MATEMATIQKI VESNIK

58 (2006), 19–29

UDK 517.984

originalni nauqni rad research paper

DIRECT AND INVERSE THEOREMS FOR SZÂSZ-LUPAS TYPE OPERATORS IN SIMULTANEOUS APPROXIMATION

Naokant Deo

Abstract. In this paper we give the direct and inverse theorems for Szâsz-Lupas operators and study the simultaneous approximation for a new modification of the Szâsz operators with the weight function of Lupas operators.

1. Introduction

Let f be a function defined on the interval [0, ∞) with real values. For f ∈ [0, ∞) and n ∈ N, the Szâsz operator S_{n} (f, x) is defined as follows:

S_{n} (f, x) = ∑_{k=0}^{∞} s_{n,k}(x)f(k/n), where s_{n,k}(x) = \frac{e^{−nx}(nx)^{k}}{k!} .

The Szâsz-type operator L_{n} (f, x) is defined by

L_{n} (f, x) = ∑_{k=0}^{∞} s_{n,k}(x)ϕ_{n,k}(f),

where

ϕ_{n,k}(f) = \{ \begin{matrix}f(0), & for k = 0 \\ n ∫_{0}^{∞} s_{n,k}(t)f(t)dt, & for k = 1, 2, … \end{matrix}

In [10], Mazhar and Totik introduced the Szâsz-type operator and showed some approximation theorems. Lupas proposed a family of linear positive operators mapping C [0, ∞) into C [0, ∞), the class of all bounded and continuous functions on [0, ∞) namely,

(B_{n}f) (x) = ∑_{k=0}^{∞} p_{n,k}(x)f(k/n), where p_{n,k}(x) = \binom{n + k − 1}{k}\frac{x^{k}}{(1 + x)^{n+k}} .

AMS Subject Classification: 41 A 35

Keywords and phrases: Linear positive operators, linear combination, integral modulus of smoothness, Steklow means.

Motivated by the integration of Bernstein polynomials of Derriennic [4], Sahai and Prasad [11] modified the operators B_{n} for function integrable on [0, ∞) as

(M_{n}f) (x) = (n − 1) ∑_{k=0}^{∞} p_{n,k}(x) ∫_{0}^{∞} p_{n,k}(t)f(t) dt.

Now we consider another modification of operators with the weight function of Lupas operators, which are defined as

(V_{n}f) (x) = (n − 1) ∑_{k=0}^{∞} s_{n,k}(x) ∫_{0}^{∞} p_{n,k}(t)f(t) dt.  (1.1)

The norm ∥.∥_{C_{α}} on the space C_{α} [0, ∞) = {f ∈ C [0, ∞) : |f(t)| ≤ Kt^{α} for some α > 0 and K > 0} is defined by

∥f∥_{C_{α}} = sup_{0≤t<∞} |f(t)| t^{−α}.

To improve the saturation order O(n^{−1}) for the operator (1.1), we use the technique of linear combination as described below:

V_{n} (f, k, x) = ∑_{j=0}^{k} C(j, k)V_{d_{j}n}(f, x),

where

C(j, x) = ∏_{{i=0 \atop i≠j}}^{k} \frac{d_{j}}{d_{j} − d_{i}} for k ≠ 0 and C(0, 0) = 1

and d_{0}, d_{1}, d_{2}, …, d_{k} are (k+1) arbitrary, fixed and distinct positive integers. For our convenience we shall write the operator (1.1) as

V_{n}(f, x) = ∫_{0}^{∞} W (n, x, t)f(t) dt,

where

W (n, x, t) = (n − 1) ∑_{k=0}^{∞} s_{n,k}(x)p_{n,k}(t).

The function f is said to belong to the generalized Zygmund class Liz (α, k, a, b) if there exists a constant M such that

ω_{2k}(f, η, a, b) ≤ Mη^{αk}, η > 0,

where ω_{2k}(f, η, a, b) denotes the modulus of continuity of 2k-th order of f(x) on the interval [a, b]. The class Liz (α, 1, a, b) is more commonly denoted by Lip*(α, a, b).

