Lp approximation with rates by multivariate generalized discrete singular operators

Applied Mathematics and Computation 265 (2015) 652–666

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Lp approximation with rates by multivariate generalized discrete singular operators George A. Anastassiou∗, Merve Kester Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152, U.S.A.

a r t i c l e

i n f o

a b s t r a c t

Keywords: Multivariate discrete singular operator Lp modulus of smoothness Lp convergence

Here we give the approximation properties with rates of multivariate generalized discrete versions of Picard, Gauss–Weierstrass, and Poisson–Cauchy singular operators over RN , N ≥ 1. We treat both the unitary and non-unitary cases of the operators above. We derive quantitatively Lp convergence of these operators to the unit operator by involving the Lp higher modulus of smoothness of an Lp function. © 2015 Elsevier Inc. All rights reserved.

1. Introduction This article is motivated mainly by [6], where Favard in 1944 introduced the discrete version of Gauss–Weierstrass operator

1 (Fn f )(x ) = √

∞ 

π n ν =−∞

f

ν  n



exp −n

ν

n

−x

2 

,

(1)

n ∈ N, which has the property that (Fn f)(x) converges to f(x) pointwise for each x ∈ R, and uniformly on any compact subinterval 2 of R, for each continuous function f ( f ∈ C (R )) that fulfills | f (t )| ≤ AeBt , t ∈ R, where Aand B are positive constants. We are also motivated by [1]–[5]. In this article, we study Lp approximation properties of the multivariate generalized discrete versions of the Picard, Gauss– Weierstrass, and Poisson–Cauchy singular operators. We cover the unitary and non-unitary cases and their interconnections. 2. Background In [1], for r ∈ N, m ∈ Z+ , the author defined

⎧  ⎪ ⎨(−1 )r− j rj j−m , if j = 1, 2, . . . , r, α [m] := r r

j,r ⎪ ⎩1 − (−1 )r− j j j−m , if j = 0,

(2)

j=1

where r 

α [m] = 1, j,r

j=0

∗

Corresponding author. Tel.: +9017513553. E-mail addresses: [email protected] (G. A. Anastassiou), [email protected] (M. Kester).

http://dx.doi.org/10.1016/j.amc.2015.05.073 0096-3003/© 2015 Elsevier Inc. All rights reserved.

(3)

G. A. Anastassiou, M. Kester / Applied Mathematics and Computation 265 (2015) 652–666

653

and [m] δk,r :=

r 

α [m] jk , k = 1, 2, . . . , m ∈ N. j,r

(4)

j=1

Moreover, the author stated

ru f (x ) := ru1 ,u2 ,...,uN f (x1 , . . . , xN ),  r  r (−1 )r− j f (x1 + ju1 , x2 + ju2 , . . . , xN + juN ), :=

(5)

j

j=0

and the modulus of smoothness of order r as

ωr ( f ; h ) p := sup ru ( f ) p ,

(6)

u2 ≤h

where p ≥ 1 and h > 0. Then, in [1], the author introduced the multiple smooth singular integral operators as [m] θr,n ( f ; x1 , . . . , xN ) :=

r 

α [m] j,r

j=0

 RN

f (x1 + s1 j, x2 + s2 j, . . . , xN + sN j )dμξn (s ),

(7)

where μξn be a probability Borel measure on RN , N ≥ 1, ξ n > 0, n ∈ N, s := (s1 , . . . , sN ), x := (x1 , . . . , xN ) ∈ RN ; r ∈ N, m ∈ Z+ , f : RN → R is a Borel measurable function, and also (ξn )n∈N is a bounded sequence of positive real numbers. In [1], for f ∈  j

) + C m (RN ),m ∈ Z + ,with fα ∈ L p (RN ),|α| = m ∈ Z + ,p ≥ 1; wherefα denotes the mixed partial ∂ fα(1·,ldots,· α ,α j ∈ Z , j = 1, . . . , N : |α| := ∂ x1 ...∂ xNN N αj =  j, j = 1, . . . , m, the author gave the following results: j=1

Theorem 1. Let f ∈ C m (RN ), m ∈ N, N ≥ 1, with fα ∈ L p (RN ), |α| = m, x ∈ RN . Let p, q > 1 : ity measure on m that we have



 RN

 |si |αi

i=1

s  2 1+ ξn



Then

R

1 q

= 1. Here μξn is a Borel probabilN i = 1, . . . , N, |α| := αi =

Z+ ,

i=1

r  p dμξn (s ) < ∞.

(8)

For  j = 1, . . . , m, and α := (α1 , . . . , αN ), αi ∈ Z+ , i = 1, . . . , N, |α| :=

cα ,n,j :=

+

for ξ n > 0, (ξn )n∈N bounded sequence. Assume for all α := (α1 , . . . , αN ), αi ∈

RN

N 

1 p

N i=1

αi =  j, call

N  sαi i dμξn (s ). N

(9)

i=1

 ⎛ ⎞     m   ⎜ ⎟   c f ( x )  α α ,n, j [m] [m] [m] ⎜  ⎟   ⎠ Er,n  p = θr,n ( f ; x ) − f (x ) − δj,r ⎝  N     j=1 |α|= j αi !   i=1 p,x ⎞ ⎛     1p   r  p N ⎜ 1 ⎟   m  s  2 ⎟· ⎜ ≤ |si |αi 1 + d μ ξn ( s ) ω r ( f α , ξ n ) p . 1 ⎠ N ξn RN (q(m − 1 ) + 1 ) q ⎝|α|=m  i=1 αi !

(10)

i=1

As n → ∞ and ξ n → 0, by (10), we obtain that Er,n  p → 0 with rates. One also gets by (10) that [m]

⎞

⎛

[m] θr,n ( f ; x ) − f (x ) p,x ≤

⎟ ⎜  |cα,n,j | |δ[m] |⎜  fα  p ⎟ ⎠ + R.H.S.(10 ), j,r ⎝ N   j=1 |α|= j αi !

m 

(11)

i=1

[m] j,  j = 1, . . . , m. Assuming that cα ,n,j → 0, ξ n → 0, as n → ∞, we get θr,n ( f ) − f  p → 0, that is given that fα p < ∞, |α| =  [m] θr,n → I the unit operator, in Lp norm, with rates.

