This paper aims to two-dimensional extension of some univariate positive approximation processes expressed by series. To be easier to use, we also modify this extension into finite sums. With respect to these two new classes designed, we investigate their approximation properties in polynomial weighted spaces. The rate of convergence is established, and special cases of our construction are highlighted.

1. Introduction

The approximation of functions by using linear positive operators is currently under research. Usually, two types of positive approximation processes are used: the discrete, respectively, continuous form. In the first case, they often are designed through a series. Since the construction of such operators requires an estimation of infinite sums, this restricts the operator usefulness from the computational point of view. In this respect, in order to approximate a function, it is interesting to consider partial sums, the number of terms considered in sum depending on the function argument. Roughly speaking, these discrete operators are truncated fading away their “tails.” Thus, they become usable for generating software approximation of functions. Among the pioneers who approached this direction we mention Gróf [1] and Lehnhoff [2]. In the same direction a class of univariate linear positive operators is investigated in [3].

This work focuses on a general bivariate class of discrete positive linear operators expressed by infinite sums. This class acts in polynomial weighted spaces of continuous functions of two variables defined by ℝ₊ × ℝ₊, where ℝ₊ = [0, ∞). By using a certain modulus of smoothness we give theorems on the degree of approximation. Further, we replace the infinite sum by a truncated one, and we study the approximation properties of the new defined family of operators. Compared to what has been done so far, the strengths of this paper consist in using general classes of two-dimensional discrete operators, implying an arbitrary network of nodes. Finally we present some particular classes of operators that can be obtained from our family.

2. The Operators

Our linear and positive operators have the role to approximate functions defined on ℝ₊ × ℝ₊. Therefore, on this domain we define for every (m, n) ∈ ℕ × ℕ a net of form Δ_m,n = Δ_1,m × Δ_2,n, where Δ_1,m (0 = x_m,0 < x_m,1 < ⋯) and Δ_2,n (0 = y_n,0 < y_n,1 < ⋯). Set ℕ₀ = {0} ∪ ℕ, and C(ℝ₊) stands for the space of all real-valued continuous functions on ℝ₊.

Products of parametric extensions of two univariate operators are appropriate tools to approximate functions of two variables. For this reason, the starting point is given by the following one-dimensional operators:

(1)

where a_m,i, b_n,j are nonnegative functions belonging to C(ℝ₊), (i, j) ∈ ℕ₀ × ℕ₀, such that the following identities

(2)

take place. These conditions mean that the operators A_m and B_n preserve the monomial e₀, e₀(t) = 1, a property often seen at classical linear positive operators.

For each z ∈ ℝ₊, define the function φ_z by φ_z(t) = t − z, t ∈ ℝ₊. Also, for each r ∈ ℕ, set

(3)

representing the rth central moment of the specified operators.

For a simplified writing, we will use the common notation L_s, where L_s = A_s for all s ∈ ℕ or L_s = B_s for all s ∈ ℕ.

For our purposes, relative to the central moments, we require additional conditions. For each r ∈ ℕ, ℳ_r(L_s; t) as a function of t is bounded by a polynomial of degree at most r. Moreover, s^r/2ℳ_r(L_s; ·) is bounded with respect to s. These requirements can be brought together and redrafted in the following way: for each r ∈ ℕ, a polynomial Γ_r exists such that

(4)

Apparently is a tough condition, but the examples that we give in the last section show that it is carried out by different classes of operators.

In what follows we specify the function spaces in which the operators act.

For univariate operators A_s, B_s we consider the space C_p(ℝ₊), p ∈ ℕ₀ fixed, consisting of all real-valued functions f continuous on ℝ₊ such that w_pf is uniformly continuous and bounded on ℝ₊, where the weight w_p is defined as follows:

(5)

The space is endowed with the norm ∥·∥_p, ∥f∥_p = sup _x≥0 w_p(x) | f(x)|.

