PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 96 (110) (2014), 159–168 DOI: 10.2298/PIM1410159G

LAGRANGE-TYPE OPERATORS ASSOCIATED WITH U_{n}^{ϱ}

Heiner Gonska, Ioan Raşa, and Elena-Dorina Stănilă

Abstract. We consider a class of positive linear operators which, among others, constitute a link between the classical Bernstein operators and the genuine Bernstein–Durrmeyer mappings. The focus is on their relation to certain Lagrange-type interpolators associated to them, a well known feature in the theory of Bernstein operators. Considerations concerning iterated Boolean sums and the derivatives of the operator images are included. Our main tool is the eigenstructure of the members of the class.

1. Introduction

The present note continues the authors’ research on a class U_{n}^{ϱ} of Bernsteintype operators which constitute a link between the classical Bernstein operators B_{n}(ϱ → ∞) and the so-called genuine Bernstein–Durrmeyer operators U_{n}(ϱ = 1). This class of operators was introduced by Păltănea in [12] and has since then been the subject of several papers dealing with various aspects of the matter. A detailed description is given in the next section. Here we present their relationship to the Lagrange interpolation using the eigenstructure of the U_{n}^{ϱ}, thus extending in a natural way results known for B_{n}. The eigenstructure is also useful to describe the convergence behavior of iterated Boolean sums based on a single mapping U_{n}^{ϱ}, ϱ and n fixed. In the final Section 4 a relationship between certain divided differences used in Section 2 and the representation of the derivatives (U_{n}^{ϱ})^{(j)} is established.

2. Lagrange type operators associated with U_{n}^{ϱ}

2.1. A first description of L_{n}^{ϱ}. Let ϱ > 0 and n ⩾ 1 be fixed. Consider the functionals F_{n,k}^{ϱ} : C[0, 1] → ℝ, k = 0, 1, …, n, defined by

2010 Mathematics Subject Classification: Primary: 41A36; Secondary: 41A05.

Key words and phrases: Bernstein operators, genuine Bernstein–Durrmeyer operators, Păltănea operators, Lagrange interpolation, eigenstructure, iterated Boolean sum, representation of derivatives.

Dedicated to Professor Giuseppe Mastroianni on the occasion of his 75th birthday.

F_{n,0}^{ϱ} (f) = f(0), F_{n,n−1}^{ϱ} (f) = f(1),
F_{n,k}^{ϱ} (f) = ∫_{0}^{1} \frac{t^{kϱ−1} (1 − t)^{(n−k)ϱ−1}}{B(kϱ, (n − k)ϱ)} f(t) dt, k = 1, …, n − 1.

The operator U_{n}^{ϱ} : C[0, 1] → Π_{n} (the space of algebraic polynomials of degree no greater than n) is given by U_{n}^{ϱ}(f; x) := ∑ _{k=0}^{n} F_{n,k}^{ϱ} (f)p_{n,k} (x), f ∈ C[0, 1], where p_{n,k} (x) = \binom{n}{k}x^{k}(1 − x)^{n−k} , x ∈ [0, 1]. With a slight abuse of notation consider also the operator U_{n}^{ϱ} : Π_{n} → Π_{n}. Its eigenvalues λ_{n,k}^{(n)} and eigenfunctions p_{n,k}^{(n)} , k = 0, 1, …, n, are described in [4]; in particular,

1 = λ_{ϱ,0}^{(n)} = λ_{ϱ,1}^{(n)} > λ_{ϱ,2}^{(n)} > λ_{ϱ,3}^{(n)} > ⋯> λ_{ϱ,n}^{(n)} > 0,

which means that U_{n}^{ϱ} : Π_{n} → Π_{n} is invertible. Consider the inverse operator (U_{n}^{ϱ})^{−1} : Π_{n} → Π_{n} (note the domain of definition here!) and define L_{n}^{ϱ} : C[0, 1] → Π_{n} by

(2.1)  L_{n}^{ϱ} = (U_{n}^{ϱ})^{−1} ∘ U_{n}^{ϱ}.

Then U_{n}^{ϱ}(L_{n}^{ϱ}f) = U_{n}^{ϱ}(f), f ∈ C[0, 1], which leads to

(2.2)  F_{n,k}^{ϱ} (L_{n}^{ϱ}f) = F_{n,k}^{ϱ} (f), f ∈ C[0, 1], k = 0, 1, …, n.

