Return to Article Details Approximation operators constructed by means of Sheffer sequences

REVUE D'ANALYSE NUMÉRIQUE ET DE THÉORIE DE L'APPROXIMATION

Rev. Anal. Numér. Théor. Approx., vol. 30 (2001) no. 2, pp. 135-150
ictp.acad.ro/jnaat

APPROXIMATION OPERATORS
CONSTRUCTED BY MEANS OF SHEFFER SEQUENCES

MARIA CRǍCIUN

Abstract

In this paper we introduce a class of positive linear operators by using the "umbral calculus", and we study some approximation properties of it. Let $Q$ be a delta operator, and $S$ an invertible shift invariant operator. For $f \in C [0, 1]$ we define $(L_{n}^{Q, S} f) (x) = \frac{1}{s_{n} (1)} \sum_{k = 0}^{n} (\binom{n}{k}) p_{k} (x) s_{n - k} (1 - x) f (\frac{k}{n}),$ where ${(p_{n})}_{n \geq 0}$ is a binomial sequence which is the basic sequence for $Q$ , and ${(s_{n})}_{n \geq 0}$ is a Sheffer set, $s_{n} = S^{- 1} p_{n}$ . These operators generalize the binomial operators of T. Popoviciu.

MSC 2000. 41A36, 05A40.

1. INTRODUCTION

Let

P

be the linear space of all polynomials with real coefficients, and

P_{n}

the linear space of all polynomials of degree at most

n

We will consider some linear operators defined on

P

. We will denote by

I

the identity and by

D

the derivative. The shift operator

E^{a} : P \to P

is defined by

E^{a} p (x) = p (x + a)

A linear operator

T

which commutes with all shift operators is called a shift invariant operator. In symbols,

E^{a} T = T E^{a}

, for all real

a

Let us remind that if

T_{1}

and

T_{2}

are shift invariant operators, then

T_{1} T_{2} = T_{2} T_{1}

Definition 1. A shift invariant operator for which

Q x =

const

\neq 0

is called a delta operator.

By a polynomial sequence we shall denote a sequence of polynomials

p_{n} (x)

n = 0, 1, 2, \dots

where

p_{n} (x)

is of degree exactly

n

for all

n

A sequence of binomial type is a polynomial sequence

{(p_{n})}_{n \geq 0}

with

p_{0} (x) = 1

and satisfying the identities

p_{n} (x + y) = \sum_{k = 0}^{n} (\binom{n}{k}) p_{k} (x) p_{n - k} (y),

for all

x, y

and

n = 0, 1, 2, \dots

.
Definition 2. Let

Q

be a delta operator and

{(p_{n} (x))}_{n \geq 0}

a polynomial sequence. If
i)

p_{0} (x) = 1

,
ii)

p_{n} (0) = 0, n = 1, 2, \dots

,
iii)

Q p_{n} = n p_{n - 1}, n = 1, 2, \dots

,
then

(p_{n})

is called the sequence of basic polynomials for

Q

.
Proposition 1. 8].
i) Every delta operator has a unique sequence of basic polynomials.
ii) If

p_{n} (x)

is a basic sequence for some delta operator

Q

, then it is binomial.
iii) If

p_{n} (x)

is a binomial sequence, then it is a basic sequence for some delta operator

Q

Let

X

be the multiplication operator defined as

(X p) (x) = x p (x)

for every polynomial

p

For any operator

T

defined on

P

, the operator

T^{'} = T X - X T

is called the Pincherle derivative of the operator

T

Proposition 2. 8].
i) If

T

is a shift invariant operator, then its Pincherle derivative is also a shift invariant operator.
ii) If

Q

is a delta operator, then its Pincherle derivative

Q^{'}

is an invertible operator.

Proposition 3. [8], [11]. If

{(p_{n} (x))}_{n \geq 0}

is a sequence of basic polynomials for the delta operator

Q

then
i)

p_{n} (x) = X {(Q^{'})}^{- 1} p_{n - 1} (x), n = 1, 2, \dots

,
ii)

p_{n} (x) = x \sum_{k = 0}^{n - 1} (\binom{n - 1}{k}) p_{n - 1 - k} (x) p_{k + 1}^{'} (0), n = 1, 2, \dots

Definition 3. A polynomial sequence

{(s_{n} (x))}_{n \geq 0}

is called a Sheffer set relative to the delta operator

Q

if:
i)

s_{0} (x) =

const

\neq 0

ii)

Q s_{n} = n s_{n - 1}, n = 1, 2, \dots

An Appel set is a Sheffer set relative to the derivative

D

Proposition 4. 11. Let

Q

be a delta operator with basic polynomial set

{(p_{n} (x))}_{n \geq 0}

and

{(s_{n} (x))}_{n \geq 0}

a polynomial sequence. The next statements are equivalent:
i)

s_{n} (x)

is a Sheffer set relative to

Q

.
ii) There exists an invertible shift invariant operator

S

such that

s_{n} (x) = S^{- 1} p_{n} (x)

.
iii) For all

x, y \in R

and

n = 0, 1, 2, \dots

, the following identity holds:

s_{n} (x + y) = \sum_{k = 0}^{n} (\binom{n}{k}) p_{k} (x) s_{n - k} (y)

From the previous Proposition it results that the pair (

Q, S

) gives us a unique Sheffer set.

