Results Math Online First c 2017 Springer International Publishing AG  DOI 10.1007/s00025-017-0759-4

Results in Mathematics

Quantitative Estimates in Lp-Approximation by Bernstein–Durrmeyer–Choquet Operators with Respect to Distorted Borel Measures Sorin G. Gal and Sorin Trifa Abstract. For the multivariate Bernstein–Durrmeyer operator Dn,µ , written in terms of the Choquet integral with respect to a distorted probability Borel measure µ on the standard d-dimensional simplex S d , quantitative Lp -approximation results, 1 ≤ p < ∞, in terms of a K functional are obtained. Mathematics Subject Classification. 41A35, 41A25, 28A12, 28A25. Keywords. Bernstein–Durrmeyer–Choquet operator, monotone and submodular set function, Choquet integral, Lp -approximation, K-functional, distorted Borel measure.

1. Introduction The approximation properties of the multivariate Bernstein–Durrmeyer linear operator defined with respect to a nonnegative, bounded Borel measure μ : BS d → R+ , by   f (t)Bα (t)dμ(t) Sd Mn,μ (f )(x) = · Bα (x) B (t)dμ(t) Sd α |α|=n  c(α, μ) · Bα (x), x ∈ S d , n ∈ N, (1) := |α|=n

where BS d denotes the sigma algebra of all Borel measurable subsets in the power set P(S d ) and f is supposed to be μ-integrable on the standard simplex S d = {(x1 , . . . , xd );

0 ≤ x1 , . . . , xd ≤ 1, 0 ≤ x1 + · · · + xd ≤ 1},

S. G. Gal and S. Trifa

Results Math

were studied in, e.g., the recent papers [1–3,10]. Note that in (1), it is used the notation n! αd 1 (1 − x1 − x2 − · · · − xd )α0 · xα 1 · · · · · xd α0 ! · α1 ! · · · · · αd ! n! · Pα (x), =: α0 ! · α1 ! · · · · · αd !  where α = (α0 , α1 , . . . , αd ), αj ∈ N {0}, j = 0, . . . , d, |α| = α0 + α1 + · · · + αd = n. In the very recent paper [8], it was proved that the pointwise and uniform approximation results of qualitative kind in [1,2], remain valid for the more general case when μ is a monotone, normalized and submodular set function on S d and the integrals used in (1) are the  nonlinear Choquet integrals with respect to μ, denoted in general, by (C) S d f dμ. In another very recent paper [9], for the pointwise and uniform approximation obtained in [8], we proved quantitative estimates in terms of the modulus of continuity and in terms of a K-functional, in the case of more general multivariate Bernstein–Durrmeyer–Choquet polynomial operators defined by  Dn,Γn,x (f )(x) = c(α, μn,α,x ) · Bα (x), x ∈ S d , n ∈ N, (2) Bα (x) =

|α|=n

where c(α, μn,α,x ) =

  (C) S d f (t)Bα (t)dμn,α,x (t) (C) S d f (t)Pα (t)dμn,α,x (t)   = (C) S d Bα (t)dμn,α,x (t) (C) S d Pα (t)dμn,α,x (t)

and for every n ∈ N and x ∈ S d , Γn,x = (μn,α,x )|α|=n is a family of bounded, monotone, submodular and strictly positive set functions on BS d . Note that if Γn,x reduces to one element (i.e. μn,α,x = μ for all n, x and α), then the operator given by (2) reduces to the operator considered in [8]. The main aim of the present paper is to study quantitative Lp -approximation results, 1 ≤ p < ∞, for the case when Γn,x reduces to one element μ, which is a normalized, monotone and submodular set function, i.e. for the Bernstein–Durrmeyer–Choquet operators given by  Dn,μ (f )(x) = c(α, μ) · Bα (x), x ∈ S d , n ∈ N, |α|=n

where

  (C) S d f (t)Bα (t)dμ(t) (C) S d f (t)Pα (t)dμ(t)   c(α, μ) = = . (C) S d Bα (t)dμ(t) (C) S d Pα (t)dμ(t)  In the classical case when μ is a Borel measure, since the integral S d f dμ is linear, it easily implies the main known tool in obtaining quantitative estimates in Lp -approximation, that is the inequality Dn,μ (f )Lpµ ≤ Cf Lpµ , for all f ∈ Lpμ , with C > 0 independent of n and f (actually with C = 1). This is a consequence of applying twice the additivity of integral with respect to the