Relation (2.2) expresses an interpolatory property with respect to the functionals F_{n,0}^{ϱ} , …, F_{n,n}^{ϱ} ; more precisely, given f ∈ C[0, 1], L_{n}^{ϱ}f is the unique polynomial in Π_{n} satisfying (2.2). In particular, L_{n}p = p, for all p ∈ Π_{n}. It is known [3] that

(2.3)  lim_{ϱ→∞} F_{n,k}^{ϱ} (f) = f(k/n), f ∈ C[0, 1], k = 0, 1, …, n.

This entails

(2.4)  lim_{ϱ→∞} U_{n}^{ϱ}(f) = B_{n}f, uniformly on [0, 1],

for all f ∈ C[0, 1]; here B_{n} denotes the classical Bernstein operator on C[0, 1]. Let L_{n} be the Lagrange operator on C[0, 1] based on the nodes 0, \frac{1}{n} , …, \frac{n−1}{n} , 1. It is easy to see that

(2.5)  L_{n} = B_{n}^{−1} ∘ B_{n},

where C[0, 1] −⟶^{B_{n}} Π_{n} −−−⟶^{(B_{n} )^{−1}} Π_{n}. Similar to the above, the symbol (B_{n})^{−1} means the inverse of B_{n} : Π_{n} → Π_{n}. We will see that

(2.6)  lim_{ϱ→∞} L_{n}^{ϱ}(f) = L_{n}f, uniformly on [0, 1],

for all f ∈ C[0, 1]. If we interpret (2.4) by saying that U_{n}^{∞} = B_{n}, then (2.6) can be interpreted as L_{n}^{∞} = L_{n}. On the other hand, one has (see [4, (3.3)])

(2.7)  U_{n}^{ϱ}f = ∑_{k=0}^{n} λ_{ϱ,k}^{(n)} p_{ϱ,k}^{(n)} μ_{ϱ,k}^{(n)} (f), f ∈ C[0, 1],

LAGRANGE-TYPE OPERATORS ASSOCIATED WITH U_{n}^{ϱ} 161

where μ_{ϱ,k}^{(n)} are the dual functionals of p_{ϱ,k}^{(n)} . This leads to

L_{n}^{ϱ}(f) = (U_{n}^{ϱ})^{−1} (U_{n}^{ϱ}f) = ∑_{k=0}^{n} λ_{ϱ,k}^{(n)} \frac{1}{λ_{ϱ,k}^{(n)}} p_{ϱ,k}^{(n)} μ_{ϱ,k}^{(n)} (f), i.e.,

L_{n}^{ϱ}(f) = ∑_{k=0}^{n} p_{ϱ,k}^{(n)} μ_{ϱ,k}^{(n)} (f), f ∈ C[0, 1]. (2.8)

So the relationship between U_{n}^{ϱ} and L_{n}^{ϱ}, expressed by (2.7) and (2.8), is similar to the relationship between B_{n} = U_{n}^{∞} and L_{n} = L_{n}^{∞} , described in [1, Sect. 6].

To conclude this section let us recall that U_{n}^{ϱ} = B_{n} ∘ \overline{𝔹}_{nϱ} where

\overline{𝔹}_{r}(f; x) = \{ \begin{matrix}f(0), & x = 0; \\ \frac{1}{B(rx, r − rx)} ∫_{0}^{1} t^{rx−1} (1 − t)^{r−rx−1} f(t) dt , & 0 < x < 1; \\ f(1), & x = 1. \end{matrix}

is the Lupaş–Mühlbach Beta operator (see [7, p. 63], [11]). From (2.1) and (2.5) it follows

(2.9)  L_{n}^{ϱ} = (\overline{𝔹}_{nϱ})^{−1} ∘ L_{n} ∘ \overline{𝔹}_{nϱ} , ϱ > 0,

i.e., the operators L_{n}^{ϱ} and L_{n} are similar.

