ON APPROXIMATION PROPERTIES OF TWO VARIABLES OF MODIFIED KANTOROVICH-TYPE OPERATORS

(1)

C om mun. Fac. Sci. U niv. A nk. Ser. A 1 M ath. Stat.

Volum e 68, N umb er 1, Pages 187–196 (2019) D O I: 10.31801/cfsuasm as.443703

ISSN 1303–5991 E-ISSN 2618-6470

http://com munications.science.ankara.edu.tr/index.php?series= A 1

ON APPROXIMATION PROPERTIES OF TWO VARIABLES OF MODIFIED KANTOROVICH-TYPE OPERATORS

MÜZEYYEN ÖZHAVZALI AND ALI OLGUN

Abstract. In the present paper, we introduce certain modi…cation of Szász- Mirakyan-Kantorovich-type operators in polynomial weighted spaces of continuous functions of two variables. Then we research some approximation properties of these operators. We give some inequalities for the operators by means of the weighted modulus of continuity and also obtain a Voronovskaya-type theorem. Furthermore, in the paper we show that our operators give better degree of approximation of functions belonging to weighted spaces than classical Szász-Mirakyan operators.

1. Introduction

In 1930, Kantorovich [7] introduced the following operators for f 2 L¹[0; 1] and x 2 [0; 1]:

Kn(f ; x) := (n + 1) X1 k=0

n

k x^k(1 x)^{n k}

k+1

Zn+1

k n+1

f (s) ds, n 2 N: (1.1)

In many papers various modi…cations of operators Kn(f ) were introduced and many authors studied their approximation properties in di¤erent function spaces (see [4, 5, 9, 12, 13, 14, 15, 18]).

In papers [1, 2, 8, 11, 14, 15, 16, 17, 21] Szász-Mirakyan operators Sn(f ; x) := e ^nx

X1 k=0

(nx)^k k! f k

n , x 2 R⁰= [0; 1) , (1.2)

Received by the editors: September 13, 2017, Accepted: November 20, 2017.

2010 Mathematics Subject Classi…cation. 41A25, 41A36.

Key words and phrases. Kantorovich-type operators, modulus of continuity, weighted spaces, Voronovskaya-type theorem.

This paper is supported by the Scienti…c Research Project Coordination Unit(BAP) in K¬r¬kkale University(Grant code: 2015/7).

c 2 0 1 8 A n ka ra U n ive rsity.

C o m m u n ic a tio n s Fa c u lty o f S c ie n c e s U n ive rs ity o f A n ka ra -S e rie s A 1 M a t h e m a tic s a n d S t a tis tic s . C o m m u n ic a tio n s d e la Fa c u lté d e s S c ie n c e s d e l’U n ive rs ité d ’A n ka ra -S é rie s A 1 M a t h e m a tic s a n d S t a tis t ic s .

187

(2)

were studied for f 2 Cp where C_p with …xed p 2 N0 := f0; 1; 2; :::g denotes the polynomial weighted space generated by the weight function

!0(x) := 1; !p(x) := (1 + x^p) ¹, p 1; (1.3) i.e. the space Cp is the set of all real-valued functions f continuous on R0for !pf is uniformly continuous and bounded on R0. The norm in Cp is also de…ned by

jjfjjp:= jjf (:)jjp:= sup

x2R⁰

!p(x) jf (x)j : (1.4) The degree of approximation of f 2 C^p by the operators (1.2) were studied and it was proved that

nlim!1Sn(f ; x) = f (x) (1.5) for every f 2 Cp, p 2 N0and x 2 R0: Moreover, the convergence in (1.5) is uniform on every interval [x₁; x₂], x₂> x₁ 0.

In [19] Szász-Mirakyan-Kantorovich operators were de…ned as

Tn(f ; x) := ne ^nx X1 k=0

n k

(nx)^k k!

k+1

Zn

k n

f (t) dt (1.6)

for x 2 R⁰, p 2 N⁰ and f 2 L¹[0; 1)(see also some modi…ed analogues of these operators [3, 6, 10, 15, 21]).