Let f ∈ C_{α} [0, ∞) and 0 < a_{1} < a_{2} < a_{3} < b_{3} < b_{2} < b_{1} < ∞. Then for m ∈ N the Steklov mean f_{η,m} of the m-th order corresponding to f, for sufficiently small values of η > 0 is defined by

f_{η,m}(x) = η^{−m}(∫_{−η/2}^{η/2})^{m}{f(x) + (−1)^{m−1}△_{∑_{i=1}^{m} x_{i}}^{m}f(x)} ∏_{i=1}^{m} dx_{i},  (1.2)

where x ∈ [a_{1}, b_{1}] and △_{η}^{m}f(x) is the m-th order forward difference with step length η.

The direct results in ordinary and simultaneous approximation for such type of modified Szâsz-Mirakyan operators were studied by many researchers see e.g. [2], [5], [6] and [12].

2. Auxiliary results

In this section, we shall give some basic results, which will be useful in proving the main results.

Lemma 2.1. [9] For m ∈ N ∪ {0}, let the m-th order moment for the Szâsz operator be defined by

μ_{n,m}(x) = ∑_{k=0}^{∞} s_{n,k}(x) (\frac{k}{n} − x)^{m} .

Then we have μ_{n,0}(x) = 1, μ_{n,1}(x) = 0 and

nμ_{n,m+1}(x) = x (μ′_{n,m}(x) + mμ_{n,m−1}(x)) , for n ∈ N.

Consequently, (i) μ_{n,m}(x) is a polynomial in x of degree [m/2];

(ii) for every x ∈ [0, ∞) , μ_{n,m}(x) = O (n^{−[(m+1)/2]}), where [β] denotes the integral part of β.

Lemma 2.2. Let the m-th moment for the Szâsz operator be defined by

μ_{n,m}(x) = (n − 1) ∑_{k=0}^{∞} s_{n,k}(x) ∫_{0}^{∞} p_{n,k}(t)(t − x)^{m} dt.

Then

(i) μ_{n,0}(x) = 1, μ_{n,1}(x) = \frac{(1+2x)}{(n−2)}, n > 2;

(ii) (n − m − 2)μ_{n,m+1}(x) = x [μ′_{n,m}(x) + m(2 + x)μ_{n,m−1}(x)] + (m + 1) × (1 + 2x)μ_{n,m}(x)

(iii) μ_{n,m}(x) = O (n^{−[(m+1)/2]}) for all x ∈ [0, ∞).

Proof. By the definition of μ_{n,m}(x), we can easily obtain (i). Now the proof of (ii) goes as follows:

xμ′_{n,m}(x) = (n − 1) ∑_{k=0}^{∞} xs′_{n,k}(x) ∫_{0}^{∞} p_{n,k}(t)(t − x)^{m} dt − mxμ_{n,m−1}(x).

Using relations t(1+t)p′_{n,k}(t) = (k−nt)p_{n,k}(t) and xs′_{n,k}(x) = (k−nx)s_{n,k}(x), we get

x [μ′_{n,m}(x) + mμ_{n,m−1}(x)]

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

22 N. Deo

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

= (n − 1) ∑_{k=0}^{∞} s_{n,k}(x) ∫_{0}^{∞} t (1 + t) p′_{n,k}(t)(t − x)^{m} dt + nμ_{n,m+1}(x)

= (n − 1) ∑_{k=0}^{∞} s_{n,k}(x) ∫_{0}^{∞} [(1 + 2x)(t − x) + (t − x)^{2} + x(1 + x)] p′_{n,k}(t)(t − x)^{m} dt

+ nμ_{n,m+1}(x) = −(m + 1)(1 + 2x)μ_{n,m}(x) − (m + 2)μ_{n,m+1}(x) − mx(1 + x)μ_{n,m−1}(x)

+ nμ_{n,m+1}(x) This leads to proof of (ii). The proof of (iii) easily follows from (i) and (ii).

Lemma 2.3. If f is differentiable r times (r = 1, 2, 3, …) on [0, ∞), then we have

(V_{n}^{(r)}f) (x) = \frac{n^{r}(n − r − 1)!}{(n − 2)!} ∑_{k=0}^{∞} s_{n,k}(x) ∫_{0}^{∞} p_{n,k}(t)f^{(r)}(t) dt.

