View of Some approximation properties of modified Szasz-Mirakyan-Kantorovich operators

Rev. Anal. Num´er. Th´eor. Approx., vol. 38 (2009) no. 1, pp. 77–86 ictp.acad.ro/jnaat

SOME APPROXIMATION PROPERTIES OF MODIFIED SZASZ-MIRAKYAN-KANTOROVICH OPERATORS

GRZEGORZ NOWAK^∗and ANETA SIKORSKA-NOWAK^†

Abstract. In this paper we consider the modified Szasz-Mirakyan-Kantorovich operators for functionsfintegrable in the sense of Denjoy-Perron. Moreover, we estimate the rate of pointwise convergence ofMnf(x) at the Lebesgue-Denjoy pointsxoff.

MSC 2000. 41A25.

Keywords. Szasz-Mirakyan operator, rate of convergence, Lebesgue-Denjoy point.

1. PRELIMINARIES

In [6], Z. Walczak introduced and considered some approximation properties of the modifed Szasz-Mirakyan operators defined by

Mnf(x) =

∞

X

k=0

pn,k(x;an)f(k/bn) (x∈R0, n∈N),

where pn,k(x;an) = exp(−a_nx)^(an_k!^x)k,R0 = [0;∞), N ={1,2, ...} and (an)^∞₁ , (b_n)^∞₁ are given increasing and unbounded sequences of positive numbers, such that

(1) limn→∞ 1

bn = 0 and ^a_bⁿ

n = 1 +O

1 bn

.

If an=bn =n, for all n∈N, then Mn become the classical Szasz-Mirakyan operator examined for continuous and bounded function in [4]. Some approx- imation properties of the operatorsMn, for continuous functionsf onR0, can be found e.g. in [1] and [6].

Denote byM_n^∗the Kantorovich type modification of operatorMnand define it for measurable functionf on R₀ as:

(2) M_n^∗f(x) =b_n

∞

X

k=0

p_n,k(x;a_n)

Z (k+1)/bn

k/bn

f(t)dt x∈R₀, n∈N,

∗University of Marketing and Management, Ostroroga 9a 64-100 Leszno, Poland, e-mail:

[email protected].

†Faculty of Mathematics and Computer Science Adam Mickiewicz University, Umul- towska 87, 61-614 Pozna´n, Poland, e-mail: [email protected].

where the integral is taken in the sense of Lebesgue or Denjoy-Perron. Con- vergence Theorems for the classical Szasz-Mirakyan-Kantorovich operators (an = bn = n) are presented in [2], [3]. Some approximation properties of M_n^∗f can be found in [7].

The aim of this paper is to examine the rate of the convergence of operators M_n^∗f, mainly, at those pointsx∈R₀ at which

limh→0 1 h

Z h 0

(f(x+t)−f(x)dt= 0.

The general estimate is expressed in terms of the quantity w_x(δ;f) = sup_0<|h|≤δ

1 h

Z h 0

(f(x+t)−f(x))dt

(δ >0).

Clearly, iff is locally integrable in the sense of Denjoy-Perron onR0, then lim_δ→0+w_x(δ;f) = 0 for almost everyx.

In view of this property, we deduce that in some classes of functions, limn→∞M_n^∗f(x) =f(x) almost everywhere.

Moreover, using some other properties of w_x(δ;f), we present the estimate of the rate of M_n^∗f(x) the pointwise convergence of in terms of the weighted moduli of continuity.

Throughout the paper, the symbol K(...), Kj(...), (j = 1,2, ...) will mean some positive constants, not necessarily the same at each occurrence, depend- ing only on the parameters indicated in parentheses.

2. AUXILIARY ESTIMATES

It is easy to see, that for every x∈R0 and for all integersn∈N , (3)

∞

X

k=0

pn,k(x;an) = 1,

(4)

∞

X

k=0

k bn −x

p_n,k(x;a_n) =x

an

bn −1 ,

∞

X

k=0

k an −x

p_n,k(x;a_n) = 0.