2.2. A concrete approach to L_{n}^{ϱ} . In order to obtain other representations of the operators L_{n}^{ϱ} we shall use a classical method described, for example, in [13, Sect. 1.2], [2], [8, Sect. 1.3]. Let n ⩾ 1, ϱ > 0 and f ∈ C[0, 1] be fixed. Then L_{n}^{ϱ}f ∈ Π_{n} has the form L_{n}^{ϱ}f = c_{0}e_{0} +c_{1}e_{1} +⋯ +c_{n}e_{n}, where e_{j} (x) = x^{j}, x ∈ [0, 1], j ⩾ 0, and c_{j} ∈ ℝ. According to (2.2), the coefficients c_{0}, …, c_{n} satisfy the system of equations

L_{n}^{ϱ}f = c_{0}e_{0} + c_{1}e_{1} + ⋯+ c_{n}e_{n}
F_{n,0}^{ϱ} (f) = c_{0}F_{n,0}^{ϱ} (e_{0}) + c_{1}F_{n,0}^{ϱ} (e_{1}) + ⋯+ c_{n}F_{n,0}^{ϱ} (e_{n})
…
F_{n,n}^{ϱ} (f) = c_{0}F_{n,n}^{ϱ} (e_{0}) + c_{1}F_{n,n}^{ϱ} (e_{1}) + ⋯+ c_{n}F_{n,n}^{ϱ} (e_{n}).

By eliminating c_{0}, …, c_{n}, we get

(2.10)  |\begin{matrix}L_{n}^{ϱ}f & e_{0} & e_{1} & … & e_{n} \\ F_{n,0}^{ϱ} (f) & F_{n,0}^{ϱ} (e_{0}) & F_{n,0}^{ϱ} (e_{1}) & … & F_{n,0}^{ϱ} (e_{n}) \\ … & … & … & … & … \\ F_{n,n}^{ϱ} (f) & F_{n,n}^{ϱ} (e_{0}) & F_{n,n}^{ϱ} (e_{1}) & … & F_{n,n}^{ϱ} (e_{n}) \end{matrix}| = 0

Since F_{n,j}^{ϱ} (e_{m}) = (iϱ)\overline{m} /(nϱ)\overline{m}, where by x\overline{k} = x(x + 1) · · · ·· (x + k − 1) we have denoted the rising factorial, from (2.10) we get after elementary computations (2.11)

L_{n}^{ϱ}f = −V ( 0, \frac{1}{n}, …, \frac{n − 1}{n} , 1)^{−1} |\begin{matrix}0 & e_{0} & \frac{(nϱ)¯1}{(nϱ)} e_{1} & … & \frac{(nϱ)n̄}{(nϱ)^{n}} e_{n} \\ F_{n,0}^{ϱ} (f) & 1 & \frac{(0ϱ)¯1}{(nϱ)} & … & \frac{(0ϱ)n̄}{(nϱ)^{n}} \\ F_{n,1}^{ϱ} (f) & 1 & \frac{(1ϱ)¯1}{(nϱ)} & … & \frac{(1ϱ)n̄}{(nϱ)^{n}} \\ … & … & … & … & … \\ F_{n,n}^{ϱ} (f) & 1 & \frac{(nϱ)¯1}{(nϱ)} & … & \frac{(nϱ)n̄}{(nϱ)^{n}} \end{matrix} |

where V is the Vandermonde determinant. Now we are in the position to prove (2.6).

Theorem 2.1. For each f ∈ C[0, 1] we have

lim_{ϱ→∞} L_{n}^{ϱ}f = L_{n}f, uniformly on [0, 1].

Proof. Let us remark that

(2.12)  lim_{ϱ→∞} \frac{(jϱ)k̄}{(nϱ)^{k}} = ( \frac{j}{n})^{k}.

From (2.3), (2.11), and (2.12) we deduce (2.13)

lim_{ϱ→∞} L_{n}^{ϱ}f = −V ( 0, \frac{1}{n}, …, \frac{n − 1}{n} , 1)^{−1} |\begin{matrix}0 & e_{0} & e_{1} & … & e_{n} \\ f(0) & 1 & 0 & … & 0 \\ f(\frac{1}{n}) & 1 & \frac{1}{n} & … & (\frac{1}{n})^{n} \\ … & … & … & … & … \\ f(\frac{n−1}{n} ) & 1 & \frac{n−1}{n} & … & (\frac{n−1}{n} )^{n} \\ f(1) & 1 & 1 & … & 1 \end{matrix} | .

