Dunkl analogue of Szász-mirakjan operators of blending type

Lemma 2.1

The operators ℒ_n,ρsatisfy the following inequalities:

1. ℒ_n,ρ(1; x) =1,

2. |Ln,ρ(u;x)−x|≤2νn,

3. |Ln,ρ(u2;x)−x2|≤x(1+6νρ+ρ)nρ+4ν2ρ−2νn2ρ,

4. |Ln,ρ(u3;x)−x3|≤3x2(1+ρ)nρ+xn2ρ2(2+(3−6ν)ρ+(1+4ν2−2ν)ρ2),

5. |Ln,ρ(u4;x)−x4|≤x3nρ+x2n2ρ2(11+18ρ+(7+16ν2+10ν)ρ2)+xn3ρ3(6+(11+18ν)ρ)+(6+24ν2+18ν)ρ2+(1−34ν+4ν2+17ν3)ρ3+1n4ρ3(32ν4ρ3+48ν3ρ2+4ν2ρ−12ν).

Consequently, for the operator ℒ_n,ρ(.; x) , we have the following inequalities:

1. |Ln,ρ(u−x;x)|≤2νn,

2. Ln,ρ((u−x)2;x)≤x(1+ρ+10νρ)nρ+4ν2ρ−2νn2ρ,

3. Ln,ρ((u−x)4;x)≤24x3(1+ρ+2νρ)nρ+x2n2ρ2(19+(30−36ν)ρ+(11+2ν+56ν2)ρ2)+xn3ρ3(6+(11+34ν)ρ+(6+18ν+72ν2)ρ2+(1−34ν+4ν2+49ν3)ρ3)+1n4ρ3(32ν4ρ3+48ν3ρ2+4ν2ρ−12ν).

Proof.

1. Ln,ρ(1;x)=1eν(nx)∑k=1∞(nx)kγν(k)∫0∞nρΓ(kρ)e−nρt(nρt)kρ−1.1 dt+1eν(nx),=1eν(nx)∑k=1∞(nx)kγν(k)Γ(kρ)∫0∞e−ttkρ−1 dt+1eν(nx)=1eν(nx)∑k=0∞(nx)kγν(k)=1.

2. Using (1), we have

   Ln,ρ(u;x)=1eν(nx)∑k=1∞(nx)kγν(k)∫0∞nρΓ(kρ)e−nρt(nρt)kρ−1 u dt,=1n eν(nx)[∑k=0∞(nx)k+1γν(k)−2ν∑k=1∞θk(nx)kγν(k)]=1n eν(nx)[nx eν(nx)−2ν∑k=1∞θk(nx)kγν(k)]=x−2νn eν(nx)∑k=1∞θk(nx)kγν(k)

Hence

|Ln,ρ(u;x)−x|≤2νn eν(nx)∑k=0∞(nx)kγν(k)=2νn.

Using similar calculations, one can easily prove (iii) − (v), therefore the details are omitted. The consequences (a) − (c) are straightforward, hence we skip the proofs.

Remark 2.2

Using Lemma 2.1 and choosing

C(ν,ρ)=Max((1+ρ+10νρ)ρ,4ν2ρ−2νρ),we obtain

Theorem 3.1

Let 𝔥 ∈ C_γ (ℝ⁺). Then,

uniformly on each compact subset 𝒜 of [0,∞).

Proof. In view of Lemma 2.1,

Ln,ρ(ui;x)→xi, as n→∞,as n →∞ , uniformly on 𝒜, for i = 0,1,2.

Hence, the required result follows from applying the Bohman-Korovkin criterion [11].

Let C_B[0, ∞) denote the space of bounded and uniformly continuous functions on [0,∞) endowed with the sup norm, ||f||=supx∈[0,∞)|f(x)|.

The first and second order modulus of continuity are respectively defined as

and

Theorem 3.2

Let 𝔥∈C_B [0, ∞) and ω(𝔥;δ), δ > 0, be its first order modulus of continuity. Then the operator ℒ_n,ρ (.;) satisfies the inequality

Proof. By definition of ω(𝔥;δ), Lemma 2.1 and Cauchy-Schwarz inequality, we may get

Now, choosing δ = n^−1/2 , we immediately obtain the result.

Corollary 3.3

Ifh∈LipM(α), 0<α≤1,then

For f ∈ C_B [0, ∞), the Steklov mean is defined as

Lemma 3.4

([12]). The Steklov mean f_h(x) satisfies the following properties:

1. || f_h − f || ≤ ω₂(f, h),

2. fh′,fh′′∈CB[0,∞) and

Theorem 3.5

For 𝔥^' ∈ C_B[0, ∞), we have

where M is some positive constant, ψ_n,ν,ρ(x) is as defined in Remark 2.2 and ω(𝔥^';δ) denotes the modulus of continuity of 𝔥^'.

Proof. Since 𝔥^' ∈ C_B[0,∞) ∃ M > 0 such that |𝔥^' (x) |≤ M, ∀ x ≥ 0.

Using mean value theorem, one may write

where ζ lies between u and x.

Now, applying the operator ℒ_n,ρ (.; x) on both sides of the above equality and using Lemma 2.1, we get

Now, from (5), and Cauchy-Schwarz inequality, we can get

Choosing δ = ψ_n,ν,ρ(x) and combining (7)-(8), we arrive to conclusion.

