Introduction. The Bernstein–Durrmeyer polynomial of order n is de- fined by

POLONICI MATHEMATICI LVII.2 (1992)

A saturation theorem for combinations of Bernstein–Durrmeyer polynomials

by P. N. Agrawal and Vijay Gupta† (Roorkee)

Abstract. We prove a local saturation theorem in ordinary approximation for com- binations of Durrmeyer’s integral modification of Bernstein polynomials.

Introduction. The Bernstein–Durrmeyer polynomial of order n is de- fined by

M

(f, x) =

1

R

0

W (n, x, t)f (t) dt , f ∈ L

[0, 1] , where

W (n, x, t) = (n + 1)

n

X

ν=0

p

(x)p

(t) ,

p

(x) being (

)x

(1−x)

, x ∈ [0, 1]. These operators were introduced by Durrmeyer [5] by replacing f (ν/n) in B

(f, x), the Bernstein polynomials, by (n+1) R

0

p

(t)f (t) dt. Several authors (see [1]–[4], [6], [8], [9]) have stud- ied the operators M

and obtained direct and inverse results both in sup- norm and L

-norm. In this paper we study the saturation behaviour of the linear combination M

(f, k, x) [7]. It turns out that even though Bernstein–

Durrmeyer polynomials are not exponential type operators [7] yet their sat- uration behaviour is similar to that of the operators of exponential type.

The linear combination M

(f, k, x) of M

(f, x), j = 0, 1, . . . , k, is de- fined by

M

(f, k, x) =

k

X

j=0

C(j, k)M

(f, x) ,

1991 Mathematics Subject Classification: 41A30, 41A36.

Key words and phrases: linear combinations, compact support, inner product.

† Research supported in part by Grant No. 5829-11-61 from Council of Scientific and

Industrial Research, India.

where

C(j, k) =

k

Y

i=0 i6=j

d

d

− d

for k 6= 0 , C(0, 0) = 1 ,

and d

, d

, . . . , d

are k + 1 arbitrary but fixed distinct positive integers.

R e m a r k. The definition of C(j, k) can also be related to the formula x

=

k

X

j=0

C(j, k)

k

Y

i=0 i6=j

(x − d

) ,

which is a simple rephrasing of the interpolation formula x

=

k

X

j=0

C(j, k)l

(x) ,

where l

, l

, . . . , l

are the Lagrange fundamental polynomials corresponding to the knots d

, d

, . . . , d

. Therefore, by simple computation, we get the relation

k

X

j=0

C(j, k) = 1 .

Throughout this paper, let C

denote the set of continuous functions on [0, 1] having a compact support and C

the subset of C

of k times continuously differentiable functions. The spaces A.C.[a, b] and L

[0, 1] are defined as the class of absolutely continuous functions on [a, b] for every a, b satisfying 0 < a < b < 1 and the class of bounded and integrable functions on [0, 1] respectively. ha, bi ⊂ [0, 1] stands for an open interval containing the closed interval [a, b].

2. Preliminary results. In the following result we obtain an estimate of the degree of approximation by M

(·, k, x) for smooth functions.

Theorem 2.1. Let 0 ≤ p ≤ 2k + 2, f ∈ L

[0, 1], and suppose f

exists and is continuous on ha, bi ⊂ [0, 1]. Then for all n sufficiently large,

kM

(f, k, ·) − f (·)k

≤ max{C

n

ω(f

, n

), C

n

} where C

= C

(k, p), C

= C

(k, p, f ) and ω(f

, δ) denotes the modulus of continuity of f

on ha, bi.

P r o o f. For x ∈ [a, b], writing F (t, x) = f (t) −

p

X

j=0

f

(x)

j! (t − x)

,

we have

|F (t, x)| ≤ |t − x|

p!