For bivariate operators we consider the space

(6)

associated to the weighted function w_p,q(x, y) = w_p(x)w_q(y), (p, q) ∈ ℕ₀ × ℕ₀. The norm of this space is denoted by ∥·∥_p,q and is defined by

(7)

These spaces are ordered with respect to inclusion as follows: if (p, q)≤(p^′, q^′), then

. Moreover,

for any

. Examining the weight w_p,q, it is easy to verify the inequality

(8)

for any x ≥ 0 and t ≥ 0.

Indeed, based on the first mean value theorem for integration, between x and t, a point ξ_x,t exists such that

(9)

Starting from (1), for each (m, n) ∈ ℕ × ℕ we introduce a linear positive operator in polynomial weighted space

as follows:

(10)

If the function

can be decomposed in the following manner f(x, y) = f₁(x)f₂(y),

, then one has

(11)

For example, we get L_m,nw_p,q = (A_mw_p)(B_nw_q).

In order to present the rate of convergence for our bivariate operators, we use a modulus of smoothness associated to any function f belonging to

. It is given by the formula

(12)

where

(13)

for (x, y) and (u, v) belonging to

. Alternative notation is ω(f; x, y). More information about moduli of smoothness can be found in the monograph [4].

Further, we indicate a truncated variant of operators defined at (10). Let (u_s) _s≥1, (v_s) _s≥1 be strictly increasing sequences of positive numbers such that

(14)

Taking in view the net Δ_1,m, we partitioned the set ℕ₀ into two parts

(15)

Similarly, via the network Δ_2,n, we introduce J(y, v_n) and .

For each (m, n) ∈ ℕ × ℕ and any

we define the linear positive operators

(16)

3. Auxiliary Results

Throughout the paper, by c(·) we denote different real constants, in the brackets specifying the parameter(s) that the indicated constant depends.

At first we collect some useful results relative to the one-dimensional operators L_s where L_s = A_s (s ∈ ℕ) or L_s = B_s (s ∈ ℕ).

Lemma 1. Let p ∈ ℕ₀, and let the weight w_p be given by (5). The operator L_s satisfies

(17)

(18)

where t ≥ 0 and s ∈ ℕ.

Proof. Let p ∈ ℕ₀ and s ∈ ℕ be fixed.

(i) By using (5), (2), (3), and (4) we can write

(19)

Since deg (t^p−kΓ_k(t)) ≤ p, the previous expression is bounded with respect to t ∈ ℝ₊, and inequality (17) follows.

(ii) To be more explicit we consider L_s = A_s

(20)

By using (17) we obtain the first inequality of the relation (18). Further, applying sup _t≥0 , the second inequality is proved.

Lemma 2. Let p ∈ ℕ₀, and let the weight w_p be given by (5). For any s ∈ ℕ the operator L_s satisfies

(21)

(22)

where φ_t(x) = x − t and

. The polynomials Γ_ν, ν ≥ 2, are introduced by (4).

Proof. Let p ∈ ℕ₀ and s ∈ ℕ be fixed.

(i) Using the identity
(23)
we can write the following:
(24)

During the previous relations we used notation (3) and hypothesis (4). Considering the significance of , relation (21) follows.

(ii) Taking into consideration relation (1), we apply the Cauchy-Schwarz inequality, and this allows us to write
(25)
Relations (17) and (21) imply (22).

We mention that with the help of the same Cauchy-Schwarz inequality, relations (2) and (4) lead us to the following inequality:

(26)

Lemma 1 leads to the following result.

Lemma 3. Let (p, q) ∈ ℕ₀ × ℕ₀. For any (m, n) ∈ ℕ × ℕ, the operator L_m,n given by (10) verifies

(27)

(28)

Proof. The first statement follows immediately from the definition of the weight w_p,q and relations (11), (17).

Regarding the second statement, based on (10), we get

(29)

Applying and taking into account (27), one obtains (28).

In our investigation we appeal to the Steklov function. This can be used to approximate continuous functions by smoother functions. The Steklov function associated with

is given as follows:

(30)

where h > 0 and δ > 0. By using (13) we deduce

(31)

In the next lemma we have gathered some known properties of Steklov function f_h,δ, where . These properties establish connections between f_h,δ and the modulus ω_f indicated at (12). For the sake of completeness we present the proofs of these inequalities.