Proof. By Leibnitz’s theorem in (1.1)

(V_{n}^{(r)}f) (x) = (n − 1) ∑_{i=0}^{r} ∑_{k=i}^{∞} \binom{r}{i}\frac{(−1)^{r−i}n^{i}e^{−nx}(nx)^{k−i}}{(k − 1)!} ∫_{0}^{∞} p_{n,k}(t)f(t) dt

= (n − 1) ∑_{k=0}^{∞} ∑_{i=0}^{r}(−1)^{r−i}\binom{r}{i}n^{r}s_{n,k}(x) ∫_{0}^{∞} p_{n,k+i}(t)f(t) dt

= (n − 1) ∑_{k=0}^{∞} s_{n,k}(x) ∫_{0}^{∞}(−1)^{r}\{∑_{i=0}^{r} \binom{r}{i}(−1)^{i}n^{r}p_{n,k+i}(t)(t)\}f(t) dt.

Again using Leibnitz’s theorem

p_{n−r,k+r}^{(r)}(t) = \frac{(n − 1)!}{(n − r − 1)!} ∑_{i=0}^{r}(−1)^{i}\binom{r}{i}p_{n,k+i}(t),

(V_{n}^{(r)}f) (x) = \frac{n^{r}(n − r − 1)!}{(n − 2)!} ∑_{k=0}^{∞} s_{n,k}(x) ∫_{0}^{∞}(−1)^{r}p_{n−r,k+r}^{(r)}(t)f(t) dt,

integrating by parts r times, we get the required result.

Lemma 2.4. For the function f_{η,m}(x) defined in (1.2), there hold:

(i) f_{η,m} ∈ C[a_{1}, b_{1}]; (ii) ∥f_{η,m}^{(r)}∥_{C[a_{2},b_{2}]} ≤ M_{r}η^{−r}ω_{r}(f, η, a_{1}, b_{1}), r = 1, 2, …, m; (iii) ∥f − f_{η,m}∥_{C[a_{2},b_{2}]} ≤ M_{m+1}ω_{m}(f, η, a_{1}, b_{1}); (iv) ∥f_{η,m}∥_{C[a_{2},b_{2}]} ≤ M_{m+2}∥f∥_{C[a_{2},b_{2}]} ≤ M′∥f∥_{C_{α}},

where Mi^{′} are certain constants independent of f and η.

For the proof of the above properties of the function f_{η,m}(x) we refer to [12, page 167].

Lemma 2.5. [8, 9] There exist polynomials q_{i,j,r}(x) independent of n and k such that

x^{r} \frac{d^{r}}{dx^{r}} [e^{−nx}(nx)^{k}] = ∑_{{2i+j≤r \atop i,j≥0}} n^{i}|k − nx|^{j}q_{i,j,r}(x) [e^{−nx}(nx)^{k}]

Lemma 2.6. Let f ∈ C_{α} [0, ∞). If f^{(2k+2)} exists at a point x ∈ (0, ∞), then

lim_{n→∞} n^{k+1} {V_{n}(f, k, x) − f(x)} = ∑_{p=0}^{2k+2} Q(p, k, x)f^{(p)}(x),

where Q(p, k, x) is a certain polynomial in x of degree p.

The proof of Lemma 2.6 follows along the lines of [7].

Lemma 2.7. Let δ and γ be any two positive numbers and [a, b] ⊂ [0, ∞). Then, for any m > 0 there exists a constant M_{m} such that

∥∫_{|t−x|≥δ} V_{n}(f, x)t^{γ}dt∥_{C[a,b]} ≤ M_{m}n^{−m}.

The proof of this result follows easily by using Schwarz inequality and Lemma 2.7 from [1].

3. Main results

Theorem 3.1. (Direct Theorem) Let f ∈ C_{α} [0, ∞). Then, for sufficiently large n, there exists a constant M independent of f and n such that

∥V_{n}(f, k, .) − f∥_{C[a_{2},b_{2}]} ≤ max {C_{1}ω_{2k+2}(f; n^{−1/2}, a_{1}, b_{1})C_{2}n^{−(k+1)}∥f∥_{C_{α}}} ,

where C_{1} = C_{1}(k) and C_{2} = C_{2}(k, f).

Proof. By linearity property

∥V_{n}(f, k, .) − f∥_{C[a_{2},b_{2}]} ≤ ∥V_{n} ((f − f_{2k+2,η}) , k, .) ∥_{C[a_{2},b_{2}]}

+ ∥V_{n} (f_{2k+2,η}, k, .) − f_{2k+2,η}∥_{C[a_{2},b_{2}]} + ∥f − f_{2k+2,η}∥_{C[a_{2},b_{2}]}
= A_{1} + A_{2} + A_{3}, say.

