Approximation of functions of two variables by modified Szasz-Mirakyan operators

Monika Herzog

Approximation of functions of two variables by modified Szasz-Mirakyan operators

Abstract. In this paper we study approximative properties of modified Szasz-Mirakyan operators for functions of two variables from polynomial weight spaces. We present some direct theorems giving a degree of approximation for these operators.

2000 Mathematics Subject Classification: 41A36.

Key words and phrases: Linear positive operators, Bessel function, Modulus of con- tinuity, Degree of approximation.

1. Introduction. Let us denote by C(R⁰) a set of all real-valued functions continuous on R⁰ = [0; +∞) and let N⁰ = N ∪ {0}. In paper [1] we investigated operators of Szasz-Mirakyan type defined as follows

A^ν_n(f; x) =

( 1

Iν(nx)P∞

k=0 (^nx₂ )^2k+ν

Γ(k+1)Γ(k+ν+1)f (^2k_n), x > 0;

f (0), x = 0,

where Γ is the Euler-gamma function and Iν the modified Bessel function defined by the formula ([6], p. 77)

Iν(z) = X∞ k=0

(^z₂)^2k+ν Γ(k + 1)Γ(k + ν + 1). We studied the operators in polynomial weight spaces

Cp= {f ∈ C(R⁰) : wpf is uniformly continuous and bounded onR⁰}, where wp was the polynomial weight function defined as follows

wp(x) =

 1, p = 0;

1+x1^p, p∈ N,

for x ∈ R⁰.

In the present paper we will consider the bivariate version of the operator A^ν_n in appropriate weight spaces.

Let C(R²0) be the set of all real-valued functions continuous onR²0. Similarly as in [1] we define the polynomial weight space

(1) Cp,q = {f ∈ C(R²0) : wp,qf is uniformly continuous and bounded onR²0}, where wp,q is the polynomial weight function defined as follows

(2) wp,q(x, y) =









1, p = q = 0;

1+x1^p, q = 0, p∈ N;

1+y1^q, p = 0, q∈ N;

(1+x^p)(1+y1 ^q), p, q∈ N, for (x, y) ∈ R²0. The space Cp,q is a normed space with the norm (3) kfk^p,q = sup{w^p,q(x, y)|f(x, y)|; (x, y) ∈ R²0}.

Moreover, we consider the modulus of continuity

(4) ω(f, Cp,q; t, s) = sup{k∆^h,dfk^p,q; h ∈ [0, t], d ∈ [0, s]}, where

∆h,df (x, y) = f (x + h, y + d)− f(x, y) for (x, y) ∈ R²0, h, d ∈ R⁰.

The note was inspired by the results of [2]-[5]. We introduce the modified Szasz- Mirakyan operator for functions f ∈ C^p,q in the following way

(5) A^ν,µ_n,m(f; x, y) =













1 Iν(nx) 1

Iµ(my)P∞ k=0P∞

j=0

(^nx₂)^2k+ν Γ(k+1)Γ(k+ν+1)

(^my₂ )^2j+µ

Γ(j+1)Γ(j+µ+1)f (^2k_n,^2j_m), x > 0, y > 0;

1 Iν(nx)P∞

k=0

(^nx₂ )^2k+ν

Γ(k+1)Γ(k+ν+1)f (^2k_n, 0), x > 0, y = 0;

1 Iµ(mx)P∞

j=0 (^mx₂ )^2j+ν

Γ(j+1)Γ(j+µ+1)f (0,^2j_m), y > 0, x = 0;

f (0, 0), x = y = 0.

where p, q ∈ N⁰, ν, µ ∈ R⁰, n, m ∈ N.

We shall present direct approximation theorems for these operators. The main re- sults of the paper are theorems giving a degree of approximation of function f ∈ C^p,q by operators A^ν,µ_n,m.

2. Auxiliary results. In this section we will show some basic properties of the operator A^ν,µ_n,m.

At the beginning we will recall preliminary results from paper [1] for the operator A^ν_n, which we shall apply to the proofs of the main theorems.