For the case of m = 0 and p > 1, the author gave

654

G. A. Anastassiou, M. Kester / Applied Mathematics and Computation 265 (2015) 652–666

Theorem 2. Let f ∈ (C (RN ) ∩ L p (RN )); N ≥ 1; p, q > 1 : bounded. Also suppose





s  2 1+ ξn

RN

1 p

+

1 q

= 1 . Assume μξn probability Borel measures on RN , (ξn )n∈N > 0 and

r p

dμξn (s ) < ∞.

(12)

Then

θ ( f ) − f  p ≤ [0] r,n

 

s  2 1+ ξn

RN

r p

 1p d μ ξn ( s )

ωr ( f, ξn ) p .

(13)

As ξ n → 0, when n → ∞, we obtain θr,n ( f ) − f  p → 0, i.e. θr,n → I, the unit operator, in Lp norm. [0]

[0]

Next, the case of m = 0 and p = 1 followed as Theorem 3. Let f ∈ (C (RN ) ∩ L1 (RN )), N ≥ 1. Assume μξn probability Borel measures on RN , (ξn )n∈N > 0 and bounded. Also suppose

 RN



1+

s  2 ξn

r

dμξn (s ) < ∞.

Then [0] θr,n ( f ) − f 1 ≤

  RN

1+

(14)

s  2 ξn

r d μ ξn ( s )

 ωr ( f, ξn )1 .

(15)

As ξ n → 0, we get θr,n → I in L1 norm. [0]

Finally, the case of m ∈ N and p = 1 covered as Theorem 4. Let f ∈ C m (RN ), m, N ∈ N , with fα ∈ L1 (RN ), |α| = m, x ∈ RN . Here μξn is a Borel probability measure on RN for ξ n >

0, (ξn )n∈N is a bounded sequence. Assume for all α := (α1 , . . . , αN ), αi ∈ Z+ , i = 1, . . . , N, |α| := N i=1 αi = m that we have

 RN



N 



|si |αi

i=1

s  2 1+ ξn

r 

dμξn (s ) < ∞.

For  j = 1, . . . , m, and α := (α1 , . . . , αN ), αi ∈ Z+ , i = 1, . . . , N, |α| :=

 cα ,n,j := Then

R

(16)

N i=1

αi =  j, call

N  sαi i dμξn (s ). N

(17)

i=1

 ⎛ ⎞     m   ⎜ ⎟   c f ( x )  α α ,n, j [m] [m] [m] ⎜   ⎟ Er,n 1 = θr,n ( f ; x ) − f (x ) − δj,r ⎝ ⎠ N      j=1 |α|= j αi !   i=1 1,x ⎛ ⎞   r  N   ⎜ 1 ⎟ s  2 α ⎜ ⎟ωr ( fα , ξn )1 ≤ | si | i 1+ d μ ξn ( s ) . ⎝ ⎠ N ξn RN i=1 |α|=m αi !

(18)

i=1

As ξ n → 0, we get Er,n 1 → 0 with rates. From (18) we get [m]

⎛

[m] θr,n f − f 1 ≤

⎞

⎜  |cα,n,j | ⎟ ⎜ |δ[m] |  fα 1 ⎟ + R.H.S.(18 ), ⎠ j,r ⎝ N   j=1 |α|= j αi !

m 

(19)

i=1

[m] [m] given that fα 1 < ∞, |α| =  j,  j = 1, . . . , m. As n → ∞, assuming ξ n → 0 and cα ,n,j → 0, we get θr,n f − f 1 → 0, that is θr,n → I in L1 norm, with rates.

On the other hand, in [4], the authors defined the following operators: Let μξn be a Borel measure on RN , N ≥ 1, 0 < ξ n ≤ 1, n ∈ N. Assume that ν := (ν1 , . . . , νN ), x := (x1,..., xN ) ∈ RN and f : RN → R is a Borel measurable function.

G. A. Anastassiou, M. Kester / Applied Mathematics and Computation 265 (2015) 652–666

655

(i) When N

μ ξn ( ν ) =

e ∞

ν1 =−∞

...

− i=1ξ

|νi |

n

,

N

∞

νN =−∞

e

(20)

|νi |

− i=1ξ n

they defined generalized multiple discrete Picard operators as: ∞

ν1 =−∞

∗[m] Pr,n ( f ; x1 , . . . , x N ) =



∞

...

νN =−∞



r

α

j=0

[m] j,r

∞

ν1 =−∞

f ( x 1 + j ν 1 , . . . , xN + j ν N ) e

...

N

∞

νN =−∞

e

N

− i=1ξ

|νi |

n

.

|νi |

(21)

− i=1ξ n

(ii) When N

μ ξn ( ν ) =

e ∞

ν1 =−∞

...

− i=1ξ

νi2

n N

∞

νN =−∞

e

− i=1ξ

ν2

,

(22)

i

n

they defined generalized multiple discrete Gauss–Weierstrass operators as: ∞

∗[m] Wr,n ( f ; x1 , . . . , xN ) =

ν1 =−∞

...



∞

νN =−∞



r

α

j=0

[m] j,r

∞

ν1 =−∞

N

f ( x 1 + j ν 1 , . . . , xN + j ν N ) e

...

N

∞

νN =−∞

e

− i=1ξ

νi2

n

.

ν2

− i=1ξ n

(23)

i

(iii) Let αˆ ∈ N and β > α1ˆ . When N 

μ ξn ( ν ) =

i=1 ∞

ν1 =−∞

...

(νi2αˆ + ξn2αˆ )−β ∞

N 

νN =−∞ i=1

,

(24)

(νi2αˆ + ξn2αˆ )−β

they defined the generalized multiple discrete Poisson–Cauchy operators as: ∞

ν1 =−∞

∗[m] Qr,n ( f ; x1 , . . . , x N ) =

...