1 + |t − x|

n



ω(f

, n

)

for t ∈ ha, bi. Thus if χ(t) denotes the characteristic function of ha, bi, by [3, Prop. II.3] and the Schwarz inequality,

M

(|F (t, x)|χ(t), x) ≤ C

n

ω(f

, n

) ,

where C

= C

(p). Similarly, for some constant C

and all n sufficiently large, we have

M

(|F (t, x)|(1 − χ(t)), x) ≤ C

[M

((t − x)

, x)]

≤ C

n

. Hence,

|M

(|F (t, x)|, x)| ≤ C

n

ω(f

, n

) + C

n

. But by [6, Prop.]

M



X

j=1

f

(x)

j! (t − x)

, k, x



= O(n

) uniformly in x ∈ [a, b]. Hence for all n sufficiently large

kM

(f, k, ·) − f (·)k

≤ C

n

ω(f

, n

) + C

n

, where C

does not depend on f , from which the required result is immediate.

In the following lemma, the inner product hh(·), g(·)i is defined as R

0

h(x)g(x) dx.

Lemma 2.2. Let 0 < a < b < 1. If f ∈ C[0, 1] and g ∈ C

with supp g ⊂ (a, b), then

|n

hM

(f, k, ·) − M

(f, k, ·), g(·)i| ≤ Kkf k

, where K is a constant independent of f and n.

P r o o f. We write

M

(f, k, x) − M

(f, k, x) =

2k+2

X

j=1

α(j, k)M

(f, x) ,

where e

∈ {d

, d

, . . . , d

, 2d

, 2d

, . . . , 2d

}. By [7, Lemma 3.5] it follows that

2k+2

X

j=1

α(j, k)e

= 0 , m = 0, 1, . . . , k .

(2.1)

Next, by using [3, Prop. II.3], we have n

hM

(f, k, ·) − M

(f, k, ·), g(·)i

= n

1

R

0 1

R

0

n

X

j=1

α(j, k)W (e

n, x, t)f (t)g(x) o

dt dx

= n

R

supp g 1

R

0

{. . .} dt dx

= n

R

supp g b

R

a

{. . .} dt dx + o(1)kf k

= n

1

R

0 b

R

a

{. . .} dt dx + o(1)kf k

.

Now, using Fubini’s theorem and expanding g(x) by Taylor’s theorem, we get

(2.2) n

hM

(f, k, ·) − M

(f, k, ·), g(·)i

= n

b

R

a 1

R

0 2k+2

X

i=0 2k+2

X

j=1

α(j, k)

i! W (e

n, x, t)f (t)g

(t)(x − t)

dx dt

+ n

b

R

a 1

R

0 2k+2

X

j=1

α(j, k)W (e

n, x, t)f (t)ε(x, t)(x − t)

dx dt + o(1)kf k

=

2k+2

X

i=0

n

b

R

a 1

R

0 2k+2

X

j=1

α(j, k)

i! W (e

n, x, t)f (t)g

(t)(x − t)

dx dt

+ n

1

R

0 b

R

a 2k+2

X

j=1

α(j, k)W (e

n, x, t)f (t)ε(x, t)(x − t)

dt dx + o(1)kf k

= J

+ J

+ o(1)kf k

, say ,

where ε(x, t)(x − t)

is the remainder term corresponding to the partial Taylor expansion of g.

Since, for ξ lying between t and x,

|ε(x, t)| = |g

(ξ) − g

(x)|

(2k + 2)! ≤ 2

(2k + 2)! kg

k

< ∞ ,

using [3, Prop. II.3], it follows that J

= O(1)kf k

.

To estimate J

, we proceed as follows: J

may be rewritten as J

= n

2k+2

X

i=0

1 i!

2k+2

X

j=1

α(j, k)

b

R

a

 R

0

W (e

n, x, t)(x − t)

dx 

f (t)g

(t) dt . Now, we note that W (n, x, t) = W (n, t, x), therefore using [3, Prop. II.3], after interchanging the variables t and x, together with equation (2.1) it fol- lows that J

= O(1)kf k

. Combining the estimates for J

, J

and (2.2), we obtain the required result.