Quantitative Estimates in Lp -Approximation

Borel μ. For example, when p = 1 and d = 1, denoting pn,k (x) = n k measure n−k x (1 − x) , we easily get k Dn,μ (f )L1µ



|f (t)|pn,k (t)dμ(t) ≤ pn,k (x) ·  1 dμ(x) p (t)dμ(t) 0 k=0 0 n,k 1 n  1  |f (t)|pn,k (t)dμ(t) pn,k (x)dμ(x) · 0  1 = p (t)dμ(t) k=0 0 0 n,k n  1  |f (t)|pn,k (t)dμ(t) = 

1

k=0  1

= 0

1

n 

0

0

|f (t)|

n  k=0

 pn,k (t)dμ(t) =

0

1

|f (t)|dμ(t) = f L1µ .

But, in the  1general case of μ monotone and submodular, the only information on (C) 0 f dμ which can be derived is that this integral is sublinear, 1  1 n

n that is (C) 0 [ k=1 fk (x)]dμ(x) ≤ k=0 (C) 0 fk (x)dμ(x). Due to this fact, even in the simplest case p = 1, d = 1, for f ∈ L1μ 1 (meaning that f is B[0,1] -measurable and f L1µ = (C) 0 |f (t)|dμ(t) < ∞), we can get only 1  1  n (C) 0 |f (t)|pn,k (t)dμ(t) Dn,μ (f )L1µ ≤ (C) pn,k (x) · dμ(x) 1 (C) 0 pn,k (t)dμ(t) 0 k=0 1  1 n  (C) 0 |f (t)|pn,k (t)dμ(t) (C) pn,k (x)dμ(x) · ≤ 1 (C) 0 pn,k (t)dμ(t) 0 k=0  1 n  (C) |f (t)|pn,k (t)dμ(t) ≤ (n + 1) · f L1µ , n ∈ N, = k=0

0

since 0 ≤ pn,k (t) ≤ 1, for all t ∈ [0, 1], n ∈ N and k = 0, 1, . . . , n. This fact suggests that in the most general case for μ, quantitative estimates in Lp -approximation by Bernstein–Durrmeyer–Choquet operators cannot be obtained by using the tool in the linear case and therefore, seems to remain an open question by using eventually another method, but which may depend on μ. However, the above inequality is not a representative one for the entire class of monotone and submodular set functions, because for a large subclass of normalized, monotone and submodular set functions called distorted probability Borel measures, in the present paper we show that the tool in the linear case is applicable and we obtain quantitative Lp -approximation results, 1 ≤ p < +∞.

S. G. Gal and S. Trifa

Results Math

The finding of the maximal subclass of normalized, monotone and submodular set functions for which the tool in the linear case is applicable, remains as an interesting open question. Section 2 contains some preliminaries on the Choquet integral. In Sect. 3, quantitative estimates in terms of a K-functional for the Lp approximation, 1 ≤ p < ∞, are obtained.