By property (iii) of Steklov mean, we get

A_{3} ≤ C_{1}ω_{2k+2}(f, η, a_{1}, b_{1}).

Next, by Lemma 2.6, we get

A_{2} ≤ C_{2}n^{−(k+1)}∥f_{2k+2,η}∥_{C[a_{1},b_{1}]}.

Using the interpolation property [5] and properties of Steklov mean,

A_{2} ≤ C_{3}n^{−(k+1)}{∥f∥_{C_{α}} + η^{−(2k+2)}ω_{2k+2}(f, η)}.
To estimate A_{1}, we choose a_{2}, b_{2} such that
0 < a_{1} < a_{2} < a_{3} < b_{3} < b_{2} < b_{1} < ∞.

Also let ψ(t) be the characteristic function of the interval [a_{2}, b_{2}], then

A_{1} ≤ ∥V_{n} (ψ(t) (f(t) − f_{2k+2,η}(t)) , k, .) ∥_{C[a_{3},b_{3}]}
+ ∥V_{n} ((1 − ψ(t)) (f(t) − f_{2k+2,η}(t)) , k, .) ∥_{C[a_{3},b_{3}]}
= A_{4} + A_{5}, say.

We note that in order to estimate A_{4} and A_{5}, it is sufficient to consider their expressions without the linear combination. It is clear that by Lemma 2.3, we obtain

V_{n} (ψ(t) (f(t) − f_{2k+2,η}(t)) , x)

= (n − 1) ∑_{k=0}^{∞} s_{n,k}(x) ∫_{0}^{∞} p_{n,k}(t)ψ(t) (f(t) − f_{2k+2,η}(t)) dt.

Hence,

∥V_{n} (ψ(t) (f(t) − f_{2k+2,η}(t)) , .) ∥_{C[a_{1},b_{1}]} ≤ C_{4}∥f − f_{2k+2,η}∥_{C[a_{2},b_{2}]}.

Now for x ∈ [a_{3}, b_{3}] and t ∈ [0, ∞) / [a_{2}, b_{2}] we can choose an η_{1} satisfying |t − x| ≥ η_{1}. Therefore by Lemma 2.5 and Schwarz inequality, we have

I ≡ |V_{n} ((1 − ψ(t)) (f(t) − f_{2k+2,η}(t)) , x) |

≤ (n − 1) ∑_{{2i+j≤r \atop i,j≥0}} n^{i} \frac{|ϕ_{i,j,r}(x)|}{x^{r}} ∑_{k=0}^{∞} s_{n,k}(x)|k − nx|^{j}×

× ∫_{0}^{∞} p_{n,k}(t) (1 − ψ(t)) |f(t) − f_{2k+2,η}(t)| dt

≤ C_{5}∥f∥_{C_{α}}(n − 1) ∑_{{2i+j≤r \atop i,j≥0}} n^{i} ∑_{k=0}^{∞} s_{n,k}(x)|k − nx|^{j} ∫_{|t−x|≥η_{1}} p_{n,k}(t) dt

≤ C_{5}η_{1}^{−2s}∥f∥_{C_{α}}(n − 1) ∑_{{2i+j≤r \atop i,j≥0}} n^{i} ∑_{k=0}^{∞} s_{n,k}(x)|k − nx|^{j}×

× (∫_{0}^{∞} p_{n,k}(t) dt)^{1/2}(∫_{0}^{∞} p_{n,k}(t)(t − x)^{4s} dt)^{1/2}

≤ C_{5}η_{1}^{−2s}∥f∥_{C_{α}} ∑_{{2i+j≤r \atop i,j≥0}} n^{i}\{ ∑_{k=0}^{∞} s_{n,k}(x)(k − nx)^{2j}\}^{1/2}×

× ((n − 1) ∑_{k=0}^{∞} s_{n,k}(x) ∫_{0}^{∞} p_{n,k}(t)(t − x)^{4s} dt)^{1/2}.

Hence, by Lemma 2.1 and Lemma 2.2, we have

I ≤ C_{6}∥f∥_{C_{α}} ∑ n^{(i+ \frac{j}{2} −s)} ≤ C_{6}n^{−q}∥f∥_{C_{α}}

where q = (s − m/2). Now choose s > 0 such that q ≥ k + 1. Then I ≤ C_{6}n^{−(k+1)}∥f∥_{C_{α}}.