Since the right-hand side of (2.13) is L_{n}f (see, e.g., [15, Sect. 3.1], [8, Sect. 1.3]), the proof is complete. □

2.3. The associated divided difference. The coefficient of e_{n} in the expression of L_{n}f is the divided difference of f at the nodes 0, \frac{1}{n} , …, \frac{n−1}{n} , 1, and is given by (see e.g. [15, Sect. 2.6]):

(2.14) [0, \frac{1}{n}, …, \frac{n − 1}{n} , 1; f] = V ( 0, \frac{1}{n}, …, \frac{n − 1}{n} , 1)^{−1}

× |\begin{matrix}1 & 0 & 0 & … & 0 & f(0) \\ 1 & \frac{1}{n} & (\frac{1}{n})^{2} & … & (\frac{1}{n})^{n−1} & f(\frac{1}{n}) \\ … & … & … & … & … & … \\ 1 & \frac{n−1}{n} & (\frac{n−1}{n} )^{2} & … & (\frac{n−1}{n} )^{n−1} & f(\frac{n−1}{n} ) \\ 1 & 1 & 1 & … & 1 & f(1) \end{matrix} | .

Let us denote by [F_{n,0}^{ϱ} , F_{n,1}^{ϱ} , …, F_{n,n}^{ϱ} ; f] the coefficient of e_{n} in L_{n}^{ϱ}f.

LAGRANGE-TYPE OPERATORS ASSOCIATED WITH U_{n}^{ϱ} 163

Theorem 2.2. For each f ∈ C[0, 1] we have

(2.15) [F_{n,0}^{ϱ} , F_{n,1}^{ϱ} , …, F_{n,n}^{ϱ} ; f] = \frac{(nϱ)n̄}{(nϱ)^{n}} V ( 0, \frac{1}{n}, …, \frac{n − 1}{n} , 1)^{−1}

× |\begin{matrix}1 & 0 & 0 & … & 0 & F_{n,0}^{ϱ} (f) \\ 1 & \frac{1}{n} & (\frac{1}{n})^{2} & … & (\frac{1}{n})^{n−1} & F_{n,1}^{ϱ} (f) \\ … & … & … & … & … & … \\ 1 & \frac{n−1}{n} & (\frac{n−1}{n} )^{2} & … & (\frac{n−1}{n} )^{n−1} & F_{n,n−1}^{ϱ} (f) \\ 1 & 1 & 1 & … & 1 & F_{n,n}^{ϱ} (f) \end{matrix} | .

Proof. From (2.11) we get immediately [F_{n,0}^{ϱ} , F_{n,1}^{ϱ} , …, F_{n,n}^{ϱ} ; f]

= \frac{(nϱ)n̄}{(nϱ)^{n}} V ( 0, \frac{1}{n}, …, \frac{n − 1}{n} , 1) ^{−1} |\begin{matrix}1 & \frac{(0ϱ)¯1}{nϱ} & … & \frac{(0ϱ)\overline{n−1}}{(nϱ)^{n−1}} & F_{n,0}^{ϱ} (f) \\ 1 & \frac{(1ϱ)¯1}{nϱ} & … & \frac{(1ϱ)\overline{n−1}}{(nϱ)^{n−1}} & F_{n,1}^{ϱ} (f) \\ … & … & … & … & … \\ 1 & \frac{(nϱ)¯1}{nϱ} & … & \frac{(nϱ)\overline{n−1}}{(nϱ)^{n−1}} & F_{n,n}^{ϱ} (f) \end{matrix} |

= \frac{(nϱ)n̄/(nϱ)^{n}}{(nϱ)^{\frac{n(n−1)}{2}} V (0, \frac{1}{n} , …, \frac{n−1}{n} , 1)} |\begin{matrix}1 & (0ϱ)¯1 & … & (0ϱ)\overline{n−1} & F_{n,0}^{ϱ} (f) \\ 1 & (1ϱ)¯1 & … & (1ϱ)\overline{n−1} & F_{n,1}^{ϱ} (f) \\ … & … & … & … & … \\ 1 & (nϱ)¯1 & … & (nϱ)\overline{n−1} & F_{n,n}^{ϱ} (f) \end{matrix} |

