PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 96 (110) (2014), 23–29 DOI: 10.2298/PIM1410023B

ON L^{p} -CONVERGENCE OF BERNSTEIN–DURRMEYER OPERATORS WITH RESPECT TO ARBITRARY MEASURE

Elena E. Berdysheva and Bing-Zheng Li

Abstract. We consider Bernstein–Durrmeyer operators with respect to arbitrary measure on the simplex in the space ℝ^{d}. We obtain estimates for rate of convergence in the corresponding weighted L^{p}-spaces, 1 ⩽ p < ∞.

1. Introduction

We consider Bernstein–Durrmeyer operators with respect to arbitrary measure. These are positive linear operators defined for functions on a d-dimensional simplex. We start with notation. Let

𝕊^{d} := {x = (x_{1}, …, x_{d}) ∈ ℝ^{d} : 0 ⩽ x_{1}, …, x_{d} ⩽ 1, 0 ⩽ x_{1} + ⋯+ x_{d} ⩽ 1}

denote the standard simplex in ℝ^{d}. We denote by ∂𝕊^{d} the boundary of 𝕊^{d}. We will also use barycentric coordinates on the simplex which we denote by the boldface symbol x = (x_{0}, x_{1}, …, x_{d}), x_{0} := 1−x_{1}−⋯ −x_{d}. We will use standard multiindex notation such as

x^{α} := x_{0}^{α_{0}} x_{1}^{α_{1}} ⋯ x_{d}^{α_{d}}  and  \frac{α}{n} := ( \frac{α_{0}}{n} , \frac{α_{1}}{n} , ⋯, \frac{α_{d}}{n} )

for x = (x_{0}, x_{1}, …, x_{d}), α = (α_{0}, α_{1}, …, α_{d}) ∈ ℝ^{d+1} , n ∈ ℕ. Functions defined on 𝕊^{d} are understood as functions of a point that can be given alternatively in cartesian or in barycentric coordinates.

The spaces L^{p}(𝕊^{d}, ρ), 1 ⩽ p < ∞, are defined in the standard way as spaces of (equivalence classes) of real-valued functions f for which |f|^{p} is integrable with respect to a measure ρ with the norm

∥f∥_{L^{p} (𝕊^{d} ,ρ)} := ( ∫_{𝕊^{d}} |f(x)|^{p}dρ(x)) ^{1/p}.

The space L^{∞} (𝕊^{d}, ρ) is the space of essentially bounded functions with the norm ∥f∥_{L^{∞} (𝕊^{d} ,ρ)} := ess sup_{x∈𝕊^{d}} |f(x)|. We will also consider the space C(𝕊^{d}) of continuous bounded functions on 𝕊^{d} with the norm ∥f∥_{C(𝕊^{d})} := max_{x∈𝕊^{d}} |f(x)|.

2010 Mathematics Subject Classification: Primary 41A36, 41A63.

Key words and phrases: positive operators, Bernstein type operators, rate of convergence.

The paper is dedicated to Giuseppe Mastroianni on the occasion of his retirement.

An important building stone of our construction are the Bernstein basis polynomials of degree n ∈ ℕ on the simplex

B_{α} (x) := ( {n \atop α}) x^{α} = \frac{n!}{α_{0}!α_{1}! ⋯ α_{d}!} (1 − x_{1} − ⋯− x_{d})^{α_{0}} x_{1}^{α_{1}} ⋯ x_{d}^{α_{d}} ,

with α = (α_{0}, α_{1}, …, α_{d}), where α_{0}, α_{1}, …, α_{d} are nonnegative integers and |α| := α_{0} + α_{1} + ⋯+ α_{d} = n. Here, and in similar expressions later, 0^{0} means 1. The Bernstein basis polynomials are nonnegative on 𝕊^{d}, and

∑_{|α|=n} B_{α} (x) = 1.

The polynomials {B_{α} }_{|α|=n} constitute a basis of the space of real algebraic polynomials in d variables of total degree at most n.

Definition 1.1. Let ρ be a nonnegative bounded Borel measure on 𝕊^{d} such that

(1.1)  suppρ ∖ ∂𝕊^{d} ≠ ∅.

The Bernstein–Durrmeyer operator with respect to the measure ρ is defined for f ∈ C(𝕊^{d}) or f ∈ L^{p}(𝕊^{d}, ρ), 1 ⩽ p ⩽ ∞, by

(1.2)  M_{n,ρ} f := ∑_{|α|=n} \frac{∫_{𝕊^{d}} fB_{α} dρ}{∫_{𝕊^{d}} B_{α} dρ} B_{α} ,  n ∈ ℕ.