2. THE OPERATORS CONSTRUCTED BY MEANS OF SHEFFER POLYNOMIALS AND THEIR CONVERGENCE

In 1931 in [9] Tiberiu Popoviciu has used binomial sequences in order to construct some operators of the form

\begin{matrix} (1) & (L_{n} f) (x) = \frac{1}{p_{n} (1)} \sum_{k = 0}^{n} (\binom{n}{k}) p_{k} (x) p_{n - k} (1 - x) f (\frac{k}{n}) \end{matrix}

where

f \in C [0, 1]

and

x \in [0, 1]

. These operators are called binomial operators.
Such operators and their generalizations have been studied by the Romanian mathematicians as: D. D. Stancu, A. Lupaş, L. Lupaş, G. Moldovan, C. Manole, O. Agratini, A. Vernescu, and others.

Let

Q

be a delta operator and

S

an invertible shift invariant operator. Let

{(p_{n} (x))}_{n \geq 0}

be the sequence of basic polynomials for

Q

, and

{(s_{n} (x))}_{n \geq 0}

a Sheffer set relative to

Q, s_{n} = S^{- 1} p_{n}

with

s_{n} (1) \neq 0

for any positive integer

n

In this note we want to study the operators

L_{n}^{Q, S} : C [0, 1] \to C [0, 1]

\begin{matrix} (2) & (L_{n}^{Q, S} f) (x) = \frac{1}{s_{n} (1)} \sum_{k = 0}^{n} (\binom{n}{k}) p_{k} (x) s_{n - k} (1 - x) f (\frac{k}{n}) \end{matrix}

Because

p_{k} (0) = δ_{k, 0}

(from the definition of basic polynomials), we have

(L_{n}^{Q, S} f) (0) = f (0)

In order to evaluate expression

(L_{n}^{Q, S} e_{m}) (x)

, where

e_{m} (x) = x^{m}

we shall make use of C. Manole's method for binomial operators (see [5]) which we have adapted to our purposes.

Let us introduce the polynomials

\begin{matrix} (3) & S_{m} (x, y, n) = \sum_{k = 0}^{n} (\binom{n}{k}) p_{k} (x) s_{n - k} (y) {(\frac{k}{n})}^{m} \end{matrix}

From Proposition 4 iii) we have

S_{0} (x, y, n) = s_{n} (x + y)

.
In the following we consider that

x

is the variable. Let us denote

θ = X {(Q^{'})}^{- 1}

From Proposition 3 i) it results that

θ p_{k} (x) = p_{k + 1} (x)

and consequently the linear operator

θ

is called the shift operator for the sequence

{(p_{n})}_{n \geq 0}

(see [10]). Therefore

θ Q p_{k} (x) = θ (k p_{k - 1} (x)) = k p_{k} (x)

; consequently

k

is an eigenvalue for the operator

θ Q

, with its eigenvector

p_{k} (x)

. We have

\begin{matrix} (4) & (θ Q)^{m} = k^{m} p_{k} (x) \end{matrix}

for every positive integer

m

, and then

\begin{aligned} S_{m} (x, y, n) & = \frac{1}{n^{m}} (θ Q)^{m} \sum_{k = 0}^{n} (\binom{n}{k}) p_{k} (x) s_{n - k} (y) \\ = \frac{1}{n^{m}} (θ Q)^{m} S_{0} (x, y, n) = \frac{1}{n^{m}} (θ Q)^{m} s_{n} (x + y) \end{aligned}

In this way we obtain

\begin{matrix} (5) & S_{m} (x, y, n) = \frac{1}{n^{m}} (θ Q)^{m} E^{y} s_{n} (x) \end{matrix}

Using the operational formula (see for instance [10])

(θ Q)^{m} = \sum_{k = 0}^{n} S (m, k) θ^{k} Q^{k}

where

S (m, k) = [0, 1, \dots, k; e_{m}]

are the Stirling numbers of the second kind, relation (5) becomes:

\begin{matrix} (6) & S_{m} (x, y, n) = \frac{1}{n^{m}} \sum_{k = 0}^{n} S (m, k) θ^{k} Q^{k} E^{y} s_{n} (x) \end{matrix}

Because

Q

is shift invariant and

Q^{k} s_{n} (x) = n (n - 1) \dots (n - k + 1) s_{n - k} (x) = n^{[k]} s_{n - k} (x)

we obtain

\begin{matrix} (7) & S_{m} (x, y, n) = \frac{1}{n^{m}} \sum_{k = 0}^{n} (\binom{n}{k}) k! S (m, k) θ^{k} E^{y} s_{n - k} (x), \forall m \in N^{*} \end{matrix}

Theorem 1. If

L_{n}^{Q, S}

is the linear operator defined by (2) then

\begin{aligned} (L_{n}^{Q, S} e_{0}) (x) = e_{0} (x) \\ (8) & (L_{n}^{Q, S} e_{1}) (x) = a_{n} e_{1} (x) \\ (L_{n}^{Q, S} e_{2}) (x) = b_{n} x^{2} + x (a_{n} - b_{n} - c_{n}), \end{aligned}

where

\begin{aligned} a_{n} = \frac{[{(Q^{'})}^{- 1} s_{n - 1}] (1)}{s_{n} (1)} \\ (9) & b_{n} = \frac{n - 1}{n} \frac{[{(Q^{'})}^{- 2} s_{n - 2}] (1)}{s_{n} (1)} \\ c_{n} = \frac{n - 1}{n} \frac{[{(Q^{'})}^{- 2} {(S^{- 1})}^{'} S s_{n - 2}] (1)}{s_{n} (1)} \end{aligned}

Proof. Using the notation (3) we can write

\begin{matrix} (10) & (L_{n}^{Q, S} e_{m}) (x) = S_{m} (x, 1 - x, n) / s_{n} (1) \end{matrix}

Because

S_{0} (x, 1 - x, n) = s_{n} (1)

we have

(L_{n}^{Q, S} e_{0}) (x) = e_{0} (x)

.
As we have

\begin{matrix} (11) & θ E^{y} s_{n - 1} (x) = X {(Q^{'})}^{- 1} E^{y} s_{n - 1} (x) = X E^{y} {(Q^{'})}^{- 1} s_{n - 1} (x) \end{matrix}

we obtain from (7):

S_{1} (x, 1 - x, n) = x [{(Q^{'})}^{- 1} s_{n - 1}] (1)

; consequently we get:

(L_{n}^{Q, S} e_{1}) (x) = \frac{[{(Q^{'})}^{- 1} s_{n - 1}] (1)}{s_{n} (1)} x .

