RATES OF CONVERGENCE OF

(1)

RATES OF CONVERGENCE OF

CHLODOVSKY-KANTOROVICH POLYNOMIALS IN CLASSES OF LOCALLY INTEGRABLE FUNCTIONS

Paulina Pych-Taberska

Faculty of Mathematics and Computer Science Adam Mickiewicz University

Umultowska 87, 61–614 Pozna´ n, Poland e-mail: ppych@amu.edu.pl

Dedicated to Professor Micha l Kisielewicz on his 70

^th

birthday

Abstract

In this paper we establish an estimation for the rate of pointwise convergence of the Chlodovsky-Kantorovich polynomials for functions f locally integrable on the interval [0, ∞). In particular, corresponding estimation for functions f measurable and locally bounded on [0, ∞) is presented, too.

Keywords and phrases: Chlodovsky polynomial, Kantorovich polynomial, rate of convergence.

2000 Mathematics Subject Classification: 41A25.

1. Introduction

Let f be a function defined on the interval [0, ∞) and let N = {1, 2 . . .}.

The Bernstein-Chlodovsky polynomials C

_n

f of the function f are defined as C

_n

f (x) :=

n

X

k=0

f kb

_n

n

P

_n,k

x b

_n

for x ∈ [0, b

n

], n ∈ N, (1)

where P

_n,k

(t) :=

ⁿ_k

t

^k

(1−t)

^n−k

for t ∈ [0, 1] and (b

n

) is a positive increasing sequence satisfying the properties

n→∞

lim b

_n

= ∞, lim

n→∞

b

_n

n = 0.

(2)

These polynomials were first introduced by I. Chlodovsky in 1937 as a generalization of the classical Bernstein polynomials

B

_n

f (x) :=

n

X

k=0

f k n

P

_n,k

(x), 0 ≤ x ≤ 1,

of functions f defined on the interval [0, 1] (see [5] or [8], Chap. II). The well-known Chlodovsky theorem states that if

n→∞

lim sup

0≤t≤bⁿ

|f(t)| exp

−α n b

_n

= 0 for every α > 0, (3)

then lim

_n→∞

C

_n

f (x) = f (x) at every point x of continuity of f . In 1960 J. Albrycht and J. Radecki [1] proved the Voronovskaya-type theorem for operators (1). Some other approximation properties of the Chlodovsky polynomials can be found e.g. in [3, 7].

For functions f Lebesgue-integrable on the interval [0, 1] the classical Kantorovich polynomial of order n is defined as

B

^?_n

f (x) := d

dx B

n+1

F (x) ≡ (n + 1)

n

X

k=0

P

_n,k

(x) Z

_n+1^k+1

k n+1

f (t)dt, 0 ≤ x ≤ 1,

where F is an indefinite integral of f . It is well known that lim

_n→∞

B

_n^?

f (x) = f (x) at any point x of (0, 1) where f is the derivative of its indefinite integral (see e.g. [8], Chap. II).

In this paper we consider the Kantorovich-type modification of the Chlodovsky operators (1). Namely, assuming that f ∈ L

loc

[0, ∞), that is f is locally integrable on [0, ∞), and denoting

F (x) = Z

_x

0

f (t)dt for x > 0,

we define the Chlodovsky-Kantorovich polynomial of degree n − 1 as K

_n−1

f (x) := d

dx C

_n

F (x), n ∈ N.

It is easy to verify that

K

_n−1

f (x) = n b

_n

n−1

X

k=0

P

_n−1,k

x b

_n

Z

^(k+1)bn

n

kbn n

f (t)dt, 0 ≤ x ≤ b

n

,

(4)

(3)

(see [3], Section 4).

In order to formulate our first result let us consider those points x ∈ (0, ∞) at which

h→0

lim 1 h

Z

_h

0

(f (x + t) − f(x)) dt = 0 (5)

and let us introduce the pointwise characteristic w

_x

(δ; f ) := sup

0<|h|≤δ

1 h

Z

_h

0

(f (x + t) − f(x)) dt

, δ > 0.

(6)

Clearly, w

_x

(δ; f ) is a non-decreasing function of δ > 0 and lim

_δ→0+

w

_x

(δ; f )

= 0 almost everywhere on [0, ∞), that is at every point x ∈ (0, ∞) at which (5) is satisfied.

