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* Institute of Mathematics, Pedagogical University of Cracow, Kraków, Poland; gkrech@up.krakow.pl.

GRAŻYNA KRECH*

AN INVESTIGATION OF THE APPROXIMATION OF FUNCTIONS OF TWO VARIABLES BY THE POISSON

INTEGRAL FOR HERMITE EXPANSIONS

APROKSYMACJA FUNKCJI DWÓCH ZMIENNYCH CAŁKĄ POISSONA ZWIĄZANĄ Z WIELOMIANAMI HERMITE’A

A b s t r a c t

This paper presents a study of the approximation properties of the Poisson integral for Hermite expansions in the space Lp. The rate of convergence of functions of two variables by these integrals is established.

Keywords: rate of convergence; Poisson integral; Hermite expansions, positive linear operators S t r e s z c z e n i e

Artykuł poświęcony jest własnościom aproksymacyjnym całek Poissona związanych z wielo- mianami Hermite’a. Udowodniono twierdzenie o rzędzie zbieżności funkcji dwóch zmiennych w przestrzeni Lp tymi operatorami.

Słowa kluczowe: promień zbieżności, całka Poissona, wielomiany Hermite’a, dodatnie opera- tory liniowe

TECHNICAL TRANSACTIONS FUNDAMENTAL SCIENCES

3-NP/2014

CZASOPISMO TECHNICZNE NAUKI PODSTAWOWE

(2)

1. Introduction

Let 1 ≤ ≤ ∞p , we denote by Lp(R2) the set of all the Lebesgue measurable functions f defined on R2 such that

−∞

−∞

∫ ∫

| ( , ) |f t t1 2 pdt dt1 2< ∞ if 1 ≤ < ∞p , and if p = ∞ we require f to be bounded almost everywhere on R2.

In this paper, we present approximation properties of the Poisson integral A in the space Lp(R2), 1 ≤ ≤ ∞p defined by:

A f r y y( ; , , ) =1 2 r K r y z K r y z f z z dz dz( , , ) ( , , ) ( , )1 1 2 2 1 2 1 2, 0

−∞

−∞

∫ ∫

<< < 1,r

where:

K r y z r h y h z

n

n n n

( , , ) = ( ) ( )

=0

= (11 ) 12 11

2

1 ,

2 1 2

2

2 2 2

π − 2

( )

− ⋅

+

(

+

)

+



r 

r

r y z ryz

exp r

h xn( ) = 2 !nn x H xn

2 ( )

1

2 2

(

π

)

exp 

and Hn is the n th Hermite polynomial (see [11]). The norm in Lp(R2) is given by:

|| ||

s

f f t t dt dt p

p

p p

t t

= | ( , ) |1 2 1 2 , 1 < ,

1

( 1, 2 ) 2

−∞

−∞

∫ ∫



 ≤ ∞

R

uup ess f t t| ( , ) |,1 2 p= .∞





Some convergence theorems, the Voronovskaya formula, and a boundary value problem for the integral A were presented in [5]. The following result was proved (see [5]):

Theorem 1 Let y=( , )y y1 2 ∈R2 and f = +f1 f2,where f1∈ R , fL1( )2 2L( )R .2 If f is continuous at y, then:

lim ( ; , ) ( ),

( , ) ( , )r y y A f r y f y

→ − =

1 y= ( , ).y y1 2

In this paper we shall give an order of approximation of functions belonging to Lp(R2) by the operator A. It is worth mentioning that approximation properties of Poisson integrals for orthogonal expansions and their various modifications were also studied in [4, 12, 6–10], in one and two dimensions.

Some auxiliary results, which will be needed in the next part of this paper, are now presented. It is clear that A f r y y( ; , , ) = ( ; , ) ( ; , )1 2 rA f r y A f r y1 1 2 2 for f f1, 2Lp( )R and such thatf z z( , ) = ( ) ( )1 2 f z f z1 1 2 2 , where A f r y( )( , ) = ( ; , ) =A f r y K r y z f z dz( , , ) ( ) , 0 < < 1.r

−∞

(3)

The operator A is linear and positive. Basic facts on positive linear operators and its applications can be found in [2, 3].

In paper [7], we can find the following equalities:

A r y

r

r r y (1; , ) = 2

1

1 2

1

1 ,

2

1/2 2

2 2

+

 

 − ⋅ −

+



 exp 

A r y

r

r r y ( m y; , ) = 2

1

1 2

1

, 2 1

1/2 2

2 2

φ  +

 

 − ⋅ −

+



 exp 

× 





− ⋅ −

+



 − −





p m

p

m p

p

m p

m p

r r

r

=0

2 2

2

( )!