Note that ρ is regular (being a nonnegative bounded Borel measure on a metric space), and thus polynomials are dense in the spaces L^{p}(𝕊^{d}, ρ), 1 ⩽ p ⩽ ∞. Condition (1.1) guarantees that ∫_{𝕊^{d}} B_{α} dρ > 0 for all Bernstein basis polynomials B_{α}.

The operator M_{n,ρ} is linear and positive, and it reproduces constant functions. It is a variant of the Bernstein polynomial operator B_{n} for integrable functions. The latter is defined as follows.

Definition 1.2. The Bernstein operator is defined for f ∈ C(𝕊^{d}) by

(1.3)  B_{n} f := ∑_{|α|=n} f( \frac{α}{n}) B_{α} ,  n ∈ ℕ.

This is a linear positive operator that reproduces linear functions. The operator B_{n} was introduced by Bernstein [7] in the one-dimensional case in order to give a constructive proof of the Weierstrass Approximation Theorem. Many variants and generalizations of operator (1.3) were studied in hundreds of papers.

The operator M_{n,ρ} without weight (i.e., when ρ is the Lebesgue measure) was defined in [12, 17] and studied in [8, 9]. In the special case when ρ is the Jacobi weight, M_{n,ρ} was introduced in [18, 6]. It is very well understood; see, e.g., [11]. See also [5] for properties and further references.

Operators (1.2) in full generality were for the first time systematically studied in [4], to our knowledge. The motivation came from learning theory; Jetter and Zhou [14] used the univariate Bernstein–Durrmeyer operators of type (1.2) to obtain bias-variance estimates for support vector machine classifiers. In [16], the second named author applied multivariate operators (1.2) as a tool for proving learning rates of least-square regularized regression with polynomial kernels.

In this paper, we continue our investigations on convergence of operators (1.2). In [2], the first named author considered uniform convergence of operators M_{n,ρ} . She proved that

lim_{n→∞} ∥f − M_{n,ρ} f∥_{C(𝕊^{d})} = 0 for every f ∈ C(𝕊^{d})

if and only if ρ is strictly positive on 𝕊^{d} (i.e., suppρ = 𝕊^{d}). In [3], she considered pointwise convergence on the support of the measure. She showed that (M_{n,ρ} f)(x) → f(x) as n → ∞ at each point x ∈ suppρ if f is bounded on suppρ and continuous at x. Moreover, the convergence is uniform on any compact set in the interior of suppρ. Her method does not lead to estimates for rates of convergence.

The second named author studied the weighted L^{p}-convergence of operators (1.2). In [16], she proved that

lim_{n→∞} ∥f − M_{n,ρ} f∥_{L^{p} (𝕊^{d} ,ρ)} = 0

for every f ∈ L^{p}(𝕊^{d}, ρ), 1 ⩽ p < ∞. Note that no additional assumptions on ρ are required. Moreover, she obtained estimates for the rate of convergence in the spaces L^{p}(𝕊^{d}, ρ), 1 ⩽ p < ∞, in terms of the following K-functional. Let C^{1}(𝕊^{d}) be the space of functions g ∈ C(𝕊^{d}) with continuous partial derivatives ∂_{i}g := \frac{∂g}{∂x_{i}} , i = 1, …, d, endowed with the seminorm

∥∇g∥_{C(𝕊^{d})} := max_{i=1,...,d} ∥∂_{i}g∥_{C(𝕊^{d})}.

The K-functional used in [16] is defined by

K(f, t)_{p} := inf_{g∈C^{1} (𝕊^{d})} {∥f − g∥_{L^{p} (𝕊^{d} ,ρ)} + t ∥∇g∥_{C(𝕊^{d})}}, 1 ⩽ p ⩽ ∞.

The following estimates were proved in [16]. If f ∈ L^{p}(𝕊^{d}, ρ), 1 ⩽ p < ∞, then

(1.4)  ∥f − M_{n,ρ} f∥_{L^{p} (𝕊^{d} ,ρ)} ⩽ 2dK(f, n^{−1/2p} [ρ(𝕊^{d})]^{1/p} )_{p},  1 ⩽ p < 2,
(1.5)  ∥f − M_{n,ρ} f∥_{L^{p} (𝕊^{d} ,ρ)} ⩽ 2dK(f, n^{−1/p} [ρ(𝕊^{d})]^{1/p})_{p},  2 ⩽ p < ∞.