Theorem 1. Let f ∈ L

loc

[0, ∞) and let at a fixed point x ∈ (0, ∞) condition (5) be fulfilled. Then, for all integers n such that b

_n

> 2x, pn/b

_n

≥ 3, we have

|K

n−1

f (x) − f(x)|

≤ c(q) 1 + ϕ

^q/2n

(x) x

^q

! b

n

^q−1₂ ^[n/bⁿ^]

X

k=1

k

^q−3²

w

_x

x

√ k ; f

+ c(r)

x

^r

ϕ

^r/2_n

(x) b

n

r/2

|f(x)|

+ c(r) x

^r

s b

_n

x(b

n

− x) ϕ

^r/2_n

(x) b

n

^r−1₂

Z

bn

0

|f(t)|dt exp

− nx 8b

n

, where q, r are arbitrary positive integers, c(q) and c(r) are positive numbers depending only on the indicated parameter q and r, respectively, ϕ

_n

(x) = x(1 −

_b^x_n

) +

^b_nⁿ

and [n/b

_n

] denotes the greatest integer not greater than n/b

_n

. Taking into acount fundamental assumptions (2) and choosing in Theorem 1, q = 3, r = 2 we easily get

Corollary 1. If f ∈ L

loc

[0, ∞) and if

n→∞

lim Z

bn

0

|f(t)|dt exp

−α n b

n

= 0 for every α > 0,

(4)

then

n→∞

lim K

_n−1

f (x) = f (x) almost everywhere on [0, ∞).

Now, let us consider the subclass M

_loc

[0, ∞) consisting of all measurable functions f locally bounded on [0, ∞). In this case

w

_x

(δ; f ) ≤ osc(f; I

x

(δ)) ≡ sup

u,v∈Ix(δ)

|f(u) − f(v)|, where 0 ≤ δ ≤ x, I

x

(δ) := [x − δ, x + δ].

Theorem 2. Let f ∈ M

loc

[0, ∞) and let at a fixed point x ∈ (0, ∞) the one- sided limits f (x+), f (x−) exist. Then, for all integers n such that b

n

> 2x, pn/b

n

≥ 3, we have

|K

n−1

f (x) − 1

2 (f (x+) + f (x−)) |

≤ c(q) 1 + ϕ

^q/2n

(x) x

^q

! b

_n

n

^q−1₂ ^[n/bⁿ^]

X

k=1

k

^q−3²

osc

g

_x

; I

_x

x

√ k

+ c

s b

_n

x(b

_n

− x) ϕ

^1/2_n

(x)M (b

_n

; f ) exp

− nx 8b

_n

+ 2 r b

n

n s

b

n

x(b

_n

− x) |f(x+) − f(x−)|,

where M (b

n

; f ) = sup

_0≤t≤b_n

|f(t)|, ϕ

ⁿ

(x) = x

1 −

_b^x_n

+

^b_nⁿ

,

g

_x

(t) :=



 

 

f (t) − f(x+) if t > x,

0 if t = x,

f (t) − f(x−) if 0 ≤ t < x, (7)

q is an arbitrary positive integer, c(q) is a positive constant depending only

on q and c is a positive absolute constant.

(5)

The function g

_x

is continuous at x. Hence lim

_δ→0+

osc(g

_x

; I

_x

(δ)) = 0. Con- sequently, Theorem 2 yields the following

Corollary 2. If f ∈ M

loc

[0, ∞) and if at a fixed point x ∈ (0, ∞) the limits f (x+), f (x−) exist, then under the Chlodovsky assumption (3), we have

n→∞

lim K

_n−1

f (x) = 1

2 (f (x+) + f (x−)) . (8)

Remark. In particular, let us consider the class BV

_Φ

[0, ∞) of functions of bounded variation in the Young sense on the interval [0, ∞) (for the definition see e.g. [4, 10]). If f ∈ BV

Φ

[0, ∞), then M(b

n

; f ) ≤ M (M = const.). The estimation given in Theorem 2 and the relation (8) hold true at every point x ∈ (0, ∞).

2. Auxiliary results

We now present certain results which will be used in the proof of our main theorems. For this, let us introduce the notation: given any fixed x ∈ [0, b

n

] and any non-negative integer q, we will write

µ

_n,q

(x) :=

n

X

k=0

kb

_n

n − x

q

P

_n,k

x b

_n

,

|µ

n,q

|(x) :=

n

X

k=0

kb

_n

n − x

q

P

_n,k

x b

_n

.