Using the Pincherle derivative of the shift operator

E^{y}

\begin{matrix} (12) & {(E^{y})}^{'} = y E^{y} = E^{y} X - X E^{y} \end{matrix}

we can write

θ E^{y} s_{n - k} (x) = E^{y} X {(Q^{'})}^{- 1} s_{n - k} (x) - y E^{y} {(Q^{'})}^{- 1} s_{n - k} (x)

Then

\begin{matrix} (13) & θ^{2} E^{y} s_{n - k} (x) = X E^{y} {(Q^{'})}^{- 1} E^{y} X {(Q^{'})}^{- 1} s_{n - k} (x) - y X E^{y} {(Q^{'})}^{- 2} s_{n - k} (x) \end{matrix}

Because

s_{n - k} = S^{- 1} p_{n - k}, X S^{- 1} = S^{- 1} X - {(S^{- 1})}^{'}

(from the definition of Pincherle derivative) and

{(Q^{'})}^{- 1} p_{n - k} (x) = p_{n - k + 1} (x) / x

(from Proposition 3 i), we obtain

\begin{aligned} (14) & θ^{2} E^{y} s_{n - k} (x) = & X E^{y} {(Q^{'})}^{- 1} s_{n - k + 1} (x) - X E^{y} {(Q^{'})}^{- 2} {(S^{- 1})}^{'} S s_{n - k} (x) - \\ - y X E^{y} {(Q^{'})}^{- 2} s_{n - k} (x) \end{aligned}

Replacing (11) and (14) in (7) we can write

\begin{aligned} S_{2} (x, y, n) = & x E^{y} {(Q^{'})}^{- 1} s_{n - 1} (x) - \frac{n - 1}{n} [x E^{y} {(Q^{'})}^{- 2} {(S^{- 1})}^{'} S s_{n - 2} (x) - \\ - y x E^{y} {(Q^{'})}^{- 2} s_{n - 2} (x)] \end{aligned}

From (10) and the previous relation one obtains expression

L_{n}^{Q, S} e_{2}

from theorem's conclusion.

Lemma 1. Let

Q

be a delta operator and

S

an invertible shift invariant operator. Let

{(p_{n} (x))}_{n \geq 0}

be the sequence of basic polynomials for

Q

and

{(s_{n} (x))}_{n \geq 0}

a Sheffer set relative to

Q, s_{n} = S^{- 1} p_{n}

with

s_{n} (1) \neq 0

for any positive integer

n

. If

p_{k}^{'} (0) \geq 0

and

s_{k} (0) \geq 0

for

n = 0, 1, 2, \dots

then the operator

L_{n}^{Q, S}

defined by (2) is positive.

Proof. If

p_{k}^{'} (0) \geq 0

using Proposition 3 ii), it is easy to prove by induction that

p_{k} (x) \geq 0, \forall k \in N

and

\forall x \in [0, 1]

If we consider

x = 0

in Proposition 4 iii) we obtain

s_{n} (x) = \sum_{k = 0}^{n} (\binom{n}{k}) p_{k} (x) s_{n - k} (0);

accordingly, for

s_{k} (0) \geq 0

and

p_{k} (x) \geq 0, \forall k \in N, \forall x \in [0, 1]

, we have

s_{k} (x) \geq 0, \forall k \in N

and

\forall x \in [0, 1]

. Therefore the operator

L_{n}^{Q, S}

is positive.

Lemma 2. If the operator

L_{n}^{Q, S}

is positive, then

a_{n} \in [0, 1], b_{n} \leq 1

and

0 \leq c_{n} \leq min {\frac{1 - b_{n}}{2}, a_{n} - a_{n}^{2}}, \forall n \in N

, where

a_{n}, b_{n}

and

c_{n}

are defined by (9).

Proof. Since

0 \leq e_{1} (t) \leq 1, \forall t \in [0, 1]

and the operator

L_{n}^{Q, S}

is positive, we have

0 \leq (L_{n}^{Q, S} e_{1}) (x) \leq 1, \forall x \in [0, 1]

, and as

(L_{n}^{Q, S} e_{1}) (x) = a_{n} x

, we get

a_{n} \in [0, 1]

From

t (1 - t) \geq 0

it results that

(L_{n}^{Q, S} e_{1}) (x) - (L_{n}^{Q, S} e_{2}) (x) \geq 0

, which leads to

x (1 - x) b_{n} + x c_{n} \geq 0, \forall x \in [0, 1]

and choosing

x = 1

, we get

c_{n} \geq 0

Since

t^{2} - t + 1 / 4 \geq 0

, we obtain

(L_{n}^{Q, S} e_{2}) (x) - (L_{n}^{Q, S} e_{1}) (x) + (L_{n}^{Q, S} e_{0}) (x) / 4 \geq 0, \forall x \in [0, 1]

, relation equivalent to

x^{2} b_{n} - x b_{n} - x c_{n} + 1 / 4 \geq 0, \forall x \in [0, 1]