Lemma 4. Let f belong to , and let f_h,δ be defined by (30). The following relations take place:

(32)

(33)

where h > 0, δ > 0, and ω_f is defined by (12).

Proof. Let h > 0 and δ > 0 be arbitrarily fixed.

(i) For u ∈ [0, h] and v ∈ [0, δ] we deduce

(34)

On the other hand,

(35)

Keeping in mind (31), we consequently obtain

(36)

and (32) is completed.

(ii) We justify only the first inequality, and the second inequality can be proven in the same manner.

Occurs

(37)

(see (13)). Further, we can write

(38)

Since 0 ≤ v ≤ δ, it is clear that , and we obtain

(39)

In the same manner we show I₂ ≤ (1/h)ω_f(h, δ). Returning at (38), the proof is ended.

In the following we denote by the space of all functions having the first order partial derivatives such that the functions ∂g/∂x, ∂g/∂y, and g belong to .

Lemma 5. Let (p, q) ∈ ℕ₀ × ℕ₀. If , then for any (m, n) ∈ ℕ × ℕ the operator L_m,n given by (10) verifies

(40)

where

(41)

the polynomials Γ_ν,

, being indicated at (4).

Proof. Let and (m, n) ∈ ℕ² be arbitrarily fixed. Since , for any we can write

(42)

Since L_m,n is linear monotone and reproduces the constants, from the previous identity we obtain

(43)

Further, one has

(44)

see (8). Applying L_m,n and by using successively (11), (26), (17), and (22) we have

(45)

where c(p, q) is a suitable constant. Following the same pathway, we find

(46)

Considering the increases established for J₁, J₂ and returning to the relation (43), the inequality (40) is completely proven.

4. Main Results

The rate of convergence for L_m,n operator will be read as follows.

Theorem 6. Let (p, q) ∈ ℕ₀ × ℕ₀. For any (m, n) ∈ ℕ × ℕ, the operator L_m,n given by (10) satisfies

(47)

, where Φ is given by (41) and c(p, q) is a suitable constant.

Proof. Setting

(48)

we can write

(49)

We establish upper bounds for these three quantities. Relations (28) and (32) imply that

(50)

For T₂ we use Lemma 5 choosing ; see definition (31).

One has

(51)

(see also (33)). Finally, inequality (32) implies

(52)

Setting and coming back to (49), we can affirm that a certain constant c(p, q) exists such that (47) holds.

Knowing that the modulus ω_f enjoys the property , from Theorem 6 we deduce the following result.

Theorem 7. Let (p, q) ∈ ℕ₀ × ℕ₀, and let the operators L_m,n, (m, n) ∈ ℕ × ℕ, be defined by (10).

For any the pointwise convergence takes place

(53)

If K₁, K₂ are compact intervals included in ℝ₊, then (53) holds uniformly on the domain K₁ × K₂.

Theorem 8. Let (p, q) ∈ ℕ₀ × ℕ₀, and let the operators , (m, n) ∈ ℕ × ℕ, be defined by (16). For any the pointwise convergence takes place

(54)

If K₁, K₂ are compact intervals included in ℝ₊, then (54) holds uniformly on the domain K₁ × K₂.

Proof. Let be arbitrarily fixed. Taking in view the partitions of ℕ₀ (see (15)), we use the following decomposition:

(55)

Setting

(56)

we can write

(57)

If we show , k ∈ {1,2, 3}, then, based on (53), our statement (54) follows, and the proof is ended.

Further, we prove the previous limit only for k = 1, other two following similar routes.

Since , a constant M_f exists such that

(58)

Based on the classical inequality (a + b) ^s ≤ 2^s−1(a^s + b^s), a ≥ 0, b ≥ 0, s ∈ ℕ₀, we deduce

(59)

Consequently, (58) implies that

(60)

where

Using this relation we have

(61)

We establish an upper bound for S₁. Since , clearly

(62)

and we can write

(63)

(see (26) and (4)). Arguing similarly we get

(64)

Considering (14), relation (61) leads to the claimed result.