Moreover, we will use the notation c

_j

(p), j = 1, 2, . . . , for positive constants, not necessarily the same at each occurrence, depending only on the indicated parameter p.

Lemma 1. Let n ∈ N, x ∈ [0, b

n

].

(i) µ

_n,0

(x) = 1, µ

_n,1

(x) = 0, µ

_n,2

(x) = b

_n

n x

1 − x

b

_n

. (ii) If s ∈ N, n ≥ 2, then

µ

_n,2s

(x) ≤ c

1

(s) b

n

s

x

1 − x

b

n

x(1 − x b

n

) + b

_n

n

_s−1

.

(6)

(iii) If q ∈ N, q ≥ 2, n ≥ 2, then

|µ

n,q

|(x) ≤ c

2

(q) b

n

^q₂

x

1 − x

b

_n

x

1 − x

b

_n

+ b

_n

n

^q₂₋₁

. P roof. Formulas (i) follow by easy calculation. Suppose s > 1 and put y := x/b

_n

. Then y ∈ [0, 1] and

µ

_n,2s

(x) = b

_n

n

2s n

X

k=0

(k − ny)

^2s

P

_n,k

(y).

Applying the known represetation formula for the above sum (see [6], Lemma 3.6 with c = −1) we obtain

µ

_n,2s

(x) = b

n

2s s

X

j=1

β

_j,s

(ny(1 − y))

^j

,

where β

_j,s

are real numbers independent of y and bounded uniformly in n.

Now, let us observe that for y ∈ [0,

¹_n

] or y ∈ [1 −

¹_n

, 1] one has ny(1 − y) ≤

n−1n

< 1 and

s

X

j=1

β

_j,s

(ny(1 − y))

^j

≤ ny(1 − y)

s

X

j=1

|β

j,s

|.

If y ∈

₁

n

, 1 −

_n¹

then (ny(1 − y))

⁻¹

≤

_n−1ⁿ

≤ 2 and

s

X

j=1

β

j,s

(ny(1 − y))

^j

≤ (ny(1 − y))

^s

s

X

j=1

|β

^j,s

| (ny(1 − y))

^j−s

≤ (ny(1 − y))

^s

s

X

j=1

2

^s−j

|β

j,s

|.

Consequently, for all y ∈ [0, 1] (that is for all x ∈ [0, b

n

]) we have µ

_n,2s

(x) ≤ c

1

(s) b

_n

n

2s

ny(1 − y) (1 + ny(1 − y))

^s−1

with

c

₁

(s) ≥

s

X

j=1

2

^s−j

|β

j,s

|.

(7)

Inequality (ii) follows by taking y = x/b

_n

. The same estimation holds true for |µ

n,q

|(x) with even q (q = 2s). If q is odd (q = 2s + 1), then

|µ

n,q

|(x) ≤ (µ

n,4s

(x))

¹²

(µ

_n,2

(x))

¹²

by Cauchy-Schwarz inequality, and the proof is complete.

Lemma 2. If n ∈ N, 0 < x < b

n

, then x

b

_n

1 − x

b

_n

K

_n−1

f (x)

= n b

²_n

n

X

k=0

kb

n

n − x

P

_n,k

x b

_n

Z

^kbn

n −x 0

f (x + t)dt.

(9)

P roof. By (4) and by partial summation, we find that

K

_n−1

f (x) = n b

_n

n−1

X

k=0

P

_n−1,k

x b

_n

Z

^(k+1)

n bn

kbn n

f (t)dt =

= n

b

_n

P

_n−1,n−1

x b

_n

Z

bn

0

f (t)dt

+ n b

_n

n−1

X

k=1

P

_n−1,k−1

x b

_n

− P

n−1,k

x b

_n

Z

^kbn

n

0

f (t)dt.

Putting y = x/b

_n

and observing that

y(1 − y) (P

n−1,k−1

(y) − P

n−1,k

(y)) = k n − y

P

_n,k

(y)

for k = 1, 2, . . . , n − 2 and

y(1 − y)n P

n−1,n−1

(y) = yP

_n,n−1

(y) = n(1 − y)P

n,n

(y), we easily get

x b

_n

1 − x

b

_n

K

_n−1

f (x) = n b

_n

n

X

k=0

k n − x

b

_n

P

_n,k

x b

_n

Z

^kbn

n

0

f (t)dt.