∞

νN =−∞

 N  α [m] f ( x + j ν , . . . , x + j ν ) (νi2αˆ + ξn2αˆ )−β 1 1 N N j,r j=0 i=1   . ∞ ∞ N

 ... (νi2αˆ + ξn2αˆ )−β r

ν1 =−∞

νN =−∞

(25)

i=1

(iv) When N

μ ξn ( ν ) =

e

− i=1ξ

|νi |

n 1

( 1 + 2 ξ n e − ξn ) N

,

(26)

they defined the generalized multiple discrete non-unitary Picard operators as: ∞

[m] Pr,n ( f ; x1 , . . . , x N ) =

ν1 =−∞

...

∞

νN =−∞



r

j=0

 α

[m] j,r

f ( x 1 + j ν 1 , . . . , xN + j ν N ) e 1

( 1 + 2 ξ n e − ξn ) N

N

− i=1ξ

|νi |

n

.

(27)

656

G. A. Anastassiou, M. Kester / Applied Mathematics and Computation 265 (2015) 652–666

(v) When N

μ ξn ( ν ) = 

π ξn



e

− i=1ξ

νi2

n





1 − erf √1

ξn

+1

N ,

(28)

they defined the generalized multiple discrete non-unitary Gauss–Weierstrass operators as: ∞

ν1 =−∞

[m] Wr,n ( f ; x1 , . . . , x N ) =

...



∞

νN =−∞



r

j=0

 α

[m] j,r

f ( x 1 + j ν 1 , . . . , xN + j ν N ) e



  N π ξn 1 − erf √1 +1

N

− i=1ξ

νi2

n

,

(29)

ξn

where erf(x ) =

x

√2

π

2

e−t dt with erf(∞ ) = 1.

0

Additionally, in [4], article they assumed that 00 = 1. In [4], for k = 1, . . . , n, the authors defined the sums ∞

...

ν1 =−∞

cα ,n, j˜ :=

νN =−∞

∞

ν1 =−∞ ∞

pα ,n, j˜ :=

ν1 =−∞

...

...

|ν |



e

,

(30)

,

(31)

|νi |

− i=1ξ n



N

ν2

i N i=1  νiαi e− ξn

i=1

N

∞

νN =−∞

e

ν2

− i=1ξ n

i

αi +r+1 , they introduced 2αˆ

and for αˆ ∈ N and β >

ν1 =−∞

N

νN =−∞

...

N

i=1

νN =−∞

ν1 =−∞

qα ,n, j˜ :=



i N i=1  νiαi e− ξn

∞

∞

∞

∞



∞

...



∞

νN =−∞

∞

...

ν1 =−∞

N  νiαi (νi2αˆ + ξn2αˆ )−β

i=1

∞

N 

νN =−∞ i=1

(νi2αˆ + ξn2αˆ )−β

 .

(32)

Furthermore, they proved that

∀ξn ∈ (0, 1],

cα ,n, j˜, pα ,n, j˜, qα ,n, j˜ < ∞,

(33)

and for αi ∈ N, as ξ n → 0 when n → ∞, the authors showed that

cα ,n, j˜, pα ,n, j˜,

and

qα ,n,j → 0.

In [4], they also proved

mξn ,P =

⎛ ∞ |νi | e − ξn N  ⎜ νi =−∞ ⎝

i=1

and

1 + 2 ξn e

⎛ mξn ,W =

i=1

⎜ ⎝

⎞

− ξ1

n

⎟ ⎠ → 1 as ξn → 0+ ,

∞

N ⎜ 

1+



(34)

νi =−∞

⎞

ν2

i e− ξ



(35)



π ξn 1 − erf √1

⎟ +  ⎟ ⎠ → 1 as ξn → 0 .

(36)

ξn

Moreover, in [4], the authors defined the following error quantities: [0] [0] En,P ( f ; x ) := Pr,n ( f ; x ) − f ( x ), [0] En,W

( f ; x ) :=

[0] Wr,n

( f ; x ) − f ( x ).

(37)

G. A. Anastassiou, M. Kester / Applied Mathematics and Computation 265 (2015) 652–666

Furthermore, they introduced the errors (n ∈ N ):

⎛

[m] [m] En,P ( f ; x ) := Pr,n ( f ; x ) − f (x ) −

m   j=1

⎜ ⎜ δ[m] j,r ⎝

and

⎞  α1 ,...,αN ≥0: |α|= j

c˜α ,n,j fα (x ) ⎟ ⎟, ⎠ N  αi !

m   j=1

where ∞

ν1 =−∞

c˜α ,n,j :=

...



∞

νN =−∞



⎞

⎜ ⎜ δ[m] j,r ⎝



α1 ,...,αN ≥0: |α|= j

N

(38)

i=1

⎛ [m] [m] En,W ( f ; x ) := Wr,n ( f ; x ) − f (x ) −

657

p˜ α ,n,j fα (x ) ⎟ ⎟, ⎠ N  αi !

(39)

i=1

|ν |

i N i=1  νiαi e− ξn

i=1

(40)

1

( 1 + 2 ξ n e − ξn ) N

and ∞

ν1 =−∞



p˜ α ,n,j :=

...



∞

νN =−∞



i=1



N

ν2

i N i=1  νiαi e− ξn





π ξn 1 − erf √1

+1

ξn

N .

(41)

3. Main results We start with Proposition 5. Let ν := (ν1 , . . . , νN ), α := (α1,..., αN ) ∈ RN , αi ∈ Z+ , i = 1, . . . , N ∈ N, |α| := ∃K1 > 0 such that ∞

ν1 =−∞

SPp,m ∗ ,ξ :=

...



∞

νN =−∞

n

N 

i=1

∞

ν1 =−∞

p

|νi

...

|αi



N

r p − i=1 i 2 1 + ν e ξn ξn N

∞

νN =−∞

e

N

i=1

αi = m ∈ Z+ , and p ≥ 1. Then,

|ν |

|νi |

≤ K1 < ∞,

(42)

− i=1ξ n

for all ξ n ∈ (0, 1], n ∈ N. Proof. Since ∞ 

∞ 

...

ν1 =−∞

N

e

− i=1ξ

|νi |

n

> 1,

(43)

νN =−∞

and

ν2 =



ν12 + · · · + νN2 ≤

N 

|νi |,

(44)

i=1

we have

SPp,m ≤ ∗ ,ξ n

∞ 

= 2

N

∞ 

ν1 =1

:= RP∗ ,ξn .