By properties (1), we have that exists a positive, absolute constants K, K1, such that

(5)

an

bn −1 ≤ ^K_b

n, _a¹

n ≤ _b^K

n and ^a_bⁿ

n ≤K1

for all n∈N.

Forq ∈N0 =N ∪ {0},l∈N0,n∈N we define (6) S_q,l⁽ⁿ⁾(x;bn) =bn

∞

X

k=0

k bn −x

q

k an −x

l

pn,k(x;an).

Lemma 1. Suppose that q ∈ N and (an)^∞₁ ,(bn)^∞₁ are fixed. Then, for n∈N, x∈R₀ we have

(7) S⁽ⁿ⁾_q,0(x;bn) =xan q−2

X

j=0 q−1

j

₁

b^q−jn

S_j,0⁽ⁿ⁾(x;bn) +x an

bn −1

S_q−1,0⁽ⁿ⁾ (x;bn),

(8) S_q,1⁽ⁿ⁾(x;b_n) =x

q−1

X

j=0 q j

₁

b^q−jn

S_j,0⁽ⁿ⁾(x;b_n),

S_q,2⁽ⁿ⁾(x;bn) = _a^x

n

q

X

j=0 q j

₁

b^q−jn S_j,0⁽ⁿ⁾(x;bn) (9)

+x²

q−1

X

j=1 q j

₁

b^q−jn

j−1

X

i=0 j i

₁

b^j−in

S_i,0⁽ⁿ⁾(x;b_n).

(Note that the symbol P−1

j=0...denotes zero.) Proof. In view of the definition (6)

S_q,0⁽ⁿ⁾(x;bn) =

∞

X

k=0

exp(−a_nx)^(an_k!^x)k k

bn −x q

= ^a_b^nx

n

∞

X

k=1

exp(−a_nx)^(a_(k−1)!^nx)k−1 k

bn −x q−1

−xS_q−1,0⁽ⁿ⁾ (x;bn)

= ^a_b^nx

n

∞

X

k=0

exp(−a_nx)^(an_k!^x)k

k+1

bn −xq−1

−xS_q−1,0⁽ⁿ⁾ (x;b_n)

=

q−1

X

j=0 q−1

j

_anx

b^q−1n

∞

X

k=0

p_n,k(x;an) k

bn −x j

−xS_q−1,0⁽ⁿ⁾ (x;bn)

=xa_n

q−2

X

j=0 q−1

j

₁

b^q−jn

S_j,0⁽ⁿ⁾(x;b_n) +x

an

bn −1

S_q−1,0⁽ⁿ⁾ (x;b_n).

This mean that (7) is true.

Next, assuming (6), we have S_q,1⁽ⁿ⁾(x;b_n) =

∞

X

k=0

exp(−a_nx)^(an_k!^x)k

k

bn −xq

k an −x

=

∞

X

k=0

exp(−a_nx)^(anx)_k!^k+1_a¹

n

k+1 bn −x

q

−xS_q,0⁽ⁿ⁾(x;bn)

=x

q

X

j=0 q j

₁

b^q−jn

S_j,0⁽ⁿ⁾(x;b_n)−xS_q,0⁽ⁿ⁾(x;b_n)

=x

q−1

X

j=0 q j

₁

b^q−jn S_j,0⁽ⁿ⁾(x;bn).

So (8) is true. Hence we get S_q,2⁽ⁿ⁾(x;bn) =

∞

X

k=0

exp(−a_nx)^(an_k!^x)k k

bn −x q

k an −x

2

=x

∞

X

k=0

exp(−a_nx)^(an_k!^x)k

k+1

bn −xq

k+1 an −x

−xS_q,1⁽ⁿ⁾(x;bn)

=x

q

X

j=0 q j

₁

b^q−jn

S_j,1⁽ⁿ⁾(x;b_n) +_a¹

nS_j,0⁽ⁿ⁾(x;b_n)

−xS_q,1⁽ⁿ⁾(x;b_n)

=x

q−1

X

j=0 q j

₁

b^q−jn

S_j,1⁽ⁿ⁾(x;bn) +_a¹

nS_j,0⁽ⁿ⁾(x;bn)

+_a^x

nS_q,0⁽ⁿ⁾(x;bn).