. If we consider

x = 1 / 2

, it results that

c_{n} \leq (1 - b_{n}) / 2

and because

c_{n} \geq 0

we get

b_{n} \leq 1

Finally, from the Schwarz's inequality,

{[(L_{n}^{Q, S} e_{1}) (x)]}^{2} \leq (L_{n}^{Q, S} e_{2}) (x) (L_{n}^{Q, S} e_{0}) (x),

we have

a_{n}^{2} x^{2} \leq b_{n} x^{2} + x (a_{n} - b_{n} - c_{n}), \forall x \in [0, 1]

. For

x = 1

that implies

c_{n} \leq a_{n} - a_{n}^{2}

Theorem 2. Let

Q

be a delta operator and

S

an invertible shift invariant operator. Let

{(p_{n} (x))}_{n \geq 0}

be the sequence of basic polynomials for

Q

, with

p_{n}^{'} (0) \geq 0, \forall n \in N

, and

{(s_{n} (x))}_{n \geq 0}

a Sheffer set relative to

Q, s_{n} = S^{- 1} p_{n}

with

s_{n} (1) \neq 0

and

s_{n} (0) \geq 0, \forall n \in N

. If

f \in C [0, 1]

and

lim_{n \to \infty} a_{n} = lim_{n \to \infty} b_{n} = 1

, where

a_{n}

and

b_{n}

are defined by (9), then the operator

L_{n}^{Q, S}

converges to the function

f

, uniformly on the interval

[0, 1]

Proof. If

lim_{n \to \infty} a_{n} = 1

then

lim_{n \to \infty} (L_{n}^{Q, S} e_{1}) (x) = e_{1} (x)

. From Lemma

2, c_{n} \leq a_{n} - a_{n}^{2}

so we have

lim_{n \to \infty} c_{n} = 0

, and as

lim_{n \to \infty} b_{n} = 1

, we get

lim_{n \to \infty} (L_{n}^{Q, S} e_{2}) (x) = e_{2} (x)

. Therefore

lim_{n \to \infty} (L_{n}^{Q, S} e_{i}) (x) = e_{i} (x)

for

i = 0, 1, 2

so we can use the convergence criterion of Bohman-Korokvin.

3. REPRESENTATIONS OF THE OPERATOR $L_{n}^{Q, S}$

Theorem 3. The operator

L_{n}^{Q, S}

can be represented in the form

\begin{matrix} (15) & (L_{n}^{Q, S} f) (x) = \sum_{k = 0}^{n} \frac{k!}{n^{k}} (\binom{n}{k}) [0, \frac{1}{n}, \dots, \frac{k}{n}; f] d_{k, n} (x) \end{matrix}

where

d_{k, n} (x) = \frac{1}{s_{n} (1)} (θ^{k} E^{1 - x} s_{n - k}) (x)

Moreover

L_{n}^{Q, S} (P_{m}) \subseteq P_{m}, \forall m \in N

.
Proof. From the Newton interpolation formula we have

f (\frac{k}{n}) = \sum_{j = 0}^{k} \frac{j!}{n^{j}} (\binom{k}{j}) [0, \frac{1}{n}, \dots, \frac{j}{n}; f]

If we denote

w_{k, n} (x, y) = (\binom{n}{k}) p_{k} (x) s_{n - k} (y)

then

\sum_{k = 0}^{n} w_{k, n} (x, y) f (\frac{k}{n}) = \sum_{k = 0}^{n} \frac{k!}{n^{k}} [0, \frac{1}{n}, \dots, \frac{k}{n}; f] \sum_{j = k}^{n} (\binom{j}{k}) w_{j, n} (x, y)

But

\begin{aligned} \sum_{j = k}^{n} (\binom{j}{k}) w_{j, n} (x, y) & = (\binom{n}{k}) \sum_{j = k}^{n} (\binom{n - k}{j - k}) p_{j} (x) s_{n - j} (y) \\ = (\binom{n}{k}) \sum_{j = 0}^{n - k} (\binom{n - k}{j}) p_{j + k} (x) s_{n - k - j} (y) \\ = (\binom{n}{k}) θ^{k} \sum_{j = 0}^{n - k} (\binom{n - k}{j}) p_{j} (x) s_{n - k - j} (y) \\ = (\binom{n}{k}) θ^{k} E^{y} s_{n - k} (x) \end{aligned}

Since

(L_{n}^{Q, S} f) (x) = \frac{1}{s_{n} (1)} \sum_{k = 0}^{n} w_{k, n} (x, 1 - x) f (\frac{k}{n})

we obtain (15).
In order to show that

L_{n}^{Q, S} (P_{m}) \subseteq P_{m}

we shall prove that

\deg (d_{k, n} (x)) = k

.
We remind that if (

p_{n}

) is a basic sequence for

Q = q (D)

and

h (t)

is the compositional inverse of

q (t)

, then the generating function for

(p_{n})

\begin{matrix} (16) & \sum_{k = 0}^{\infty} p_{k} (x) \frac{t^{k}}{k!} = e^{x h (t)} \end{matrix}

and if

s_{n} = S^{- 1} p_{n}

, with

S = s (D)

then

\begin{matrix} (17) & \sum_{k = 0}^{\infty} s_{k} (x) \frac{t^{k}}{k!} = \frac{1}{s (h (t))} e^{x h (t)} \end{matrix}

If we differentiate the relation (16)

m

times with respect to

t

, we get
(18)

\sum_{k = 0}^{\infty} p_{k + m} (x) \frac{t^{k}}{k!} = \frac{d^{m}}{d x^{m}} (e^{x h (t)}) = (x h_{1} (t) + x^{2} h_{1} (t) + \dots + x^{m} h_{m} (t)) e^{x h (t)}

,
where every

h_{i} (t)

is a product of derivatives of

h (t)