Therefore by property (iii) of Steklov mean, we get

A_{1} ≤ C_{7}∥f − f_{2k+2,η}∥_{C[a_{2},b_{2}]} + C_{6}n^{−(k+1)}∥f∥_{C_{α}}
≤ C_{8}ω_{2k+2}(f, η, a_{1}, b_{1}) + C_{6}n^{−(k+1)}∥f∥_{C_{α}}

Hence with η = n^{−1/2}, the theorem follows.

Theorem 3.2. (Inverse Theorem) If 0 < α < 2 and f ∈ C_{α} [0, ∞) then in the following statements (i) ⇒ (ii):

(i) ∥V_{n}(f, k, x) − f(x)∥_{C[a_{1},b_{1}]} = O (n^{−α(k+1)/2}), where f ∈ C_{α}[a, b],

(ii) f ∈ Liz(α, k + 1, a_{2}, b_{2}).

Proof. Let us choose points a′, a′′, b′, b′′ in such a way that a_{1} < a′ < a′′ < a_{2} < b_{2} < b′′ < b′ < b_{1}. Also suppose g ∈ C_{0}^{∞} with supp(g) ⊂ [a′′, b′′] and g(x) = 1 for x ∈ [a_{a}, b_{2}]. It is sufficient to show that

∥V_{n}(fg, k, .) − fg∥_{C[a′,b′]} = O (n^{−α(k+1)/2}) ⇒ (ii).  (3.1)

Using F in place of fg for all the values of r > 0, we get

∥△_{r}^{2k+2}F ∥_{C[a′′,b′′]} ≤ ∥△_{r}^{2k+2}(F − V_{n}(F, k, .))∥_{C[a′′,b′′]} + ∥△_{r}^{2k+2}V_{n}(F, k, .)∥_{C[a′′,b′′]}

(3.2)

By the definition of △_{r}^{2k+2},

∥△_{r}^{2k+2}V_{n} (F, k, .) ∥_{C[a′′,b′′]}

= ∥ ∫_{0}^{r}⋯ ∫_{0}^{r} V_{n}(F, k, x + ∑_{i=1}^{2k+2} x_{i}) dx_{1} … dx_{2k+2}∥_{C[a′′,b′′]}

≤ r^{2k+2}∥V_{n}^{(2k+2)} (F, k, .) ∥_{C[a′′,b′′+(2k+2)r]}

≤ r^{2k+2}{∥V_{n}^{(2k+2)} (F − F_{η,2k+2}, k, .) ∥_{C[a′′,b′′+(2k+2)r]}

+ ∥V_{n}^{(2k+2)} (F_{η,2k+2}, k, .) ∥_{C[a′′,b′′+(2k+2)r]}}, (3.3)

where F_{η,2k+2} is the Steklov mean of (2k + 2)-th order corresponding to F. By Lemma 3 from [1], we get

∫_{0}^{∞} |\frac{∂^{2k+2}}{∂x^{2k+2}} W_{n}(t, x) dt|

≤ ∑_{{2i+j≤2k+2 \atop i,j≥0}}(n − 1) ∑_{k=0}^{∞} n^{i}|k − nx|^{j} \frac{|q_{i,j,2k+2}(x)|}{x^{2k+2}}s_{n,k}(x) ∫_{0}^{∞} p_{n,k}(t) dt.

26 N. Deo

Since ∫_{0}^{∞} p_{n,k}(t) dt = \frac{1}{n−1}, by Lemma 2.1,

∑_{k=0}^{∞} s_{n,k}(x)(k − nx)^{2j} = n^{2j} ∑_{k=0}^{∞} s_{n,k}(x) (\frac{k}{n} − x)^{2j} = O(n^{j})  (3.4)

Using Schwarz inequality and Lemma 2.1, we obtain

∥V_{n}^{(2k+2)} (F − F_{η,2k+2}, k, .) ∥_{C[a′′,b′′+(2k+2)r]} ≤ K_{1}n^{k+1}∥F −F_{η,2k+2}∥_{C[a′′,b′′]} (3.5)

By Lemma 2 from [1], we get

∫_{0}^{∞} [\frac{∂^{k}}{∂x^{k}} W_{n}(t, x)] (t − x)^{i}dt = 0, for k > i.  (3.6)