= \frac{(nϱ)n̄/(nϱ)^{n}}{(nϱ)^{\frac{n(n−1)}{2}} V (0, \frac{1}{n} , …, \frac{n−1}{n} , 1)} |\begin{matrix}1 & 0 & … & 0 & F_{n,0}^{ϱ} (f) \\ 1 & ϱ & … & ϱ^{n−1} & F_{n,1}^{ϱ} (f) \\ … & … & … & … & … \\ 1 & nϱ & … & (nϱ)^{n−1} & F_{n,n}^{ϱ} (f) \end{matrix} |

and this leads to (2.15). □

Remark 2.1. From (2.3), (2.14) and (2.15) we derive

lim_{ϱ→∞} [F_{n,0}^{ϱ} , F_{n,1}^{ϱ} , …, F_{n,n}^{ϱ} ; f] = [0, \frac{1}{n}, …, \frac{n − 1}{n} , 1; f]

for all f ∈ C[0, 1]. Moreover, let f ∈ C[0, 1] and Φ_{n} a (Lagrange type) polynomial with Φ_{n} ∈ Π_{n}, Φ_{n}(\frac{j}{n}) = F_{n,j}^{ϱ} (f), j = 0, …, n. From (2.14) and (2.15) it is easy to deduce

[F_{n,0}^{ϱ} , F_{n,1}^{ϱ} , …, F_{n,n}^{ϱ} ; f] = \frac{(nϱ)n̄}{(nϱ)^{n}} [0, \frac{1}{n}, …, \frac{n − 1}{n} , 1; Φ_{n}].

The last (classical) divided difference can be computed by recurrence; see [15].

Remark 2.2. Using (2.8), we see that

(2.16)  μ_{ϱ,n}^{(n)} = [F_{n,0}^{ϱ} , F_{n,1}^{ϱ} , …, F_{n,n}^{ϱ} ; ·].

For the Bernstein operator (i.e., for ϱ → ∞), (2.16) can be found in [1, p. 164] .

Remark 2.3. u_{n+1}^{ϱ} := e_{n+1} − L_{n}^{ϱ}e_{n+1} is the unique monic polynomial in Π_{n+1}

such that L_{n}^{ϱ}u_{n+1}^{ϱ} = 0. For example, u_{n+1}^{∞} = x(x − 1)(x − \frac{1}{n}) · · · ·· (x − \frac{n−1}{n} ). Moreover, u_{n+1}^{1} (x) = x(x−1)J_{n−1} (x), where J_{0}(x), J_{1}(x), … are the monic Jacobi polynomials, orthogonal on [0, 1] with respect to the weight function x(1 − x). Indeed, F_{n,0}^{1} (u_{n+1}^{1} ) = F_{n,n}^{1} (u_{n+1}^{1} ) = 0, and

∫_{0}^{1} t^{k−1} (1 − t)^{n−k−1} u_{n+1}^{1} (t) dt = ∫_{0}^{1} t^{k−1} (1 − t)^{n−k−1} t(t − 1)J_{n−1} (t) dt = 0

(since for all k = 1, …, n − 1, t^{k−1} (1 − t)^{n−k−1} is a polynomial of degree n − 2). This implies F_{n,k}^{1} (u_{n+1}^{1} ) = 0, k = 1, …, n − 1, and so L_{n}^{1}(u_{n+1}^{1} ) = 0. For general values of ϱ we have not found such compact representations.

Now we shall prove a general result.

Theorem 2.3. The polynomial u_{n+1}^{ϱ} has n + 1 distinct roots in [0, 1].

Proof. By using Remark 2.3 and (2.9) we get (\overline{𝔹}_{nϱ}^{−1} ∘ L_{n} ∘ \overline{𝔹}_{nϱ})(u_{n+1}^{ϱ} ) = 0, which entails L_{n}(\overline{𝔹}_{nϱ} u_{n+1}^{ϱ} ) = 0. Now the same Remark 2.3 yields

\overline{𝔹}_{nϱ} u_{n+1}^{ϱ} = \frac{(nϱ)^{n+1}}{(nϱ)\overline{n+1}} u_{n+1}^{∞} .

So \overline{𝔹}_{nϱ} u_{n+1}^{ϱ} has n + 1 distinct roots in [0, 1]. According to [6], u_{n+1}^{ϱ} has at least n + 1 distinct roots in [0, 1]; to finish the proof, it suffices to remark that u_{n+1}^{ϱ} is a polynomial of degree n + 1. □

Now let us recall the representation of L_{n} in terms of the fundamental Lagrange polynomials:

L_{n}f(x) = ∑_{k=0}^{n} l_{n,k} (x)f( \frac{k}{n}) , f ∈ C[0, 1], x ∈ [0, 1].

Using (2.9) we infer that L_{n}^{ϱ} has a similar representation, namely

L_{n}^{ϱ}f(x) = ∑_{k=0}^{n} l_{n,k}^{ϱ} (x)F_{n,k}^{ϱ} (f),

where

(2.17)  l_{n,k}^{ϱ} := \overline{𝔹}_{nϱ}^{−1} (l_{n,k} ), k = 0, 1, …, n.