.
Let us denote

r (k, m, x) = \sum_{j = 0}^{k} (\binom{k}{j}) p_{j + m} (x) s_{k - j} (1 - x)

. Expanding

\frac{1}{s (h (t))} h_{i} (t) e^{h (t)} = \sum_{k \geq 0} α_{i k} \frac{t^{k}}{k!}

, from (17) and (18) we get

r (k, m, x) = x α_{1 k} + x^{2} α_{2 k} + \dots + x^{m} α_{m k}

Because

d_{k, n} (x) = r (n - k, k, x) / s_{n} (1)

we obtain

\deg (d_{k, n} (x)) = k

.
Suppose that

p \in P_{m}

. Then

[0, \frac{1}{n}, \dots, \frac{k}{n}; p] = 0

for

k \geq m + 1

, and using (15) we get

L_{n}^{Q, S} (P_{m}) \subseteq P_{m}

Remark 1. For

Q = D

(it means that

s_{n}

is an Appell set

A_{n}

) we have

θ = X

, therefore in this case

d_{k, n} = \frac{A_{n - k} (1)}{A_{n} (1)} x^{k}

. This representation for operators constructed with Appell sequences was given by C. Manole in [5].

Theorem 4. Suppose that all the assumptions of Theorem 2 are true, then there exists

θ_{1 n}, θ_{2 n}, θ_{3 n} \in [0, 1]

such that

\forall x \in [0, 1]

and

\forall f \in C [0, 1]

we have

(L_{n}^{Q, S} f) (x) = f (a_{n} x) + α (x, n) [θ_{1 n}, θ_{2 n}, θ_{3 n}; f]

where

α (x, n) = x^{2} (b_{n} - a_{n}^{2}) + x (a_{n} - b_{n} - c_{n})

.
Proof. First we shall prove that

f (a_{n} x) \leq (L_{n}^{Q, S} f) (x)

for every convex function

f

Let us denote

c_{k} = \frac{1}{s_{n} (1)} (\binom{n}{k}) p_{k} (x) s_{n - k} (1 - x)

and

x_{k} = \frac{k}{n}, k = 0, 1, \dots, n

.
We have

c_{k} \geq 0, \sum_{k = 0}^{n} c_{k} = 1

and

x_{k} > 0, \forall k \in N

. If

f

is a convex function then

f (\sum_{k = 0}^{n} c_{k} x_{k}) \leq \sum_{k = 0}^{n} c_{k} f (x_{k})

; but

\sum_{k = 0}^{n} c_{k} x_{k} = (L_{n}^{Q, S} e_{1}) (x) = a_{n} x and \sum_{k = 0}^{n} c_{k} f (x_{k}) = (L_{n}^{Q, S} f) (x)

therefore we get

f (a_{n} x) \leq (L_{n}^{Q, S} f) (x)

.
If we consider the formula

f (a_{n} x) = (L_{n}^{Q, S} f) (x) + (R_{n} f) (x)

we have

(R_{n} f) \leq 0

for every convex function

f

.
Since

(R_{n} e_{i}) (x) = 0

for

i = 0, 1

, the degree of exactness of the previous formula is one and then there exist

θ_{1 n}, θ_{2 n}, θ_{3 n} \in [0, 1]

such that the remainder can be represented in the following form

(R_{n} f) (x) = (R_{n} e_{2}) (x) [θ_{1 n}, θ_{2 n}, θ_{3 n}; f]

where

(R_{n} e_{2}) (x) = x^{2} (a_{n}^{2} - b_{n}) + x (b_{n} + c_{n} - a_{n})

, so we obtain the conclusion.

4. EXAMPLES

If $S = I$ then $s_{n} = p_{n}$ and in this case the operator defined by (2) becomes the binomial operator (1) introduced by Tiberiu Popoviciu in 9 .
1.1. For $Q = D$ the basic sequence is $p_{n} (x) = x^{n}$ and $L_{n}^{D, I}$ is the Bernstein operator $B_{n}$ .
1.2. If $Q$ is Abel operator $A = E^{- β} D$ we have $p_{n} (x) = x (x + n β)^{n - 1}$ and $L_{n}^{A, I}$ is the second operator introduced by Cheney and Sharma in [1,

\begin{aligned} (L_{n}^{A, I} f) (x) = \\ = \frac{1}{(1 + n β)^{n - 1}} \sum_{k = 0}^{n} (\binom{n}{k}) x (x + k β)^{k - 1} (1 - x) (1 - x + (n - k) β)^{n - k - 1} f (\frac{k}{n}) \end{aligned}

1.3. For Laguerre delta operator

L = \frac{D}{D + I}

the basic sequence is

l_{n} (x) = \sum_{k = 0}^{n} (\binom{n}{k}) \frac{(n - 1)!}{(k - 1)!} x^{k}

and the coresponding binomial operator has been considered by T. Popoviciu.
1.4. The delta operator

Q = \frac{1}{α} \nabla_{α} = \frac{1}{α} (I - E^{- α})

has the basic sequence

p_{n} (x) = x^{[n, - α]} = x (x + α) \dots (x + (n - 1) α)

and in this case we obtain the operator

(S_{n} f) (x) = \frac{1}{1^{[n, - α]}} \sum_{k = 0}^{n} (\binom{n}{k}) x^{[k, - α]} (1 - x)^{[n - k, - α]} f (\frac{k}{n})

which has been introduced and investigated in detail by D. D. Stancu in [14], [16] and other papers.
1.5. The exponential polynomials

t_{n} (x) = \sum_{k = 0}^{n} S (n, k) x^{k} = e^{- x} \sum_{k = 0}^{\infty} \frac{k^{n} x^{k}}{k!}

, where

S (n, k)

denote the Stirling numbers of the second kind, are basic polynomials for the delta operator