Applying (8) the desired equality (9) follows.

Lemma 2. For every n∈N, q ∈N and q ∈R0 the following inequality is true

(10) |S_q,2⁽ⁿ⁾(x;b_n)| ≤K(q)

x²(1 +x)^{q−1 1}

b^[(q+3)/2]n

+x(x+ 1) ¹

b^q+1n

,

where (a_n)^∞₁ ,(b_n)^∞₁ are fixed.

Proof. Using (4), (7) and the method of induction one can easily verify that for all n∈N,q∈N,x∈R0 there holds

(11) |S_q,0⁽ⁿ⁾(x;bn)| ≤K(q)x(1 +x)^{q−1 1}

b^[(q+1)/2]n

Simple calculation, (3), (4) give us S_0,2⁽ⁿ⁾(x;b_n) =

∞

X

k=0

exp(−a_nx)^(an_k!^x)k

k an −x2

= _a¹

n

∞

X

k=1

exp(−a_nx)^(a_(k−1)!^nx)k

k an −x

−xS_0,1⁽ⁿ⁾(x;b_n)

=x

∞

X

k=0

exp(−a_nx)^(an_k!^x)k k+1

an −x

−xS⁽ⁿ⁾_0,1(x;bn)

= _a^x

nS_0,0⁽ⁿ⁾(x;b_n) = _a^x

n,

and

S_1,2⁽ⁿ⁾(x;b_n) =

∞

X

k=0

exp(−a_nx)^(an_k!^x)k

k

an −x2

k bn −x

= _b¹

n

∞

X

k=1

exp(−a_nx)^(a_(k−1)!^nx)k

k

an −x2

−xS_0,2⁽ⁿ⁾(x;b_n)

= ^xa_bⁿ

n

∞

X

k=0

exp(−a_nx)^(an_k!^x)k

k an −x

2

+_a¹2 n

+ _a²

n

k

an −x

−xS_0,2⁽ⁿ⁾(x;b_n)

= ^xa_bⁿ

n

S⁽ⁿ⁾_0,2(x;b_n) +_a¹2

nS_0,0⁽ⁿ⁾(x;b_n) + _a²

nS_0,1⁽ⁿ⁾(x;b_n)

−xS_0,2⁽ⁿ⁾(x;bn)

= ^xa_bⁿ

n

x an +_a¹2

n

−x_a^x

n = _a^x2

n

an

bn −1 + _a^x

nbn. Applying (5) we obtained (10) for q= 1. Hence

S_2,2⁽ⁿ⁾(x;b_n) = _a^x

n

2

X

j=0 2 j

₁

b^2−jn

S_j,0⁽ⁿ⁾(x;b_n) +x^{2 2}_b

n

1

bnS_0,0⁽ⁿ⁾(x;b_n)

= _a^x

n

1

b²_nS_0,0⁽ⁿ⁾(x;bn) +_b²

nS_1,0⁽ⁿ⁾(x;bn) +S_2,0⁽ⁿ⁾(x;bn)

+^2x_b2²

n S_0,0⁽ⁿ⁾(x;b_n).

Clearly, in view of (3), (5) and (11)

|S⁽ⁿ⁾_2,2(x;bn)| ≤K_a^x

n

1 b²_n +_b^x2

n +x(1 +x)_b¹

n

+^2x_b2²

n

≤K₁n

x²(1 +x)_b¹

n

1 an + _b¹

n

+^x(1+x)_a

nb²_n

o

≤K₂n

x²(1 +x)_b¹2

n +^x(1+x)_b3 n

o .