In 2003, Walczak [20] introduced modi…cation of the operators (1.2) with two variables. In the paper he considered the space Cp;q, associated with the weight function

!_p;q(x; y) := !_p(x)!_q(y), p; q 1, (x; y) 2 R²0= R₀ R₀, (1.7) and composed of all real-valued functions f continuous on R0, for wp;qf is uniformly continuous and bounded on R²₀. The norm on Cp;q is de…ned as

jjfjjp;q := jjf (:; :)jjp;q := sup

(x;y)2R²0

!_p;q(x; y) jf (x; y)j . (1.8) Similarly, the modulus of continuity of f 2 Cp;qis de…ned as usual by the formula

! (f ; Cp;q; t; s) = !p;q(x; y) := sup

0 h t; 0 sjj ^h; f (:; :)jjp;q, 8t; s 0, (1.9) where h; f (x; y) := f (x + h; y + ) f (x; y) for (x + h; y + ) 2 R²0. In addition C_p;q¹ is the set of all functions f 2 C^p;q, which whose …rst partial derivatives belong also to C_p;q. From (1.9) it follows that

lim

t;s!0⁺! (f ; Cp;q; t; s) = 0

(3)

for every f 2 Cp;q and p; q 2 N0. In [20] Walczak introduced a modi…ed Szász- Mirakyan operators on C_p;q for m, n, r, s 2 N, > 0 and (x; y) 2 R²0

A_m;n(f ; r; s; ; x; y) =_{g((m x+1)}2;r)g((n y+1)¹ ²;s)

X1 j=0

X1 k=0

(m x+1)^2j (j+r)!

(n y+1)^2k

(k+s)! f _{m x+1}^j+r ;_{n y+1}^k+s ,

(1.10)

where

g (t; r) :=

X1 k=0

t^k

(k + r)!, t 2 R0 (1.11)

i.e.

g (0; r) = 1

r!, g (t; r) = 1 t^r

0

@e^t

r 1

X

j=0

t^j j!

1 A , t > 0:

If f 2 C^p;q and f (x; y) = f1(x) f2(y), then

A_m;n(f ; r; s; ; x; y) = A_m(f₁; r; ; x) A_n(f₂; s; ; y) (1.12) for all (x; y) 2 R²0 and m, n; r; s 2 N.

Also he gave the theorems on the degree of approximation of functions from polynomial and exponential weighted spaces by the operators (1.10). In his work degree of these operators for approximation is similar but in some cases it is better than for aproximation in [19].

The purpose of this paper is to introduce a modi…ed Kantorovich-type of (1.10) with two variables and also study convergence properties of the operators for functions on Cp;q and C_p;q² by using the methods in [6, 20, 21].

2. Auxiliary Results

In the sequel we shall need several lemmas, which are necessary to prove the main theorems. Firstly we will give the moments of the operators. For this purpose we introduce the following class of operators on Cp;q:

De…nition 1. Let m; n; r; s 2 N and p; q 2 N0 and (m ) ; (n ) be positive se- quences such that lim

m!1m = lim

n!1n = 1 for > 0. Then for f 2 Cp;q we de…ne the modi…ed Szász-Mirakyan-Kantorovich operators as

A_m;n(f ; r; s; ; x; y) = A_m;n(f ; x; y) := _{g((m x+1)}2;r)g((n y+1)^mn ²;s)

X1 j=0

X1 k=0

(m x+1)^2j (j+r)!

(n y+1)^2k (k+s)!

j+r+1

Rm j+r

m k+s+1

Rn k+s

n

f _{m x+1}^t ;_{n y+1}^u dtdu;

(2.1)

where (1.11) holds.

(4)

In this paper we use short notation

g (m x + 1)²; r g (n y + 1)²; s = gm;r(x) gn;s(y):

Also, we will denote by Mk, k = 1; 2; :::; the suitable positive constants depending only on the parameters p; q and r; s.

It is known that A_m;nare positive linear operators acting from C_p;qto C_p;q a nd we have

A_m;n(1; x; y) = 1 (2.2)

for p; q 2 R0, m; n; r; s 2 N, > 0 and (x; y) 2 R0².