Theorem 2.3 [2]. Let f ∈ C[0, 1], 0 < a

< a

< a

< b

< b

< b

< 1 and 0 < α < 2. Then, in the following, the implications (i)⇒(ii)⇔(iii)⇒(iv) hold :

(i) kM

(f, k, ·) − f (·)k

= O(n

).

(ii) f ∈ Liz(α, k + 1, a

, b

).

(iii) (a) For m < α(k + 1) < m + 1, m = 0, 1 . . . , 2k + 1, f

exists and is is in Lip(α(k + 1) − m, a

, b

).

(b) For α(k + 1) = m + 1, m = 0, 1, . . . , 2k, f

exists and is in Lip

(1, a

, b

).

(iv) kM

(f, k, ·) − f (·)k

= O(n

).

Here Liz(α, k, a, b) denotes the class of functions for which ω

(f, h, a, b)

≤ M h

; when k = 1, Liz(α, 1) reduces to the Zygmund class Lip

α.

3. The saturation result

Theorem 3.1. Let f ∈ C[0, 1] and 0 < a

< a

< a

< b

< b

< b

< 1.

Then, in the following statements, the implications (i)⇒(ii)⇒(iii) and (iv)⇒(v)⇒(vi) hold true:

(i) n

kM

(f, k, ·) − f (·)k

= O(1);

(ii) f

∈ A.C.[a

, b

] and f

∈ L

[a

, b

];

(iii) n

kM

(f, k, ·) − f (·)k

= O(1);

(iv) n

kM

(f, k, ·) − f (·)k

= o(1);

(v) f ∈ C

[a

, b

] and

2k+2

X

j=1

Q(j, k, x)

j! f

(x) = 0, x ∈ [a

, b

], where Q(j, k, x) are the polynomials occurring in [6, Th. 2];

(vi) n

kM

(f, k, ·) − f (·)k

= o(1),

where all O(1) and o(1) terms are with respect to n, as n → ∞.

P r o o f. First assume (i); then in view of (i)⇒(iii) of Theorem 2.3, it follows that f

exists and is continuous on (a

, b

). Moreover, the statement

(3.1) kM

(f, k, ·) − f (·)k

= O(n

) is equivalent to

(3.2) kM

(f, k, ·) − M

(f, k, ·)k

= O(n

) .

Indeed, trivially (3.1)⇒(3.2). Also, assuming (3.2), since lim

M

(f, k, x)

= f (x), we can write

f (x) = M

(f, k, x) + [M

(f, k, x) − M

(f, k, x)]

+ [M

(f, k, x) − M

(f, k, x)] + . . . + [M

(f, k, x) − M

(f, k, x)] + . . . Hence,

kf (·) − M

(f, k, ·)k

≤ kM

(f, k, ·) − M

(f, k, ·)k

+ kM

(f, k, ·) − M

(f, k, ·)k

+ . . . + kM

(f, k, ·) − M

(f, k, ·)k

+ . . .

= K

 1

n

+ 1

2

n

+ . . . + 1

(2

)

n

+ . . .



= K

n

1

1 − 2

= K

n

,

where K

= K

/(1 − 2

), showing that (3.1) holds.

Thus, we may assume that {n

(M

(f, k, ·) − M

(f, k, ·))} is bounded as a sequence in C[a

, b

] and hence in L

[a

, b

]. Since L

[a

, b

] is the dual space of L

[a

, b

], it follows by Alaoglu’s theorem that there exists h ∈ L

[a

, b

] such that for some subsequence {n

}

of natural numbers and for every g ∈ C

with supp g ⊂ (a

, b

)

(3.3) lim

ni→∞

n

hM

(f, k, ·) − M

(f, k, ·), g(·)i = hh(·), g(·)i . Now, since C

[a

, b

]∩C[0, 1] is dense in C[0, 1] there exists a sequence {f

}

in C

[a

, b

] ∩ C[0, 1] converging to f in k · k

-norm. Then, for any g ∈ C

with supp g ⊂ (a

, b

) and each function f

, by [6, Th. 2]

we have

(3.4) lim

ni→∞

n

hM

(f

, k, ·) − M

(f

, k, ·), g(·)i

=

−(1 − 2

)