Hence for q= 2 the inequality (10) is true.

Analogously we have (10) forq = 3 S_3,2⁽ⁿ⁾(x;b_n) =

=

x an

3

X

j=0 3 j

₁

b^3−jn

S_j,0⁽ⁿ⁾(x;b_n) +x²

2

X

j=1 3 j

₁

b^3−jn

j−1

X

i=0 j i

₁

b^j−in

S_i,0⁽ⁿ⁾(x;b_n)

=

x an

1

b³_nS_0,0⁽ⁿ⁾(x;bn) + _b³2

nS_1,0⁽ⁿ⁾(x;bn) +_b³

nS_2,0⁽ⁿ⁾(x;bn) +S_3,0⁽ⁿ⁾(x;bn)

+x^{2 3}_b2 n

1

bnS_0,0⁽ⁿ⁾(x;bn) +x^{2 3}_b

n

1

b²_nS_0,0⁽ⁿ⁾(x;bn) +_b²

nS_1,0⁽ⁿ⁾(x;bn)

≤ _a^x

n

1

b³_n +Kx_b¹3

n +x(1 +x)_b¹2

n +x(1 +x)^{2 1}_b2 n

+K

x^{2 1}_b3

n +x^{3 1}_b3 n

≤K_1x2(1+x)²

b³_n +^x(1+x)_b4 n

.

Ifq ≥4 then, in view of (9),

|S_q,2⁽ⁿ⁾(x;b_n)| ≤

≤ _a^x

n

q

X

j=0 q j

₁

b^q−jn

|S_j,0⁽ⁿ⁾(x;b_n)|+x²

q−1

X

j=1

v ¹

b^q−jn

j−1

X

i=0 j i

₁

b^j−in

|S_i,0⁽ⁿ⁾(x;b_n)|

= _a^x

n



_b¹q

n|S_0,0⁽ⁿ⁾(x;bn)|+ ^q

b^q−1n

|S_1,0⁽ⁿ⁾(x;bn)|+

q

X

j=2 q j

₁

b^q−jn

|S_j,0⁽ⁿ⁾(x;bn)|





+x²

q b^q−1n

1

bn|S_0,0⁽ⁿ⁾(x;b_n)|+ ^q_{2 1}

b^q−2n

1

b²_n|S_0,0⁽ⁿ⁾(x;b_n)|+_b²

n|S_1,0⁽ⁿ⁾(x;b_n)|

+

q−1

X

j=3 q j

₁

b^q−jn

j−1

X

i=2 j i

₁

b^j−in

|S_i,0⁽ⁿ⁾(x;b_n)|

+¹

b^jn

|S_0,0⁽ⁿ⁾(x;bn)|+j ¹

b^j−1n

|S_1,0⁽ⁿ⁾(x;bn)|

≤ _a^x

n



_b¹q n + ^q

b^q−1n

x

an

bn −1 +

q

X

j=2 q j

₁

b^q−jn

K(j)x(1 +x)^{j−1 1}

b^[(j+1)/2]n





+x²K1(q)

1 b^qn + ¹

b^q−1n

x

an

bn −1

+

q−1

X

j=3 1 b^q−jn

j−1

X

i=2 1 b^j−1n

x(1 +x)^{i−1 1}

b^[(i+1)/2]n

+ ¹

b^jn

+x ¹

b^j−1n

an

bn −1

!

.