By Taylor’s expansion, we obtain

F_{η,2k+2}(t) = ∑_{i=0}^{2k+1} \frac{F_{η,2k+2}^{(i)}(x)}{i!}(t − x)^{i} + F_{η,2k+2}^{(2k+2)}(ξ)\frac{(t − x)^{2k+2}}{(2k + 2)!} ,  (3.7)

where t < ξ < x. By (3.6) and (3.7), we get

∥\frac{∂^{2k+2}}{∂x^{2k+2}} V_{n} (F_{η,2k+2}, k, .) ∥_{C[a′′,b′′+(2k+2)r]}

≤ ∑_{j=0}^{k} \frac{|C(j, k)|}{(2k + 2)!}∥F_{η,2k+2}^{(2k+2)}∥_{C[a′′,b′′]}∥ ∫_{0}^{∞} [\frac{∂^{2k+2}}{∂x^{2k+2}} W_{d_{j}n}(t, x)] (t−x)^{2k+2} dt∥_{C[a′′,b′′]}.

Again applying Schwarz inequality for integration and summation and Lemma 3 from [1], we obtain

I ≡ ∫_{0}^{∞} |\frac{∂^{2k+2}}{∂x^{2k+2}} W_{n}(t, x)|(t, x)^{2k+2} dt

≤ (n − 1) ∑_{{2i+j≤2k+2 \atop i,j≥0}} ∑_{k=0}^{∞} n^{i}s_{n,k}(x)|k − nx|^{j} \frac{|q_{i,j,2k+2}(x)|}{x^{2k+2}} ∫_{0}^{∞} p_{n,k}(t)(t − x)^{2k+2} dt

≤ ∑_{{2i+j≤2k+2 \atop i,j≥0}} n^{i} \frac{|q_{i,j,2k+2}(x)|}{x^{2k+2}}\{ ∑_{k=0}^{∞} s_{n,k}(x)(k − nx)^{2j}\}^{1/2}×

× {(n − 1) ∑_{k=0}^{∞} s_{n,k}(x) ∫_{0}^{∞} p_{n,k}(t)(t − x)^{4k+4}dt}^{1/2}.  (3.8)

Using Lemma 2 from [1],

(n − 1) ∑_{k=0}^{∞} s_{n,k}(x) ∫_{0}^{∞} p_{n,k}(t)(t − x)^{4k+4} dt = T_{n,4k+4}(x) = O (n^{−(2k+2)}) . (3.9)

Using (3.4) and (3.9) in (3.8), we obtain

I ≤ ∑_{{2i+j≤2k+2 \atop i,j≥0}} n^{i}^{\frac{|q_{i,j,2k+2}(x)|}{x^{k+1}}}O(n^{j/2})O (n^{−(k+1)}) = O(1).

Hence

∥W_{n}^{(2k+2)} (F_{η,2k+2}, k, .) ∥_{C[a′′,b′′+(2k+2)r]} ≤ K_{2}∥F_{η,2k+2}^{(2k+2)}∥_{C[a′′,b′′]}.  (3.10)

On combining (3.2), (3.3), (3.5) and (3.10) it follows ∥△_{r}^{2k+2}F ∥_{C[a′′,b′′]} ≤ ∥△_{r}^{2k+2} (F − V_{n}(F, k, .)) ∥_{C[a′′,b′′]}

+ K_{3}r^{2k+2}(n^{k+1}∥F − F_{η,2k+2}∥_{C[a′′,b′′]} + ∥F_{η,(2k+2)}^{(2k+2)}∥_{C[a′′,b′′]}).

Since for small values of r the above relation holds, it follows from the properties of F_{η,2k+2} and (3.1) that

ω_{2k+2}(F, h, [a′′, b′′])

≤ K_{4} {n^{−α(k+1)/2} + h^{2k+2} (n^{k+1} + η^{−2k+2})_{ω_{2k+2}} (F, η, [a′′, b′′])} .

Choosing η is such a way that n < η^{−2} < 2r and following Berens and Lorentz [3], we obtain

w_{2k+2}(F, h, [a′′, b′′]) = O(h^{α(k+1)}).  (3.11)

Since F (x) = f(x) in [a_{2}, b_{2}], from (3.11) we have w_{2k+2}(f, h, [a_{2}, b_{2}]) = O(h^{α(k+1)}), i.e., f ∈ Liz(α, k + 1, a_{2}, b_{2}).

