TECHNICAL TRANSACTIONS FUNDAMENTAL SCIENCES

CZASOPISMO TECHNICZNE NAUKI PODSTAWOWE

3-NP/2014

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

Abstract 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 Streszczenie Artykuł poświęcony jest własnościom aproksymacyjnym całek Poissona związanych z wielomianami 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 operatory liniowe *

Institute of Mathematics, Pedagogical University of Cracow, Kraków, Poland; [email protected].

32 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 , t ) | 1

2

p

dt1 dt2 < ∞ 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 , y1 , y2 ) =

∫ ∫ r K (r , y , z ) K (r , y , z ) f ( z , z ) dz dz , 1

∞

K (r , y, z ) = ∑r hn ( y ) hn ( z ) n

=

2

1

(π(1 − r )) 2

n =0



1

2

1

2

1

2

0 < r < 1,

−∞ −∞

where:

∞ ∞

(

hn ( x) = 2n n! π

)

−

1 2

1 2

 1 1+ r2 2 2ryz  exp  − ⋅ y + z2 + , 2 1 − r 2   2 1− r

(

)

 x2  exp  −  H n ( x)  2

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



1  ∞∞ p  p   ∫ ∫ | f (t1 , t2 ) | dt1 dt2  ,  || f || p =   −∞ −∞  sup ess | f (t , t ) |, 1 2  (t ,t )∈R2  12

1 ≤ p < ∞, 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]): ∞ 2 1 2 2 Theorem 1 Let y = ( y1 , y2 ) ∈R and f = f1 + f 2 , where f1 ∈ L (R ), f 2 ∈ L (R ). If f is continuous at y, then:

lim

( r , y ) → (1− , y )

A( f ; r , y ) = f ( y ), y = ( y , y ). 1 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 , y1 , y2 ) = rA( f1 ; r , y1 ) A( f 2 ; r , y2 ) for f1 , f 2 ∈ Lp (R) and such that f ( z1 , z2 ) = f1 ( z1 ) f 2 ( z2 ) , where A( f )(r , y ) = A( f ; r , y ) =

∞

∫ K (r , y, z ) f ( z )dz,

−∞

0 < r < 1.

33 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: 1/ 2



 2  A(1; r , y ) =   1 + r 2 



 2  A(φ m, y ; r , y ) =   1 + r 2 

 1 1− r2 2  y , exp  − ⋅  2 1 + r 2 

1/ 2

m  2 

 m  (m − p )! ×∑   p p =0  p  ( m − 2 p )!2



 1 1− r2 2  y exp  − ⋅  2 1 + r 2  p

 1 − r 2   (1 − r ) 2 ⋅ −  1 + r 2   1+ r 2

 y 

m−2 p

m for 0 < r < 1, y ∈ R, where [a] denotes the integral part of a ∈ R and ϕ m, y ( z ) = ( z − y ) . From the above, we have the following result in the bivariate case. n Lemma 1 Let ϕ n , yi ( z1 , z2 ) = ( zi − yi ) , yi , zi ∈ R, i = 1, 2, n ∈ N . It holds

(

A ϕ1, yi ; r , y1 , y2



)

2r ≤ 1+ r2

1

 1 1− r2 2   1 − r 2 (r − 1) 4 2  2 y y1 + y22  (1) ⋅ + exp − ⋅ i 2 2 2 2   (1 + r )  2 1+ r  1 + r 

(

)

for 0 < r < 1. Proof. Using Hölder’s inequality, we get:

(

A φ1, y1 ; r , y1 , y2



)

1

∞ ∞ 2 ≤  ∫ ∫ rK (r , y1 , z1 ) K (r , y2 , z2 ) dz1dz2   −∞ −∞  1

1 1 ∞ ∞ 2 ×  ∫ ∫ rK (r , y1 , z1 ) K (r , y2 , z2 ) | z1 − y1 |2 dz1dz2  = A (1; r , y1 , y2 ) 2 ⋅ A φ 2, y1 ; r , y1 , y2 2  −∞ −∞  (2)

