PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 101(115) (2017), 247–259 https://doi.org/10.2298/PIM1715247V

APPROXIMATION BY GENERALIZED SRIVASTAVA–GUPTA OPERATORS BASED ON CERTAIN PARAMETER

D. K. Verma

Abstract. We establish some direct results in simultaneous approximation for a generalization of the Srivastava–Gupta operators. We establish pointwise convergence, Voronovskaja type asymptotic formula and an error estimate in terms of modulus of continuity of the function.

1. Introduction

In the year 2003, Srivastava and Gupta [18] introduced a general family of summation-integral type operators which include some well-known operators as special cases. They estimated the rate of convergence for functions of bounded variation. After that several researchers studied different approximation properties of these important general form of operators, we mention some of them as [10,19, 22], etc. Also, very recently in order to modify the Phillips operators, Pǎltǎnea [15] proposed a generalization of the well known Phillips operators [17] (see also [5]) based on the parameter ρ > 0, which provide a link with classical Szász operators as ρ → ∞. As the operators introduced in [18] include Phillips operators as special case, this motivated us to study in this direction and here we consider the generalization of the Srivastava–Gupta operators.

For f ∈ C_{γ}[0, ∞) = {f ∈ C[0, ∞) : f(t) = O(t^{γ}), γ > 0}, we define the following generalization of Srivastava–Gupta operators based on certain parameter ρ > 0 in the following way:

(1.1)  L_{n,ρ} (f; x) = ∑_{k=1}^{∞} p_{n,k} (x, c) ∫_{0}^{∞} Θ_{n,k}^{ρ} (t, c)f(t) dt + p_{n,0} (x, c)f(0),

p_{n,k} (x, c) = \frac{(−x)^{k}}{k!} ϕ_{n,c}^{(k)} (x), Θ_{n,k}^{ρ} (t, c) = \{ \begin{matrix}\frac{nρ}{Γ(kρ)} e^{−nρt} (nρt)^{kρ−1} , & c = 0 \\ \frac{Γ(nρ /c+kρ)}{Γ(kρ)Γ(nρ /c)} \frac{c^{kρ} t^{kρ−1}}{(1+ct)^{nρ /c+kρ}} , & c = 1, 2, 3, … \end{matrix}

2010 Mathematics Subject Classification: 41A30; 41A35.

Key words and phrases: Phillips operators, modulus of continuity, pointwise convergence, Voronovskaja type asymptotic formula.

Communicated by Gradimir Milovanović.

ϕ_{n,c} (x) = \{ \begin{matrix}e^{−nx} , & c = 0, \\ (1 + cx)^{−n/c} , & c = 1, 2, 3, … \end{matrix}

Here {ϕ_{n,c} (x)}_{n=1}^{∞} is a sequence of functions defined on the closed interval [0, b], b > 0, satisfying the following properties. For each n ∈ ℕ and k ∈ ℕ^{0} := ℕ ∪ {0}:

(i) ϕ_{n,c} ∈ C^{∞} [a, b], b > a ⩾ 0, (ii) ϕ_{n,c} (0) = 1, (iii) ϕ_{n,c} is completely monotone so that (−1)^{k}ϕ_{n,c}^{(k)} (x) ⩾ 0, x ∈ [0, b], (iv) there exists an integer c such that

ϕ_{n,c}^{(k+1)} (x) = −nϕ_{n+c,c}^{(k)} (x), n > max{0, −c}; x ∈ [0, b],

(see [18,19]). It can be easily verified that the operators L_{n,ρ} (f, x) are well defined for f ∈ C_{γ}[0, ∞).

For ρ = 1, operators (1.1) reduce to the Srivastava–Gupta operators [18]. When c = 0, ρ > 0, we immediately get the generalized operators due to Pǎltǎnea [15]. In the case c = 0, he studied some approximation properties in simultaneous of these operators in [16]. Also, for c = 0 and ρ = 1 operators (1.1) reduce to Phillips operators [5, 17].

Remark 1.1. It is observed that the general form of operators (1.1) provide link with the classical Szász and Baskakov operators. For f ∈ Π̄, where Π̄ be the closure of the space of polynomials, we have

lim_{ρ→∞} L_{n,ρ} (t^{r}, x) = ∑_{k=0}^{∞} p_{n,k} (x, c)( \frac{k}{n})^{r}, uniformly for all x ∈ [0, ∞).

Obviously,

∫_{0}^{∞} Θ_{n,k}^{ρ} (t, c)t^{r}dt = \frac{Γ(kρ + r)}{Γ(kρ)} \frac{1}{∏ _{i=1}^{r} (nρ − ic)} ,

lim_{ρ→∞} \frac{Γ(kρ + r)}{Γ(kρ)} \frac{1}{∏ _{i=1}^{r} (nρ − ic)} = ( \frac{k}{n})^{r}.