Theorem 3.6

Let 𝔥∈C_B[0, ∞). Then for each x ∈[0,∞), we have

Proof. Applying Lemma 2.1 and Lemma 3.4, one has

Since fh″∈CB[0,∞), by Taylor’s expansion,

Applying operator ℒ_n,ρ (.; x) on the above equality, we get

Hence, using Lemma 2.1, we have

Finally, choosing h = n^−1/2 , the required result is obtained.

Next, we define some weighted spaces on [0,∞) to obtain the weighted approximation results for the operators defined by (4).

and

where σ(x) =1+ x² is a weight function and ℳ_f is a constant depending only on the function f . From [13], it is noted that C_σ(ℝ⁺) is a normed linear space endowed with the norm ||f||σ:=supx≥0|f(x)|σ(x).

It is well known that the classical modulus of continuity ω(f;δ) does not tend to zero if f is continuous on an infinite interval. Therefore, in order to study the approximation of functions in the weighted space Cσk(R+), Ispir and Atakut [13] introduced the following weighted modulus of continuity

and proved that limδ→0+Ω(f;δ)=0 and

Theorem 3.7

For eachh∈Cσk(R+),the sequence of linear positive operators {ℒ_n,ρ}, satisfies the following equality

Proof. From Lemma 2.1, clearly limn→∞||Ln,ρ(1;x)−1||σ=0.

Now,

Therefore, limn→∞||Ln,ρ(u;x)−x||σ=0. Again,

we obtain limn→∞||Ln,ρ(u2;x)−x2||σ=0. Hence, applying weighted Korovkin-type theorem given by Gadzhiev [14], we reach the desired result.

Theorem 3.8

Leth∈Cσk(R+).Then the following inequality is verified

where 𝒦 is a constant not dependent on 𝔥 and n.

Proof. Using (5), definition of Ω(f; δ) , Lemma 2.1 and Cauchy-Schwarz inequality, one can easily see that

Now, choosing δ=1n, we arrive to conclusion immediately.

In our next result, we shall discuss a direct result with the help of unified Ditzian-Totik modulus of smoothness ωϕλ(h;t), 0≤λ≤1.In 2007, Guo et al. [15] discussed the direct, inverse and equivalence approximation results by means of unified modulus. We consider ϕ2(x)=1+x and f∈CB[0,∞). The modulus ωϕλ(h,t),0≤λ≤1, is defined as

and the corresponding K -functional is given by

where Wλ={g:g∈ACloc[0,∞),||ϕλg′||<∞},AC_loc is defined as the space of locally absolutely continuous functions on [0,∞).

From [16], there exists a constant 𝒞 > 0 such that

Theorem 3.9

For each 𝔥∈C_B[0, ∞) and sufficiently large n, we have

where 𝒜 is some constant not dependent on 𝔥 and n.

Proof. By the definition of Kϕλ(h;t), for fixed λ, n and x∈[0,∞), we can find a g=gn,x,λ∈Wλ such that

From the representation of g as g(u)=g(x)+∫xug′(s)ds, it follows that

Applying Hölder’s inequality,

Hence, in view of Cauchy-Schwarz inequality and Remark 2.2,

We may write,

Now, using (10)-(13), one can easily obtain

Choosing 𝒜 = Max(2, 2^λ (C(ν ,ρ)^1/2) , using (10) and the relation given in (9), we arrive at the required result.

Next, we discuss the degree of approximation for functions having derivatives of bounded variation. Let H[0,∞) denote the space of all f ∈C₂[0,∞) such that f^' is equivalent to a function locally of bounded variation.

If f ∈ H[0,∞), we may write

where g is locally of bounded variation on [0,∞).

Lemma 3.10

Let x ∈(0,∞) . Then for sufficiently large n , we have

1. ϑn,ρ(x,t)=∫0tW(x,n,ρ,u)du≤C(ν,ρ)(x−t)2(1+x)n,0≤t<x,

2. 1−ϑn,ρ(x,t)=∫t∞W(x,n,ρ,u)du≤C(ν,ρ)(x−t)2(1+x)n,x≤t<∞,

where C(ν ,ρ) is a positive constant depending on ν and ρ.

Proof.

1. Using Remark 2.2, for sufficiently large n, we have

   ϑn,ρ(x,t)=∫0tW(x,n,ρ,u)du≤∫0t(x−ux−t)2W(x,n,ρ,u)du≤1(x−t)2Ln,ρ((u−x)2;x)≤C(ν,ρ)(x−t)2(1+x)n.

2. By a similar reasoning, one can easily prove (ii). Hence the details are omitted.

In the following theorem, it is shown that the points x∈(0,∞), where the left hand and right hand derivatives of f^' exist, Ln,ρ(f;x)→f(x), as n→∞.

Theorem 3.11

Let 𝔥∈H[0,∞) . Then, for each x ∈(0,∞) and sufficiently large n , we

have

where C(ν , ρ) > 0, is a constant and ⋁cdf is the total variation of f on [c, d] and fx′ is defined by

Proof. Let f ∈ H[0,∞). Then from (14), we can easily write

where

Using (5), for each x ∈(0,∞), we have