∞ 

...

ν1 =−∞



νN =−∞

...

∞ 

νN =1



N 

 |νi |αi p

i=1

N  ν αi p i

i=1



⎛

⎛

⎞r p N

|ν | |νi | i ⎟ ⎜ ⎜1 + i=1 ⎟ e− i=1ξn ⎝ ξn ⎠ N

N

⎞r p

νi ⎟ νi ⎜ ⎜1 + i=1 ⎟ e− i=1ξn ⎝ ξn ⎠ N

(45)

658

G. A. Anastassiou, M. Kester / Applied Mathematics and Computation 265 (2015) 652–666

On the other hand, for ν i ≥ 1, we realize that



N i=1

1+

ξn

νi

r p ≤ 2r p ξn−r p Nr p

N 

νir p

i=1

≤ 2r p ξn−Nr p Nr p

N 

νir p .

(46)

i=1

Thus, by (45) and (46), we have

RP∗ ,ξn ≤

2N+r p n−Nr p Nr p

ξ

= 2N+r p Nr p

N  i=1

≤2

N+r p

≤2



...

ν1 =1 ∞ 

νi =1

∞ 



νN =1

νiαi p

i

ξn

 ∞ N   N νiαi p 1 + N

rp

N  i=1



νi =1

 ∞ 

αi p νi =1

N  ν αi p

νi



N 

i

i=1

 ν r p

rp

i=1 N+r p



∞ 



 ν

rp i

N

e

− i=1 ξ

νi

n

i=1

νi

e − ξn

νi r p − ξνni e ξn



 νi  r p − ξνni 1+ e , ξn

(47)

where . is the ceiling of the number. In [2], for αi p ∈ N, the authors showed that

M1, αi p :=

 ∞ 

αi p νi =1

νi

1+

νi  r p − ξνni e ξn

≤ 22 r p ( r p )! × [λ αi p + (2 αi p + 1 ) αi p e−

2 αi p+1 2

+ ( αi p )!2 αi p+1 ],

(48)

where

λ αi p :=

2 αi p



νi =1

νi

νi αi p e− 2 < ∞,

(49)

for all ξ n ∈ (0, 1]. Hence, by (48) and (49), we obtain

M1, αi p < ∞,

(50)

for all ξ n ∈ (0, 1], and αi p ∈ N. When αi p = 0, we get

νi  r p − ξνni e ξn νi =1  ∞  ν  r p − ξνni ≤ νi 1 + i e := M1,1. ξn ν =1

M1,0 =

∞  

1+

(51)

i

Then, by (50) and (51), we have

M1,0 < ∞,

(52)

for all ξ n ∈ (0, 1]. Thus, by (47), (48), (50), and (52), we get

SPp,m ∗ ,ξ ≤ RP ∗ ,ξn < ∞, n

for all ξ n ∈ (0, 1].  Next, we have

(53)

G. A. Anastassiou, M. Kester / Applied Mathematics and Computation 265 (2015) 652–666

Proposition 6. Let ν := (ν1 , . . . , νN ), α := (α1,..., αN ) ∈ RN , αi ∈ Z+ , i = 1, . . . , N ∈ N, |α| := ∃K2 > 0 such that ∞

p,m SW := ∗ ,ξ

ν1 =−∞

...



∞

νN =−∞

n

p

N  i=1

∞

ν1 =−∞

|νi

...

|αi



N

νN =−∞

e

i=1

αi = m ∈ Z+ , and p ≥ 1. Then,

r p − i=1 i 2 1 + ν e ξn ξn N

∞

N

659

− i=1ξ

ν2

ν2 i

n

≤ K2 < ∞,

(54)

for all ξ n ∈ (0, 1], n ∈ N. Proof. We observe that ∞ 

...

ν1 =−∞

N

∞ 

e

− i=1ξ

νi2

> 1.

n

Thus, by (44) and (55), we have p,m SW ∗ ,ξ n

(55)

νN =−∞



p

r p N νi2 i=1 ν2 ≤ ... |νi 1+ e − ξn ξn ν1 =−∞ νN =−∞ i=1 ⎛ ⎞r p N

  N

ν i ∞ N ∞ ⎜ ⎟ − i=1 νi2    i=1 αi p ⎜ N ⎟ ξn ≤ 2 ... νi ⎝1 + ξ n ⎠ e ν =1 ν =1 i=1 ∞ 

∞ 

1

N 

|αi

N

:= RW ∗ ,ξn .

(56)

Additionally, for ν i ≥ 1, we get

νi2 ≥ νi ,

(57)

which gives us N

e

− i=1ξ

N

νi2

n

≤e

− i=1 ξ

νi

.

n

(58)

Therefore, by (53), (56) and (58), we have p,m SW ∗ ,ξ ≤ RW ∗ ,ξn . ≤ RP ∗ ,ξn < ∞,

(59)

n

for all ξ n ∈ (0, 1], n ∈ N.  We demonstrate also Proposition 7. Let ν := (ν1 , . . . , νN ), α := (α1,..., αN ) ∈ RN , αi ∈ Z+ , i = 1, . . . , N ∈ N, |α| := ∃ K3 > 0 such that ∞

SQp,m := ∗ ,ξ

ν1 =−∞

...



∞

νN =−∞

N 

i=1

∞

n

ν1 =−∞

p

|νi

...

|αi



r p 2 1 + ν ξn



∞

νN =−∞

N 

i=1



N  i=1

N i=1



2αˆ i

2αˆ −β n

(ν + ξ ) 

(νi2αˆ + ξn2αˆ )−β

≤ K3 < ∞,

(60)

for all ξ n ∈ (0, 1] where αˆ , n ∈ N , β > max{ Proof. Since ∞ 

ν1 =−∞

...

∞ 



νN =−∞

N 

n

1+ αi p+ r p 2+ r p , 2αˆ } 2αˆ

 (ν

2αˆ i

+ξ

i=1

then, by (44), (46) and (61), we get

SQp,m ≤ RW ∗ ,ξn . ∗ ,ξ

αi = m ∈ Z+ , and p ≥ 1. Then,

2αˆ −β n

)

=

N ∞   i=1 νi =−∞

for all i.