Other moments of Am;n(t^k; r; s; x; y) can be obtained easily for k = 1; 2. From (2.2) and (1.12) we get the following lemmas:

Lemma 1. Let m; n; r; s 2 N be …xed numbers. Then for all (x; y) 2 R²0; f 2 Cp;q

and > 0 we have

Am;n((t x) ; r; s; :; :) = _m¹ +_{2m (m x+1)}¹ +m (m x+1)(r¹ 1)!gm;r(x); (2.3) Am;n (t x)²; r; s; :; : =_m²2 +^(r ^1)!g^m;r(x)+3(r+1)+3(m x+1)²

6m² (m x+1)²(r 1)!gm;r(x)

1 2m² (m x+1)

(m x+1)³(r 1)! m² (m x+1)²

m² (m x+1)⁴(r 1)! +^{(m x+1)}_m2 (m x+1)²^(r ^1)!⁴ :

(2.4)

Lemma 2. Let m; n; r; s 2 N be …xed numbers. Then there exist s;j(r) = r^j ¹ and _;j(s) = s^j ¹ depending only j; r; s such that

Am;n t^u+1+ z ⁺¹; x; y

= x+_m¹ ^u+1 8<

:

u+1X

j=1 1

(m x+1)^2(j ¹⁾ '_u;j+_{(m x+1)}^j 2 +_{(m x+1)}2^u;j(r^(r)1)!gm;r(x)

9=

;

+ y+_n¹ ⁺¹ 8<

: X+1 j=1

1

(n y+1)^2(j ¹⁾ ' _;j+_{(n y+1)}^j 2 +_{(n y+1)}2(s 1)!g^;j^(s) n;s(y)

9=

; (2.5) for all f 2 C^p;q; > 0; 1 j r; 1 j s and (x; y) 2 R²0. Also _u;1(:) ; _;1(:) and '_u;j; ' _;j; _j are positive constants and the others are equal to one.

Lemma 3. Let p; q 2 N0 and m; n; r; s 2 N be …xed numbers. Then for given positive constants M₂; M₃ we have

A_m;n 1

!p;q(t; z); r; s; :; :

p;q

M₂; m; n 2 N (2.6) and for all f 2 C^p;q we obtain

jjA^m;n(f ; r; s; :; :)jjp;q M3jjfjjp;q; m; n 2 N. (2.7)

(5)

Lemma 4. Let p; q 2 N0 and m; n; r; s 2 N be …xed numbers. Then for given positive constants M₄; M₅ we have

Am;n

(t :)²

!p;q(t; z); r; s; :; :

!

p;q

M4

m² + M5

n² ; m; n 2 N (2.8) for all f 2 C^p;q:

The methods used to prove the above Lemmas are similar to modi…ed Szász- Mirakyan operators for f in [14, 15, 20]. Thus their proofs are very obvious.

3. Approximation Behaviour of Operators

Our …rst main result is the following theorem for approximation behaviour of Am;n.

Theorem 1. Let f 2 Cp;q¹ ; > 0 be with p; q 2 N⁰ and r; s 2 N. Then for a given positive constant M6 we have

jjAm;n(f ; :; :) f (:; :)jj_p;q M₆ 1

m jjfx⁰jjp;q+ 1 n f_y⁰

p;q ; m; n 2 N. (3.1) Proof. Let (x; y) 2 R0² be a …xed point. Then for f 2 Cp;q¹ and (t; z) 2 R0², t x,

> 0 we get

f (t; z) f (x; y) = Zt

x

f_u⁰ (u; z) du + Zz

y

f_v⁰(x; v) dv. (3.2)

By linearity of A_m;n, (3.2) we obtain

Am;n(f (t; z); x; y) f (x; y)) = Am;n

Rt x

f_u⁰ (u; z) du; x; y +Am;n

Rz y

f_v⁰(x; v) dv; x; y

!