Theorem 2.4. For each k = 0, 1, …, n, the polynomial l_{n,k}^{ϱ} has n distinct roots in [0, 1].

Proof. Since, according to (2.17), \overline{𝔹}_{nϱ} (l_{n,k}^{ϱ} ) = l_{n,k} , the proof is similar to that of Theorem 2.3 and we omit it. □

In what follows we shall establish mean value theorems for the generalized divided difference and for the remainder R_{n}^{ϱ}f := f − L_{n}^{ϱ}.

Theorem 2.5. Let n ⩾ 1, ϱ > 0 and f ∈ C[0, 1] be given. Then there exist 0 = t_{0} < t_{1} < ⋯< t_{n} = 1 such that

(2.18)  R_{n}^{ϱ}f(t_{i}) = 0, i = 0, 1, …, n.
Proof. According to (2.2), F_{n,k}^{ϱ} (R_{n}^{ϱ}f) = 0, k = 0, 1, …, n, i.e.

R_{n}^{ϱ}f(0) = R_{n}^{ϱ}f(1) = 0, (2.19)

∫_{0}^{1} t^{kϱ−1} (1 − t)^{(n−k)ϱ−1} R_{n}^{ϱ}f(t) dt = 0, k = 1, …, n − 1. (2.20)

Set x := ( \frac{t}{1−t} )^{ϱ}, j := k − 1, and h(x) := R_{n}^{ϱ}f( \frac{x^{1/ϱ}}{1+x^{1/ϱ}} ), x ⩾ 0. Then (2.20) becomes

(2.21)  ∫_{0}^{∞} (1 + x^{1/ϱ})^{−nϱ} x^{j} h(x) dx = 0, j = 0, 1, …, n − 2.

Suppose that the number of the roots of h in (0, +∞) is at most n − 2, i.e.,

{x ∈ (0, +∞) : h(x) = 0} = {x_{1}, …, x_{r}}, r ⩽ n − 2.

Then there exists a polynomial p ∈ Π_{n−2} such that {x ∈ (0, +∞) : p(x) = 0} ⊂ {x_{1}, …, x_{r}} and, moreover,

(2.22)  ∫_{0}^{∞} (1 + x^{1/ϱ})^{−nϱ} p(x)h(x) dx > 0.

Obviously (2.22) contradicts (2.21), which means that h has at least n − 1 roots in (0, +∞). It follows that R_{n}^{ϱ}f has at least n − 1 roots in (0, 1). Together with (2.19), this proves the theorem. □

Corollary 2.1. Let n ⩾ 1, ϱ > 0 and f ∈ C^{n}[0, 1] be given. Then there exists ξ ∈ (0, 1) such that

[F_{n,0}^{ϱ} , F_{n,1}^{ϱ} , …, F_{n,n}^{ϱ} ; f] = \frac{f^{(n)} (ξ)}{n!} .

Proof. According to Theorem 2.5, R_{n}^{ϱ}f has at least n + 1 roots in [0, 1]. It follows that (R_{n}^{ϱ}f)^{(n)} has at least a root ξ ∈ (0, 1). Thus

0 = (R_{n}^{ϱ}f)^{(n)} (ξ) = f^{(n)} (ξ) − n![F_{n,0}^{ϱ} , F_{n,1}^{ϱ} , …, F_{n,n}^{ϱ} ; f],

and the proof is finished. □

Let now n ⩾ 1, ϱ > 0 and f ∈ C^{n+1} [0, 1] be given. Consider the points t_{0}, t_{1}, …, t_{n} satisfying (2.18), and let ω(t) = (t − t_{0}) ⋯ (t − t_{n}).

Corollary 2.2. Let x ∈ [0, 1] ∖ {t_{0}, t_{1}, …, t_{n}}. Under the above assumption there exists η_{x} ∈ (0, 1) such that

R_{n}^{ϱ}f(x) = ω(x)\frac{f^{(n+1)} (η_{x})}{(n + 1)!} .