(νi2αˆ + ξn2αˆ )−β ≥

N  i=1

ˆ −2Nαβ ˆ ξn−2αβ = ξn ,

(61)

660

G. A. Anastassiou, M. Kester / Applied Mathematics and Computation 265 (2015) 652–666

 r p  N  ν 2 := ξn ... |νi |αi p 1 + (νi2αˆ + ξn2αˆ )−β ξn ν1 =−∞ νN =−∞ i=1 i=1    r p  N ∞ N ∞     ν2 ˆ αi p N 2Nαβ 2αˆ 2αˆ −β = 2 ξn ... νi 1+ ( ν i + ξn ) ξn ν1 =1 νN =1 i=1 i=1    ∞ ∞ N    νi r p 2αˆ ˆ αi p N+r p r p 2Nαβ 2αˆ −β ≤ 2 N ξn ... νi 1 + ( ν i + ξn ) ξn ν1 =1 νN =1 i=1    r p ∞ N   ν 2 αβ ˆ i α p = 2r p N r p 2 ξn νi i 1 + (νi2αˆ + ξn2αˆ )−β ξn νi =1 i=1    N ∞   νi  r p 2αˆ 2αβ ˆ

αi p rp rp 2αˆ −β ≤ 2 N 2 ξn νi 1+ ( ν i + ξn ) . ξn ν =1 i=1 ∞ 

∞  2Nαβ ˆ





N 

(62)

i

In [2], for αi ∈ N, β >

1+ αi p+ r p , 2αˆ

2αβ ˆ

M2,αi : = 2ξn

the authors showed

 ∞ 

αi p

νi =1

νi

1+

νi  r p 2αˆ (νi + ξn2αˆ )−β ξn

 ˆ − αi p− r p ∞   1 2αβ

≤ 2 r p+1

νi =1

νi

< ∞,

(63)

for all ξ n ∈ (0, 1]. Thus, we have rp rp SQp,m N ∗ ,ξ ≤ 2

N 

n

M2,αi < ∞.

(64)

i=1

for all ξ n ∈ (0, 1], n ∈ N. When αi = 0, we get

νi  r p 2αˆ (νi + ξn2αˆ )−β ξ n νi =1  ∞  ν  r p 2αˆ 2αβ ˆ ≤ 2 ξn νi 1 + i (νi + ξn2αˆ )−β ξ n ν =1 2αβ ˆ

M2,0 = 2ξn

∞  

1+

i

= M2,1 .

(65)

Therefore, by (63) and (65), we get

M2,0 < ∞, for β >

2+ r p . 2αˆ

(66)

Hence, we have

SQp,m ∗ ,ξ < ∞,

(67)

n

for all ξ n ∈ (0, 1], n ∈ N where, β > max{

1+ αi p+ r p 2+ r p , 2αˆ }. 2αˆ



∗[m]

Next, we state our results for the operators Pr,n

Theorem 8. Let f ∈ C m (RN ), m ∈ N, N ≥ 1, with fα ∈ L p (RN ), |α| = m, x ∈ RN , p, q > 1 :

 p∗[m] Pr,n

 ⎛ ⎞     m  ∗[m]  [m] ⎜  cα ,n,j fα (x ) ⎟  ⎜ ⎟   :=  P ( f ; x ) − f ( x ) − δ   r,n ⎠ j,r ⎝ N     j=1 |α|= j αi !   i=1 p,x ⎞ ⎛   ⎜  1 ⎟ p,m 1 m ⎟ ( S ∗ ) p ω r ( f α , ξn ) p . ⎜ ≤ 1 ⎠ P ,ξn N (q(m − 1 ) + 1 ) q ⎝|α|=m  αi ! i=1

1 p

+

1 q

= 1, and 0 < ξ n ≤ 1, n ∈ N. Then

(68)

G. A. Anastassiou, M. Kester / Applied Mathematics and Computation 265 (2015) 652–666

As n → ∞ and ξ n → 0, by (68), we obtain that 

⎛ ∗[m] Pr,n ( f ; x ) − f (x ) p,x ≤

p ∗[m]

Pr,n

661

→ 0 with rates. One also gets by (68) that

⎞

⎟ ⎜  |cα,n,j | |δ[m] |⎜  fα  p ⎟ ⎠ + R.H.S.(68 ), j,r ⎝ N   j=1 |α|= j αi !

m 

(69)

i=1

∗[m] ∗[m] j,  j = 1, . . . , m. Then, for αi ∈ N, as n → ∞, we get Pr,n ( f ; x ) − f (x ) p → 0, that is Pr,n → I the unit given that fα p < ∞, |α| =  operator, in Lp norm, with rates.

Proof. Theorem 1, (33), (34), and Proposition 5.  Next, we give our result for the case of m = 0 and p > 1. Theorem 9. Let f ∈ (C (RN ) ∩ L p (RN )); N ≥ 1; p, q > 1 :

1 p

+

1 q

= 1 , and 0 < ξ n ≤ 1, n ∈ N. Then

∗[0] p Pr,n ( f ) − f  p ≤ (SPp,0 ∗ ,ξ ) ωr ( f, ξn ) p . n 1

(70)

∗[0]

∗[0]

As ξ n → 0, when n → ∞, we obtain Pr,n ( f ) − f  p → 0, i.e. Pr,n → I, the unit operator, in Lp norm. Proof. Theorem 2 and Proposition 5.



For the case of m = 0 and p = 1, we have Theorem 10. Let f ∈ (C (RN ) ∩ L1 (RN )), N ≥ 1, and 0 < ξ n ≤ 1, n ∈ N.Then ∗[0] Pr,n ( f ) − f 1 ≤ SP1,0 ∗ ,ξ ωr ( f, ξn )1 n

(71)

∗[0]

As ξ n → 0, we get Pr,n → I in L1 norm. Proof. Theorem 3 and Proposition 5. Our final result for the operators



∗[m] Pr,n

is for the case of m ∈ N and p = 1

Theorem 11. Let f ∈ C m (RN ), m ∈ N, N ≥ 1, with fα ∈ L1 (RN ), |α| = m, x ∈ RN , and 0 < ξ n ≤ 1, n ∈ N. Then

1P∗[m] r,n

 ⎛ ⎞     m   ∗[m] ⎜ ⎟   c f ( x ) α ,n, j α [m] ⎜  ⎟ = P ( f ; x ) − f ( x ) − δ    r,n ⎠ j,r ⎝ N    j=1 |α|= j ( αi ! )    i=1 1,x ⎛ ⎞ ≤

⎜  1 ⎟ 1,m ⎜ ⎟S ∗ ωr ( fα , ξn )1 . ⎝ ⎠ P ,ξn N  |α|=m αi !