( 2 )!2 1 1

(1 )22

2 2

1+





r y

m p

for 0 < r < 1, y R∈ , where [a] denotes the integral part of a R∈ and ϕm y, ( ) = (z z y− )m. From the above, we have the following result in the bivariate case.

Lemma 1 Let ϕn yi, ( , ) = (z z1 2 z yii)n, y zi, ∈ ,i R i = 1, 2, n N∈ . It holds

A

(

ϕ1,yi ; , ,r y y1 2

)

1+2rr211+rr22+(1( 1)r+r2 2)4 yi212exp− ⋅12 11+rr22

(

y122+y22

)

 (1)

for 0 < r < 1.

Proof. Using Hölder’s inequality, we get:

A

(

φ1, 1y ; , ,r y y1 2

)

 rK r y z K r y z dz dz( , , ) ( , , )1 1 2 2 1 2

−∞

−∞

∫ ∫

11 2

× −



−∞ 

−∞

∫ ∫

rK r y z K r y z( , , ) ( , , ) |1 1 2 2 z1 y1|2dz dz1 2

1

2= 1; , ,1 2 12 2, 1; , ,1 2

1

A r y y

( )

A y r y y 2

( )

( (

φ

) )

(2) for ( , )y y1 2R2 and 0 < r < 1. We have (see [5]):

A r y y r

r

r

r y y (1; , , ) = 2

1

1 2

1

1 ,

1 2 2

2 2 12

22

+ − ⋅ −

+

(

+

)



 exp 

A

(

φ2,yi; , ,r y y1 2

)

= 21 11 1

1

1 2

1

2 1

2 2

4 2 2

2 2

2 12

r r

r r

r

r y r

r y + i

+ +

(

)

(

+

)



 − ⋅ −

+ +

exp

(

yy22

)

,i = 1, 2.

From this and (2) we obtain (1) for i = 1. Analogously, we calculate (1) for i = 2.

(4)

2. Rate of convergence

In this section, we give an order of approximation of function of two variables in the space Lp.

We achieve this using the modulus of continuity ω( ; , )f δ δ1 2 , δ δ1, 2> 0 of f L R∈ ( )p 2 defined as follows:

ω δ δ

δ δ

( ; , ) =1 2 ( , )

0< 1 1

0< 2 2 ( 1, 2 ) 2 1 1 2 2

f f y h y h f

h

h y y

{

+ +

sup sup

R

(( , ) .y y1 2

}

First, we prove the following lemma, which we will use in the proof of the approximation theorem.

We shall apply the method used in [12].

Lemma 2 Let f C R1( )2L Rp( )2 , 1 ≤ ≤ ∞p . Therefore

A f r y y

(

; , ,1 2

)

f y y A r y y( , ) 1; , ,1 2

(

1 2

)

1+2 2 − ⋅12 11+

(

+

)



2 2 12

22

r r

r

r y y exp

× −

+ + −

+





∂ 1

1

( 1) (1 )

( , )

2 2

4 12 2 2

1 2

( 1, 2 ) 2

1 2

r r

r y

r

f y y

y y R y sup

11





+ −

+ + −

+





∂ 1

1

( 1) (1 )

( , )

2 2

4 22 2 2

1 2

( 1, 2 ) 2

1 2

r r

r y

r

f y y

y y R y sup

22

.





for 0 < r < 1 and all ( , )y y1 2 ∈ .R2

Proof. Let ( , )y y1 2 ∈ .R2 be a fixed point and f C R1( )2L Rp( ).2 We have:

f z z f y y

u f u z du

v f y v dv

y z

y z

( , )1 2 ( , ) =1 2 ( , ) ( , )

1 1

2 2 2

− ∂ 1

∂ + ∂

∫ ∫

for all ( , )z z1 2R2. Let us denote:

λy τ

y z

y

y z

z z u f u z du z z

v f y v dv

1 1 2

1 1

2 2 1 2

2 2

( , ) =

( , ) , ( , ) =

( , ) .1

Observe that:

λy

y y

z z z y f y y

y

1 1 2 1 1

( 1, 2 ) 2

1 2 1

( , ) ≤ − ∂ ( , ),

∈R

sup τy

y y

z z z y f y y

y

2 1 2 2 2

( 1, 2 ) 2

1 2 2

( , ) ≤ − ∂ ( , ).