2. Preliminaries Some known concepts and results concerning the Choquet integral can be summarized by the following. Definition 2.1. Suppose Ω = ∅ and let C be a σ-algebra of subsets in Ω. (i) (see, e.g., [12], p. 63) The set function μ : C → [0, +∞] is called a monotone set function (or capacity) if μ(∅) = 0 and μ(A) ≤ μ(B) for all A, B ∈ C, with A ⊂ B. Also, μ is called submodular if  μ(A B) + μ(A B) ≤ μ(A) + μ(B), for all A, B ∈ C. μ is called bounded if μ(Ω) < +∞ and normalized if μ(Ω) = 1. (ii) (see, e.g., [12], p. 233, or [5]) If μ is a monotone set function on C and if f : Ω → R is C-measurable (that is, for any Borel subset B ⊂ R it follows f −1 (B) ∈ C), then for any A ∈ C, the concept of Choquet integral is defined by  +∞  0      μ Fβ (f ) A − μ(A) dβ, f dμ = μ Fβ (f ) A dβ + (C) A

0

−∞

where we used the notation Fβ (f ) = {ω ∈ Ω; f (ω) ≥ β}. Notice that if f ≥ 0 0 on A, then in the above formula we get −∞ = 0.  The function f will be called Choquet integrable on A if (C) A f dμ ∈ R. In what follows, we list some known properties of the Choquet integral. Remark 2.2. If μ : C → [0, +∞] is a monotone set function, then the following properties hold :   (i) For all a ≥ 0 we have (C) A af dμ = a · (C) A f dμ (if f ≥ 0 then see, e.g., [12], Theorem 11.2, (5), p. 228 and if f is of arbitrary sign, then see, e.g., [6], p. 64, Proposition 5.1, (ii)). (ii) For all c ∈ R and f of arbitrary sign, we have (see, e.g., [12], pp. 232–233, or [6], p. 65) (C) A (f + c)dμ = (C) A f dμ + c · μ(A). If μ is submodular too, then for all f, g of arbitrary sign and lower bounded, we have (see, e.g., [6], p. 75, Theorem 6.3)    f dμ + (C) gdμ. (C) (f + g)dμ ≤ (C) A

A

A

Quantitative Estimates in Lp -Approximation

  (iii) If f ≤ g on A then (C) A f dμ ≤ (C) A gdμ (see, e.g., [12], p. 228, Theorem 11.2, (3) if f, g ≥ 0 and  p. 232 if f, g are of arbitrary sign). (iv) Let f ≥ 0. If A ⊂ B then (C) A f dμ ≤ (C) B f dμ. In addition, if μ is finitely subadditive, then    f dμ + (C) f dμ. (C)  f dμ ≤ (C) A

B



A

B

(v) It is immediate that (C) A 1 · dμ(t) = μ(A). (vi) The formula μ(A) = γ(M (A)), where γ : [0, 1] → [0, 1] is an increasing and concave function, with γ(0) = 0, γ(1) = 1 and M is a probability measure (or only finitely additive) on a σ-algebra on Ω (that is, M (∅) = 0, M (Ω) = 1 and M is countably additive), gives simple examples of normalized, monotone and submodular set functions (see, e.g., [6], pp. 16–17, Example 2.1). Such of set functions μ are also called distorsions of countably normalized, additive measures (or distorted measures). For 2t . a simple example, we can take γ(t) = 1+t If the above γ function is increasing, concave and satisfies only γ(0) = 0, then for any bounded Borel measure m, μ(A) = γ(m(A)) gives a simple example of bounded, monotone and submodular set function. (vii) If μ is a countably additive bounded measure, then the Choquet integral (C) A f dμ reduces to the usual Lebesgue type integral (see, e.g., [6], p. 62, or [12], p. 226).