From (1.4) and (1.5) we have Zt

x

f_u⁰(u; z) du jjfx⁰jjp;q

1

!p;q(t; z)+ 1

!p;q(x; z) jt xj , (x; y) 2 R²0. (3.3) By (3.3) it follows that

!p;q(x; y) jA^m;nf (t; z); x; y) f (x; y)j jjfx⁰jjp;q!_p;q(x; y)n

A_m;n _!^{jt xj}

p;q(x;z); x; y + A_m;n _!^{jt xj}

p;q(t;z); x; y o (3.4)

(6)

for m; n 2 N. Using the Hölder inequality, by Lemmas 1, 3, 4 and (2.2) we obtain

A_m;n(jt xj ; x; y) n

A_m;n (t x)²; x; y o¹²

fAm;n(1; x; y)g

1 2

M7

m :

Applying for the last inequality by (1.9), we get

!p;q(x; y) Am;n jt xj

!p;q(t;z); x; y n

!p;q(x; y) Am;n (t x)²

!p;q(t;z); x; y o¹2

n

!p;q(x; y) Am;n 1

!p;q(t;z); x; y o¹2

M8

m

(3.5)

for every (x; y) 2 R²0implying

!p;q(x; y) Am;n

Rt x

f_u⁰ (u; z) du; x; y ^M_m⁹jjfx⁰jjp;q, m; n 2 N. (3.6)

Analogously we have

w_p;q(x; y) A_m;n Rz y

f_v⁰(x; v) dv; x; y

!

M10

n f_y⁰

p;q, m; n 2 N. (3.7) We combine (3.6) and (3.7) and derive from (3.3) that (3.1) is satis…ed.

Now, we compute the rate of convergence of Am;n by means of the weighted modulus of continuity given by (1.9).

Theorem 2. Let f 2 Cp;q¹ and p; q 2 N⁰; r; s 2 N and > 0. Then there exists a positive constant M11 such that

jjAm;n(f ; r; s; :; :) f (:; :)jj_p;q M₁₁!₁ f ; C_p;q; 1 m ; 1

n ; m; n 2 N. (3.8) Proof. Let f_h; be the Steklov means of function f 2 Cp;q¹ de…ned by the formula

f_h; (x; y) := 1 h

Zh

0

du Z

0

f (x + u; y + v) dv; (x; y) 2 R²0; h; > 0. (3.9)

(7)

From (3.9) we get

@

@xf_h; (x; y) = (f_h; )⁰_x(x; y) = 1 h

Zh

0

h;0f (x + u; y) du,

@

@yf_h; (x; y) = (fh; )⁰_y(x; y) = 1 h

Z

0

o; f (x; y + v) dv, which imply f_h; (x; y) 2 Cp;q¹ for every …xed h; > 0. Also we have

f_h; f _p;q ! (f ; Cp;q; h; ) ; (3.10) (fh; )⁰_x _p;q 2h ¹! (f ; Cp;q; h; ) ; (3.11) (fh; )⁰_y

p;q 2 ¹! (f ; Cp;q; h; ) : (3.12) Hence by the last inequalities we can write

!_p;q(x; y) j(Am;n(f ; r; x; y) f (x; y))j

!_p;q(x; y) fjAm;n(f (t; z)) f_h; (t; z); x; y)j + jAm;n(f_h; (t; z); x; y) f_h; (x; y)j + jfh; (x; y) f (x; y)jg := L1+ L₂+ L₃

(3.13) for every m; n 2 N; h; > 0 and (x; y) 2 R0². For L₁and L₃; by using Lemma 3 and (3.10), we get

kL1kp;q M₁₂kf f_h; kp;q M₁₂! (f ; C_p;q; h; ) , kL³kp;q ! (f ; Cp;q; h; ) .

Similarly, by Theorem 1 and (3.11),(3.12) we have kL2kp;q M₁₃ _m¹ (f_h; )⁰_{x p;q}+_n¹ (f_h; )⁰_y

p;q

2M14! (f ; Cp;q; h; ) _{m h}¹ +_n¹ ; h; > 0; m; n 2 N:

(3.14)

Hence, from (3.14) for (3.13) it follows that

kA^m;n(f ; r; s; ; :; :) f (:; :)kp;q M15 1 + 1 m h+ 1

n ! (f ; Cp;q; h; ) . Now, for …xed m; n 2 N, substitution of h = _m¹ and = _n¹ in the last inequality, we obtain the desired result of (3.8). This completes the proof of Theorem 2.