) ( (

(

))

2 for ( y1 , y2 ) ∈ R and 0 < r < 1. We have (see [5]):

A(1; r , y1 , y2 ) =





(

)

A φ 2, yi ; r , y1 , y2 =

2r 1+ r2

 1 1− r2 2  2r − ⋅ y1 + y22  , exp 2 2  1+ r  2 1+ r 

(

)

4    1 1− r2 2  1 − r 2 ( r − 1) 2   exp − ⋅ + y y1 + y22  , i = 1, 2. i 2 2 2   2 1+ r    1 + r 1+ r2

(

)

(

)

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

34 2. Rate of convergence In this section, we give an order of approximation of function of two variables in the space Lp. p 2 We achieve this using the modulus of continuity ω ( f ; δ1 , δ 2 ), δ1 , δ 2 > 0 of f ∈ L ( R ) defined as follows:

ω ( f ; δ1 , δ 2 ) = sup

{

0< h1 ≤ δ1 0< h2 ≤ δ 2

sup

( y1 , y2 ) ∈R2

f ( y1 + h1 , y2 + h2 ) − f ( y1 , y2 )

}.

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 ∈ C1 ( R 2 ) ∩ Lp ( R 2 ) , 1 ≤ p ≤ ∞. Therefore

A ( f ; r , y1 , y2 ) − f ( y1 , y2 ) A (1; r , y1 , y2 )

1  4 2   1 − r 2 (r − 1) y1  2 ×  + 2 (1 + r 2 ) 2   1 + r 

≤

 1 1− r2 2  2r exp  − ⋅ y1 + y22  2 1+ r  2 1+ r2 

(

)

1

sup

( y1 , y2 ) ∈R 2

∂f ( y1 , y2 )  1 − r 2 (r − 1) 4 y22  2 + + ∂y1  1 + r 2 (1 + r 2 ) 2 

sup

( y1 , y2 ) ∈R 2

 ∂f ( y1 , y2 )  . ∂y2  

for 0 < r < 1 and all ( y1 , y2 ) ∈ R 2 . Proof. Let ( y1 , y2 ) ∈ R 2 . be a fixed point and f ∈ C1 ( R 2 ) ∩ Lp ( R 2 ). We have: f ( z1 , z2 ) − f ( y1 , y2 ) =



z1

∫

y1

∂ f (u , z2 )du + ∂u

z2

∂

∫ ∂v f ( y , v)dv 1

y2

for all ( z1 , z2 ) ∈ R 2 . Let us denote:

z1

∂ λ y1 ( z1 , z2 ) = ∫ f (u , z2 )du , τ y2 ( z1 , z2 ) = ∂ u y1

z2

∂

∫ ∂v f ( y , v)dv. 1

y2

Observe that: λ y1 ( z1 , z2 ) ≤ z1 − y1

sup

( y1 , y2 ) ∈R2

∂f ( y1 , y2 ) , τ y2 ( z1 , z2 ) ≤ z2 − y2 ∂y1

sup

( y1 , y2 ) ∈R2

∂f ( y1 , y2 ) . (3) ∂ y2

From (3) and Lemma 1, we obtain:



(

) (

A λ y1 ; r , y1 , y2 ≤ A ϕ1, y1 ; r , y1 , y2

)

sup

( y1 , y2 ) ∈R2

∂f ( y1 , y2 ) ∂y1



35 2r ≤ 1+ r2

1

 1 − r 2 (r − 1) 4 y12  2  1 1− r2 2 2  exp +  1 + r 2 (1 + r 2 ) 2   − 2 ⋅ 1 + r 2 y1 + y2 

(

(

A τ y2 ; r , y1 , y2



)

∂f ( y1 , y2 ) , ∂y1

1

)

(

sup

( y1 , y2 ) ∈R2

Hence:

sup

( y1 , y2 ) ∈R2

)