In this paper, we improve the rates given in estimates (1.4) and (1.5). Namely, by a modification of the method of [16], we obtain the following result.

Theorem 1.1. Let ρ be a nonnegative bounded Borel measure on 𝕊^{d} such that suppρ ∖ ∂𝕊^{d} ≠ ∅, and let f ∈ L^{p}(𝕊^{d}, ρ), 1 ⩽ p < ∞. Then

∥f − M_{n,ρ} f∥_{L^{p} (𝕊^{d} ,ρ)} ⩽ 2K(f, C_{p} n^{−1/2} d[ρ(𝕊^{d})]^{1/p} )_{p}, 1 ⩽ p < ∞,

where C_{p} is a constant that depends only on p. It holds C_{p} ⩽ Cp̃ for p ⩽ p̃. Moreover, one can take C_{p} = \frac{1}{2} for 1 ⩽ p ⩽ 2.

2. Proof of Theorem 1.1

Denote φ_{i}(x) := x_{i}, i = 1, …, d, and for 1 ⩽ p ⩽ ∞

Δ_{n,p} := ∑_{i=1}^{d} ∥M_{n,ρ} (|φ_{i}(x) − φ_{i}(·)|)∥_{L^{p} (𝕊^{d} ,ρ)} .

It is easy to see that

(2.1)  ∥M_{n,ρ} f − f∥_{L^{p} (𝕊^{d} ,ρ)} ⩽ 2K(f, Δ_{n,p} /2)_{p}

for f ∈ L^{p}(𝕊^{d}, ρ), 1 ⩽ p ⩽ ∞ (see [4, Theorem 4.5] or [16, Theorem 2.1]). Thus, the key to proving estimates for the rate of convergence of the operator M_{n,ρ} is to study the behaviour of Δ_{n,p}.

We were able to obtain estimates for Δ_{n,p} in case when 1 ⩽ p < ∞. Theorem 1.1 is a direct consequence of the lemma given below.

Lemma 2.1. Let ρ be a nonnegative bounded Borel measure on 𝕊^{d} such that suppρ ∖ ∂𝕊^{d} ≠ ∅, and let f ∈ L^{p}(𝕊^{d}, ρ), 1 ⩽ p < ∞. Then

(2.2) ∥M_{n,ρ} (|φ_{i}(x) − φ_{i}(·)|)∥_{L^{p} (𝕊^{d} ,ρ)} ⩽ c_{p} n^{−1/2} [ρ(𝕊^{d})]^{1/p},  i = 1, …, d,

where c_{p} is a constant that depends only on p. It holds c_{p} ⩽ cp̃ for p ⩽ p̃. Moreover, one can take c_{p} = 1 for 1 ⩽ p ⩽ 2.

Proof. Denote θ_{α} := ∫_{𝕊^{d}} B_{α} dρ. Following [16], we write

M_{n,ρ} (|φ_{i}(x) − φ_{i}(·)|)(x) = ∑_{|α|=n} \frac{1}{θ_{α}} ∫_{𝕊^{d}} |φ_{i}(x) − φ_{i}(t)| B_{α} (t) dρ(t) B_{α} (x)

= ∑_{|α|=n} |φ_{i}(x) − \frac{α_{i}}{n} | B_{α} (x) + ∑_{|α|=n} \frac{1}{θ_{α}} ∫_{𝕊^{d}} | \frac{α_{i}}{n} − φ_{i}(t)| B_{α} (t) dρ(t) B_{α} (x)

= B_{n}(|φ_{i}(x) − φ_{i}(·)|)(x) + I(x) ,

where B_{n} is the Bernstein operator (1.3), and

I(x) := ∑_{|α|=n} \frac{1}{θ_{α}} ∫_{𝕊^{d}} | \frac{α_{i}}{n} − φ_{i}(t)| B_{α} (t) dρ(t) B_{α} (x).

By Cauchy–Schwarz inequality for positive operators (e.g., [13]), we have

B_{n}(|φ_{i}(x) − φ_{i}(·)|)(x) ⩽ (B_{n} ([φ_{i}(x) − φ_{i}(·)]^{2}) (x))^{1/2} (B_{n}(1)(x))^{1/2} .