From this the result follows immediately. For c = 0 the link with the classical Szász operators was proved in [15] in more general conditions.

Thus in case ρ → ∞ we have the following links with the generalized Srivastava– Gupta operators: For c = 0 the operator L_{n,ρ} provides the link with the classical Szász operators (see [15]) and for c = 1 the operators L_{n,ρ} (f, x) reduce to the well known Baskakov operators.

The study in simultaneous approximation began with a remarkable result for the Bernstein polynomials B_{n}(f) owing to Wigert [21] after that Lorentz [12], who proved that B_{n}^{(r)} (f) → f(r), whenever the latter exists at the particular point x ∈ [0, 1], r = 1, 2, 3, … being arbitrary. For further study in this direction, we refer the reader to [2–4, 6, 8, 9, 11, 13, 14].

The aim of this paper is to study some results in simultaneous approximation by the operators L_{n,ρ} . First, we establish the basic pointwise convergence theorem, Voronovskaja type asymptotic formula and then proceed to study the degree of this approximation.

GENERALIZED SRIVASTAVA–GUPTA OPERATORS 249

2. Preliminaries

Lemma 2.1. [7] Let m ∈ ℕ ∪ 0. If the m-th order is defined as

T_{n,m} (x) = ∑_{k=0}^{∞} p_{n,k} (x, c)( \frac{k}{n} − x)^{m},

then T_{n,0} (x) = 1, T_{n,1} (x) = 0 and also there holds the recurrence relation:

nT_{n,m+1} (x) = x(1 + cx)[T′_{n,m} (x) + mT_{n,m−1} (x)].

Consequently, we have T_{n,m} (x) = O(n^{−[(m+1)/2]} ).

Remark 2.1. For Θ_{n,k}^{ρ} (t, c) the following identity holds:

(1 + ct)[tΘ_{n,k}^{ρ} (t, c)]′ = ρ(k − nt)Θ_{n,k}^{ρ} (t).

Taking the derivative of [tΘ_{n,k}^{ρ} (t, c)] with respect to t, we get

[tΘ_{n,k}^{ρ} (t)]′ = \frac{Γ( \frac{nρ}{c} + kρ)}{Γ(kρ)Γ( \frac{nρ}{c} )} c^{kρ}[ \frac{kρt^{kρ−1}}{(1 + ct)^{\frac{nρ}{c} +kρ}} − \frac{(\frac{nρ}{c} + kρ)t^{kρ}}{(1 + ct)^{\frac{nρ}{c} +kρ+1}} c],

simplifying the expression, identity is immediate.

Lemma 2.2. If we define the central moments as

μ_{n,m} (x) = L_{n,ρ} ((t − x)^{m}, x)

= ∑_{k=1}^{∞} p_{n,k} (x, c) ∫_{0}^{∞} Θ_{n,k}^{ρ} (t, c)(t − x)^{m}dt + p_{n,0} (x, c)(−x)^{m}, m ∈ ℕ

Then, μ_{n,0} (x) = 1, μ_{n,1} (x) = \frac{cx}{nρ−c} and for nρ > (m + 1)c we have the following recurrence relation:

(nρ − (m + 1)c)μ_{n,m+1} (x) = ρx(1 + cx)μ′_{n,m} (x) + [cx + m(1 + 2cx)]μ_{n,m} (x)
+ mx(1 + cx)(1 + ρ)μ_{n,m−1} (x).

Proof. Taking the derivative of μ_{n,m} , we have

μ′_{n,m} (x) = − m ∑_{k=1}^{∞} p_{n,k} (x, c) ∫_{0}^{∞} Θ_{n,k}^{ρ} (t, c)(t − x)^{m−1} dt − mp_{n,0} (x, c)(−x)^{m−1}

+ ∑_{k=1}^{∞} p′_{n,k} (x, c) ∫_{0}^{∞} Θ_{n,k}^{ρ} (t, c)(t − x)^{m}dt + p′_{n,0} (x, c)f(−x)^{m}

= − mμ_{n,m−1} (x) + ∑_{k=1}^{∞} p′_{n,k} (x, c) ∫_{0}^{∞} Θ_{n,k}^{ρ} (t, c)(t − x)^{m}dt

+ p′_{n,0} (x, c)(−x)^{m}.

x(1 + cx)[μ′_{n,m} (x) + mμ_{n,m−1} (x)]

= ∑_{k=1}^{∞} x(1 + cx)p′_{n,k} (x, c) ∫_{0}^{∞} Θ_{n,k}^{ρ} (t, c)(t − x)^{m}dt + x(1 + cx)p′_{n,0} (x, c)(−x)^{m}

250 VERMA

using x(1 + cx)p′_{n,k} (x, c) = (k − nx)p_{n,k} (x, c), we get

(2.1) x(1 + cx)[μ′_{n,m} (x) + mμ_{n,m−1} (x)]