∂f ( y1 , y2 ) . ∂y2

A ( f ; r , y1 , y2 ) − f ( y1 , y2 ) A (1; r , y1 , y2 )

1  4 2   1 − r 2 (r − 1) y1  2 ×  + 2 (1 + r 2 ) 2   1 + r 

( y1 , y2 ) ∈R2

 1 1− r2 2  2r  1 − r 2 (r − 1) 4 y22  2 exp  − ⋅ y1 + y22  ≤ + 2 2  2 2 2  2 1 + r 1 + r (1 + r )  1+ r  





sup

≤

 1 1− r2 2  2r exp y1 + y22  − ⋅ 2 2  1+ r  2 1+ r 

(

)

1  ∂f ( y1 , y2 )  ∂f ( y1 , y2 )  1 − r 2 (r − 1) 4 y22  2 + + sup  ∂y2 ∂y1  1 + r 2 (1 + r 2 ) 2  ( y1 , y2 )∈R2  

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 2 ) ∩ Lp ( R 2 ), 1 ≤ p ≤ ∞ . Therefore

A( f ; r , y1 , y2 ) − f ( y1 , y2 ) A(1; r , y1 , y2 )



 1 − r 2 (r − 1) 4 2 1 − r 2 (r − 1) 4 2  ≤ 6ω  f ; + y1 , + y2  1 + r 2 (1 + r 2 ) 2 1 + r 2 (1 + r 2 ) 2  

2 for 0 < r < 1 and all ( y1 , y2 ) ∈R . 2 Proof. Let ( y1 , y2 ) ∈R . be a fixed point and f δ1 ,δ2 be the Steklov mean defined by:



f δ1 ,δ2 ( y1 , y2 ) =

1 δ1δ 2

δ1 δ 2

∫ ∫ f (y

1

+ u , y2 + v)dudv

for

( y1 , y2 ) ∈ R 2 , δ1 , δ 2 > 0.

0 0

From this definition, we conclude that:



f δ1 ,δ2 ( y1 , y2 ) − f ( y1 , y2 ) = ∂ 1 f δ1 ,δ2 ( y1 , y2 ) = ∂y1 δ1δ 2

1 δ1δ 2

δ2

∫ (∆ 0

δ1 , v

δ1 δ 2

∫ ∫∆

u ,v

f ( y1 , y2 )dudv,

0 0

)

f ( y1 , y2 ) − ∆ 0,v f ( y1 , y2 ) dv,



36 ∂ 1 f δ1 ,δ2 ( y1 , y2 ) = ∂y2 δ1δ 2



δ1

∫ (∆ 0

u , δ2

)

f ( y1 , y2 ) − ∆ u ,0 f ( y1 , y2 ) du ,

where

∆ u ,v f ( y1 , y2 ) = f ( y1 + u , y2 + v) − f ( y1 , y2 ). 1 2 p 2 Hence, if f ∈ C ( R 2 ) ∩ Lp ( R 2 ), then f δ1 ,δ2 ∈ C ( R ) ∩ L ( R ). Moreover





f δ1 ,δ2 ( y1 , y2 ) − f ( y1 , y2 ) ≤ ω ( f ; δ1 , δ 2 ),

sup

( y1 , y2 ) ∈R 2

sup

( y1 , y2 ) ∈R 2

∂ f δ ,δ ( y1 , y2 ) ≤ 2δ1−1ω ( f ; δ1 , δ 2 ), ∂y1 1 2

(4)

for all δ1 , δ 2 > 0. Observe that



A( f ; r , y1 , y2 ) − f ( y1 , y2 ) A(1; r , y1 , y2 ) ≤ A( f − f δ1 ,δ2 ; r , y1 , y2 ) + A( f δ1 ,δ2 ; r , y1 , y2 ) − f δ1 ,δ2 ( y1 , y2 ) A(1; r , y1 , y2 ) + f δ1 ,δ2 ( y1 , y2 ) − f ( y1 , y2 ) ⋅ A(1; r , y1 , y2 ), ( y1 , y2 ) ∈ R 2 , δ1 , δ 2 > 0.