∈R

sup (3)

From (3) and Lemma 1, we obtain:

A r y y A r y y f y y

y y y

y y

λ1 1 2 ϕ1, 1 1 2

( 1, 2 ) 2

1 2 1

; , , ; , , ( , )

( )

( )

sup∈R

(5)

≤ +

+ + −

+



 − ⋅ − + 2

1 1 1

( 1) (1 )

1 2

1

2 1

2 2

4 12 2 2

1

2 2

2 12

r r

r r

r y

r

r r y

exp

(

++y22

)

 ( 1, 2 ) 2y ysupR f y y( , )y11 2 ,

A

(

τy2 ; , ,r y y1 2

)

1+2 2 11+ +( 1)(1+ )  − ⋅12 11+ 2

2

4 22 2 2

1

2 2

2 12

r r

r r

r y

r

r r y exp

(

++y22

)



( 1, 2 ) 2

1 2

2

( , ).

y y

f y y

y

R ∂ sup Hence:

A f r y y

(

; , ,1 2

)

f y y A r y y( , ) 1; , ,1 2

(

1 2

)

+ − ⋅ −

+

(

+

)





2 1

1 2

1

2 1

2 2 12

22

r r

r

r y y exp

× −

+ + −

+





∂ 1

1

( 1) (1 )

( , )

2 2

4 12 2 2

1 2

( 1, 2 ) 2

1 2

r r

r y

r

f y y

y y R y sup

11





+ −

+ + −

+





∂ 1

1

( 1) (1 )

( , )

2 2

4 22 2 2

1 2

( 1, 2 ) 2

1 2

r r

r y

r

f y y

y y R y sup

22





and the proof of the lemma is completed.

We are now in a position to prove the approximation theorem.

Theorem 2 Let f C R∈ ( )2L Rp( )2 , 1 ≤ ≤ ∞p . Therefore A f r y y( ; , , )1 2f y y A r y y( , ) (1; , , )1 2 1 2

≤ −

+ + −

+

+ + −

+

 6  ; 1

1

( 1)

(1 ) , 1 1

( 1) (1 )

2 2

4

2 2 12 2

2

4 2 2 22

ω f r

r r

r y r

r r

r y

 

 for 0 < r < 1 and all ( , )y y ∈R .1 2 2

Proof. Let ( , )y y ∈R .1 2 2 be a fixed point and fδ δ1, 2 be the Steklov mean defined by:

fδ δ y y f y u y v dudv y y R

δ δ

δ δ δ

1, 2 1 2

1 2 0 1 0

2

1 2 1 2 2

( , ) = 1

∫ ∫

( + , + ) for ( , ) , 11,δ2> 0.

From this definition, we conclude that:

fδ δ y y f y y u vf y y dudv

δ δ

δ δ

1, 2 1 2 1 2

1 2 0 1 0 2

, 1 2

( , )− ( , ) = 1

∫ ∫

( , ) ,

y f y y

(

vf y yvf y y dv

)

1 1, 2 1 2

1 2 0 2

1, 1 2 0, 1 2

( , ) = 1 ( , ) ( , ) ,

δ δ

δ

δ δ ∆δ

(6)

y f y y

(

u f y yu f y y du

)

2 1, 2 1 2

1 2 0 1

, 2 1 2 ,0 1 2

( , ) = 1 ( , ) ( , ) ,

δ δ

δ

δ δ ∆ δ

where

u v, f y y( , ) = (1 2 f y u y v1+ , 2+ −) f y y( , ).1 2

Hence, if f C R∈ ( )2L Rp( )2 , then fδ δ1, 2C R1( )2L Rp( )2 . Moreover

( 1, 2 ) 2 1, 2( , )1 2 ( , )1 2 ( ; , ),1 2 y y R

f y y f y y f

− ≤

sup δ δ ω δ δ

( 1, 2 ) 2 1 1, 2 1 2 11

1 2

( , ) 2 ( ; , ),

y y R y f y y f

∂ ≤

sup δ δ δ ω δ δ (4)

for all δ δ1, 2 > 0. Observe that

A f r y y( ; , , )1 2f y y A r y y( , ) (1; , , )1 2 1 2

A f f( − δ δ1, 2; , , )r y y1 2 + A f( δ δ1, 2; , , )r y y1 2fδ δ1, 2( , ) (1; ,y y A r1 2 yy y1, )2

+ fδ δ1, 2( , )y y1 2f y y( , )1 2A r y y(1; , , ),1 2 ( , )y y1 2R2, ,δ δ1 2 > 0.

From Lemma 2 and (4) we obtain

A f( δ δ1, 2; , , )r y y1 2fδ δ1, 2( , ) (1; , , )y y A r y y1 2 1 2

≤ + − ⋅ −

+

(

+

)





− + 2

1

1 2

1

1 2 ( ; , ) 1

1

2

2 2 12

22 11

1 2

r 2

r

r

r y y f r

exp δ ω δ δ

rr

r y

r

2

4 12 2 2

1

( 1) 2

(1 ) + −

+









+ −

+ + −

+









2 ( ; , ) 1 1

( 1) (1 )

21

1 2 2 2

4 22 2 2

1

δ ω f δ δ r 2

r

r y

r

≤ −

+ + −

+



 + −

2 ( ; , ) 1 1

( 1) (1 )

1

1 2 11 2

2

4 12 2 2

1 2

21 2

ω f δ δ δ r δ

r

r y

r

r 11

( 1)

(1 ) .