(72)

i=1

As n → ∞ and ξ n → 0, by (72), we obtain that 1∗[m] → 0 with rates. One also gets by (72) that

⎛ ∗[m] Pr,n ( f ; x ) − f (x )1,x ≤

Pr,n

⎞

⎟ ⎜  |cα,n,j | |δ[m] |⎜  f α 1 ⎟ ⎠ + R.H.S.(72 ), j,r ⎝ N   j=1 |α|= j αi !

m 

(73)

i=1

∗[m] ∗[m] given that fα 1 < ∞, |α| =  j,  j = 1, . . . , m. Then, for αi ∈ N, as n → ∞, we get Pr,n ( f ; x ) − f (x )1 → 0, that is Pr,n → I the unit operator, in L1 norm, with rates.

Proof. Theorem 4, (33), (34), and Proposition 5.  ∗[m]

Now, we state our results for the operators Wr,n . We begin with Theorem 12. Let f ∈ C m (RN ), m ∈ N, N ≥ 1, with fα ∈ L p (RN ), |α| = m, x ∈ RN , p, q > 1 :

1 p

+

1 q

= 1, and 0 < ξ n ≤ 1, n ∈ N. Then

662

G. A. Anastassiou, M. Kester / Applied Mathematics and Computation 265 (2015) 652–666

 p ∗[m] Wr,n

 ⎛ ⎞     m   ∗[m] ⎜ ⎟   p f ( x )  α α ,n, j [m] ⎜  ⎟   ⎠ := Wr,n ( f ; x ) − f (x ) − δj,r ⎝  N     j=1 |α|= j αi !   i=1 p,x ⎞ ⎛   ⎜  1 ⎟ p,m 1 m ⎟ ( S ∗ ) p ω r ( f α , ξn ) p . ⎜ ≤ 1 ⎠ W ,ξn N (q(m − 1 ) + 1 ) q ⎝|α|=m  αi !

(74)

i=1

As n → ∞ and ξ n → 0, by (74), we obtain that 

⎛

∗[m] Wr,n ( f ; x ) − f (x ) p,x ≤

p ∗[m]

Wr,n

→ 0 with rates. One also gets by (74) that

⎞

⎜  | pα,n,j | ⎟ ⎜ |δ[m] |  fα  p ⎟ + R.H.S.(74 ), ⎠ j,r ⎝ N   j=1 |α|= j αi !

m 

(75)

i=1

j,  j = 1, . . . , m. Then, for αi ∈ N, as n → ∞, we get Wr,n ( f ; x ) − f (x ) p → 0, that is Wr,n given that fα p < ∞, |α| =  unit operator, in Lp norm, with rates. ∗[m]

Proof. Theorem 1, (33), (34), and Proposition 6.

∗[m]

→ I the



Next, we present our result for the case of m = 0 and p > 1. Theorem 13. Let f ∈ (C (RN ) ∩ L p (RN )); N ≥ 1; p, q > 1 :

1 p

+

1 q

= 1 , and 0 < ξ n ≤ 1, n ∈ N. Then

∗[0] p,0 p Wr,n ( f ) − f  p ≤ (SW ∗ ,ξ ) ωr ( f, ξn ) p . n 1

(76)

∗[0]

∗[0]

As ξ n → 0, when n → ∞, we obtain Wr,n ( f ) − f  p → 0, i.e. Wr,n → I, the unit operator, in Lp norm. Proof. Theorem 2 and Proposition 6.



For the case of m = 0 and p = 1, we obtain Theorem 14. Let f ∈ (C (RN ) ∩ L1 (RN )), N ≥ 1, and 0 < ξ n ≤ 1, n ∈ N.Then ∗[0] 1,0 Wr,n ( f ) − f 1 ≤ SW ∗ ,ξ ωr ( f, ξn )1 n

(77)

∗[0]

As ξ n → 0, we get Wr,n → I in L1 norm. Proof. Theorem 3 and Proposition 6.



For case of m ∈ N and p = 1, we demonstrate Theorem 15. Let f ∈ C m (RN ), m ∈ N, N ≥ 1, with fα ∈ L1 (RN ), |α| = m, x ∈ RN , and 0 < ξ n ≤ 1, n ∈ N. Then

1 W ∗[m] r,n

 ⎛ ⎞     m   ∗[m] ⎜ ⎟   p f ( x )  α α ,n, j [m] ⎜  ⎟   ⎠ = Wr,n ( f ; x ) − f (x ) − δj,r ⎝  N     j=1 |α|= j αi !   i=1 1,x ⎛ ⎞ ≤

⎜  1 ⎟ 1,m ⎜ ⎟S ∗ ωr ( fα , ξn )1 . ⎝ ⎠ W ,ξn N  |α|=m αi !

(78)

i=1

As n → ∞ and ξ n → 0, by (78), we obtain that 1

⎛

∗[m] Wr,n ( f ; x ) − f (x )1,x ≤

∗[m]

Wr,n

→ 0 with rates. One also gets by (78) that

⎞

⎜  | pα,n,j | ⎟ |δ[m] |⎜  f α 1 ⎟ ⎠ + R.H.S.(78 ), j,r ⎝ N   j=1 |α|= j αi !

m 

i=1

(79)

G. A. Anastassiou, M. Kester / Applied Mathematics and Computation 265 (2015) 652–666

663

∗[m] ∗[m] given that fα 1 < ∞, |α| =  j,  j = 1, . . . , m. Then, for αi ∈ N, as n → ∞, we get Wr,n ( f ; x ) − f (x )1 → 0, that is Wr,n → I the unit operator, in L1 norm, with rates.