From Lemma 2 and (4) we obtain



A( f δ1 ,δ2 ; r , y1 , y2 ) − f δ1 ,δ2 ( y1 , y2 ) A(1; r , y1 , y2 ) 1   1 − r 2 (r − 1) 4 y12  2  1 1− r2 2 2r 2  −1 exp  − ⋅ y1 + y2  2δ1 ω ( f ; δ1 , δ 2 )  ≤ + 1+ r2  2 1+ r2   1 + r 2 (1 + r 2 ) 2  

(

)



1   1 − r 2 (r − 1) 4 y22  2  + 2δ ω ( f ; δ1 , δ 2 )  +   1 + r 2 (1 + r 2 ) 2   



1 1  4 2 4 2 2 2   r y ( − 1) 1 r −   −1  1 − r 2 (r − 1) y1  2 2 + . ≤ 2ω ( f ; δ1 , δ 2 ) δ1  + + δ −21  2 2 2  2 2 2   (1 + r )   (1 + r )  1 + r  1 + r  

−1 2



37 Using (4) we have: f δ1 ,δ2 ( y1 , y2 ) − f ( y1 , y2 ) ⋅ A(1; r , y1 , y2 ) ≤ A(1; r , y1 , y2 ) ω ( f ; δ1 , δ 2 ) ≤ ω ( f ; δ1 , δ 2 )

and



A( f − f δ1 ,δ2 ; r , y1 , y2 ) ≤

sup

( y1 , y2 ) ∈R2

≤

∞ ∞

∫ ∫ r K (r , y , z ) K (r , y , z ) | f ( z , z ) − f 1

1

2

2

1

2

−∞ −∞

f δ1 ,δ2 ( y1 , y2 ) − f ( y1 , y2 )

δ1 , δ 2

( z1 , z2 ) |dz1 dz2

∞ ∞

∫ ∫ r K (r , y , z ) K (r , y , z ) dz dz 1

1

2

2

1

2

−∞ −∞

≤ A(1; r , y1 , y2 ) ω ( f ; δ1 , δ 2 ) ≤ ω ( f ; δ1 , δ 2 ).

Finally, we get:

A( f ; r , y1 , y2 ) − f ( y1 , y2 ) A(1; r , y1 , y2 )



1 1   2 2 (r − 1) 4 2  2  (r − 1) 4 2  2  −1  1 − r −1  1 − r + y ≤ 2ω ( f ; δ1 , δ 2 ) 1 + δ1  + y δ + 2   1 2  1 + r 2 (1 + r 2 ) 2    1 + r 2 (1 + r 2 ) 2    

2 for all ( y1 , y2 ) ∈ R , δ1 , δ 2 > 0. Choosing: 1



 1 − r 2 (r − 1) 4 2  2 δ1 =  + y1 ,  1 + r 2 (1 + r 2 ) 2 

1

 1 − r 2 (r − 1) 4 2  2 y2 , δ2 =  +  1 + r 2 (1 + r 2 ) 2 

we obtain the desired estimation for A . From Theorem 2, we can derive the following result. 2 p 2 Corollary 1 Let f ∈ C ( R ) ∩ L ( R ), 1 ≤ p ≤ ∞ . Then it holds



 1 − r 2 (r − 1) 4 2 1 − r 2 (r − 1) 4 2  y2  + A( f ; r , y1 , y2 ) − f ( y1 , y2 ) ≤ 6ω  f ; + y1 , 1 + r 2 (1 + r 2 ) 2 1 + r 2 (1 + r 2 ) 2  



+ | f ( y1 , y2 ) | ⋅ A(1; r , y1 , y2 ) − 1

2 for 0 < r < 1 and all ( y1 , y2 ) ∈R .

38 References [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 orthogonal 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.