= ∑_{k=1}^{∞} (k − nx)p_{n,k} (x, c) ∫_{0}^{∞} Θ_{n,k}^{ρ} (t, c)(t − x)^{m}dt − nxp_{n,0} (x, c)(x)(−x)^{m}

= ∑_{k=1}^{∞} kp_{n,k} (x, c) ∫_{0}^{∞} Θ_{n,k}^{ρ} (t, c)(t − x)^{m}dt − nxμ_{n,m} (x)

= I − nxμ_{n,m} (x). We can write I as

(2.2)  I = [ ∑_{k=1}^{∞} p_{n,k} (x, c) ∫_{0}^{∞} (k − nt)Θ_{n,k}^{ρ} (t, c)(t − x)^{m}dt

+ [n ∑_{k=1}^{∞} p_{n,k} (x, c) ∫_{0}^{∞} Θ_{n,k}^{ρ} (t, c) t(t − x)^{m}dt]

= I_{1} + I_{2}, (say). To estimate I_{2} using t = [(t − x) + x], we have

(2.3) I_{2} = n[ ∑_{k=1}^{∞} p_{n,k} (x, c) ∫_{0}^{∞} Θ_{n,k}^{ρ} (t, c)(t − x)^{m+1} dt

+ x ∑_{k=1}^{∞} p_{n,k} (x) ∫_{0}^{∞} Θ_{n,k}^{ρ} (t, c)(t − x)^{m}dt]

= n[ ∑_{k=1}^{∞} p_{n,k} (x, c) ∫_{0}^{∞} Θ_{n,k}^{ρ} (t, c)(t − x)^{m+1} dt + p_{n,0} (x, c)(−x)^{m+1}

+ x ∑_{k=1}^{∞} p_{n,k} (x, c) ∫_{0}^{∞} Θ_{n,k}^{ρ} (t, c)(t − x)^{m}dt + xp_{n,0} (x, c)(−x)^{m}]

= nμ_{n,m+1} (x) + nxμ_{n,m} (x).

Next, to estimate I_{1}, using the equality, ρ(k − nt)Θ_{n,k}^{ρ} (t, c) = (1 + ct)[tΘ_{n,k}^{ρ} (t, c)]′, we have

I_{1} = \frac{1}{ρ} ∑_{k=1}^{∞} p_{n,k} (x, c) ∫_{0}^{∞} (1 + ct)[tΘ_{n,k}^{ρ} (t, c)]′(t − x)^{m}dt

= \frac{(1 + cx)}{ρ} ∑_{k=1}^{∞} p_{n,k} (x, c) ∫_{0}^{∞} [tΘ_{n,k}^{ρ} (t, c)]′(t − x)^{m}dt

+ \frac{c}{ρ} ∑_{k=1}^{∞} p_{n,k} (x, c) ∫_{0}^{∞} [tΘ_{n,k}^{ρ} (t, c)]′(t − x)^{m+1} dt

GENERALIZED SRIVASTAVA–GUPTA OPERATORS 251

Now integrating by parts for nρ > (m + 1)c, we get

I_{1} = \frac{(1 + cx)}{ρ} ∑_{k=1}^{∞} p_{n,k} (x)[tΘ_{n,k}^{ρ} (t, c)(t − x)^{m}] _{0}^{∞}

− \frac{m(1 + cx)}{ρ} ∑_{k=1}^{∞} p_{n,k} (x, c) ∫_{0}^{∞} [tΘ_{n,k}^{ρ} (t, c)](t − x)^{m−1} dt

+ \frac{c}{ρ} ∑_{k=1}^{∞} p_{n,k} (x, c)[tΘ_{n,k}^{ρ} (t, c)(t − x)^{m+1} ] _{0}^{∞}

− \frac{c(m + 1)}{ρ} ∑_{k=1}^{∞} p_{n,k} (x, c) ∫_{0}^{∞} tΘ_{n,k}^{ρ} (t, c)(t − x)^{m}dt

= −\frac{m(1 + cx)}{ρ} ∑_{k=1}^{∞} p_{n,k} (x, c) ∫_{0}^{∞} [tΘ_{n,k}^{ρ} (t, c)](t − x)^{m−1} dt

− \frac{c(m + 1)}{ρ} ∑_{k=1}^{∞} p_{n,k} (x, c) ∫_{0}^{∞} tΘ_{n,k}^{ρ} (t, c)(t − x)^{m}dt

=: J_{1} + J_{2}.

Proceeding in a similar manner and taking t = [(t − x) + x], we get J_{1} and J_{2} as

(2.4)  J_{1} = −\frac{m(1 + cx)}{ρ} μ_{n,m} (x) − \frac{mx(1 + cx)}{ρ} μ_{n,m−1} (x),
(2.5)  J_{2} = −\frac{c(m + 1)}{ρ} μ_{n,m+1} (x) − \frac{c(m + 1)x}{ρ} μ_{n,m} (x).