Proof. Theorem 4, (33), (34), and Proposition 6.  ∗[m]

We continue with the results for the operators Qr,n . Firstly, we have Theorem 16. Let f ∈ C m (RN ), m ∈ N, N ≥ 1, with fα ∈ L p (RN ), |α| = m, x ∈ RN , p, q > 1 :

β>

1+ αi p+ r p 2+ r p max{ , 2αˆ } 2αˆ

 p ∗[m] Qr,n

1 p

+

1 q

= 1, and 0 < ξ n ≤ 1, n, αˆ ∈ N ,

for all i, i = 1, . . . , N. Then

 ⎛ ⎞     m  ∗[m]  ⎜ ⎟   q f ( x )  α α ,n, j [m] ⎜  ⎟   ⎠ := Qr,n ( f ; x ) − f (x ) − δj,r ⎝  N     j=1 |α|= j αi !   i=1 p,x ⎛ ⎞   ⎜  1 ⎟ p,m 1 m ⎜ ⎟ ( S ∗ ) p ω r ( f α , ξn ) p . ≤ 1 ⎠ Q ,ξn N q (q(m − 1 ) + 1 ) ⎝|α|=m  αi !

(80)

i=1

As n → ∞ and ξ n → 0, by (80), we obtain that 

p ∗[m]

Qr,n

→ 0 with rates. One also gets by (80) that

⎛ ∗[m] Qr,n ( f ; x ) − f (x ) p,x ≤

⎞

⎜  |qα,n,j | ⎟ |δ[m] |⎜  fα  p ⎟ ⎠ + R.H.S.(80 ), j,r ⎝ N   |α|= j j=1 αi !

m 

(81)

i=1

∗[m] ∗[m] given that fα p < ∞, |α| =  j,  j = 1, . . . , m. Then, for αi ∈ N, as n → ∞, we get Qr,n ( f ; x ) − f (x ) p → 0, that is Qr,n → I the unit operator, in Lp norm, with rates.

Proof. Theorem 1, (33), (34), and Proposition 7.  Next, we present our result for the case of m = 0 and p > 1. Theorem 17. Let f ∈ (C (RN ) ∩ L p (RN )); N ≥ 1; p, q > 1 :

1 p

+

1 q

= 1 , and 0 < ξ n ≤ 1, n, αˆ ∈ N , β >

∗[0] Qr,n ( f ) − f  p ≤ (SQp,0∗ ,ξn ) p ωr ( f, ξn ) p . 1

∗[0]

2+ r p . 2αˆ

Then (82)

∗[0]

As ξ n → 0, when n → ∞, we obtain Qr,n ( f ) − f  p → 0, i.e. Qr,n → I, the unit operator, in Lp norm. Proof. Theorem 2 and Proposition 7.



For the case of m = 0 and p = 1, we obtain Theorem 18. Let f ∈ (C (RN ) ∩ L1 (RN )), N ≥ 1, and 0 < ξ n ≤ 1, n, αˆ ∈ N , β > ∗[0] Qr,n ( f ) − f 1 ≤ SQ1,0∗ ,ξn ωr ( f, ξn )1

∗[0]

As ξ n → 0, we get Qr,n → I in L1 norm. Proof. Theorem 3 and Proposition 7.



For case of m ∈ N and p = 1, we demonstrate

2+r . 2αˆ

Then (83)

664

G. A. Anastassiou, M. Kester / Applied Mathematics and Computation 265 (2015) 652–666

Theorem 19. Let f ∈ C m (RN ), m ∈ N , N ≥ 1, with fα ∈ L1 (RN ), |α| = m, x ∈ RN , and 0 < ξ n ≤ 1, n, αˆ ∈ N , β > max{ for all i. Then

1Q ∗[m] r,n

1+αi +r 2+r , 2αˆ } 2αˆ

 ⎛ ⎞     m  ∗[m]  [m] ⎜  qα ,n,j fα (x ) ⎟  ⎜ ⎟   :=  Q ( f ; x ) − f ( x ) − δ   r,n ⎠ j,r ⎝ N     j=1 |α|= j αi !   i=1 1,x ⎛ ⎞ ≤

⎜  1 ⎟ 1,m ⎜ ⎟S ∗ ωr ( fα , ξn )1 . ⎝ ⎠ Q ,ξn N  |α|=m αi !

(84)

i=1

As n → ∞ and ξ n → 0, by (84), we obtain that 1 ∗[m] → 0 with rates. One also gets by (84) that

⎛ ∗[m] Qr,n ( f ; x ) − f (x )1,x ≤

Qr,n

⎞

⎟ ⎜  |qα,n,j | |δ[m] |⎜  f α 1 ⎟ ⎠ + R.H.S.(84 ), j,r ⎝ N   j=1 |α|= j αi !

m 

(85)

i=1

∗[m] ∗[m] given that fα 1 < ∞, |α| =  j,  j = 1, . . . , m. Then, for αi ∈ N, as n → ∞, we get Qr,n ( f ; x ) − f (x )1 → 0, that is Qr,n → I the unit operator, in L1 norm, with rates.

Proof. Theorem 4, (33), (34), and Proposition 7.

 [0]

[0]

[m]

[m]

Now, we give our results for the error quantities En,P ( f ; x ), En,P ( f ; x ), and the errors En,P ( f ; x ), En,P ( f ; x ) defined as (37)–(39). We start with Remark 20. We observe that, since [0] [0] En,P ( f ; x ) = Pr,n ( f ; x ) − f (x ) − f (x )mξn ,P + f (x )mξn ,P ,

(86)

where mξn ,P is defined as (35), then we get



[0] En,P

   [0]    Pr,n ( f ; . )  ( f ) p = mξn ,P − f (. ) + f (. )(mξn ,P − 1 )  mξn ,P p  [0]   Pr,n ( f )   ≤ mξn ,P   mξn ,P − f  +  f  p |mξn ,P − 1|.

(87)

p

Since [0] Pr,n ( f ; x) ∗[0] = Pr,n ( f ; x ), mξn ,P

(88)

[0] ∗[0] En,P ( f ) p ≤ mξn ,P Pr,n ( f ) − f  p +  f  p |mξn ,P − 1|.