The following corollories are immediate consequences of Theorem 1 and 2.

(8)

Corollary 1. For every …xed numbers r; s 2 N; p; q 2 N0 and f 2 Cp;q; we have

m;nlim!1jjA^m;n(f ; r; s; :; :) f (:; :)jjp;q= 0. (3.15) Corollary 2. For every …xed numbers r; s 2 N; p; q 2 N⁰ and f 2 Cp;q¹ ; we have

jjA^m;n(f ; r; s; :; :) f (:; :)jjp;q= o 1 m ; 1

n (3.16)

as m; n ! 1.

Now we will prove the following Voronovskaya-type theorem.

Theorem 3. Let f 2 Cp;q² be with given p; q 2 N0 and r; s 2 N. Then for every (x; y) 2 R²0

nlim!1n fA^n;n(f ; r; s; x; y) f (x; y)g = x

2f_xx⁰⁰ (x; y) +y

2f_yy⁰⁰ (x; y) . (3.17) Proof. Let (x; y) be a …xed point in R²₀. Then, by the Taylor formula we can write

f (t; z) = f ( x; y) + f_x⁰ (x; y) (t x) + f_y⁰(x; y) (z y) +¹₂n

f_xx⁰⁰ (x; y) (t x)² +2f_xy⁰⁰ (x; y) (t x) (z y) + f_yy⁰⁰ (x; y) (z y)²

o

+" (t; z; x; y) n

(t x)⁴+ (z y)⁴ o¹₂

for f 2 Cp;q² , (t; z) 2 R²0 where " (:; :; x; y) " (:; :) 2 Cp;q¹ is function such that lim

(t;z)!(x0;y0)

" (t; z; x; y) = 0:

Applying (2.1) to the last equality, we get

A_n;n(f ; x; y) f (x; y) = f_x⁰(x; y) A_n;n((t x) ; x; y) +f_y⁰(x; y) A_n;n((z y) ; x; y) +¹₂

n

f (x; y)⁰⁰_xx(x; y) An;n (t x)²; x; y +2f_xy⁰⁰ (x; y) A_n;n((t x) (z y) ; x; y)

+f_yy⁰⁰ (x; y) A_n;n((z y)²; x; y)o

+A_n;n " (t; z) q

(t x)⁴+ (z y)⁴; x; y) := L₁+ L₂+ L₃+ L₄+ L₅+ L₆:

From (3.2),(3.3) and Lemma 1, the limit of the L1; L2 and L4 are equal to zero as n ! 1 and

nlim!1n L₃= x; lim

n!1n L₅= y:

(9)

For the right term in the last equation by the Hölder inequality we obtain jL⁶j 2 An;n("²(t; z); x; y)

1 2n

An;n (t x)⁴+ (z y)⁴; x; y o¹₂

By Corollary 1 and properties of " (:; :) we deduce that

nlim!1An;n "²(t; z) ; x; y = "²(x; y) = 0:

From this, the linearity of A_n;n and Lemma 1 we have

nlim!1n A_n;n q

(t x)⁴+ (z y)⁴; x; y) = 0:

Collecting these results, we immediately obtain the desired result (3.17).

In this paper, Theorem 1, 2 and Corollary 2 show that our operator A_m;n; m; n 2 N , give better degree of approximation of functions f 2 Cp;q and f 2 Cp;q¹ than classical Szász-Kantorovich operators.

AcknowledgmentsThe authors are thankful to referees for the attentive read- ing of the manuscript and for useful comments.

References

[1] Aral, A., A Generalization of Szász-Mirakyan Operators besed on q-integers, Math.Comp.

Modelling, 47(2008), 1052-1062.

[2] Becker, M., Global approximation theorems for Szász-Mirakyan and Baskakov in polynomial weighted spaces, Indian Univer. Math. J., 27 (1978), 127-14.

[3] Stancu, D. D., A new class of uniform approximating polynomial operators in two and several variables, Prooc. of Conf. on the Constr. Theory of Func. (ApproximationTheory, Bu- dapest,1969), 443-455, 1972.