3. Lp -approximation Results Recall that μ : BS d → [0, +∞) is said strictly positive if for every open set A ⊂ Rn with A ∩ S d = ∅, we have μ(A ∩ S d ) > 0. The support of μ is defined by supp(μ) = {x ∈ S d ; μ(Nx ) > 0 for every open neighborhood Nx ∈ BS d of x}. Note that the strict positivity of μ, evidently implies the condition supp(μ) \  ∂S d = ∅, which guarantees that (C) S d Bα (t)dμ(t) > 0, for all Bα . If μ : BS d → [0, +∞) is a monotone set function and 1 ≤ p < +∞, then we make the following notations:    p d d p |f (t)| dμ(t) < +∞ , Lμ (S ) = f : S → R; f is BS d -measurable, (C) Sd Lpμ,+ (S d ) = Lpμ (S d ) {f : S d → R+ }, C(S d ) = {f : S d → R; f is continuous on S d }, endowed with the norm F C(S d ) = sup{|F (x)|; x ∈ S d }, C+ (S d ) = {f ∈ C(S d ); f ≥ 0 on S d },

S. G. Gal and S. Trifa

Results Math

1 C+ (S d ) is the subspace of all functions g ∈ C+ (S d ) with continuous partial derivatives ∂g/∂xi , i ∈ {1, . . . , d},     ∂g    , ∇gC(S d ) = max i={1,...,d}  ∂xi C(S d )

K (f ; t)Lpµ =

inf

1 (S d ) g∈C+

 with the notation F Lpµ =

{f − gLpµ + t∇gC(S d ) }, t ≥ 0,



(C)

1/p |F (t)|p dμ(t) ,

Sd

IC[0, 1] = {g : [0, 1] → [0, 1] : g(0) = 0, g(1) = 1, g is concave and strictly increasing on [0, 1] and there exists g  (0) < +∞}. Also, denote by D(BS d ) the class of all set functions μ : BS d → [0, +∞) of the form μ(A) = g(M (A)), for all A ∈ BS d , where g ∈ IC[0, 1] and M is a strictly positive, probability Borel measure on BS d . In the words of Remark 2.2, (vi), any such a μ can be called distorted probability Borel measure. Remark 3.1. According to Remark 2.2, (vi), any μ ∈ D(BS d ) is a normalized, monotone, strictly positive and submodular set function. Simple examples of μ ∈ D(BS d ) are μ(A) = sin[π · m(A)/2], or μ(A) = arctan[tan(1) · m(A)], or 2m(A) e , or μ(A) = (1 − e−m(A) ) · e−1 , or μ(A) = ln[1 + (e − 1)m(A)], μ(A) = 1+m(A) for all A ∈ BS d , where m denotes the d-dimensional Lebesgue measure. The main result of the paper is the following. Theorem 3.2. Let 1 ≤ p < ∞. If μ ∈ D(BS d ), with μ = g ◦ M , g ∈ IC[0, 1] and M a strictly positive, probability Borel measure on BS d , then for all f ∈ Lpμ,+ (S d ), n ∈ N, we have   Δn,p f − Dn,μ (f )Lpµ ≤ c · K f ; , c Lp µ

d where c = 1 + g  (0)(p+1)/p , Δn,p = i=1 Dn,μ (|ϕi (x) − ϕi (·)|)Lpµ , ϕi (x) = xi for x = (x1 , . . . , xd ) ∈ S d . Proof. Firstly, we note that by g(0) = 0, g(1) = 1 and by the concavity of g, since all the points of the segment passing through the points (0, g(0)) and (1, g(1)) are below the graph of g and since all the points of the tangent to the graph of g at (0, g(0)) are above the graph of g, we immediately get the double inequality x ≤ g(x) ≤ g  (0)x, for all x ∈ [0, 1]. This immediately implies M (A) ≤ μ(A) ≤ g  (0)M (A), for all A ∈ BS d .

(3)

Quantitative Estimates in Lp -Approximation

We divide the proof into three steps. Step 1. For all f ∈ Lpμ,+ (S d ) we have Dn,μ (f )Lpµ ≤ [g  (0)](p+1)/p · f Lpµ .

(4)

Indeed, by Lemma 2.2 in [10] we get Dn,M (f )LpM ≤ f LpM , which combined with Remark 2.2, (vii) and with (3), implies f LpM ≤ f Lpµ and therefore Dn,M (f )LpM ≤ f Lpµ .