(89)

we get

Similarly we obtain the inequalities [0] ∗[0] En,W ( f ) p ≤ mξn ,W Wr,n ( f ) − f  p +  f  p |mξn ,W − 1|,

[m] ∗[m] En,P ( f ; x ) p ≤ mξn ,P Pr,n (f) − f −

 j=1

⎛ ⎜ ⎝

×⎜

m 

(90)

δ[m] j,r

⎞



cα ,n,j fα ⎟ N 

α1 ,...,αN ≥0: αi ! |α|= j i=1

⎟ p +  f  p |mξn ,P − 1|, ⎠

(91)

G. A. Anastassiou, M. Kester / Applied Mathematics and Computation 265 (2015) 652–666

665

and [m] ∗[m] En,W ( f ) p ≤ mξn ,W Wr,n (f) − f −

m   j=1

⎛

δ[m] j,r

⎞

⎜ ⎝

pα ,n,j fα ⎟



×⎜

N 

α1 ,...,αN ≥0: αi ! |α|= j i=1

⎟ p +  f  p |mξn ,W − 1|. ⎠

(92)

Next, we have Theorem 21. Let f ∈ C m (RN ), m ∈ N , N ≥ 1, with fα ∈ L p (RN ), |α| = m, x ∈ RN , p, q > 1 :



[m] En,P ( f ) p ≤ mξn ,P

1

p m(SPp,m ∗ ,ξ ) ωr ( f α , ξn ) p



n

(q (m − 1 ) + 1 )

1 q

 |α|=m

1 N  i=1

and

 [m] En,W ( f ) p ≤ mξn ,W

1

p,m p m(SW ∗ ,ξ ) ωr ( f α , ξn ) p





n

(q (m − 1 ) + 1 )

1 q

αi !

|α|=m

i=1

As n → ∞ and ξ n → 0, we obtain that

[m] En,P ( f ; x ) p

+

1 q

= 1, and 0 < ξ n ≤ 1, n ∈ N. Then

+  f  p |mξn ,P − 1|,

1 N 

1 p

αi !

+  f  p |mξn ,W − 1|.

(93)

(94)

→ 0 and En,W ( f ; x ) p → 0 with rates. [m]

Proof. By (35), (36), (91), (92) and Theorems 8, 12.  We present our result for the case of m = 0 and p > 1 as Theorem 22. Let f ∈ (C (RN ) ∩ L p (RN )); N ≥ 1; p, q > 1 :

1 p

+

1 q

= 1 , and 0 < ξ n ≤ 1, n ∈ N. Then

1

[0] p En,P ( f ) p ≤ mξn ,P (SPp,0 ∗ ,ξ ) ωr ( f, ξn ) p +  f  p |mξn ,P − 1|, n

(95)

and 1

[0] p,0 p En,W ( f ) p ≤ mξn ,W (SW ∗ ,ξ ) ωr ( f, ξn ) p +  f  p |mξn ,W − 1|. n

(96)

As ξ n → 0, when n → ∞, we obtain En,P ( f ; x ) p → 0 and En,W ( f ; x ) p → 0. [0]

[0]

Proof. By (35), (36), (89), (90) and Theorems 9, 13.



For the case of m = 0 and p = 1, we obtain Theorem 23. Let f ∈ (C (RN ) ∩ L1 (RN )), N ≥ 1, and 0 < ξ n ≤ 1, n ∈ N.Then [0] En,P ( f )1 ≤ mξn ,P SP1,0 ∗ ,ξ ωr ( f, ξn )1 +  f 1 |mξn ,P − 1|, n

(97)

[0] 1,0 En,W ( f )1 ≤ mξn ,W SW ∗ ,ξ ωr ( f, ξn )1 +  f 1 |mξn ,W − 1|. n

(98)

and

As ξ n → 0, we get En,P ( f ) → I and En,W ( f ) → I in L1 norm. [0]

[0]

Proof. By (35), (36), (89), (90) and Theorems 10, 14.  Our final result is for the case of m ∈ N and p = 1 Theorem 24. Let f ∈ C m (RN ), m ∈ N , N ≥ 1, with fα ∈ L1 (RN ), |α| = m, x ∈ RN , and 0 < ξ n ≤ 1, n ∈ N. Then

⎛

⎜ [m] En,P ( f )1 ≤ mξn ,P ⎜ ⎝

|α|=m

⎞

1 N  i=1

αi !

⎟ 1,m ⎟S ∗ ωr ( fα , ξn )1 +  f 1 |mξn ,P − 1|, ⎠ P ,ξn

(99)

666

G. A. Anastassiou, M. Kester / Applied Mathematics and Computation 265 (2015) 652–666

and

⎛ ⎜ [m] En,W ( f )1 ≤ mξn ,W ⎜ ⎝

|α|=m

⎞ 1 N  i=1

αi !

⎟ 1,m ⎟S ∗ ωr ( fα , ξn )1 +  f 1 |mξn ,W − 1|. ⎠ W ,ξn

(100)

As n → ∞ and ξ n → 0, we obtain that En,P ( f ; x )1 → 0 and En,W ( f ; x )1 → 0 with rates. [m]

[m]

Proof. By (35), (36), (91), (92) and Theorems 11, 15.  References [1] G.A. Anastassiou, Approximation by Multivariate Singular Integrals, Briefs in Mathematics, Springer, New York, 2011. [2] G.A. Anastassiou, M. Kester, Quantitative uniform approximation by generalized discrete singular operators, Studia Mathematica Babes-Bolyai 60 (1) (2015) 39–60. [3] G.A. Anastassiou, M. Kester, Lp Approximation with Rates by Generalized Discrete Singular Operators, Commun. Appl. Anal., accepted for publication, 2014. [4] G.A. Anastassiou, M. Kester, Uniform Approximation with Rates by Multivariate Generalized Discrete Singular Operators, Submitted, 2015. [5] G.A. Anastassiou, R.A. Mezei, Approximation by Singular Integrals, Cambridge Scientific Publishers, Cambrige, UK, 2012. [6] J. Favard, Sur les multiplicateurs d’interpolation, J. Math. Pures Appl. IX 23 (1944) 219–247.