(8)

Now, it is enough to recall that

n

X

k=0

kb

n

n − x

P

_n,k

x b

_n

= µ

_n,1

(x) = 0

(Lemma 1 (i)). Consequently, x

b

_n

(1 − x

b

_n

)K

_n−1

f (x) = n b

²_n

n

X

k=0

kb

n

n − x

P

_n,k

x b

_n

Z

^kbn_n

x

f (t)dt

and the proof is complete.

Note that a corresponding representation like in the formula (9) for the classical Kantorovich polynomials is given in [2].

Lemma 3. If 0 < δ ≤ x < b

n

then X

|^kbnn −x|≥δ

P

_n,k

x b

n

≤ 2 exp

− nδ

²

4xb

n

for all n ∈ N such that b

ⁿ

≥

_3x−δ^3x²

.

The proof of Lemma 3 runs as in [1] and is based on the known Chlodovsky inequality ([8], Theorem 1.5.3)):

X

|k−nt|≥2z

√

nt(1−t)

P

_n,k

(t) ≤ 2 exp −z

²

,

provided that 0 ≤ t ≤ 1, 0 ≤ z ≤

³₂

pnt(1 − t).

Lemma 4. Let 0 < x < b

_n

and let n ≥ 2.

(i) If 0 ≤ k ≤ n − 1, then

P

_n−1,k

( x b

n

) ≤ 1

√ e r b

n

s b

_n

x(b

n

− x) .

(9)

(ii)

X

nx bn<k≤n

P

_n−1,k

x b

_n

− 1 2

≤ 0.82 √ 2 r b

_n

n s

b

_n

x(b

_n

− x) .

P roof. Estimation (i) follows from the result by X.M. Zeng [11] (Theo- rem 1): if 0 ≤ k ≤ n and y ∈ (0, 1), then

P

_n,k

(y) ≤ 1

√ 2e 1 pny(1 − y) .

Inequality (ii) is an immediate consequence of the Berry-Ess´een Theorem:

X

k n>y

P

_n,k

(y) − 1 2

< 0.82

pny(1 − y) , 0 < y < 1

(see e.g., [12], Lemma 2).

3. Proofs of theorems

Proof of Theorem 1. In view of Lemma 1 (i) one can write x

b

_n

1 − x

b

_n

f (x) = n

b

²_n

µ

_n,2

(x)f (x)

= n b

²_n

n

X

k=0

kb

n

n − x

P

_n,k

x b

_n

Z

^kbn_n _−x

0

dtf (x).

The above identity and the representation (9) lead to x

b

_n

1 − x

b

_n

(K

_n−1

f (x) − f(x))

= n b

²_n

n

X

k=0

kb

n

n − x

P

_n,k

x b

_n

Z

^kbn

n −x

0

(f (x + t) − f(x)) dt

≡ X

k∈Λ

+ X

k∈Ω

,

(10)

where Λ and Ω are the sets of indices k ∈ {0, 1, . . . , n} such that |

^kb_nⁿ

−x| ≤ x and

^kb_nⁿ

− x > x, respectively.

For the sake of brevity let us introduce the notation: d

_n

= pb

n

/n, m = [pn/b

_n

], w

_x

(δ; f ) = w

_x

(δ). Consider the sum P

k∈Λ

and divide the set Λ in the following manner: Λ = S

_m

j=0

Λ

_j

, where Λ

_j

are the sets of indices k such that

0 ≤

kb

n

n − x

≤ xd

n

if j = 0, jxd

_n

<

kb

_n

n − x

≤ (j + 1)xd

n

if j = 1, 2, . . . , m − 1, mxd

_n

<

kb

_n

n − x

≤ x if j = m.

In view of definition (6),

X

k∈Λ

≤

m−1

X

j=0

T

_n,j

(x)w

_x

((j + 1)xd

_n

) + T

_n,m

(x)w

_x

(x),

where

T

_n,j

(x) := n b

²_n

X

k∈Λj

kb

_n

n − x

2

P

_n,k

x b

_n

.