(5)

On the other hand, by the same inequalities (3), we can write ⎡ ⎤p ⎛ ⎞1/p    f (t)B (t)dM (t) α d ⎣ ⎦ dM (x)⎠ Dn,M (f )LpM = ⎝ Bα (x) · S B (t)dM (t) Sd Sd α |α|=n

⎛ ≥

1 · ⎝(C) g  (0)1/p ⎛

≥ =

1 · ⎝(C) g  (0)1/p

⎡



⎣ Sd

⎡



⎣ Sd





Bα (x) ·

|α|=n



|α|=n

⎤p

⎞1/p

f (t)Bα (t)dM (t) ⎦ dμ(x)⎠ B (t)dM (t) Sd α

Sd

⎤p ⎞1/p f (t)B (t)dμ(t) 1 α d ⎦ dμ(x)⎠ · S Bα (x)·  g (0) B (t)dμ(t) α d S 

1 · Dn,μ (f )Lpµ . [g  (0)](p+1)/p

Therefore, combined this with (5), we immediately obtain (4). Step 2. For n ∈ N and |α| = n arbitrary fixed, let us consider Tn,α : Lpμ,+ (S d ) → R+ defined by  Tn,α (f ) = (C) f (t)Bα (t)dμ(t), f ∈ Lpμ,+ (S d ). Sd p d Lμ,+ (S ) ⊂

L1μ,+ (S d ), for all 1 ≤ p < +∞. Firstly, we note that  Then, by 0 ≤ Bα (x) ≤ 1 for x ∈ S d , we get 0 ≤ (C) S d |f (t)Bα (t)|p dμ(t)  ≤ (C) S d |f (t)|p dμ(t) < ∞, for any f ∈ Lpμ,+ (S d ), i.e. f · Bα ∈ Lpμ,+ (S d ). Now, by Remark 3.1, μ is a monotone and submodular set function. Based also on the Remark 2.2, (i), (ii), (iii) and reasoning exactly as in the proof of Lemma 3.1 in [8], we get |Tn,α (f ) − Tn,α (g)| ≤ Tn,α (|f − g|). Then, since Tn,α is positively homogeneous, sublinear and monotonically increasing, we immediately get that Dn,μ keeps the same properties and as a consequence it follows |Dn,μ (f )(x) − Dn,μ (g)(x)| ≤ Dn,μ (|f − g|)(x), f, g ∈ Lpμ,+ (S d ),

(6)

Dn,μ (λf ) = λDn,μ (f ), Dn,μ (f + g) ≤ Dn,μ (f ) + Dn,μ (g) and that f ≤ g on S d implies Dn,μ (f ) ≤ Dn,μ (g) on S d , for all λ ≥ 0, f, g ∈ Lpμ,+ (S d ), n ∈ N. Then, (6) immediately implies Dn,μ (f ) − Dn,μ (g)Lpµ ≤ Dn (|f − g|)Lpµ .

(7)

S. G. Gal and S. Trifa

Results Math

Step 3. Let f, g ∈ Lpμ,+ (S d ). We will apply the Minkowski’s inequality in the case of Choquet integral with respect to μ (see. e.g., Theorem 3.7 in [11] or Theorem 2 in [4]). Note here that the proof of Minkowski’s inequality in [11] or [4] is based on the H¨ older’s inequality for Choquet integral,  1/p  1/q    p q (C) |f g|dμ ≤ (C) |f | dμ · (C) |g| dμ , 1/p + 1/q = 1, Sd

Sd

Sd

where the proof cited two papers under the supposi  is performed in the above tion that (C) S d |f |p dμ = 0 and (C) S d |g|q dμ = 0. But from the inequalities in (3), older’s inequality immediately holds even if  the H¨  it easily follows that (C) S d |f |p dμ = 0 or (C) S d |g|q dμ = 0. Concluding, under the hypothesis in the statement, the Minkowski’s inequality holds in its full generality. Therefore, we obtain f − Dn,μ (f )Lpµ = (f − g) + (g − Dn,μ (g)) + (Dn,μ (g) − Dn,μ (f ))Lpµ ≤ f − gLpµ + g − Dn,μ (g)Lpµ + Dn,μ (g) − Dn,μ (f )Lpµ . (8) By applying now (7) and (4), we get Dn,μ (g) − Dn,μ (f )Lpµ ≤ [g  (0)](p+1)/p · f − gLpµ .