It is well known and easy to prove that

(2.3)  B_{n} ([φ_{i}(x) − φ_{i}(·)]^{2}) (x) = \frac{φ_{i}(x) (1 − φ_{i}(x))}{n} ⩽ \frac{1}{4n},

and B_{n}(1) = 1. Thus, B_{n}(|φ_{i}(x) − φ_{i}(·)|)(x) ⩽ \frac{1}{2 \sqrt{n}} , and

(2.4)  ∥B_{n}(|φ_{i}(x) − φ_{i}(·)|)∥_{L^{p} (𝕊^{d} ,ρ)} ⩽ \frac{1}{2\sqrt{n}}[ρ(𝕊^{d})]^{1/p}.

Next we obtain an estimate for ∥I∥_{L^{p} (𝕊^{d} ,ρ)} . Take q such that \frac{1}{p} + \frac{1}{q} = 1. Applying the Hölder inequality two times, we obtain

∥I∥_{L^{p} (𝕊^{d} ,ρ)}^{p} = ∫_{𝕊^{d}} \{ ∑_{|α|=n} \frac{1}{θ_{α}} ∫_{𝕊^{d}} | \frac{α_{i}}{n} − φ_{i}(t)|B_{α} (t) dρ(t) B_{α}^{\frac{1}{p} + \frac{1}{q}} (x)\}^{p}dρ(x)

⩽ ∫_{𝕊^{d}} ∑_{|α|=n} \frac{1}{θ_{α}^{p}} ( ∫_{𝕊^{d}} | \frac{α_{i}}{n} − φ_{i}(t)|B_{α} (t) dρ(t)) ^{p} B_{α} (x) dρ(x)

= ∑_{|α|=n} \frac{1}{θ_{α}^{p−1}} ( ∫_{𝕊^{d}} | \frac{α_{i}}{n} − φ_{i}(t)|B_{α}^{\frac{1}{p} + \frac{1}{q}} (t) dρ(t)) ^{p}

⩽ ∑_{|α|=n} \frac{1}{θ_{α}^{p−1}} ∫_{𝕊^{d}} | \frac{α_{i}}{n} − φ_{i}(t)|^{p} B_{α} (t) dρ(t) θ_{α}^{p/q}

= ∫_{𝕊^{d}} ∑_{|α|=n} | \frac{α_{i}}{n} − φ_{i}(t)|^{p} B_{α} (t) dρ(t)

= ∥B_{n}(|φ_{i}(x) − φ_{i}(·)|^{p})∥_{L^{1} (𝕊^{d} ,ρ)} .

First suppose that p ⩾ 1 is an even integer. In this case, the expression in the last line of the previous formula is the L^{1}(𝕊^{d}, ρ)-norm of a moment of the Bernstein operator (1.3), namely, of B_{n}([φ_{i}(x) − φ_{i}(·)]^{p})(x). First we note that the value of this moment is independent of the dimension d. To see this, consider without loss of generality i = 1. Then

B_{n}([φ_{1}(x) − φ_{1}(·)]^{p})(x) = ∑_{|α|=n} ( x_{1} − \frac{α_{1}}{n} )^{p} B_{α} (x)

= ∑_{|α|=n} ( x_{1} − \frac{α_{1}}{n} )^{p}( {n \atop α_{1}} ) x_{1}^{α_{1}} (1 − x_{1})^{n−α_{1}}

× \frac{(n − α_{1})!}{α_{2}! ⋯ α_{0}!} ( \frac{x_{2}}{1 − x_{1}} )^{α_{2}} ⋯ ( \frac{x_{0}}{1 − x_{1}} )^{α_{0}}

= ∑_{α_{1} =0}^{n} ( x_{1} − \frac{α_{1}}{n} )^{p}( {n \atop α_{1}} ) x_{1}^{α_{1}} (1 − x_{1})^{n−α_{1}}

× ∑_{|(α_{2} ,...,α_{d} ,α_{0} )|=n−α_{1}} B_{(α_{2} ,...,α_{d} ,α_{0} )} ( \frac{x_{2}}{1 − x_{1}} , ⋯, \frac{x_{d}}{1 − x_{1}} )

= ∑_{α_{1} =0}^{n} ( x_{1} − \frac{α_{1}}{n} )^{p}( {n \atop α_{1}} ) x_{1}^{α_{1}} (1 − x_{1})^{n−α_{1}}

which is the p-th moment of the one-dimensional Bernstein operator. Estimates for these moments are well known an can be found, e.g., in [10, Chapter 10, §1]. It follows from Corollary to Theorem 1.1 of this chapter that there is a constant A_{p} depending only on p such that

∑_{α_{1} =0}^{n} ( x_{1} − \frac{α_{1}}{n} )^{p}( {n \atop α_{1}} ) x_{1}^{α_{1}} (1 − x_{1})^{n−α_{1}} ⩽ A_{p} n^{−p/2} .