Combining (2.1)–(2.5), we obtain

x(1 + cx)[μ′_{n,m} (x) + mμ_{n,m−1} (x)] = − \frac{c(m + 1)}{ρ} μ_{n,m+1} (x) − \frac{c(m + 1)x}{ρ} μ_{n,m} (x)

− \frac{m(1 + cx)}{ρ} μ_{n,m} (x) − \frac{mx(1 + cx)}{ρ} μ_{n,m−1} (x)

+ nμ_{n,m+1} (x) + nxμ_{n,m} (x) − nxμ_{n,m} (x).

Hence,

(nρ − (m + 1)c)μ_{n,m+1} (x) = ρx(1 + cx)μ′_{n,m} (x) + [cx + m(1 + 2cx)]μ_{n,m} (x)
+ mx(1 + cx)(1 + ρ)μ_{n,m−1} (x). □

Remark 2.2. From Lemma 2.2, we get that L_{n,ρ} (t^{m}, x) is a polynomial in x of degree exactly m, for all m ∈ ℕ^{0} and we can write as

L_{n,ρ} (t^{m}, x) = \frac{n(n + c) …[n + (m − 1)c]}{(nρ − c) …(nρ − mc)} (ρx)^{m}

+ \frac{m(m − 1)(1 + ρ)}{2} \frac{n(n + c) …[n + (m − 2)c]}{(nρ − c) …(nρ − mc)} (ρx)^{m−1} + O(n^{−2} ).

Lemma 2.3. There exist the polynomials q_{i,j,r} (x, c) independent of n and k such that

[x(1 + cx)]^{r} \frac{d^{r}}{dx^{r}} [p_{n,k} (x, c)] = ∑_{\begin{matrix}2i+j⩽r \\ i,j⩾0 \end{matrix}} n^{i}(k − nx)^{j} q_{i,j,r} (x, c)p_{n,k} (x, c).

The proof of the Lemma follows along the line of proof in [11].

3. Main Results

In this section, we state and prove the main results in simultaneous approximation.

Theorem 3.1. If r ∈ ℕ, f ∈ C_{γ}[0, ∞) for some γ > 0 and f^{(r)} exists at a

point x ∈ (0, ∞), then

(3.1)  lim_{n→∞} L_{n,ρ}^{(r)} (f, x) = f^{(r)} (x).

Further, if f^{(r)} exists and continuous on (a − η, b + η) ⊂ (0, ∞), η > 0, then (3.1) holds uniformly in x ∈ [a, b].

Proof. By Taylor’s expansion of f, we have

f(t) = ∑_{i=0}^{r} \frac{f^{(i)} (x)}{i!} (t − x)^{i} + ε(t, x)(t − x)^{r},

where ε(t, x) → 0 as t → x.

L_{n,ρ}^{(r)} (f, x) = ∑_{i=0}^{r} \frac{f^{(i)} (x)}{i!} L_{n,ρ}^{(r)} ((t − x)^{i}, x) + L_{n,ρ}^{(r)} (ε(t, x)(t − x)^{r}, x) =: I_{1} + I_{2}.

In view of Remark 2.2, we have

I_{1} = ∑_{i=0}^{r} \frac{f^{(i)} (x)}{i!} ∑_{j=0}^{i} \binom{i}{j} (−x)^{i−j} L_{n,ρ}^{(r)} (t^{j} , x)

= \frac{f^{(r)} (x)}{r!} ( \frac{n(n + c) …(n + (r − 1)c)}{(nρ − c) …(nρ − rc)} r!ρ^{r})

= f^{(r)} (x)( \frac{n(n + c) …(n + (r − 1)c)}{(nρ − c) …(nρ − rc)} ρ^{r}) → f^{(r)} (x) as n → ∞.

Next, we estimate I_{2} by using Lemma 2.3, we have

I_{2} = L_{n,ρ}^{(r)} (ε(t, x)(t − x)^{r}, x)

= ∑_{\begin{matrix}2i+j⩽r \\ i,j⩾0 \end{matrix}} n^{i} \frac{q_{i,j,r} (x, c)}{x^{r}(1 + cx)^{r}} ∑_{k=1}^{∞} p_{n,k} (x, c)(k − nx)^{j} ∫_{0}^{∞} Θ_{n,k}^{ρ} (t, c)ε(t, x)(t − x)^{r}dt

+ (−1)^{r} \frac{Γ(\frac{n}{c} + r)}{Γ(\frac{n}{c})} (1 + cx)^{− \frac{n}{c} −r} ε(0, x)(−x)^{r}

|I_{2}| ⩽ ∑_{\begin{matrix}2i+j⩽r \\ i,j⩾0 \end{matrix}} n^{i} \frac{|q_{i,j,r} (x, c)|}{x^{r}(1 + cx)^{r}} ∑_{k=1}^{∞} p_{n,k} (x, c)|k − nx|^{j} ∫_{0}^{∞} Θ_{n,k}^{ρ} (t, c)|ε(t, x)||t − x|^{r}dt

+ \frac{Γ(\frac{n}{c} + r)}{Γ(\frac{n}{c})} (1 + cx)^{− \frac{n}{c} −r} |ε(0, x)||x|^{r} =: I_{3} + I_{4}.