(9)

1 (S d ). Thus, by Dn,μ (e0 )(x) = Now, let us estimate g − Dn,μ (g)Lpµ for g ∈ C+ e0 (x) = 1 and (6), we get

|g(x) − Dn,μ (g)(x)| = |Dn,μ (g(x))(x) − Dn,μ (g(t))(x)| ≤ Dn,μ (|g(x) − g(·)|)(x). 1 (S d ) and x = (x1 , . . . , xd ), t = (t1 , . . . , td ) ∈ S d , it follows Since for g ∈ C+ (see, e.g., [10], formula (2.5))

|g(x) − g(t)| ≤ ∇gC(S d )

d 

|xi − ti | = ∇gC(S d )

i=1

d 

|ϕi (x) − ϕi (t)|,

i=1

applying here Dn,μ , since it is subadditive as function of f , it easily follows

d Dn,μ (|g(x) − g(·)|)(x) ≤ ∇gC(S d ) · i=1 Dn,μ (|ϕi (x) − ϕi (·)|)(x). Therefore, taking to the power p, integrating above with respect to x and μ and applying the Minkowski’s inequality, we immediately obtain g − Dn,μ (g)

Lp µ

≤ ∇gC(S d ) ·

n 

Dn,μ (|ϕi (x) − ϕi (·)|)Lpµ .

(10)

i=1

In conclusion, replacing the inequalities (9) and (10) in (8) and denoting c = 1 + g  (0)(p+1)/p , we obtain   f − Dn,μ (f )Lpµ ≤ c f − gLpµ + ∇gC(S d ) · Δn,p /c , which immediately proves the statement of the theorem.



Quantitative Estimates in Lp -Approximation

Remark 3.3. The positivity of function f in Theorem 3.2 is necessary because of the positive homogeneity of the Choquet integral used in the proof. However, if f is of arbitrary sign and lower bounded on S d with f (x) − m ≥ 0, for all x ∈ S d , then the statement of Theorem 3.2 can be restated for the slightly modified Bernstein–Durrmeyer operator defined by ∗ (f )(x) = Dn,μ (f − m)(x) + m, Dn,μ ∗ (f )(x) − f (x) = Dn,μ (f − m)(x) − (f (x) − m). Note that where we have Dn,μ we may consider here that m < 0 and we immediately get the relations

K(f − m; t)Lpµ = = =

inf

{f − (g + m)Lpµ + t∇gC(S d ) }

inf

{f − (g + m)Lpµ + t∇(g + m)C(S d ) }

1 (S d ) g∈C+ 1 (S d ) g∈C+

inf

h∈C 1 (S d ), h≥m

{f − hLpµ + t∇hC(S d ) }.

