Inequalities for a class of multivariate operators

This paper introduces and studies a class of generalized multivariate Bernstein operators defined on the simplex. By means of the modulus of continuity and so-called Ditzian-Totik’s modulus of function, the direct and inverse inequalities for the operators approximating multivariate continuous functions are simultaneously established. From these inequalities, the characterization of approximation of the operators follows. The obtained results include the corresponding ones of the classical Bernstein operators.

MSC: 41A25, 41A36, 41A60, 41A63.

Let N be the set of natural numbers, and { s n } n = 1 ∞ ( s n ≥ 1 , s n ∈ N ) be a sequence. In [3], Cao introduced the following generalized Bernstein operators defined on [ 0 , 1 ] :

where x ∈ [ 0 , 1 ] , f ∈ C [ 0 , 1 ] , and

Clearly, when s n = 1 , L n f reduce to the classical Bernstein operators, B n f , given by

Furthermore, Cao [3] proved that the necessary and sufficient condition of convergence for the operators is lim n → ∞ ( s n / n ) = 0 , and he also proved that for n ∈ Q = { n : n ∈ N ,  and  0 < ( s n − 1 ) / n + 1 / n ≤ 1 } the following estimate of approximation degree holds:

Here, ω ( f , t ) is the modulus of continuity of first order of the function f. In [4], some approximation properties for the operators were further investigated.

In this paper, we will introduce and study the multivariate version defined on the simplex of the generalized Bernstein operators given by (1). The main aim is to establish the direct and inverse inequalities of approximation, which will imply the characterization of approximation of the operators.

For convenience, we denote by bold letter the vector in R d . Let

denote the canonical unit vector in R d , i.e., its ith component is 1 and the others are 0, and let

be the simplex in R d . For x ∈ T , k : = ( k 1 , k 2 , … , k d ) ∈ N 0 d , we denote as usual

Then the well-known Bernstein basis function on T is given by

By means of the basis function, we define the multivariate generalized Bernstein operators on the simplex T as

Obviously, when d = 1 , these operators reduce to the univariate operators defined by (1), and when s n = 1 they are just the well-known multivariate Bernstein operators on the simplex T, B n , d , defined by

Let C ( T ) denote the space of continuous functions on T with the norm defined by ∥ f ∥ : = max x ∈ T | f ( x ) | , f ∈ C ( T ) . For arbitrary vector e ∈ R d , we write for the rth symmetric difference of a function f in the direction of e

Then the Ditzian-Totik’s modulus of function f ∈ C ( T ) is defined by (see [1])

where the weighted functions

and

Define differential operators:

then the weighted Sobolev space can be defined by

where T ∘ is inner of T, and the Peetre K-functional on C ( T ) is given by

Berens and Xu [1] proved that K φ r ( f , t r ) is equivalent to ω φ r ( f , t ) , i.e.,

here and in the following C denotes a positive constant independent of f and n, but its value may be different at a different occurrence.

We also need the usual modulus of continuity of function f ∈ C ( T ) defined by (see [8])

where h = ( h 1 , h 2 , … , h d ) ∈ R d and | h | 2 : = ( ∑ i = 1 d h i 2 ) 1 / 2 , and another K -functional given by (see [8])

It is shown in [8] that

Now we state the main results of this paper as follows.

Theorem 1.1Let f ∈ C ( T ) , then for n ∈ Q = { n : n ∈ N , and  0 < s n − 1 n + 1 n ≤ 1 } , there holds

Theorem 1.2If f ∈ C ( T ) and lim n → ∞ ( s n / n ) = 0 , then

Theorem 1.3If f ∈ C ( T ) and lim n → ∞ ( s n / n ) = 0 , then there hold

and

From Theorem 1.2 and Theorem 1.3, we easily obtain the following corollaries, which characterize the approximation feature of the multivariate operators L n , d given by (8).

Corollary 1.1Let f ∈ C ( T ) , 0 < α < 1 . Then, for the Bernstein operators given by (9), the necessary and sufficient condition for which

is ω φ 2 ( f , t ) = O ( t 2 α ) ( t → 0 ).