T = \ln (I + D)

. The approximation operator construct by means of the exponential polynomials

(L_{n}^{T} f) (x) = \frac{1}{t_{n} (1)} \sum_{k = 0}^{n} (\binom{n}{k}) t_{k} (x) t_{n - k} (1 - x) f (\frac{k}{n})

was studied by C. Manole in [5].
1.6. If we take the delta operator

Q = G = \frac{1}{α} E^{- β} \nabla_{α} = \frac{1}{α} (E^{- β} - E^{- α - β}

) its basic sequence is

p_{n} (x) = x (x + α + n β)^{[n - 1, - α]}

and the operator

\begin{aligned} (L_{n}^{G} f) (x) = \frac{1}{(1 + n β)^{[n, - α]}} \\ \cdot \sum_{k = 0}^{n} (\binom{n}{k}) x (x + α + k β)^{[k - 1, - α]} (1 - x) (1 - x + (n - k) β)^{[n - k, - α]} f (\frac{k}{n}) \end{aligned}

was investigated by D. D. Stancu, G. Moldovan. In [18] D. D. Stancu and M. R. Occorsio have studied this operator with the nodes

\frac{k + γ}{n + δ}, 0 \leq γ \leq δ

.
2. If

Q = D

and

S

is an invertible shift invariant operator then

p_{n} (x) = x^{n}

and

s_{n} = A_{n} = S^{- 1} x^{n}

is an Appell set. The operator of the form

(L_{n}^{D, S} f) (x) = \frac{1}{A_{n} (1)} \sum_{k = 0}^{n} (\binom{n}{k}) x^{k} A_{n - k} (1 - x) f (\frac{k}{n})

was introduced and investigated by C. Manole in [5].
2.1. If

S = (I + D)^{- 1}

the coresponding Appell set is

A_{n} (x) = x^{n} + n x^{n - 1}

and then

(L_{n}^{D, (I + D)^{- 1}} f) (x) = \frac{1}{n + 1} \sum_{k = 0}^{n} (\binom{n}{k}) x^{k} (1 - x)^{n - k} (n - k + 1 - x) f (\frac{k}{n})

If we take $Q = A = E^{- β} D$ and $S = E^{β} Q^{'} = I - β D$ then $p_{n} (x) = x (x + n β)^{n - 1}$ is the basic sequence for $Q$ and $s_{n} (x) = (x + n β)^{n}$ a Sheffer set for $Q$ we obtain the first operator introduced by Cheney and Sharma in [1]:

(L_{n}^{A, I - β D} f) (x) = \frac{1}{(1 + n β)^{n}} \sum_{k = 0}^{n} (\binom{n}{k}) x (x + k β)^{k - 1} (1 - x + (n - k) β)^{n - k} f (\frac{k}{n})

For $Q = \frac{1}{α} E^{- β} \nabla_{α} = \frac{1}{α} (E^{- β} - E^{- α - β})$ and $S = E^{α + β} Q^{'} = \frac{1}{α} ((α + β) I - β E^{α})$ we have $p_{n} (x) = x (x + α + n β)^{[n - 1, - α]}$ and $s_{n} (x) = (x + n β)^{[n, - α]}$ therefore the operator $L_{n}^{Q, S}$ in this case is

\begin{aligned} (L_{n}^{[α, β]} f) (x) = \\ = \frac{1}{(1 + n β)^{[n, - α]}} \sum_{k = 0}^{n} (\binom{n}{k}) x (x + α + k β)^{[k - 1, - α]} (1 - x + (n - k) β)^{[n - k, - α]} f (\frac{k}{n}) \end{aligned}

If we replace

x

with

s (x)

we obtain a operator which has been studied by G . Moldovan in [6]. He has found the value of this operator for the monomials

e_{i}

for

i = 1, 2

using some generalized identities of Vandermonde type.

We want to find the sequences

a_{n}, b_{n}, c_{n}

, which appears in

(L_{n}^{[α, β]} e_{i}) (x)

, using relations (9).

The Pincherle derivative of

Q

Q^{'} = - \frac{β}{α} E^{- β} + (1 + \frac{β}{α}) E^{- α - β} = E^{- α - β} (I - \frac{β}{α} Δ_{α})

{(Q^{'})}^{- 1} = E^{α + β} \sum_{k \geq 0} β^{k} {(\frac{Δ_{α}}{α})}^{k}

Since

s_{n - 1} (x) = (x + (n - 1) β)^{[n - 1, - α]} = E^{(n - 2) α + (n - 1) β} x^{[n - 1, α]}

and

x^{[n, α]} = x (x - 1) \dots (x - (n - 1) α)

is the basic sequence for the delta operator

\frac{Δ_{α}}{α}

, we have

{(\frac{Δ_{α}}{α})}^{k} s_{n - 1} (x) = E^{(n - 2) α + (n - 1) β} {(\frac{Δ_{α}}{α})}^{k} x^{[n - 1, α]} = (n - 1)^{[k]} E^{(n - 2) α + (n - 1) β}

\cdot x^{[n - 1 - k, α]}

. Because

a_{n} = \frac{[{(Q^{'})}^{- 1} s_{n - 1}] (1)}{s_{n} (1)}

we get

a_{n} = \sum_{k = 0}^{n - 1} (\binom{n - 1}{k}) \frac{k! β^{k}}{(1 + n β)^{[k + 1, - α]}}