Since ε(t, x) → 0 as t → x, for a given ε > 0 there exists δ > 0 such that |ε(t, x)| < ϵ whenever |t − x| < δ, further if λ is any integer > max{γ, r} then we find a constant K > 0 such that |ε(t, x)||t − x|^{r} ⩽ K|t − x|^{λ}. Thus

I_{3} = C_{1} ∑_{\begin{matrix}2i+j⩽r \\ i,j⩾0 \end{matrix}} n^{i} ∑_{k=1}^{∞} p_{n,k} (x, c)|k − nx|^{j} \{ ∫_{|t−x|<δ} εΘ_{n,k}^{ρ} (t, c)|t − x|^{r}dt

+ ∫_{|t−x|⩾δ} KΘ_{n,k}^{ρ} (t, c)|t − x|^{γ}dt}

=: I_{5} + I_{6}.

Applying the Schwarz inequality for the integration and summation we have

I_{5} ⩽ εC_{1} ∑_{\begin{matrix}2i+j⩽r \\ i,j⩾0 \end{matrix}} n^{i} ∑_{k=1}^{∞} p_{n,k} (x, c)|k − nx|^{j} ( ∫_{0}^{∞} Θ_{n,k}^{ρ} (t, c) dt) ^{1/2}

× ( ∫_{0}^{∞} Θ_{n,k}^{ρ} (t, c)(t − x)^{2r} dt) ^{1/2}

⩽ εC_{1} ∑_{\begin{matrix}2i+j⩽r \\ i,j⩾0 \end{matrix}} n^{i}( ∑_{k=1}^{∞} p_{n,k} (x, c)(k − nx)^{2j} ) ^{1/2}

× ( ∑_{k=1}^{∞} p_{n,k} (x, c) ∫_{0}^{∞} Θ_{n,k}^{ρ} (t, c)(t − x)^{2r} dt) ^{1/2} ,

as ∫_{0}^{∞} Θ_{n,k}^{ρ} (t, c) dt = 1. Making use of Lemma 2.2, we get

(3.2)  ∑_{k=1}^{∞} p_{n,k} (x, c)(k − nx)^{2j}

= n^{2j} [ ∑_{k=0}^{∞} p_{n,k} (x, c)( \frac{k}{n} − x)^{2j} − (1 + cx)^{− \frac{n}{c}} (−x)^{2j} ]

= n^{2j} [O(n^{−j} ) + O(n^{−s} )] = O(n^{−j} ) for any s > 0.

Also, by using Lemma 2.2 and arguing as above, we have

(3.3)  ∑_{k=1}^{∞} p_{n,k} (x, c) ∫_{0}^{∞} Θ_{n,k}^{ρ} (t, c)(t − x)^{2r} dt = O(n^{−r} ).

Thus

I_{5} ⩽ εC_{1} ∑_{\begin{matrix}2i+j⩽r \\ i,j⩾0 \end{matrix}} n^{i} · O(n^{j/2} ) · O(n^{−r/2} ) = εO(1).

Next, using the Schwarz inequality for the integration and summation, in view of (3.2) and (3.3), we have

I_{6} ⩽ C_{2} ∑_{\begin{matrix}2i+j⩽r \\ i,j⩾0 \end{matrix}} n^{i} ∑_{k=1}^{∞} p_{n,k} (x, c)|k − nx|^{j} ∫_{|t−x|⩾δ} Θ_{n,k}^{ρ} (t, c)|t − x|^{γ} dt

⩽ C_{2} ∑_{\begin{matrix}2i+j⩽r \\ i,j⩾0 \end{matrix}} n^{i} ∑_{k=1}^{∞} p_{n,k} (x, c)|k − nx|^{j} ( ∫_{|t−x|⩾δ} Θ_{n,k}^{ρ} (t, c) dt)^{1/2}

× ( ∫_{|t−x|⩾δ} Θ_{n,k}^{ρ} (t, c)(t − x)^{2γ} dt)^{1/2}

⩽ C_{2} ∑_{\begin{matrix}2i+j⩽r \\ i,j⩾0 \end{matrix}} n^{i}( ∑_{k=1}^{∞} p_{n,k} (x, c)(k − nx)^{2j} ) ^{1/2}

× ( ∑_{k=1}^{∞} p_{n,k} (x, c) ∫_{0}^{∞} Θ_{n,k}^{ρ} (t, c)(t − x)^{2γ} dt) ^{1/2}