Lemma 2.1 For each n ∈ N, ν ∈ R⁰ and x ∈ R⁰

A^ν_n(1; x) = 1, A^ν_n(t; x) = xIν+1(nx) Iν(nx) , A^ν_n(t²; x) = x²Iν+2(nx)

Iν(nx) + x2 n

Iν+1(nx) Iν(nx) , A^ν_n(t − x; x) = x

Iν+1(nx) Iν(nx) −1

 ,

A^ν_n((t − x)²; x) = x²

Iν+2(nx)

Iν(nx) −2Iν+1(nx) Iν(nx) + 1

 + x2

n

Iν+1(nx) Iν(nx) .

Lemma 2.2 For each ν ∈ R⁰there exists a positive constant M(ν) such that for all n∈ N and x ∈ R⁰ we have Iν+1(nx)

Iν(nx) 

 ¬ M(ν),

nx 

Iν+1(nx) Iν(nx) −1

 ¬ M(ν).

Hence we immediately get

Lemma 2.3 For each ν ∈ R0there exists a positive constant M(ν) such that for all n∈ N and x ∈ R0 we have

|A^νn(t − x; x)| ¬ M (ν)

n , |A^νn((t − x)²; x)| ¬ M(ν)x n.

Lemma 2.4 For all p ∈ N⁰and ν ∈ R⁰there exists a positive constant M(p, ν) such that for each n ∈ N we have

kA^νn(1/wp; ·)k^p ¬ M(p, ν).

Theorem 2.5 For all p ∈ N0 and ν ∈ R0 there exists a positive constant M(p, ν) such that for each n ∈ N we have

kA^νn(f; ·)k^p¬ M(p, ν)kfk^p.

Lemma 2.6 For all p ∈ N⁰and ν ∈ R⁰there exists a positive constant M(p, ν) such that for all x ∈ R⁰ and n ∈ N we have

wp(x)A^ν_n

(t − x)² wp(t) ; x



¬ M(p, ν)x + 1 n .

The definition of the operator A^ν,µ_n,m implies

(6) A^ν,µ_n,m(f; x, y) = A^ν_n(f₁; x)A^µ_m(f₂; y) for all functions f ∈ C^p,q of the form f(x, y) = f1(x)f2(y).

In particular we get

A^ν,µ_n,m(1; x, y) = 1,

A^ν,µ_n,m(1/wp,q; x, y) = A^ν_n(1/wp; x)A^µ_m(1/wq; y).

From the above facts and Lemma 2.4 we derive

Lemma 2.7 For all p, q ∈ N⁰ and ν, µ ∈ R⁰ there exists a positive constant M (p, q, ν, µ) such that for all n, m ∈ N we have

kA^ν,µn,m(1/wp,q; ·)k^p,q ¬ M(p, q, ν, µ).

Theorem 2.8 For all p, q ∈ N0 and ν, µ ∈ R0 there exists a positive constant M (p, q, ν, µ) such that for any f ∈ C^p,q and n, m ∈ N we have

kA^ν,µn,m(f; ·)k^p,q¬ M(p, q, ν, µ)kfk^p,q. Proof Applying equation (6) and definition (3) we get

wp,q(x, y)|A^ν,µn,m(f(t, s); x, y)| ¬ w^p,q(x, y)A^ν,µ_n,m(|f(t, s)|; x, y) =

wp,q(x, y)A^ν,µ_n,m



wp,q(t, s)f(t, s) 1

wp,q(t, s); x, y



¬

kfk^p,qwp(x)wq(y)A^ν_n

 1

wp(t); x

 A^µ_q

 1

wq(s); y



¬ M(p, ν)M(q, µ)kfk^p,q. Hence the operator A^ν,µ_n,mtransforms the space Cp,q into Cp,q. _

3. Approximation theorems.

Theorem 3.1 For all p, q ∈ N0, ν, µ ∈ R0 and for each function g ∈ Cp,q¹ = {f ∈ Cp,q : f⁰ ∈ C^p,q} there exists a positive constant M(p, q, ν, µ) such that for all (x, y) ∈ R²0 and n, m ∈ N we have

wp,q(x, y)|A^ν,µn,m(g; x, y) − g(x, y)| ¬

M (p, q, ν, µ) kgx⁰k^p,q

x + 1 n

¹₂

+ kgy⁰k^p,q

y + 1 m

¹₂! .