[4] Duman, E. E., Duman, O., Integral-type generalizations of operators obtained from certain multivariate polynomials, Calcolo,Volume 45, Issue 1, pp 53–67(2008).

[5] Erençin, A. , Tunca, G.B. and Ta¸sdelen, F., Kantorovich type q-Bernstein-Stancu operators, Stud. Univ. Babe¸ s-Bolyai Math. 57 (2012), No. 1, 89–105.

[6] Firlej, B., Lesniewicz, M, Rempulska L., Approximation of Functions Two Variables by Some Operators in Weighted Spaces, Rendiconti del Seminario Matematico della Universita di Padova, Tome 101(1999), p.63-82.

[7] Kantorovich, L.V., Sur certain developments suivant les polynomes de la forme de S. Bern- stein, I,II, C.R. Acad. URSS(1930), 563-568, 595-600.

[8] Gupta P., Gupta V., Rate of Convergence on Baskakov-Szász operators type operators, Fasc.

Math., 31(2001), 37-44.

[9] Gupta V., Agrawall Ravi P., Convergence Estimates in Approximation Theory, Springer, Switzerland, (2014).

[10] Moreno, A.J.L., Delgado, F.J.M., Asymtotic Expansion of Multivariate Kantorovich Type Operators, Numerical Algorithms, (2005), 39, 237-252.

[11] Olgun, A., Some Properties of The Multivariate Szazs Operators, Comptes rendus de I’Academie Bulgare des Sciences, Tome 65, No 2 ( 2012),139-146.

[12] O. Dogru, Özalp, N., Approximation by Kantorovich type generalization of Meyer-König and Zeller operators, Glasnik Mat. (2001), No 36, 311-317.

(10)

[13] Özarslan, M.A., Duman, O. and Srivastava, H.M., Statistical Apparoximation Resullts for Kantorovich-Type Operators Involving Some Special Polynomials, Mathematical and Com- puter Modelling, 48 (3)(2008), 388-401.

[14] Özhavzal¬, M., On Properties For Some Kantorovich-Type Operators, Lambert Publishing, Germany, (2016).

[15] Özhavzal¬, M., Olgun, A., Approximation Properties For Modi…ed Kantorovich-Type Oper- ators, Journal of Mathematics and System Science, 5 (2015), 18-26.

[16] Rempulska, L., Walczak, Z., Modi…ed Szász-Mirakyan Operators, Mathematica Balkanica, New Series, vol. 18,(2001), Fasc. 1-2.

[17] Ta¸sdelen, F., Akta¸s, R. and Alt¬n, A., Kantorovich Type of Szasz Operators Including Brenke- Type Polynomials, Abstract and Applied Analysis, (2012), doi:10.1155/2012/867203.

[18] Tasdelen F., Olgun A., Tunca, G. B., Approximation of functions of two variables by certain linear positive operators, Proc. Indian Acad. Sci.,(2005), Vol.117 No.3, 387–399.

[19] Totic, V., Approximation by Szász-Mirakyan-Kantorovich Operators in L^p(p > 1), Analysis Math, 9 (1983)147–167.

[20] Walczak, Z., Approximation of Functions of Two Variables by Some Linear Positive Opera- tors, Acta Math. Univ. Comenianae, vol. LXXIV,1 (2005), pp.37–48.

[21] Walczak, Z., On Certain Positive Linear Operators in Polinomial Weight Spaces, Acta Math.

Hungar., 101(3) (2003), pp.179-191.

Current address : Müzeyyen Özhavzal¬(corresponding author): K¬r¬kkale University, Faculty of Art&Science Mathematics Department 71650 K¬r¬kkale Turkey

E-mail address : thavzalimuzeyyen@hotmail.com

ORCID Address: http://orcid.org/0000-0003-0518-3021

Current address : Ali Olgun: K¬r¬kkale University Faculty of Art&Science Mathematics De- partment 71650 K¬r¬kkale Turkey

E-mail address : aliolgun71@gmail.com

ORCID Address: http://orcid.org/0000-0001-5365-4110