Corollary 3.4. Under the hypothesis and notations in Theorem 3.2, we have the estimate   d · cp 1 ·√ , f − Dn,μ (f )Lpµ ≤ c · K f ; c n Lpµ where cp > 0 is a constant that depends only p and c is given by Theorem 3.2. Proof. Let μ = g ◦ M ∈ D(BS d ). By relation (3) we easily obtain c(α, μ) ≤ g  (0) · c(α, M ) for all α, which immediately implies Dn,μ (F ) ≤ Dn,M (F ), for any F ≥ 0. Therefore, replacing here F (t) by |ϕi (x)−ϕi (t)|, taking the inequality at the power p, integrating it with respect to μ, applying again (3) and taking into account the estimate in Lemma 2.1 in [3] too, we are easily lead to the inequality Dn,μ (|ϕi (x) − ϕi (·)|)Lpµ ≤ [g  (0)](p+1)/p · Dn,M (|ϕi (x) − ϕi (·)|)LpM ≤ [g  (0)](p+1)/p · cp · n−1/2 , which leads to Δn,p ≤ d · cp n−1/2 , for all n ∈ N, where cp > 0 depends only on p. Combining with Theorem 3.2, the conclusion is immediate.  Remark 3.5. The results in this paper show that for a large subclass of Bernstein–Durrmeyer–Choquet operators we can obtain similar orders in Lp approximation to those obtained for the classical Bernstein–Durrmeyer operators. Combined with the fact that the results of pointwise and uniform approximation in [9] (obtained without the restrictions on the monotone and submodular set function μ in Theorem 3.2 of this paper) present also the advantage that can give better approximation orders than the classical Durrmeyer kind

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Results Math

operators, we may conclude that the Bernstein–Durrmeyer–Choquet operators are of interest in approximation theory. Furthermore, together with the results obtained in [7] too, concerning the construction of approximation operators through the classical Feller scheme, but involving the Choquet integral instead of the classical integral, they suggest that the approximation properties of all classical integral operators can be extended and in many cases improved, for their Choquet kind correspondents.

References [1] Berdysheva, E.E.: Uniform convergence of Bernstein–Durrmeyer operators with respect to arbitrary measure. J. Math. Anal. Appl. 394, 324–336 (2012) [2] Berdysheva, E.E.: Bernstein–Durrmeyer operators with respect to arbitrary measure II: pointwise convergence. J. Math. Anal. Appl. 418, 734–752 (2014) [3] Berdysheva, E.E., Li, B.-Z.: On Lp -convergence of Bernstein–Durrmeyer operators with respect to arbitrary measure. Publ. Inst. Math. (Beograd) 96(110), 23–29 (2014) [4] Cerd` a, J., Mart´ın, J., Silvestre, P.: Capacitary function spaces. Collect. Math. 62, 95–118 (2011) [5] Choquet, G.: Theory of capacities. Ann. Inst. Fourier (Grenoble) 5, 131–295 (1954) [6] Denneberg, D.: Non-additive Measure and Integral. Kluwer, Dordrecht (1994) [7] Gal, S.G.: Approximation by Choquet integral operators. Ann. Matem Pure Appl. 195(3), 881–896 (2016) [8] Gal, S.G., Opris, B.D.: Uniform and pointwise convergence of Bernstein– Durrmeyer operators with respect to monotone and submodular set functions. J. Math. Anal. Appl. 424, 1374–1379 (2015) [9] Gal, S.G., Trifa, S.: Quantitative estimates in uniform and pointwise approximation by Bernstein–Durrmeyer–Choquet operators. Carpath. J. Math. 33(1), 49–58 (2017) [10] Li, B.-Z.: Approximation by multivariate Bernstein–Durrmeyer operators and learning rates of least-square regularized regression with multivariate polynomial kernel. J. Approx. Theory 173, 33–55 (2013) [11] Wang, R.S.: Some inequalities and convergence theorems for Choquet integrals. J. Appl. Math. Comput. 35, 305–321 (2011) [12] Wang, Z., Klir, G.J.: Generalized Measure Theory. Springer, New York (2009) Sorin G. Gal Department of Mathematics and Computer Science University of Oradea Universitatii Street No.1 410087 Oradea Romania e-mail: [email protected]

Quantitative Estimates in Lp -Approximation Sorin Trifa Faculty of Mathematics and Computer Science Babes-Bolyai University Str. Kogalniceanu No. 1 400084 Cluj-Napoca Romania e-mail: [email protected] Received: November 28, 2016. Accepted: September 22, 2017.