-1

Hw ( t ) t d t = 0.

w →+ ∞ R+ \ ]1 - δ,1 +δ[

If K stisfies ~K w . 1) and ~K w . 2) , we will write K < K - 1 , where - 1 denotes that the kernels are ho2 mogeneous of degree - 1. Now , we define the family of Mellin type convolution operators of the form ( Tw f ) ( s ) =

∫K ( s , t) f ( t) d t = ∫H ( st +

w

+

R

-1

w

) f ( t)

R

dt t

.

We observe that

∫H ( t +

-1

w

) t- 1d t =

R

∫K (1 , t) d t = : A +

w

w

,

R

and analogously

∫

-1

+

R \ ]1- δ,1 +δ[

-1

Hw ( t ) t d t =

∫

δ

+

R \ ]1 - δ,1 +δ[

Kw ( 1 , t ) d t = : A w .

So if K < K - 1 , then K < Kη is singular ; therefore applying the previous theory to the above class of integral operators ( Tw ) w > 0 , we obtain the following corollary. loc + Corollary 2. 　Let K < K - 1 be the f amily of kernels above defined . If f ∈Aφ ( R ) , there ex2

ists a constant λ > 0 such that

lim Vφ [λ( Tw f - f ) ] = 0.

w →+ ∞

Now , concerning the order of approximation , we assume that ) there exists constants A , B > 0 , such that for every w > 0 , i’

∫H ( t

B ≤

+

w

-1

) t - 1 d t = : Aw ≤ A ;

R

) 　　ii ’

∫H ( t

[

-1

w

+

) t - 1 d t - 1 ] = :Ωw = O (ξ( w - 1 ) ) , 　　as w →+ ∞;

R

) for every δ∈]0 ,1[ , there results that 　　iii ’

∫

-1

-1

Hw ( t ) t d t =

| t - 1| > δ

∫

| t - 1| > δ

δ

Kw (1 , t ) d t = : Aw = O (ξ( w ) ) , 　as w →+ ∞. -1

) , ii ’ ) and iii ’ ) give the ξ2singularity of the family of kernels K. So we may for2 The assumptions i’ mulate the following corollary. ' 1995-2004 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.


31

Corollary 3. 　Let γ ∈Γ andξ∈X be fixed . Suppose that K is aξ2singular f amily of kernels and that there is a δ> 0 such that 1 +δ

∫H ( z 1- δ

) z - 1γ( z ) d z = O (ξ( w - 1 ) ) , 　　　as w → ∞.

-1

w

( 8)

Then , if f ∈Lipγ (Vφ) , we have , for sufficiently smallλ> 0 Vφ [λ( Tw f - f ) ] = O (ξ( w ) ) , 　　as w →+ ∞. -1

α

α

　　Example 1. 　As an example , we consider the case of γ( z ) = | 1 - z | and ξ( z ) = z ,0 < α< 1. Then the absolute α2moment of Kw , for w > 0 is

∫K (1 , z) γ( z) d z = ∫H ( z ) z | 1 -

mα ( Kw ) =

+

w

+

w

R

-1

α

-1

z | dz.

R

Therefore , if mα ( Kw ) = O ( w

-α

) , ( 8) of the previous corollary is automatically satisfied and , under

the assumption that K is a ξ2singular family of kernels , we obtain that Vφ [λ( Tw f - f ) ] = O ( w

-α

) , 　　as w →+ ∞.

　　61111 　Moment type operators Here we will discuss a particular case of the Mellin convolution integral operators which is given by the moment or average operators. In this case the moment of average kernel is givne by Kw ( s , t ) = M n ( s , t ) = ns

Here η( z ) = z

- 1

- n n- 1

t

χ[0 , s [ ( t ) , 　n ∈ N , 　n > 1 , 　s , t ∈ R+ .

, Aw = A n =

1

∫ +

∫

Mn (1 , z) d z = n

z

n- 1

dz = 1 ,

0

R

and δ

δ

Aw = A n =

∫ +

1- δ

R \ ]1 - δ,1 +δ[

∫z

M n (1 , z) d z = n

n- 1

n d z = ( 1 - δ) →0

0

as n →+ ∞, for δ∈]0 ,1[ . Therefore , in this case the family of kernels ( M n ) n ∈N satisfies the assumptions Kw . 1) and Kw . 2) of singularity given in Section 4. Thus we may formulate the following corollary. + Corollary 4. 　Let ( M n ) n ∈N be the f amily above defined . If f ∈ACφ ( R ) , there exists a con2 loc

stant λ > 0 such that

lim Vφ [λ( M n f - f ) ] = 0.

w →+ ∞

δ

n 　　Concerning the order of approximation , we have that Aw = A n ≡1 , n ∈N , and A n = ( 1 - δ) =

O( n

-α

) for n →+ ∞ and for every α > 0.

Take now γ( z ) = | 1 - z |

α

,0 < α< 1. Then we have


Approx . Theory & its Appl . 　　　　　　　　　　17 : 4 , 2001 3　　　　　　　　　　　 2 1 +δ

1 +δ

∫

∫z n ∫z

M n ( 1 , z ) γ( z ) d z = n

1- δ

n- 1

1- δ

α

χ]0 ,1[ ( z ) | 1 - z | d z

1

=

n- 1

1- δ

α

(1 - z) d z .