5. Particular Cases

In presenting these cases, we are looking for one-dimensional linear operators that verify conditions (2) and (4).

(1) Baskakov operators [5] are given by the formula

(65)

where V_ne₀ = e₀, hence (2) fulfilled. Since V_n reproduces the monomial e₁, e₁(t) = t, its first central moment ℳ₁(V_n; ·) is null.

For r ≥ 2, the rth central moment is given as follows [6, Lemma 4]:

(66)

where δ_r = 1 if r is odd, δ_r = 0 if r is even, and b_n,r,j are positive coefficients bounded with respect to n. In particular, ℳ_r(V_n; x) is a polynomial of degree r without a constant term. These properties ensure that condition (4) is achieved.

The study of these operators in polynomial weighted spaces was carried out in [6]. Choosing in (10) A_m = V_m and B_n = V_n we obtain the Baskakov operator for functions of two variables. The net is Δ_m,n = (i/m, j/n) _i,j≥0. Our results indicated at (47) and (53) are identified with the results established by Gurdek et al. [7, Equations (22), (28)].

The univariate truncated operators has been approached in [8]. The truncated version specified in (16) coincides with the operators studied by Walczak [9, Equation (17)]. In this case I(x, u_m) from (15) becomes {i ∈ ℕ₀ : i ≤ [m(x + u_m)]}. Here [λ] indicates the largest integer not exceeding λ.

(2) Szász operators [10] are of the form:

(67)

One has S_ne₀ = e₀ and ℳ₁(S_n; x) = 0. The central moments have the following form:

(68)

where q_j, p_j are constants; see, for example, [11, Equations (9.5.10)-(9.5.11)]. For this class, conditions (2) and (4) are achieved.

The research of S_n, n ∈ ℕ, operators in polynomial weighted spaces has appeared in [6]. The truncated univariate Szász operators and another extension to functions of two variables in weighted spaces have been considered in [2] and [12], respectively. In the latter paper instead w_p,q was used the weight ρ, ρ(x, y) = 1 + x² + y².

Our theorems of the previous section lead us to two-dimensional versions of genuine Szász operators and of their truncated form. In this case the net is Δ_m,n = (i/m, j/n) _i,j≥0.

The next example comes from the world of Quantum Calculus which, in the past two decades, has gained popularity in the construction of linear approximation processes. We choose a q-analogue of Szász-Mirakjan operators recently introduced and studied by Mahmudov [13].

(3) q-Szász-Mirakjan-Mahmudov operators. Let q > 1 and n ∈ ℕ. For f : ℝ₊ → ℝ one defines the operator

(69)

where

. We recall the standard notations in q-calculus. For n ∈ ℕ consider

(70)

Also, [0] = 0 and [0]! = 1. In [13] it was proved that M_n,qe₀ = e₀ and M_n,q is a linear positive operator from C_p(ℝ₊) to C_p(ℝ₊) for any p ∈ ℕ₀. Withal, for all moments M_n,qe_r, e_r(t) = t^r, explicit formulas were given as follows:

(71)

where

(72)

represent q-analogue of Stirling numbers; see [13, Lemma 2.6]. One can see that (M_n,qe_r)(x) is a polynomial in x of degree r without a constant term, and it is bounded with respect to [n] _q. These properties are transferred to the central moments

, and consequently (4) takes place, provided to replace n with [n] _q. To construct two-dimensional operators of the form (10) and (16) we choose A_m = M_m,q, B_n = M_n,q, and the network will be

(73)

In time were carried out q-analogues of these operators not only for q > 1 but for the case q ∈ (0,1); see, for example, [14, 15]. Extensions of these classes of operators by our method also work there.

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Bivariate Positive Operators in Polynomial Weighted Spaces

Abstract

1. Introduction

2. The Operators

3. Auxiliary Results

4. Main Results

5. Particular Cases

References

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Bivariate Positive Operators in Polynomial Weighted Spaces

Abstract

1. Introduction

2. The Operators

3. Auxiliary Results

4. Main Results

5. Particular Cases

References

Citing Literature

References

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