Corollary 1.2If f ∈ C ( T ) , 0 < α < 1 , s n > 1 and lim n → ∞ ( s n / n ) = 0 , then ω φ 2 ( f , t ) = O ( t 2 α ) and ω ( f , t ) = O ( t α ) ( t → 0 ) imply

Corollary 1.3If s n > 1 and s n = O ( n 1 − ϵ ) ( n → ∞ ), 0 < ϵ ≤ 1 , then for any f ∈ C ( T ) and 0 < α < 1 , the statement

implies that ω φ 2 ( f , t ) = O ( t 2 ϵ α ) and ω ( f , t ) = O ( t ϵ α ) ( t → 0 ).

From Corollary 1.2 and Corollary 1.3, we have the following.

Corollary 1.4Let 0 < α < 1 and 1 < s n = O ( 1 ) ( n → ∞ ), then, for any f ∈ C ( T ) the necessary and sufficient condition for which

is ω φ 2 ( f , t ) = O ( t 2 α ) and ω ( f , t ) = O ( t ϵ α ) ( t → 0 ).

In this section, we prove some lemmas.

Defining the transformation T i ( i = 1 , 2 , … , d ) from T to itself, i.e.,

we have the following symmetric property for the operators L n , d , which is similar to the known one of the multivariate Bernstein operators (see [6,7]).

Lemma 2.1For the above transformation T i , i = 1 , 2 , … , d , there holds

where f i ( x ) = f ( T i ( x ) ) , u = T i ( x ) .

Proof It is sufficient to prove the case i = 1 . Let

and x = ( x 1 , x ∗ ) ∈ T , x ∗ = ( x 2 , x 3 , … , x d ) . Then, from definition (8), it follows that

The proof of Lemma 2.1 is completed. □

To prove Theorem 1.3, we need some the following lemmas. At first, similar to the estimates for the Bernstein operators (see [2,5,6]), it is not difficult to derive the following Lemma 2.2.

Lemma 2.2The following inequalities hold:

Secondly, we need prove two Bernstein type inequalities.

Lemma 2.3Let f ∈ C ( T ) , 1 ≤ i ≤ j ≤ d . Then

Proof For d = 1 , by direct computation we have (see [9])

where

Noting that

we obtain

This inequality shows that Lemma 2.3 is valid for d = 1 . For the proof of the case d > 1 , we use a decomposition technique and the induction. In fact, let

and

then we can decompose the generalized Bernstein operators as

Therefore,

Now, suppose that Lemma 2.3 is valid for d − 1 , then from (12) it follows that

So, Lemma 2.3 is true for i = 2 . From the symmetry, the proof of the cases i = 1 , 3 , 4 , … , d is the same. For the cases 1 ≤ i < j ≤ d , we use Lemma 2.1 and obtain that

So, the proof of Lemma 2.3 is complete. □

Lemma 2.4For f ∈ C 2 ( T ) , 1 ≤ i ≤ j ≤ d , one has

Proof We only need to prove the case s n > 1 because the case s n = 1 has been shown in [1]. Our approach is based on the induction. At first, for d = 1 , let h = ( n + s n − 1 ) − 1 , then by simple calculation we have

Therefore,

Let y = ( k + j ) / ( n + s n − 1 ) , then we have for 1 ≤ k ≤ n − 1 , 0 ≤ j ≤ s n − 1 ,

and for | u | ≤ h , there holds | 1 − 2 y − u | ≤ 1 . Hence,

which implies

So,

Now, assume that Lemma 2.4 is valid for d − 1 , then by (12)

Also, we can check the following inequalities:

and

Thus,

Similarly, the cases i = 1 , 3 , 4 , … , d can be proved. For the case 1 ≤ i ≤ j ≤ d , we use the transformation T i and Lemma 2.1, it is easy to verify

Hence, the proof of Lemma 2.4 is complete. □

We also need the following two interesting results related to nonnegative numerical sequence. The proof of the first result can be found in [10], and the proof of the other is similar to Lemma 2.1 of [10] where the proof of case ν 1 = 0 and C = 1 was given.