From Lemma 1 (i) one has T

_n,0

(x) ≤ n

b

²_n

µ

_n,2

(x) = x b

_n

1 − x

b

_n

.

Next, given any positive integer q, we have

T

_n,j

(x) ≤ n b

²_n

1 (jxd

_n

)

^q

n

X

k=0

kb

_n

n − x

q+2

P

_n,k

x b

_n

for j = 1, 2, . . . , m. Hence Lemma 1 (iii) yields T

_n,j

(x) ≤ c

₁

(q)

j

^q

x

^q

x b

n

1 − x

b

n

ϕ

^q/2_n

(x)

(11)

where ϕ

_n

(x) = x

1 −

_b^x_n

+

^b_nⁿ

. Consequently,

X

k∈Λ

≤ x b

_n

1 − x

b

_n

1 + c

1

(q)

x

^q

ϕ

^q/2_n

(x)





m−1

X

j=1

w

x

((j + 1)xd

n

)

j

^q

+ w

x

(x) m

^q



 . Clearly,

m−2

X

j=1

w

_x

((j + 1)xd

_n

)

j

^q

≤ 3

^q

d

^q−1_n

Z

_md_n

2dn

w

_x

(xt) t

^q

dt

≤ 3

^q

d

^q−1_n

Z

m²

1

( √

s)

^q−3

w

_x

x

√ s

ds

≤ c

²

(q)d

^q−1_n

m²−1

X

k=1

( √

k + 1)

^q−3

w

x

√ k

and w

_x

(x)

(m − 1)

^q

+ w

_x

(x)

m

^q

≤ 2

(m − 1)

^q

w

x

(x) ≤ 3

^q

d

^q−1_n

w

x

(x).

Hence

X

k∈Λ

≤ c

3

(q) x b

_n

1 − x

b

_n

1 + ϕ

^q/2n

(x) x

^q

! d

^q−1_n

m²−1

X

k=1

√ k

_q−3

w

_x

x

√ k

. (11)

Now, let us consider the sum P

k∈Ω

in formula (10). Given any positive integer r, we have

X

k∈Ω

≤ n

b

²_n

x

^r

X

k∈Ω

kb

n

n − x

r+2

P

_n,k

x b

_n

|f(x)|

+ n

b

²_n

x

^r

X

k∈Ω

kb

_n

n − x

r+1

P

_n,k

x b

_n

Z

_b_n

0

|f(t)|dt

≤ n

b

²_n

x

^r

|µ

n,r+2

|(x) |f(x)|

+ n

b

²_n

x

^r

Z

bn

0

|f(t)|dt (µ

n,2r+2

(x))

^1/2

X

k∈Ω

P

_n,k

x b

_n

!

1/2

.

(12)

Applying Lemmas 1 and 3 we then get

X

k∈Ω

≤ c

₄

(r) x

^r

x b

n

1 − x

b

n

b

n

r/2

ϕ

^r/2_n

(x)|f(x)|

+ c

₄

(r) x

^r

x b

_n

1 − x

b

_n

b

n

^r₂₋¹₂

s b

_n

x(b

_n

− x) ϕ

^r/2_n

(x) Z

_b_n

0

|f(t)|dt exp

− nx 8b

_n

.

This gives the desired conclusion when combined with (10) and (11).

Proof of Theorem 2 . Let f ∈ M

loc

[0, ∞) and let the limits f(x+), f(x−) exist at a fixed point x > 0. Consider the function g

_x

defined by (7). It is easily seen that

f (t) − f (x+) + f (x−)

2 = g

x

(t) + f (x+) − f(x−)

2 sgn

_x

(t) +

f (x) − f (x+) + f (x−) 2

δ

x

(t),

where sgn

_x

(t) = sgn(t − x), δ

x

(t) = 1 if t = x, δ

_x

(t) = 0 otherwise (see e.g.

[9]). Hence

K

_n−1

f (x) − f (x+) + f (x−) 2

= K

_n−1

g

_x

(x) + f (x+) − f(x−)

2 K

_n−1

sgn

_x

(x).