Consequently, from (5)

|S_q,2⁽ⁿ⁾(x;b_n)| ≤

≤K(q)_a^x

n

1 b^qn +_b¹q

nx+x(1 +x)^{q−1 1}

b^[(q+1)/2]n

+x²K(q)



 ¹

b^qn +_b¹q nx+

q−1

X

j=3 1 b^q−jn

x(1 +x)^{j−2 1}

b^[j/2+1]n

+ ¹

b^jn

+x ¹

b^jn





≤K1(q)

x²(1 +x)^{q−1 1}

b^[(q+3)/2]n

+x² 1+x

b^qn + ^1+x

b^q−1n + ^x(1+x)q−3

b[(q+1)/2]+1 n

+^x(1+x)

b^q+1n

≤K2(q)

x²(1 +x)^{q−1 1}

b^[(q+3)/2]n

+^x(1+x)

b^q+1n

,

and inequality (10) follows now immediately.

Identity (3), estimate (10) and the known Schwarz inequality lead to Lemma 3. Let q ∈N0, x∈R0 . Then, forn∈N

∞

X

k=0

k an −x

k bn −x

q+1

p_n,k(x;a_n)≤ (12)

≤K(q)

x(1+x)^q+1/2 b^q/2+1n

+

√

x(1+x) b^q+3/2n

,

where (a_n)^∞₁ ,(b_n)^∞₁ , are fixed.

3. MAIN RESULT

In this Section, we consider only the points x∈[0;∞) at which wx(δ;f)<

∞ for all δ >0.

Theorem4. Letf :R0→Ris integrable in the Lebesgue or Denjoy-Perron sesnse on every compact interval contained in R₀ and let n ∈ N, x ∈ R₀. Given any number q∈N, we have

|M_n^∗f(x)−f(x)| ≤K(q)

(1 +x)^q+1/2+

1+x xb^q+1n

1/2 (13)

∞

X

v=0

w_x

√v+1 bn

1 (v+1)^q.

Proof. For the sake of brevity we will write f(x+t)−f(x) = ϕx(t) and w_x(δ;f) =w_x(δ). In view of (3) we have

M_n^∗f(x)−f(x) =b_n

∞

X

k=0

p_n,k(x;a_n)

Z (k+1)/bn

k/bn

(f(t)−f(x))dt

=b_n

∞

X

k=0

p_n,k(x;a_n)

Z (k+1)/bn−x k/bn−x

ϕ_x(t)dt

=bn

∞

X

k=1

(pn,k−1(x;an)−pn,k(x;an))

Z k/bn−x 0

ϕx(t)dt

−b_nexp(−a_nx) Z −x

0

ϕx(t)dt.

It is easy to see that

x(pn,k−1(x;an)−pn,k(x;an)) =pn,k(x;an) k

an −x

.

Consequently

x(M_n^∗f(x)−f(x)) =b_n

∞

X

k=0

p_n,k(x;a_n)

k

an −xZ k/bn−x 0

ϕ_x(t)dt.

Applying the obviously inequality

Rh

0 ϕx(t)dt

≤ |h|w_x(|h|), we obtain x|M_n^∗f(x)−f(x)| ≤bn

∞

X

k=0

p_n,k(x;an)

k an −x

k bn −x

wx

k bn −x

≤

∞

X

v=0

T_v⁽ⁿ⁾(λ;x)w_x((v+ 1)λ), whereλis an arbitrary positive number and

T_v⁽ⁿ⁾(λ;x) = X

vλ<|k/bn−x|≤(v+1)λ

bn

k an −x

k bn −x

p_n,k(x;an).

We shall estimate the factors Tv⁽ⁿ⁾(λ;x) for v = 0 and v ≥ 1, respectively.

Clearly, in view of (12)

T₀⁽ⁿ⁾(λ;x) ≤b_n

∞

X

k=0

k an −x

k bn −x

p_n,k(x;a_n)

≤K

x(1 +x)^1/2+

x(1+x) bn

1/2 .

Next, if v≥1, we have, by Lemma 3, T_v⁽ⁿ⁾(λ;x) ≤ _v^bqⁿλ^q

∞

X

k=0

k an −x

k bn −x

q+1

p_n,k(x;a_n)

≤ ^K(q)_vqλ^q

_x(1+x)q+1/2

b^q/2n

+^x1/2(1+x)1/2

b^q+1/2n

.