= ∑_{\begin{matrix}2i+j⩽r \\ i,j⩾0 \end{matrix}} n^{i} · O(n^{j/2} ) · O(n^{−m/2} ) = O(n^{(r−m)/2} ) = o(1),

where m is an integer ≧ λ. Thus due to the arbitrariness of ε, it follows that I_{3} = o(1). Also, I_{4} → 0 as n → ∞ and hence I_{2} = o(1). Combining the estimates I_{1} and I_{2} we obtain the desired result (3.1). □

Theorem 3.2. Let f ∈ C_{γ}[0, ∞) for some γ > 0, admitting the derivative of order (r + 2) at a fixed x ∈ (0, ∞). Let f(t) = O(t^{γ}) as t → ∞ for some γ > 0, then we have

(3.4) lim_{n→∞} n[L_{n,ρ}^{(r)} (f, x) − f^{(r)} (x)] = ( \frac{cr(r − 1)}{2} + \frac{cr(r + 1)}{2ρ} ) f^{(r)} (x)

+ ( \frac{r(1 + ρ)(1 + 2cx)}{2ρ} + \frac{cx}{ρ}) f^{(r+1)} (x) + \frac{x(1 + cx)(1 + ρ)}{2ρ} f^{(r+2)} (x).

Proof. By Taylor’s expansion of f, we have

f(t) = ∑_{i=0}^{r+2} \frac{f^{(i)} (x)}{i!} (t − x)^{i} + ε(t, x)(t − x)^{r+2} ,

where ε(t, x) → 0 as t → x and ε(t, x) = o((t − x)^{δ}) as t → ∞ for some δ > 0, Applying the r-th derivative of L_{n,ρ} , we can write

n[L_{n,ρ}^{(r)} (f, x) − f^{(r)} (x)] = n[ ∑_{i=0}^{r+2} \frac{f^{(i)} (x)}{i!} L_{n,ρ}^{(r)} ((t − x)^{i}, x) − f^{(r)} (x)]
+ n[L_{n,ρ}^{(r)} (ε(t, x)(t − x)^{r+2} , x)] =: I_{1} + I_{2}.

By Lemma 2.2 and Remark 2.2, we have

I_{1} = n ∑_{i=0}^{r+2} \frac{f^{(i)} (x)}{i!} ∑_{j=r}^{i} \binom{i}{j} (−x)^{i−j} L_{n,ρ}^{(r)} (t^{j} , x) − nf^{(r)} (x)

= \frac{f^{(r)} (x)}{r!} n[L_{n,ρ}^{(r)} (t^{r}, x) − (r!)]

+ \frac{f^{(r+1)} (x)}{(r + 1)!} n\{(r + 1)(−x)L_{n,ρ}^{(r)} (t^{r}, x) + L_{n,ρ}^{(r)} (t^{r+1} , x)\}

+ \frac{f^{(r+2)} (x)}{(r + 2)!} n\{ \frac{(r + 2)(r + 1)}{2} x^{2}L_{n,ρ}^{(r)} (t^{r}, x)

+ (r + 2)(−x)L_{n,ρ}^{(r)} (t^{r+1} , x) + L_{n,ρ}^{(r)} (t^{r+2} , x)}

= n[ \frac{n(n + c) …[n + (r − 1)c]}{(nρ − c) …(nρ − rc)} ρ^{r} − 1]f^{(r)} (x)

+ n\frac{f^{(r+1)} (x)}{(r + 1)!} \{(r + 1)(−x)\frac{n(n + c) …[n + (r − 1)c]}{(nρ − c) …(nρ − rc)} ρ^{r}r!

+ \frac{n(n + c) …[n + rc]}{(nρ − c) …(nρ − (r + 1)c)} ρ^{r+1} (r + 1)!x

+ \frac{r(r + 1)(1 + ρ)}{2} \frac{n(n + c) …[n + (r − 1)c]}{(nρ − c) …(nρ − (r + 1)c)} ρ^{r}r!\}

+ n\frac{f^{(r+2)} (x)}{(r + 2)!} \{ \frac{(r + 2)(r + 1)}{2} x^{2} \frac{n(n + c) …[n + (r − 1)c]}{(nρ − c) …(nρ − rc)} ρ^{r}r!

− (r + 2)x( \frac{n(n + c) …[n + rc]}{(nρ − c) …(nρ − (r + 1)c)} ρ^{r+1} (r + 1)!x

+ \frac{r(r + 1)(1 + ρ)}{2} \frac{n(n + c) …[n + (r − 1)c]}{(nρ − c) …(nρ − (r + 1)c)} ρ^{r}r!)

+ \frac{n(n + c) …[n + (r + 1)c]}{(nρ − c) …(nρ − (r + 2)c)} ρ^{r+2} \frac{(r + 2)!}{2} x^{2}

+ \frac{(r + 1)(r + 2)(1 + ρ)}{2} \frac{n(n + c) …(n + rc)}{(nρ − c) …(nρ − (r + 2)c)} ρ^{r+1} (r + 1)!x\}

+ O(n^{−2} ).