Since

b_{n} = \frac{n - 1}{n} \frac{[{(Q^{'})}^{- 2} s_{n - 2}] (1)}{s_{n} (1)}

and

{(Q^{'})}^{- 2} = E^{2 α + 2 β} {(I - \frac{β}{α} Δ_{α})}^{- 2} = E^{2 α + 2 β} \sum_{k \geq 0} (k + 1) β^{k} {(\frac{Δ_{α}}{α})}^{k}

we obtain

b_{n} = \frac{n - 1}{n} \sum_{k = 0}^{n - 2} (\binom{n - 2}{k}) \frac{(k + 1)! β^{k}}{(1 + n β)^{[k + 2, - α]}}

The Pincherle derivative of

S^{- 1}

may be written in the form

\begin{aligned} {(S^{- 1})}^{'} & = - (α + β) E^{- α - β} {(Q^{'})}^{- 1} - E^{- α - β} {(Q^{'})}^{- 2} Q^{''} \\ = - E^{- α - β} {(Q^{'})}^{- 2} ((α + β) Q^{'} + Q^{''}) . \end{aligned}

Because

Q^{''} = \frac{1}{α} (β^{2} E^{- β} - (α + β)^{2} E^{- α - β})

and

(α + β) Q^{'} + Q^{''} = - β E^{- β}

this implies

\begin{aligned} {(Q^{'})}^{- 2} {(S^{- 1})}^{'} S & = β {(Q^{'})}^{- 3} E^{- β} \\ = β E^{3 α + 2 β} {(I - \frac{β}{α} Δ_{α})}^{- 3} \\ = E^{3 α + 2 β} \sum_{k \geq 0} \frac{(k + 1) (k + 2)}{2} β^{k + 1} {(\frac{1}{α} Δ_{α})}^{k} . \end{aligned}

From (9) and the previous relation we get

c_{n} = \frac{n - 1}{2 n} ((1 + n α + n β) \sum_{k = 0}^{n - 3} (\binom{n - 2}{k}) \frac{(k + 2)! β^{k + 1}}{(1 + n β)^{[k + 3, - α]}} + \frac{n! β^{n - 1}}{(1 + n β)^{[n, - α]}})

5. EVALUATION OF THE ORDERS OF APPROXIMATION

Now we establish some estimates of the order of approximation of a function

f \in C [0, 1]

by means of the operator

L_{n}^{Q, S}

, defined by (2).

According to a result of O. Shisha and B. Mond [13], we can write

| f (x) - (L_{n}^{Q, S} f) (x) | \leq [1 + \frac{1}{δ^{2}} L_{n}^{Q, S} ((t - x)^{2}; x)] ω_{1} (f; δ), δ \in R^{+}

Using the relations (8) we have

L_{n}^{Q, S} ((t - x)^{2}; x) = x^{2} (b_{n} - 2 a_{n} + 1) + x (a_{n} - b_{n} - c_{n})

so we get

| f (x) - (L_{n}^{Q, S} f) (x) | \leq [1 + \frac{1}{δ^{2}} [x^{2} (b_{n} - 2 a_{n} + 1) + x (a_{n} - b_{n} - c_{n})]] ω_{1} (f; δ)

.
One observes that if

b_{n} - 2 a_{n} + 1 < 0

then

x^{2} (b_{n} - 2 a_{n} + 1) + x (a_{n} - b_{n} - c_{n}) \leq \frac{{(a_{n} - b_{n} - c_{n})}^{2}}{4 (2 a_{n} - b_{n} - 1)}, \forall x \in [0, 1]

By choosing

δ = \frac{1}{\sqrt{n}}

we can state
Theorem 5. If

f \in C [0, 1]

and

\exists k \in N

such as

b_{n} - 2 a_{n} + 1 < 0, \forall n \geq k

, then we can give the following estimation of the order of approximation, by means of the first modulus of continuity

‖ f - L_{n}^{Q, S} f ‖ \leq (1 + \frac{n}{4} \frac{{(a_{n} - b_{n} - c_{n})}^{2}}{(2 a_{n} - b_{n} - 1)}) ω_{1} (f; \frac{1}{\sqrt{n}}), n \geq k

where

a_{n}, b_{n}, c_{n}

are defined by (9).
In the case of binomial operators of positive type defined by (1), since

S^{'} = I^{'} = O

we have

\begin{matrix} (19) & a_{n} = 1, c_{n} = 0, b_{n} = \frac{n - 1}{n} \frac{[{(Q^{'})}^{- 2} p_{n - 2}] (1)}{p_{n} (1)} \end{matrix}

Then

b_{n} - 2 a_{n} + 1 = b_{n} - 1 < 0, \forall n \in N

therefore the previous inequality reduces to

‖ f - L_{n} f ‖ \leq (\frac{5}{4} + \frac{n}{4} d_{n}) ω_{1} (f; \frac{1}{\sqrt{n}})

where

\begin{matrix} (20) & d_{n} = \frac{n - 1}{n} - b_{n} = \frac{n - 1}{n} (1 - \frac{[{(Q^{'})}^{- 2} p_{n - 2}] (1)}{p_{n} (1)}) \end{matrix}

We mention that this inequality was established by D. D. Stancu in [18].
In order to find an evaluation of the order of approximation using both moduli of smoothness

ω_{1}

and

ω_{2}

we can use a result of H. H. Gonska and R. K. Kovacheva included in the following

Lemma 3. [2]. If

I = [a, b]

is a compact interval of the real axis and

I_{1} = [a_{1}, b_{1}]

is a subinterval of it, and if we assume that

L : C (I) \to C (I_{1})

is a positive operator, such that

L e_{0} = e_{0}

and

0 \leq δ \leq \frac{1}{2} (b - a)

, then we have

\begin{aligned} | f (x) - L (f (t); x) | \leq & \frac{2}{δ} | L (t - x; x) | ω_{1} (f; δ) + \\ + \frac{3}{2} [1 + \frac{1}{δ} | L (t - x; x) | + \frac{1}{2 δ^{2}} L ((t - x)^{2}; x)] ω_{2} (f; δ) \end{aligned}