If now ξ( z ) = z ,0 < α< 1 , we have that the above family of kernels is ξ2singular and α

1 +δ

1

∫

1

∫z

α+1

M n (1 , z ) γ( z ) d z = n

ξ( n ) -1

1- δ

n- 1

1- δ

α+1

α

(1 - z ) d z ≤ n

B ( n ,α + 1) ,

where B is the Euler2Beta function. Now it is well2known ( see [ 27 ] ) that α+1

lim n

n →+ ∞

B ( n ,α + 1) = Γ(α + 1) ,

and so ( 8) of Corollary 3 is satisfied. Therefore we may formulate the following corollary. Corollary 5. 　Let be γ( z ) = | 1 - z |

α

α

+ ,ξ( z ) = z , z ∈R0 ,0 < α < 1. If f ∈Lipγ ( Vφ) , then

Vφ [λ( M n f - f ) ] = O ( n

-α

) , 　　as n →+ ∞,

for sufficiently small λ> 0.

612 　Convolution operators of fractional order In this section we consider the family of kernels which are homogeneous of degree α

- 1 ,{ Hn } n ∈N defined , for α ∈]0 ,1[ , by α

Hn ( s , t ) =

α- n Γ( n + 1 - α) - α n- 1 + + s ( s - t ) t χ]0 , s [ ( t ) , 　　( x , t ) ∈ R ×R . ( n - 1) !Γ( 1 - α)

By means of the above kernels , we may define the family of Erdelyi2Kober fractional operators ( see [ 13 ,16 ] ) of the form ( Tn f ) ( s ) =

∫H ( s , t) f ( t) d t . α

+

n

R

In this case η( t ) = t

- 1

+∞

∫

Aw = A n =

and α

Hn ( 1 , t ) d t =

0

=

Γ( n + 1 - α) ( n - 1) !Γ( 1 - α)

1

∫(1 -

t)

- α n- 1

t

dt

0

Γ( n + 1 - α) Γ( n + 1 - α) Γ( 1 - α) Γ( n) B ( 1 - α , n) = = 1. ( n - 1) !Γ( 1 - α) ( n - 1) !Γ( 1 - α) Γ( n + 1 - α) Γ( n + 1 - α) , it results ( n - 1) ! Γ( 1 - α)

Moreover , for δ∈]0 ,1[ and putting En ,α = δ

∫

δ

Aw = A n = En ,α

+

R \ ]1 - δ,1 +δ[

1- δ

∫

= En ,α

0

t

n- 1

(1 - t )

-α

t

n- 1

-α

( 1 - t ) χ]0 ,1[ ( t ) d t

dt ≤

En ,α α

δ

1- δ

∫ 0

t

n- 1

dt =

En ,α α

nδ

( 1 - δ) n .

It is well2known ( see [ 27 ] ) that En ,α is an infinite for n →+ ∞of degree 1 - α with respect to n . ' 1995-2004 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.


33

Hence lim A n = 0 and therefore the family { Hn } n ∈N is singular and we may state the following corol2 δ

α

n →+ ∞

lary. + Corollary 6. 　Let { Hn } n ∈N be the f amily above defined . If f ∈ACφ ( R ) , there exists a con2

α

loc

stant λ > 0 such that

lim Vφ [λ( Tn f - f ) ] = 0.

n →+ ∞

δ

Now we discuss the assumptions concerning the degree of approximation. In this case we have A n = O (n

-β

) , for every β> 0.

Finally , take γ( z ) = | 1 - z | 1 +δ

∫

β

,0 < β< 1. Then for 0 < δ< 1 , we have that 1 +δ

α

Hn ( 1 , z ) γ( z ) d z =

1- δ

∫H (1 , z) | 1 -

1

∫(1 -

　　 = En ,α

z)

1- δ

1- δ

α n

β- α n - 1

z

β

z | dz

d z ≤ En ,αB ( 1 + β - α, n ) .

+ Now if ξ( z ) = z , z ∈R0 ,0 < β< 1 , then from a well2known result ( see [ 27 ] ) , we have

β

β+1 - α

β

lim n En ,αB ( 1 + β - α, n ) = lim

n →+ ∞

Γ( n + 1 - α) ( β α ) Γ( 1 - α) ( n - 1) ! n1 - α B 1 + - , n n

n →+ ∞

=

Γ( 1 + β - α) , (Γ( 1 - α) ) 2

and so ( 7) of Theorem 5 holds and the family { Hn } n ∈N is ξ2singular. Hence we have the following α

Corollary 7. 　Let γ( z ) = | 1 - z |

β

β

,ξ( z ) = z , z ∈R0 ,0 < β< 1. If f ∈Lipγ ( Vφ) then +

-β

Vφ [λ( Tn f - f ) ] = O ( n ) , 　　 as n →+ ∞ for sufficiently small λ> 0. Acknowledgements . 　The authors wish to thank Prof . C. Bardaro for the interesting and helpful

discussions in the matter. The authors want to thank also Prof . P. L. Butzer for the integesting discus2 sions about the rate of convergence with respect to the variation and the referee for some historical sug2 gestion.

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Dipartimento di Matematica e Informatica Universitádegli studi di Perugia Italy