Now, takingλ= 1/b^1/2n , we easily get x|M_n^∗f(x)−f(x)| ≤

x(1 +x)^q+1/2+^x1/2(1+x)1/2

b^(q+1)/2n

· Kw_x

√1 bn

+K(q)

∞

X

v=1

w_x

v+1√ bn

1 (v+1)^q

! .

This last relation is equivalent to (13).

Remark 5. Let D_loc^∗ (R) be the class of all functions integrable in the Denjoy-Perron sense on every compact interval contained in R. Clearly, if f ∈D_loc^∗ (R), then the function

F(x) = Z x

0

f(t)dt

is ACG^∗ on every [a;b]⊂R₀ and F⁰(x) =f(x) almost everywhere [5].

Consequently,

(14) limδ→0+w_x(δ;f) = 0 a.e. on R.

Suppose that f ∈D^∗_loc(R) and that

||f|| ≡sup−∞<v<∞

sup_0≤u≤1

Z v+u v

f(t)dt

<∞.

The operators M_n^∗f(x) are well-defined for all n ∈ N. This follows at once from the inequality

Z (k+1)/n k/n

f(t)dt

≤ ||f||.

In view of (14), given any > 0 there is a δ₀ > 0 such that w_x(δ;f) <

whenever 0< δ≤δ0. In case δ≥δ0 we have wx(δ;f)< +|f(x)|+ sup_{δ0≤|h|≤δ}

1 h

Z h 0

f(x+t)dt

≤+|f(x)|+_δ¹

0||f||

provided that δ ≤1. If δ > 1, then putting µ = [δ] + 1 and x_j = x+jh/µ (j = 0,1, ..., µ), we get

Z h 0

f(x+t)dt

≤

µ−1

X

j=0

Z xj+1

xj

f(t)dt

≤µ||f|| ≤2δ||f||.

Hence

w_x(δ;f)< +|f(x)|+_δ¹

0(1 + 2δ)||f|| for all δ >0.

This inequality and the condition (14) ensure that the right-hand side of the estimate (13) (with arbitraryq ≥3) converges to zero asn→ ∞.

Letm ∈N0. Denote byCm(R0) the class of all measurable functions f on R₀, such that

||f||_m = sup_x∈R1+x^|f(x)|2m <∞.

It is easy to see, that operators M_n^∗f are well-defined for every function f ∈ Cm(R0).

Further, for continuousf ∈C_m(R₀), let us introduce the weighted modulus of continuity

ω(δ;f)_m = sup_|h|≤δ||f(·+h)−f(·)|| (δ >0).

Theorem6. Letm∈N0,x∈R0,f ∈Cm(R0)and(an)^∞₁ , (bn)^∞₁ are fixed.

Then

(1 +x^2m)⁻¹|M_n^∗f(x)−f(x)| ≤

≤K(m)

(1 +x)^2m+3+ ^1+x_x 1/2 1 b^m+5/4n

ω(b^−1/2_n ;f)m, for all n∈N.

Proof. It is easy to see, that for anyx∈R0,r∈N

|f(x+rt)−f(x)| ≤

r−1

X

v=0

|f(x+vt+t)−f(x+vt)|

1+(x+vt)^2m (1 + (x+vt)^2m)

≤ω(|t|;f)_m

r−1

X

v=0

(1 + (|x|+v|t|)^2m)

≤(1 + (|x|+ (r−1)|t|)^2mrω(|t|;f)m. Therefore, for any x∈R,δ >0,r ∈N

w_x(rδ;f)≤(1 + (2x)^2m+ (2(r−1)δ)^2m)rω(δ;f)_m.

This inequality and Theorem 4, with q= 2m+ 5/2, lead to Theorem 6.

Acknowledgement. We are grateful to the referee’s valuable suggestions.

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Received by the editors: January 20, 2009.