2

4 22 2 2

1 2

+ + −

+









 r 

r y

r

(7)

Using (4) we have:

fδ δ1, 2( , )y y1 2f y y( , )1 2A r y y(1; , , )1 2A r y y(1; , , ) ( ; , )1 2 ω f δ δ1 2 ≤≤ ω( ; , )f δ δ1 2 and

A f f( − δ δ1, 2; , , )r y y1 2

−∞

−∞

∫ ∫

r K r y z K r y z( , , ) ( , , ) | ( , )1 1 2 2 f z z1 2 fδ δ1, 2( , ) |z z dz d1 2 1 zz2

≤ −

−∞

−∞

∫ ∫

( 1, 2 ) 2 1, 2( , )1 2 ( , )1 2 ( , , ) (1 1 y y

f y y f y y r K r y z K r

R

sup δ δ ,, , )y z dz dz2 2 1 2

A r y y(1; , , ) ( ; , )1 2 ω f δ δ1 2 ≤ω( ; , )f δ δ .1 2 Finally, we get:

A f r y y( ; , , )1 2f y y A r y y( , ) (1; , , )1 2 1 2

≤ + −

+ + −

+



 + −

2 ( ; , ) 1 1 1

( 1) (1 )

1

1 2 11 2

2

4 2 2 12

1 2

21

ω f δ δ δ r δ

r r

r y rr

r r

r y

2 2

4 2 2 22

1 2

1

( 1) (1 )

+ + −

+













for all ( , )y y1 2R2, ,δ δ1 2 > 0. Choosing:

δ1 22 2 2 14 2 δ

1 2

2

2 2

= 1 4

1

( 1)

(1 ) = 1

1

( 1) (1

+ + −

+





+ + −

+ r

r r

r y r

r , r

rr2 2 y22

1 2

) ,





we obtain the desired estimation for A.

From Theorem 2, we can derive the following result.

Corollary 1 Let f C R∈ ( )2L Rp( ),2 1 ≤ ≤ ∞p . Then it holds

A f r y y f y y f r

r r

r y r

( ; , , ) ( , ) 6 ; 1 1

( 1)

(1 ) , 1

1 2 1 2 1

2 2

4

2 2 12 2

− ≤ −

+ + −

+

− ω +

rr r

r y

2

4 2 2 22

( 1) (1 )

+ −

+

 

 +| ( , ) |f y y1 2A r y y(1; , , ) 11 2

for 0 < r < 1 and all ( , )y y ∈R .1 2 2

(8)

R e f e r e n c e s

[1] Bergh J., Löfström J., Interpolation Spaces. An Introduction, Springer-Verlag, Berlin, Heidelberg, New York 1976.

[2] DeVore R.A., Lorentz G.G., Constructive Approximation, Springer, Berlin 1993.

[3] Ditzian Z., Totik V., Moduli of Smoothness, Springer, New York 1987.

[4] Gosselin J., Stempak K., Conjugate expansions for Hermite functions, Illinois J. Math.

38 1994, 177-197.

[5] Krech G., A note on the Poisson integral for Hermite expansions of functions of two variables, J. Appl. Anal. (submitted).

[6] Krech G., On the rate of convergence theorem for the alternate Poisson integrals for Hermite and Laguerre expansions, Ann. Acad. Paedagog. Crac. Stud. Math. 4, 2004, 103-110.

[7] Krech G., On some approximation theorems for the Poisson integral for Hermite expansions, Analysis Math. 40, 2014, 133-145.

[8] Krech G., Wachnicki E., Approximation by some combinations of the Poisson integrals for Hermite and Laguerre expansions, Ann. Univ. Paedagog. Crac. Stud. Math. 12, 2013, 21-29.

[9] Muckenhoupt B., Poisson integrals for Hermite and Laguerre expansions, Trans. Amer.

Math. Soc. 139, 1969, 231-242.

[10] Özarslan M.A., Duman O., Approximation properties of Poisson integrals for orthogo- nal expansions, Taiwanese J. Math. 12, 2008, 1147-1163.

[11] Szegö G., Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ. 33, Amer. Math.

Soc., Providence, R.I., 1939.

[12] Toczek G., Wachnicki E., On the rate of convergence and the Voronovskaya theorem for the Poisson integrals for Hermite and Laguerre expansions, J. Approx. Theory 116, 2002, 113-125.

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