Using the relations (8) we obtain the inequality

\begin{aligned} | f (x) - (L_{n}^{Q, S} f) (x) | \leq \frac{2}{δ} | (a_{n} - 1) x | ω_{1} (f; δ) + \\ + \frac{3}{2} [1 + \frac{1}{δ} | (a_{n} - 1) x | + \frac{1}{2 δ^{2}} [x^{2} (b_{n} - 2 a_{n} + 1) + x (a_{n} - b_{n} - c_{n})]] ω_{2} (f; δ) \end{aligned}

b_{n} - 2 a_{n} + 1 < 0

the previous inequality implies

‖ f - L_{n}^{Q, S} f ‖ \leq \frac{2}{δ} (1 - a_{n}) ω_{1} (f; δ) + \frac{3}{2} [1 + \frac{1}{δ} (1 - a_{n}) + \frac{1}{8 δ^{2}} \frac{{(a_{n} - b_{n} - c_{n})}^{2}}{(2 a_{n} - b_{n} - 1)}] ω_{2} (f; δ) .

By choosing

δ = \frac{1}{\sqrt{n}}

we get

\begin{aligned} ‖ f - L_{n}^{Q, S} f ‖ \leq & 2 \sqrt{n} (1 - a_{n}) ω_{1} (f; \frac{1}{\sqrt{n}}) + \\ + \frac{3}{2} [1 + \sqrt{n} (1 - a_{n}) + \frac{n}{8} \frac{{(a_{n} - b_{n} - c_{n})}^{2}}{(2 a_{n} - b_{n} - 1)}] ω_{2} (f; \frac{1}{\sqrt{n}}) . \end{aligned}

If we consider the binomial operator introduced by Tiberiu Popoviciu, using (19) and the previous relation, we arrive at an inequality which has found by D. D. Stancu (see [18])

‖ f - L_{n} f ‖ \leq \frac{3}{16} (9 + n d_{n}) ω_{2} (f; \frac{1}{\sqrt{n}}),

where

d_{n}

is defined by (20).

REFERENCES

[1] Cheney, E. W. and Sharma, A., On a generalization of Bernstein polynomials, Riv. Mat. Univ. Parma, 5, pp. 77-82, 1964.
[2] Gonska, H. H. and Kovacheva, R. K., The second order modulus revisited: remarks, applications, problems, Conferenze del Seminario di Matematica Univ. Bari, 257, pp. 132, 1994.
[3] Lupaş, L. and Lupaş, A., Polynomials of binomial type and approximation operators, Studia Univ. Babes-Bolyai, Mathematica, 32, pp. 61-69, 1987.
[4] Lupaş, A., Approximation operators of binomial type, Proc. IDoMAT 98, International Series of Numerical Mathematics, ISNM 132, Birkhäuser Verlag, Basel, pp. 175-198, 1999.
[5] Manole, C., Developments in series of generalized Appell polynomials, with applications to the approximation of functions, Ph.D. Thesis, Cluj-Napoca, Romania, 1984 (in Romanian).
[6] Moldovan, G., Generalizations of the S. N. Bernstein operators, Ph.D. Thesis, ClujNapoca, Romania, 1971 (in Romanian).
[7] Moldovan, G., Discrete convolutions and positive operators I, Annales Univ. Sci. Budapest R. Eötvös, 15, pp. 31-34, 1972.
[8] Mullin, R. and Rota, G. C., On the foundations of combinatorial theory III, theory of binomial enumeration, in: B. Harris, ed., Graph Theory and Its Applications, Academic Press, New York, pp. 167-213, 1970.
[9] Popoviciu, T., Remarques sur les poynômes binomiaux, Bul. Soc. Ştiinte Cluj, 6, pp 146-148, 1931.
[10] Roman, S., Operational formulas, Linear and Multilinear Algebra, 12, pp. 1-20, 1982.
[11] Rota, G. C., Kahaner, D. and Odlyzko, A., Finite operator calculus, J. Math. Anal. Appl., 42, pp. 685-760, 1973.
[12] Sablonnière, P., Positive Bernstein-Sheffer operators, J. Approx. Theory, 83, pp. 330-341, 1995.
[13] Shisha, O. and Mond, B., The degree of convergence of linear positive operators, Proc. Nat. Acad. Sci. U.S.A., 60, pp. 1196-1200, 1968.
[14] Stancu, D. D., Approximation of functions by a new class of linear positive operators, Rev. Roum. Math. Pures Appl., 13, pp. 1173-1194, 1968.
[15] Stancu, D. D., On a generalization of the Bernstein polynomials, Studia Univ. BabeşBolyai, Cluj, 14, pp. 31-45, 1969.
[16] Stancu, D. D., Approximation properties of a class of linear positive operators, Studia Univ. Babeş-Bolyai, Cluj, 15, pp. 31-38, 1970.
[17] Stancu, D. D., Approximation of functions by means of some new classes of positive linear operators, Numerische Methoden der Approximationstheorie, Proc. Conf. Oberwolfach 1971 ISNM 16, Birkhäuser-Verlag, Basel, pp. 187-203, 1972.
[18] Stancu, D. D. and Occorsio, M.R., On approximation by binomial operators of Tiberiu Popoviciu type, Rev. Anal. Numér. Théor. Approx., 27 no.1, pp. 167-181, 1998. 줄
[19] Stancu, D. D. and Cismaşiu, C., On an approximating linear positive operator of Cheney-Sharma, Rev. Anal. Numér. Théor. Approx., 26, pp. 221-227, 1997.«
Received September 12, 2000.