Let us assume (i). Putting τ = α(k + 1), we first consider the case 0 < τ ≤ 1. For x ∈ [a′, b′], we get

V_{n}(fg, k, x) − f(x)g(x) = g(x)V_{n} ((f(t) − f(x)) , k, x) +

+ ∑_{j=0}^{k} C (j, k) ∫_{a_{1}}^{b_{1}} W_{d_{j},n}(t, x)f(x) (g(t) − g(x)) dt + O (n^{−k+1)})

= I_{1} + I_{2} + O (n^{−(k+1)}) , (3.12)

where the O-term holds uniformly for x ∈ [a′, b′]. Now by assumption

∥V_{n}(f, k, .) − f∥_{C[a_{1},b_{1}]} = O (n^{−τ/2}) ,

we have

∥I_{1}∥_{C[a′,b′]} ≤ ∥g∥_{C[a′,b′]}∥V_{n}(f, k, .) − f∥_{C[a′,b′]} ≤ K_{5}n^{−τ/2}.  (3.13)

By the Mean Value Theorem, we get

I_{2} = ∑_{j=0}^{k} C(j, k) ∫_{a_{1}}^{b_{1}} W_{d_{j},n}(t, x)f(t) {g′(ξ)(t − x)} dt.

Once again applying Cauchy-Schwarz inequality and Lemma 2 from [1], we get

∥I_{2}∥_{C[a′,b′]} ≤ ∥f∥_{C[a_{1},b_{1}]}∥g′∥_{C[a′,b′]}( ∑_{j=0}^{k} |C(j, k)|)×

× max_{0≤j≤k} ∥ ∫_{0}^{∞} W_{d_{j},n}(t, x)(t − x)^{2}dt∥_{C[a′,b′]}^{1/2} = O (n^{−τ/2}) . (3.14)

Combining (3.12–3.14), we obtain

∥V_{n}(fg, k, .) − fg∥_{C[a′,b′]} = O (n^{−τ/2}) , for 0 < τ ≤ 1.

Now to prove the implication for 0 < τ < 2k + 2, it is sufficient to assume it for τ ∈ (m − 1, m) and prove if for τ ∈ (m, m + 1), (m = 1, 2, 3, …, 2k + 1). Since the result holds for τ ∈ (m − 1, m), we choose two points x_{1}, y_{1} in such a way that a_{1} < x_{1} < a′ < b′ < y_{1} < b_{1}. Then in view of assumption (i) ⇒ (ii) for the interval (m − 1, m) and equivalence of (ii) it follows that f^{(m−1)} exists and belongs to the class Lip(1 − δ, x_{1}, y_{1}) for any δ > 0.

Let g ∈ C_{0}^{∞} be such that g(x) = 1 on [a′′, b′′] and supp g ⊂ [a′′, b′′]. Then with χ_{2}(t) denoting the characteristic function of the interval [x_{1}, y_{1}], we have

∥V_{n}(f, g, k, .) − fg∥_{C[a′,b′]} ≤ ∥V_{n}(g(x)f(t) − f(x)), k, .)∥_{C[a′,b′]}+

+ ∥V_{n}(f(t)(g(t) − g(x))χ(t), k, .)∥_{C[a′,b′]} + O (n^{−(k+1)}) . (3.15)

Now

∥V_{n}(g(x)(f(t) − f(x)), k, .)∥_{C[a′,b′]} ≤ ∥g∥_{C[a′′,b′′]}∥V_{n}(f, k, .) − f∥_{C[a_{1},b_{1}]}
= O (n^{−τ/2}) .  (3.16)

Applying Taylor’s expansion of f, we have I_{3} ≡ ∥V_{n}(f(t)g(t) − g(x))χ(t), k, .)∥_{C[a′,b′]} =

∥V_{n}([ ∑_{i=0}^{m−1} \frac{f^{(i)}(x)}{i!}(t−x)^{i}+\frac{\{f^{(m−1)}(ξ) − f^{(m−1)}(x)\}}{(m − 1)!}](g(t)−g(x))χ(t), k, .)∥_{C[a′,b′]}

where ξ lies between t and x. Since f^{(m−1)} ∈ Lip(1 − δ, x_{1}, y_{1}),

|f^{(m−1)}(ξ) − f^{(m−1)}(x)| ≤ K_{6}|ξ − x|^{1−δ} ≤ K_{6}|t − x|^{1−δ},

where K_{6} is the Lip(1 − δ, x_{1}, y_{1}) constant for f^{(m−1)}, we have

I_{3} ≤ ∥V_{n}( ∑_{i=0}^{m−1} \frac{f^{(i)}(x)}{i!}(t − x)^{i}(g(t) − g(x))χ(t), k, .)∥_{C[a′,b′]}