2k+2

X

j=1

Q(j, k, ·)

j! f

(·), g(·)

= hP

(D)f

(·), g(·)i = hf

(·), P

(D)g(·)i , where P

(D) denotes the operator adjoint to P

(D) (in this case, it is simply a result of integration by parts). By Lemma 2.2, we conclude that

(3.5) lim

ni→∞

n

|hM

(f − f

, k, ·) − M

(f − f

, k, ·), g(·)i|

≤ Kkf − f

k

. Hence, by (3.5), (3.4) and (3.3) (in that order)

hf (·), P

(D)g(·)i = lim

σ→∞

hf

(·), P

(D)g(·)i

= lim

σ→∞

{ lim

ni→∞

n

hM

(f − f

, k, ·) − M

(f − f

, k, ·), g(·)i + hf

(·), P

(D)g(·)i}

= lim

ni→∞

n

hM

(f, k, ·) − M

(f, k, ·), g(·)i = hh(·), g(·)i , for all g ∈ C

with supp g ⊂ (a

, b

). Thus

(3.6) P

(D)f (x) = h(x)

as generalized functions.

Note that Q(2k + 2, k, x) 6= 0 by [6, Prop.]. Therefore, regarding (3.6) as a first order linear differential equation for f

, we deduce that f

∈ A.C.[a

, b

] and hence f

∈ L

[a

, b

]. This completes the proof of the implication (i)⇒(ii).

Now assuming (ii), it follows that f

∈ Lip

(1, a

, b

) with M = kf

k

. Hence (iii) follows by Theorem 2.1.

To prove (iv)⇒(v), assuming (iv) and proceeding in the manner of the proof of (i)⇒(ii), we get P

(D)f (x) = 0, from which in view of the non-vanishing of Q(2k + 2, k, x), (v) is clear.

The proof of (v)⇒(vi) follows from [6, Th. 2]. This completes the proof of the theorem.

Acknowledgement. The authors are extremely grateful to the referee

for the critical review of the paper.

References

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[2] —, —, Inverse theorem for linear combinations of modified Bernstein polynomials, preprint.

[3] M. M. D e r r i e n n i c, Sur l’approximation de fonctions int´ egrables sur [0, 1] par des polynˆ omes de Bernstein modifi´ es, J. Approx. Theory 31 (1981), 325–343.

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[5] J. L. D u r r m e y e r, Une formule d’inversion de la transform´ ee de Laplace: Applica- tion ` a la th´ eorie des moments, Th` ese de 3e cycle, Facult´ e des Sciences de l’Universit´ e de Paris, 1967.

[6] H. S. K a s a n a and P. N. A g r a w a l, On sharp estimates and linear combinations of modified Bernstein polynomials, Bull. Soc. Math. Belg. S´ er. B 40 (1) (1988), 61–71.

[7] C. P. M a y, Saturation and inverse theorems for combinations of a class of exponential type operators, Canad. J. Math. 28 (1976), 1224–1250.

[8] B. W o o d, L

-approximation by linear combinations of integral Bernstein-type oper- ators, Anal. Num´ er. Th´ eor. Approx. 13 (1) (1984), 65–72.

[9] —, Uniform approximation by linear combinations of Bernstein-type polynomials, J. Approx. Theory 41 (1984), 51–55.

DEPARTMENT OF MATHEMATICS

Current address of P. N. Agrawal:

UNIVERSITY OF ROORKEE DEPARTMENT OF MATHEMATICS

ROORKEE 247667, U.P., INDIA MOI UNIVERSITY

ELDORET, KENYA

Re¸ cu par la R´ edaction le 10.6.1991

R´ evis´ e le 15.2.1992