Lemma 2.5Let μ n , ν n , and ψ n are all nonnegative numerical sequence, and μ 1 = ν 1 = 0 . If for 0 < r < s and 1 ≤ k ≤ n , n ∈ N , there holds

then

Lemma 2.6Let ν n and ψ n are all nonnegative numerical sequence. If for s > 0 and 1 ≤ k ≤ n , n ∈ N , there holds ν n ≤ ( k n ) s ν k + C ψ k , then

First, we prove Theorem 1.1. By straight calculation, we have (see also [3])

Then we use the same method as Theorem 2 of [3] and obtain easily

We now prove Theorem 1.2. We use a known estimation on Bernstein operators (see [1]) as an intermediate step to deduce the direct theorem. Since

from the fact that (see [1])

we get

This completes the proof of Theorem 1.2.

Finally, we prove Theorem 1.3. Let

and ψ n = 4 ∥ ( L n , d f ) − f ∥ , then μ 1 = ν 1 = 0 and from Lemma 2.2, Lemma 2.3, and Lemma 2.4, we have for 1 ≤ k ≤ n

which implies from Lemma 2.5 that μ n ≤ C n − 1 ∑ k = 1 n ψ k , i.e.,

Let ν n = ∥ D i ( L n , d f ) ∥ and ψ n be the same as the above, then we have by Lemma 2.2

So, using Lemma 2.6 gives ν n ≤ C n − 1 ( ∑ k = 1 n ψ k + ν 1 ) , namely,

For n ≥ 2 , there is an m ∈ N , such that n / 2 ≤ m ≤ n and ∥ L m , d f − f ∥ ≤ ∥ L k , d f − f ∥ hold for 1 ≤ k ≤ n . Then

So, combining (10), (13), and (15) we see

Also, collecting (11), (13), and (15) implies

The proof of Theorem 1.3 is complete.

The authors declare that they have no competing interests.

This research was supported by the National Nature Science Foundation of China (No. 61101240, 61272023), the Zhejiang Provincial Natural Science Foundation of China (No. Y6110117), and the Science Foundation of the Zhejiang Education Office (No. Y201122002).

1. Berens, H, Xu, Y: K-moduli, moduli of smoothness and Bernstein polynomials on a simplex. Indag. Math.. 2, 411–421 (1991). Publisher Full Text

2. Cao, FL: Derivatives of multidimensional Bernstein operators and smoothness. J. Approx. Theory. 132, 241–257 (2005). Publisher Full Text

3. Cao, JD: A generalization of the Bernstein polynomials. J. Math. Anal. Appl.. 209, 140–146 (1997). Publisher Full Text

4. Ding, CM: Approximation by generalized Bernstein polynomials. Indian J. Pure Appl. Math.. 35(6), 817–826 (2004)

5. Ding, CM, Cao, FL: K-functionals and multivariate Bernstein polynomials. J. Approx. Theory. 155, 125–135 (2008). Publisher Full Text

6. Ditzian, Z: Inverse theorems for multidimensional Bernstein operators. Pac. J. Math.. 121, 293–319 (1986). Publisher Full Text

7. Ditzian, Z, Zhou, XL: Optimal approximation class for multivariate Bernstein operators. Pac. J. Math.. 158, 93–120 (1993)

8. Johnen, H, Scherer, K: On the equivalence of the K-functional and modulus of continuity and some applications. In: Schempp W, Zeller K (eds.) Constructive Theory of Functions of Several Variable, pp. 119–140. Springer, Berlin (1977)

9. Lorentz, GG: Bernstein Polynomials, Toronto University Press, Toronto (1953)

10. Wickeren, EV: Steckin-Marchaud-type inequalities in connection with Bernstein polynomials. Constr. Approx.. 2, 331–337 (1986). Publisher Full Text