Consequently,

(2.5) ∥I∥_{L^{p} (𝕊^{d} ,ρ)} ⩽ {∥B_{n}([φ_{i}(x) − φ_{i}(·)]^{p})∥_{L^{1} (𝕊^{d} ,ρ)} }^{1/p} ⩽ n^{−1/2} A_{p}^{1/p} [ρ(𝕊^{d})]^{1/p} .

For an arbitrary p ⩾ 1, take p̃ to be the smallest even integer with p ⩽ p̃. The L^{p}(𝕊^{d}, ρ)-and Lp̃(𝕊^{d}, ρ)-norms are connected by the inequality

(2.6)  ∥ · ∥_{L^{p} (𝕊^{d} ,ρ)} ⩽ ∥ · ∥_{Lp̃(𝕊^{d} ,ρ)} [ρ(𝕊^{d})]^{\frac{1}{p} − \frac{1}{p̃}}

(e.g., [15, Chapter IV, §3, Theorem 6]). This estimate and (2.5) yield

∥I∥_{L^{p} (𝕊^{d} ,ρ)} ⩽ n^{−1/2} A1/p̃ ; p̃[ρ(𝕊^{d})]^{1/p}.

Combining this with (2.4), we obtain

∥M_{n,ρ} (|φ_{i}(x) − φ_{i}(·)|)∥_{L^{p} (𝕊^{d} ,ρ)} ⩽ ∥B_{n}(|φ_{i}(x) − φ_{i}(·)|)∥_{L^{p} (𝕊^{d} ,ρ)} + ∥I∥_{L^{p} (𝕊^{d} ,ρ)}
⩽ n^{−1/2} c_{p}[ρ(𝕊^{d})]^{1/p}

with c_{p} = \frac{1}{2} + A1/p̃ ; p̃, which is (2.2).

It follows from (2.6) that for all p ⩽ p̃ it holds

∥M_{n,ρ} (|φ_{i}(x) − φ_{i}(·)|)∥_{L^{p} (𝕊^{d} ,ρ)} ⩽ n^{−1/2} cp̃[ρ(𝕊^{d})]^{1/p}.

Thus, c_{p} ⩽ cp̃ for p ⩽ p̃.

Finally, consider p = 2. In this case A_{2} = \frac{1}{4} (see (2.3)). Thus, c_{2} = 1, and we also can take c_{p} = 1 for 1 ⩽ p ⩽ 2. □

Remark 2.1. Representations of general moments of the multivariate Bernstein operator (1.3) of the form B_{n}( ∏ _{i=1}^{d} (φ_{i}(x) − φ_{i}(·))^{p_{i}} ) with nonnegative integers p_{i}, i = 1, …, d, in terms of Stirling numbers are given by Abel and Ivan [1]. They also estimated the order of these moments.

Remark 2.2. Alternatively, we could use in Theorem 1.1 the estimate

∥M_{n,ρ} f − f∥_{L^{p} (𝕊^{d} ,ρ)} ⩽ max{2, d}K (f, Δ_{n,p}) _{p}

instead of (2.1). This inequality leads to the estimate

∥M_{n,ρ} f − f∥_{L^{p} (𝕊^{d} ,ρ)} ⩽ max{2, d} K(f, n^{−1/2} c_{p} [ρ(𝕊^{d})]^{1/p} )_{p}, 1 ⩽ p < ∞,

with a constant c_{p} like in Lemma 2.1.

Remark 2.3. Our method does not lead to estimates for the rates of convergence of the operator M_{n,ρ} in the space L^{∞} (𝕊^{d}, ρ), or to pointwise estimates. These are important and interesting open questions.

Acknowledgement. The research was initiated while the first named author was visiting Mathematics Research Institute, The Ohio State University, Columbus, OH, USA. She acknowledges the hospitality of this organization and financial support provided by them.

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German University of Technology in Oman Muscat, Oman

 elena.berdysheva@gutech.edu.om

Zhejiang University Hangzhou, China

 libingzheng@zju.edu.cn