Proof. Consider the function w(t) = ω(x)R_{n}^{ϱ}f(t) − ω(t)R_{n}^{ϱ}f(x), t ∈ [0, 1]. Then x, t_{0}, …, t_{n} are roots of w, which means that there exists η_{x} ∈ (0, 1) such that w^{(n+1)} (η_{x}) = 0. Now it suffices to remark that w^{(n+1)} (t) = ω(x)f^{(n+1)} (t) − (n + 1)!R_{n}^{ϱ}f(x). □

Corollaries 2.1 and 2.2 generalize the mean value theorems for the divided difference and the remainder in the classical Lagrange interpolation; see [15, Sect. 3.1], [8, Sect. 1.4]..

3. Iterated Boolean sums of the operators U_{n}^{ϱ}

For M ⩾ 1, let ⨁ ^{M} U_{n}^{ϱ} = I − (I − U_{n}^{ϱ})^{M} be the iterated Boolean sum of U_{n}^{ϱ}; here I stands for the identity operator on C[0, 1]. Iterated Boolean sums of the classical Bernstein operator and modifications thereof were investigated by numerous authors in the past, among them Mastroiani and Occorsio [9, 10]. Some historical information on this method which may be traced to Natanson can be found in [5]. From a general result of Wenz [16, Theorem 2] it follows that lim_{M→∞} ⨁ ^{M} U_{n}^{ϱ}f = L_{n}^{ϱ}f, f ∈ C[0, 1], n ⩾ 1. With the notation from the preceding sections, we can say more, namely

Theorem 3.1. Let n ⩾ 2 and f ∈ C[0, 1] be given. Then

(3.1) lim_{M→∞} (1 − λ_{ϱ,n}^{(n)} )^{−M} ( ⨁ ^{M}U_{n}^{ϱ}f − L_{n}^{ϱ}f) = −[F_{n,0}^{ϱ} , F_{n,1}^{ϱ} , …, F_{n,n}^{ϱ} ; f]p_{ϱ,n}^{(n)} ,

uniformly on [0, 1].

Proof. We have, according to (2.7)

⨁ ^{M} U_{n}^{ϱ}f = (I − (I − U_{n}^{ϱ})^{M} )f = ∑_{i=1}^{M} (−1)^{i+1} ( {M \atop i} ) (U_{n}^{ϱ})^{i}f

= ∑_{i=1}^{M} (−1)^{i+1} ( {M \atop i} ) ∑_{k=0}^{n} (λ_{ϱ,k}^{(n)} )^{i}p_{ϱ,k}^{(n)} μ_{ϱ,k}^{(n)} (f)

= ∑_{k=0}^{n} p_{ϱ,k}^{(n)} μ_{ϱ,k}^{(n)} (f) ∑_{i=1}^{M} (−1)^{i+1} ( {M \atop i} ) (λ_{ϱ,k}^{(n)} )^{i}

= ∑_{k=0}^{n} p_{ϱ,k}^{(n)} μ_{ϱ,k}^{(n)} (f)(1 − (1 − (λ_{ϱ,k}^{(n)} )^{M} )).

Combined with (2.8) this yields

⨁ ^{M} U_{n}^{ϱ}f − L_{n}^{ϱ}f = − ∑_{k=0}^{n} p_{ϱ,k}^{(n)} μ_{ϱ,k}^{(n)} (f)(1 − λ_{ϱ,k}^{(n)} )^{M} , i.e.,

(1−λ_{ϱ,n}^{(n)} )^{−M} ( ⨁ ^{M} U_{n}^{ϱ}f −L_{n}^{ϱ}f) = −p_{ϱ,n}^{(n)} μ_{ϱ,n}^{(n)} (f)−∑_{k=0}^{n−1} μ_{ϱ,k}^{(n)} (f)μ_{ϱ,k}^{(n)} (f)( \frac{1 − λ_{ϱ,k}^{(n)}}{1 − λ_{ϱ,n}^{(n)}} )^{M}.

Since 0 < (1 − λ_{ϱ,k}^{(n)} )/(1 − λ_{ϱ,n}^{(n)} ) < 1, k = 0, …, n − 1, we get

lim_{M→∞} (1 − λ_{ϱ,n}^{(n)} )^{−M} ( ⨁ ^{M} U_{n}^{ϱ}f − L_{n}^{ϱ}f) = −μ_{ϱ,n}^{(n)} (f)p_{ϱ,n}^{(n)} .

To conclude the proof it suffices to use (2.16). □

Remark 3.1. For ϱ → ∞, (3.1) was obtained in [14, Th. 26.7].