Now the coefficients of f^{(r)} (x), f^{(r+1)} (x) and f^{(r+2)} (x) in the above expression are respectively \frac{cr(r−1)}{2} + \frac{cr(r+1)}{2ρ} , \frac{r(1+ρ)(1+2cx)}{2ρ} and \frac{x(1+cx)(1+ρ)}{2ρ} respectively, which follows by using induction hypothesis on r and taking the limits as n → ∞. Hence in order to prove (3.4), it suffices to show that I_{2} → 0 as n → ∞, which follows on proceeding along the lines in the estimation of I_{2} as done in Theorem 3.1. □

Remark 3.1. In the case c = 0 the above theorem is given in [16]. Another special case of the above asymptotic formula in simultaneous approximation for c = 0 and ρ = 1 was discussed by Agrawal and Gupta in [1].

Theorem 3.3. Let f ∈ C_{γ}[0, ∞) for some γ > 0 and r ⩽ m ⩽ r + 2. If f^{(m)}

exists and is continuous on (a − η, b + η) ⊂ (0, ∞), η > 0, then for n sufficiently large

∥L_{n,ρ}^{(r)} (f, .)−f^{(r)} ∥_{C[a,b]} ⩽ C_{1}n^{−1} ∑_{i=r}^{m} ∥f^{(i)} ∥_{C[a,b]} +C_{2}n^{−1/2} ω(f^{(m)} , n^{−1/2} )+O(n^{−2} ),

where C_{1}, C_{2} are constants independent of f and n, ω(f^{(m)} , δ) is the modulus of continuity of f^{(m)} on (a − η, b + η) and ∥.∥_{C[a,b]} denotes the sup-norm on [a, b].

Proof. By Taylor’s expansion of f, we have

f(t) = ∑_{i=0}^{m} \frac{f^{(i)} (x)}{i!} (t − x)^{i} + \frac{f^{(m)} (ξ) − f^{(m)} (x)}{m!} (t − x)^{m}χ(t) + h(t, x)(1 − χ(t)),

where ξ lies between t and x and χ(t) is the characteristic function on the interval (a − η, b + η).

Now,

L_{n,ρ}^{(r)} (f, x) − f^{(r)} (x) = \{ ∑_{i=0}^{m} \frac{f^{(i)} (x)}{i!} L_{n,ρ}^{(r)} ((t − x)^{i}, x) − f^{(r)} (x)\}

+ L_{n,ρ}^{(r)} ( \frac{f^{(m)} (ξ) − f^{(m)} (x)}{m!} (t − x)^{m}χ(t), x)

+ L_{n,ρ}^{(r)} (h(t, x)(1 − χ(t)), x) =: E_{1} + E_{2} + E_{3}.

By using Lemma 2.2 and Remark 2.2, we have

E_{1} = ∑_{i=0}^{m} \frac{f^{(i)} (x)}{i!} ∑_{j=0}^{i} \binom{i}{j} (−x)^{i−j} \frac{d^{r}}{dx^{r}} ( \frac{n(n + c) …[n + (j − 1)c]}{(nρ − c) …(nρ − jc)} (ρx)^{j}

+ \frac{j(j − 1)(1 + ρ)}{2} \frac{n(n + c) …[n + (j − 2)c]}{(nρ − c) …(nρ − jc)} (ρx)^{j−1} + O(n^{−2} )) − f^{(r)} (x).

Consequently,

∥E_{1}∥_{C[a,b]} ⩽ C_{1}n^{−1} ∑_{i=r}^{m} ∥f^{(i)} ∥_{C[a,b]} + O(n^{−2} ), uniformly on [a, b].

Next, we estimate E_{2} as follows

|E_{2}| ⩽ ∑_{k=1}^{∞} p_{n,k}^{(r)} (x, c) ∫_{0}^{∞} Θ_{n,k}^{ρ} (t, c)\{ | \frac{f^{(m)} (ξ) − f^{(m)} (x)}{m!} ||t − x|^{m}χ(t)\} dt

+ p_{n,0}^{(r)} (x, c)| \frac{f^{(m)} (ξ) − f^{(m)} (x)}{m!} ||x|^{m}

⩽ \frac{ω(f^{(m)} , δ)}{m!} [ ∑_{k=1}^{∞} |p_{n,k}^{(r)} (x, c)| ∫_{0}^{∞} Θ_{n,k}^{ρ} (t, c)( 1 + \frac{|t − x|}{δ} ) |t − x|^{m}dt

+ p_{n,0}^{(r)} (x, c)( 1 + \frac{|x|}{δ}) |x|^{m}]

⩽ \frac{ω(f^{(m)} , δ)}{m!} [ ∑_{k=1}^{∞} |p_{n,k}^{(r)} (x)| ∫_{0}^{∞} Θ_{n,k}^{ρ} (t, c)(|t − x|^{m} + δ^{−1} |t − x|^{m+1} ) dt

+ p_{n,0}^{(r)} (x, c)(|x|^{m} + δ^{−1} |x|^{m+1} )].