(12)

The function g

_x

is continuous at x and g

_x

(x) = 0. So, K

_n−1

g

_x

(x) = K

_n−1

g

_x

(x) − g

x

(x) can be estimated as in the proof of Theorem 1. Namely, using formula (10) in which f is replaced by g

x

and observing that

w

_x

(δ; g

_x

) ≤ osc (g

x

; I

_x

(δ)) for 0 < δ ≤ x we get the estimation for

P

k∈Λ

as in (11) with w

x

√x k

replaced by osc

g

x

; I

x

√x k

. Indeed, we estimate the sum P

k∈Ω

as follows:

X

k∈Ω

≤ 2n

b

²_n

M (b

_n

; f ) X

k∈Ω

kb

n

n − x

2

P

_n,k

x b

_n

,

(13)

where M (b

_n

; f ) = sup

_0≤t≤b_n

|f(t)|. Next, the Cauchy-Schwarz inequality and Lemmas 1, 3 lead to

X

k∈Ω

≤ 2M(b

n

; f ) n

b

²_n

(µ

_n,4

(x))

^1/2

2 exp

− nx 4b

n

1/2

≤ c x b

_n

1 − x

b

_n

s b

_n

x(b

_n

− x) ϕ

^1/2_n

(x)M (b

_n

; f ) exp

− nx 8b

_n

, where c is an absolute positive constant. Consequently,

|K

n−1

g

x

(x)| ≤

≤ c(q) 1 + ϕ

^q/2n

(x) x

^q

! b

n

^q−1₂ ^[n/bⁿ^]

X

k=1

( √

k)

^q−3

osc

g

_x

; I

_x

x

√ k

+ c

s b

_n

x(b

_n

− x) ϕ

^1/2_n

(x)M (b

_n

; f ) exp

− nx 8b

_n

,

where q is arbitrary positive integer, c(q) is a positive constant depending only on q and c is an absolute constant.

Now it is enough to estimate the term K

_n−1

sgn

_x

(x). Choose the integer l such that x ∈ [

_n^l

b

_n

,

^l+1_n

b

_n

). It is clear that

K

_n−1

sgn

_x

(x) = X

k>l

P

_n−1,k

x b

n

− X

k<l

P

_n−1,k

x b

n

+ n

b

_n

P

_n−1,l

x b

_n

2 l

n b

_n

+ b

_n

n − 2x

= 2 X

k>l

P

_n−1,k

x b

_n

− 1 + 2P

n−1,l

x b

_n

n b

_n

l + 1 n b

_n

− x

.

Therefore,

|K

n−1

sgn

_x

(x)| ≤ 2

X

k>l

P

_n−1,k

x b

_n

− 1 2

+ 2P

_n−1,l

x b

_n

≤ 4 r b

n

s b

_n

x(b

n

− x) ,

(14)

by Lemma 4. Combining the above estimations for |K

n−1

g

_x

(x)| and

|K

n−1

sgn

_x

(x)| with (12) we obtain the desired conclusion. Thus the proof of Theorem 2 is complete.

References

[1] J. Albrycht and J. Radecki, On a generalization of the theorem of Voronovskaya, Zeszyty Naukowe UAM, Zeszyt 2, Pozna´ n (1960), 1–7.

[2] R. Bojanic and O. Shisha, Degree of L

₁

approximation to integrable functions by modified Bernstein polynomials, J. Approx. Theory 13 (1975), 66–72.

[3] P.L. Butzer and H. Karsli, Voronovskaya-type theorems for derivatives of the Bernstein-Chlodovsky polynomials and the Sz´ asz-Mirakyan operator, Com- ment. Math., to appear.

[4] Z.A. Chanturiya, Modulus of variation of functions and its application in the theory of Fourier series, Dokl. Akad. Nauk SSSR 214 (1974), 63–66.

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[6] M. Heilmann, Direct and converse results for operators of Baskakov- Durrmeyer type, Approx. Theory Appl. 5 (1) (1989), 105–127.

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[8] G.G. Lorentz, Bernstein Polynomials, University of Toronto Press, Toronto, 1953.

[9] P. Pych-Taberska, Some properties of the B´ezier-Kantorovich type operators, J. Approx. Theory 123 (2003), 256–269.

[10] L.C. Young, General inequalities for Stieltjes integrals and the convergence of Fourier series, Math. Annalen 115 (1938), 581–612.

[11] X.M. Zeng, Bounds for Bernstein basis functions and Meyer-K¨ onig and Zeller basis functions, J. Math. Anal. Appl. 219 (2) (1998), 364–376.

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Received 12 May 2009

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