Proof Pick (x, y) ∈ R²0. For (t, s) ∈ R²0 and g ∈ Cp,q¹ we can write

g(t, s)− g(x, y) = Z t

x

g_k⁰(k, s) dk +Z s y

g_u⁰(x, u) du.

By the linearity of the operator A^ν,µ_n,mwe obtain

|A^ν,µn,m(g(t, s); x, y) − g(x, y)| = |A^ν,µn,m(g(t, s) − g(x, y); x, y)| = 

A^ν,µn,m

Z t x

g_k⁰(k, s) dk +Z s y

g⁰_u(x, u) du; x, y ¬ A^ν,µ_n,m

Z t x

g⁰_k(k, s) dk   ; x, y



+ A^ν,µ_n,m Z s

y

g_u⁰(x, u) du   ; x, y

 . Since

Z t x

g_k⁰(k, s) dk 

 ¬ kg^0xk^p,q  

Z t x

1 wp,q(k, s)dk

¬ kg⁰xk^p,q

 1

wp,q(t, s)+ 1 wp,q(x, s)



|t − x|, we get

wp,q(x, y)A^ν,µ_n,m 

Z t x

g⁰_k(k, s) dk   ; x, y



¬

kgx⁰k^p,qwp,q(x, y)

 A^ν,µ_n,m

 |t − x|

wp,q(t, s); x, y



+ A^ν,µ_n,m

 |t − x|

wp,q(x, s); x, y



=

kg⁰xk^p,qwp,q(x, y)A^µ_m

 1

wq(s); y

  A^ν_n

|t − x|

wp(t); x



+ 1

wp(x)A^ν_n(|t − x|; x)



=

kg⁰xk^p,qwq(y)A^µ_m

 1

wq(s); y

 

wp(x)A^ν_n

|t − x|

wp(t); x



+ A^ν_n(|t − x|; x)

 . Using the H¨older inequality and Lemmas 2.3 - 2.5 we obtain

wp,q(x, y)A^ν,µ_n,m Z t

x

g⁰_k(k, s) dk   ; x, y



¬

M (p, q, ν, µ)kgx⁰k^p,q

x + 1 n

¹₂ . Analogously we can write the following estimation

wp,q(x, y)A^ν,µ_n,m Z s

y

g⁰_u(x, u) du   ; x, y



¬

M (p, q, ν, µ)kg⁰yk^p,q

y + 1 m

¹₂ ,

which completes the proof. _

Theorem 3.2 For all p, q ∈ N⁰, ν, µ ∈ R⁰ and for each f ∈ C^p,q there exists a positive constant M(p, q, ν, µ) such that for all (x, y) ∈ R²0 and n, m ∈ N we have

wp,q(x, y)|A^ν,µn,m(f; x, y)−f(x, y)| ¬ M(p, q, ν, µ)ω f, Cp,q,

x + 1 n

¹₂ ,

y + 1 m

¹₂! .

Proof Pick (x, y) ∈ R²0and h, d ∈ R⁺. We define the Steklov mean of f ∈ C^p,q as follows

fh,d(x, y) = 1 hd

Z h 0

Z d

0 f (x + u, y + v) du dv fh,d(x, y) = 1

hd Z x+h

x

Z y+d y

f (u, v) du dv.

Observe that

(fh,d)⁰_x(x, y) = 1 hd

Z y+d y

f (x + h, v) dv− Z y+d

y

f (x, v) dv

!

=

1 hd

Z y+d

y (f(x + h, v) − f(x, v)) dv = 1 hd

Z d

0 (f(x + h, y + v) − f(x, y + v)) dv = 1

hd Z d

0

(∆h,vf (x, y)− ∆^0,vf (x, y)) dv.