+ \frac{K_{6}}{(m − 1)!}∥g′∥_{C[a′′,b′′]}( ∑_{j=0}^{k} |C(j, k)|)∥V_{d_{j},n}(|t − x|^{m+1−δ}χ(t), .)∥_{C[a′,b′]}

= I_{4} + I_{5} say.  (3.17)

By Taylor’s expansion of g and Lemma 2.6, we have

I_{4} = O (n^{−(k+1)}) .  (3.18)

Also, by Hölder’s expansion of g and Lemma 2 from [1], we have

I_{5} ≤ \frac{K_{6}}{(m − 1)!}∥g′∥_{C[a′′,b′′]}( ∑_{j=0}^{k} |C(j, k)|)×

× max_{0≤j≤k} ∥ ∫_{x_{1}}^{y_{1}} W_{d_{j},n}(t − x)|t − x|^{m+1−δ}dt∥_{C[a′,b′]}

≤ K_{7} max_{0≤j≤k} ∥ ∫_{x_{1}}^{y_{1}} W_{d_{j},n}(t − x)(t − x)^{2(m+1)}dt∥C[a′,b′]^{\frac{(m+1−δ)}{2(m+1)}}

= O (n^{−(m+1−δ)/2}) = O (n^{τ/2}) ,  (3.19)

by choosing δ such that 0 < δ < m + 1 − δ. Combining the estimates (3.15–3.19), we get

∥V_{n}(fg, k, .) − fg∥_{C[a′,b′]} = O (n^{τ/2}) .

This completes the proof of the Theorem 3.2.

Acknowledgment. The author is extremely thankful to the referee for his valuable comments and suggestions, leading to a better presentation of the paper. The author is also thankful to Professor Mila Mršević for special attention.

REFERENCES

[1] Agrawal, P. N. and Thamer, K. J., Approximation of unbounded functions by a new sequence of linear positive operators, J. Math. Anal. Appl., 225 (1998), 660–672.

[2] Agrawal, P. N. and Thamer, K. J., Degree of approximation by a new sequence of linear operators, Kyungpook Math. J., 41 (2001), 65–73.

[3] Berens, H. and Lorentz, G. G., Inverse theorems for Bernstein polynomials, Indiana Univ. Math. J., 21 (1972), 693–708.

[4] Derriennic, M. M., Sur l’approximation de functions integrable sur [0, 1] par des polynomes de Bernstein modifies, J. Approx. Theory, 31 (1981), 325–343.

[5] Goldberg, S. and Meir, V., Minimum moduli of differentiable operators, Proc. London Math. Soc., 23 (1971), 1–15.

[6] Gupta, V., A note on modified Szâsz operators, Bull Inst. Math. Academia Sinica, 21 (3) (1993), 275–278.

[7] Gupta, V. and Srivastava, G. S., Simultaneous approximation by Baskakov-Szâsz type operators, Bull. Math. Dela Soc. Sci Math de Roumanie (N. S.), 37 (85) (1993), 73–85.

[8] Gupta, V. and Srivastava, G. S., On the convergence of derivative by Szâsz-Mirakyan Baskakov type operators, The Math. Student, 64 (1–4) (1995), 195–205.

[9] Kasana, H. S., Prasad, G., Agrawal, P. N. and Sahai, A., Modified Szâsz operators, Conference on Mathematica Analysis and its Applications, Kuwait, Pergamon Press, Oxford, 1985, 29– 41.

[10] Mazhar, S. M. and Totil, V., Approximation by modified Szâsz operators, Acta Sci. Math. (Szeged), 49 (1985), 257–269.

[11] Sahai, A. and Prasad, G., On simultaneous approximation by modified Lupas operators, J. Approx. Theory, 45 (1985), 122–128.

[12] Timan, A. F., Theory of Approximation of Functions of Real Variable (English translation), Pergamon Press Long Island City, N. Y., 1963.

(received 07.04.2004, in revised form 29.01.2006)

Department of Applied Mathematics, Delhi College of Engineering, Bawana Road, Delhi-110042, India. Presently at: School of Information Science and Engineering, Graduate School of the Chinese Academy of Sciences, No.3 Nanyitiao, Zhongguancun Haidian District, Beijing-100080, P. R. China.

E-mail: dr naokant deo@yahoo.com