Using the Schwarz inequality for both integration and summation, we get

(3.5)  ∑_{k=1}^{∞} p_{n,k} (x, c)|k − nx|^{j} ∫_{0}^{∞} Θ_{n,k}^{ρ} (t, c)|t − x|^{m}dt

⩽ ∑_{k=1}^{∞} p_{n,k} (x, c)|k − nx|^{j} ( ∫_{0}^{∞} Θ_{n,k}^{ρ} (t, c) dt)^{1/2}

× ( ∫_{0}^{∞} Θ_{n,k}^{ρ} (t, c)(t − x)^{2m} dt)^{1/2}

⩽ ( ∑_{k=1}^{∞} p_{n,k} (x, c)(k − nx)^{2j} ) ^{1/2}

× ( ∑_{k=1}^{∞} p_{n,k} (x, c) ∫_{0}^{∞} Θ_{n,k}^{ρ} (t, c)(t − x)^{2m} dt)^{1/2}

= O(n^{j/2} ) · O(n^{−m/2} ) = O(n^{(j−m)/2} ), uniformly on[a, b].

Therefore, by Lemma 2.3 and (3.5), we get

(3.6) ∑_{k=1}^{∞} |p_{n,k}^{(r)} (x, c)| ∫_{0}^{∞} Θ_{n,k}^{ρ} (t, c)|t − x|^{m}dt

⩽ ∑_{k=1}^{∞} ∑_{\begin{matrix}2i+j⩽r \\ i,j⩾0 \end{matrix}} n^{i}|k − nx|^{j} \frac{q_{i,j,r} (x, c)}{x^{r}(1 + cx)^{r}} p_{n,k} (x, c) × ∫_{0}^{∞} Θ_{n,k}^{ρ} (t, c)|t − x|^{m}dt

⩽ ( sup_{\begin{matrix}2i+j⩽r \\ i,j⩾0 \end{matrix}} \frac{q_{i,j,r} (x, c)}{x^{r}(1 + cx)^{r}} )

× ∑_{\begin{matrix}2i+j⩽r \\ i,j⩾0 \end{matrix}} n^{i}( ∑_{k=1}^{∞} p_{n,k} (x, c)|k − nx|^{j} ∫_{0}^{∞} Θ_{n,k}^{ρ} (t, c)|t − x|^{m}dt)

= C ∑_{\begin{matrix}2i+j⩽r \\ i,j⩾0 \end{matrix}} n^{i}O(n^{(j−m)/2} ) = O(n^{(r−m)/2} ), uniformly on [a, b].

where C = sup_{2i+j⩽r; i,j⩾0} \frac{|q_{i,j,r} (x)|}{x^{r} (1+x)^{r}} , ∀x ∈ [0, ∞).

Choosing δ = n^{−1/2} and applying (3.6), we obtain

|E_{2}| ⩽ \frac{ω(f^{(m)} , n^{−1/2} )}{m!} [O(n^{(r−m)/2} ) + n^{1/2} O(n^{(r−m−1)/2} ) + O(n^{−m} )
⩽ C_{2}n^{−(r−m)/2} ω(f^{(m)} , n^{−1/2} ).

Since t ∈ [0, ∞) ∖ (a − η, b + η), we can choose δ such that |t − x| ⩾ δ for all x ∈ [a, b]. Thus by Lemma 2.3, we have

|E_{3}| ⩽ C ∑_{\begin{matrix}2i+j⩽r \\ i,j⩾0 \end{matrix}} n^{i} ∑_{k=1}^{∞} p_{n,k} (x, c)|k − nx|^{j} ∫_{0}^{∞} Θ_{n,k}^{ρ} (t, c)|h(t, x)|

+ \frac{(n + r − 1)!}{(n − 1)!} (1 + cx)^{−n−r} |h(0, x)|.

For |t − x| ⩾ δ, we can find the a constant M such that |h(t, x)| ⩽ M|t − x|^{β}, where β is an integer ≧ {γ, m}. Hence using the Schwarz inequality for both integration and summation, (3.2) and (3.3), it easily follows that I_{3} = O(n^{−s} ) for any s > 0, uniformly on [a, b].

Combing the estimates of E_{1}, E_{2}, E_{3}, the required result is immediate. □

Remark 3.2. Very recently, Verma et al. [20] proposed the Stancu-type generalization of Baskakov–Durrmeyer type operators. One can consider the similar generalization of the operators (1.1) further based on two more parameters, but that will just provide extension of work without much gain as far as the convergence is concerned.

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Department of Mathematics (Received 01 12 2015) Miranda House, University of Delhi New Delhi India

    durvesh.kv.du@gmail.com
    durvesh.verma@mirandahouse.ac.in