Similarly we can show that (fh,d(x, y))⁰_y= 1

hd Z h

0

(∆u,df (x, y)− ∆u,0f (x, y)) du.

Therefore (fh,d)⁰_x, (fh,d)⁰_y ∈ C^p,q. Moreover,

fh,d(x, y) − f(x, y) = 1 hd

Z h 0

Z d

0 ∆u,vf (x, y) du dv, hence and definitons (3), (4) we get

kf^h,d− fk^p,q ¬ ω(f, C^p,q, h, d), k(f^h,d)⁰_xk^p,q ¬ 1

h(k∆^h,vfk^p,q+ k∆0,vfk^p,q) ¬ 2

hω(f, Cp,q, h, d), k(f^h,d)⁰_yk^p,q ¬ 2

dω(f, Cp,q, h, d).

By the linearity of A^ν,µ_n,m, Theorems 2.2 and 3.1 we obtain wp,q(x, y)|A^ν,µn,m(f; x, y) − f(x, y)| ¬

wp,q(x, y)(A^ν,µn,m(|f − f^h,d|; x, y) + |A^ν,µn,m(fh,d; x, y) − f^h,d(x, y)| + |f^h,d(x, y) − f(x, y)|) ¬ M(p, q, ν, µ)(kf−f^h,dk^p,q+k(f^h,d)⁰xk^p,q_{x+ 1}

n

¹₂

+k(f^h,d)⁰yk^p,q_{y+ 1} m

¹₂

+kf^h,d−fk^p,q) ¬

M(p, q, ν, µ)ω(f, Cp,q, h, d)

 2 +2

h

_{x+ 1} n

¹₂ +2

d

_{y+ 1} m

¹₂ ,

for (x, y) ∈ R²0, n, m ∈ N and h, d ∈ R⁺. Setting, for (x, y) ∈ R²0 and n, m ∈ N, h = ^x+1_n
¹₂ and d = ^y+1_m
¹₂ we get the desired estimation. _

Theorem 3.2 implies the following corollaries.

Corollary 3.3 If ν, µ ∈ R⁰ and f ∈ C^p,q with some p, q ∈ N⁰, then for all (x, y) ∈ R²0

n,m→∞lim A^ν,µ_n,m(f; x, y) = f(x, y).

Moreover, the above convergence is uniform on every set [x1, x2] × [y¹, y2] with 0 ¬ x¹< x2, 0 ¬ y¹< y2.

Corollary 3.4 For all α, β ∈ (0; 1], p, q ∈ N⁰ and for each f ∈ Lip(C^p,q, α, β)) = {f ∈ C^p,q: ω(f, Cp,q; t, s) = O(t^α+s^β)} there exists a positive constant M(p, q, α, β) such that for all (x, y) ∈ R²0 and n, m ∈ N we have

wp,q(x, y)|A^ν,µn,m(f; x, y) − f(x, y)| ¬ M(p, q, α, β)

x + 1 n

^α₂ +

y + 1 m

^β₂! .

References

[1] M. Herzog, Approximation theorems for modified Szasz-Mirakjan operators in polynomial we- ight spaces, Le Matematiche54.1 (1999), 77-90.

[2] L. Rempulska, M. Skorupka, On some operators in weighted spaces of functions of two varia- bles, Ricerche Mat.48.1 (1999), 1-20.

[3] E. Wachnicki, Approximation by bivariate Mazhar-Totik operators, Comment. Math.50.2 (2010), 141-153.

[4] Z. Walczak, On certain modified Szasz-Mirakjan operators for functions of two variables, Demonstratio Math.33.1 (2000), 91-100.

[5] Z. Walczak, Approximation of functions of two variables by modified Szasz-Mirakyan opera- tors, Fasc. Math.34 (2004), 129-140.

[6] G. N. Watson, Theory of Bessel functions, Cambridge Univ. Press, Cambridge, 1966 Monika Herzog

Inistitute of Mathematics, Cracow University of Technology Warszawska 24, 31-155 Cracow, Poland

E-mail: mherzog@pk.edu.pl

(Received: 16.09.2011)