4. The derivatives of U_{n}^{ϱ}

In this section we show that there is a natural relationship between the derivatives of the operator images and the divided differences […; Φ_{n}] which we introduced in Remark 2.1.

Theorem 4.1. With the usual notation the following relationships hold

(i) (U_{n}^{ϱ}(f; x))′ = n ∑_{k=0}^{n−1} p_{n−1,k} (x)Δ^{1}F_{n,k}^{ϱ} (f) = ∑_{k=0}^{n−1} p_{n−1,k} (x)[ \frac{k}{n}, \frac{k + 1}{n} ; Φ_{n}];

(ii) (U_{n}^{ϱ}(f; x))^{(j)} = n(n − 1) ⋯ (n − j + 1) ∑_{k=0}^{n−j} p_{n−j,k} (x)Δ^{j} F_{n,k}^{ϱ} (f)

= n(n − 1) ⋯ (n − j + 1) ∑_{k=0}^{n−j} p_{n−j,k} (x)\frac{j!}{n^{j}} [ \frac{k}{n}, …, \frac{k + j}{n} ; Φ_{n}];

(iii) U_{n}^{ϱ}(f; x) = ∑_{k=0}^{n} ( {n \atop k}) Δ^{k}F_{n,0}^{ϱ} (f)e_{k}(x) = ∑_{k=0}^{n} ( {n \atop k}) \frac{k!}{n^{k}} [0, \frac{1}{n}, …, \frac{k}{n}; Φ_{n}]e_{k}(x);

where as before Φ_{n} ( \frac{k}{n} ) = F_{n,k}^{ϱ} (f).

Proof. (i) The forward difference was defined in [3, p. 792] by

Δ^{j} F_{n,k}^{ϱ} (f) = ∑_{i=0}^{j} ( {j \atop i} ) (−1)^{i+j} F_{n,k+i}^{ϱ} (f).

Thus we have

[ \frac{k}{n}, \frac{k + 1}{n} ; Φ_{n}] = \frac{Φ_{n}(\frac{k+1}{n} ) − Φ_{n}(\frac{k}{n})}{\frac{k+1}{n} − \frac{k}{n}} = n[F_{n,k+1}^{ϱ} (f) − F_{n,k}^{ϱ} (f)] = nΔ^{1}F_{n,k}^{ϱ} (f);

(ii) The first equality can be found in [3, p. 792]. It remains to show that

Δ^{j} F_{n,k}^{ϱ} (f) = \frac{j!}{n^{j}} [ \frac{k}{n}, …, \frac{k + j}{n} ; Φ_{n}].

We have that Δ^{j+1} F_{n,k}^{ϱ} (f) = Δ(Δ^{j} F_{n,k}^{ϱ} (f)) = Δ^{j} F_{n,k+1}^{ϱ} (f) − Δ^{j} F_{n,k}^{ϱ} (f). By using the recurrence formula for divided differences (see e.g. [15, p.104]) we get

Δ^{j} F_{n,k+1}^{ϱ} (f) − Δ^{j} F_{n,k}^{ϱ} (f) = \frac{j!}{n^{j}} · \frac{j + 1}{n} · \frac{[ \frac{k+1}{n} , …, \frac{k+j+1}{n} ; Φ_{n}] − [ \frac{k}{n} , …, \frac{k+j}{n} ; Φ_{n}]}{\frac{k+j+1}{n} − \frac{k}{n}}

= \frac{(j + 1)!}{n^{j+1}} [ \frac{k}{n}, …, \frac{k + j + 1}{n} ; Φ_{n}] = Δ^{j+1} F_{n,k}^{ϱ} (f),

U_{n}^{ϱ}(f; x) = ∑_{j=0}^{n} \frac{(U_{n}^{ϱ}f)^{(j)} (0)}{j!} x^{k}

and show that (U_{n}^{ϱ}(f; x))^{(j)} = n(n − 1) ⋯ (n − j + 1)Δ^{j} F_{n,0}^{ϱ} (f).

To this end we take x = 0 in (ii); because p_{n−j,0} (0) = 1 and for all k ⩾ 1, p_{n−j,k} (0) = 0, from ∑ _{k=0}^{n−j} only the first term remains, which concludes the proof.

□

Remark 4.1. In the case ϱ → ∞ we can find the analogues of the above relationships in [15, pp. 300–302].

References

1. S. Cooper, S. Waldron, The eigenstructure of the Bernstein operator, J. Approx. Theory 105 (2000), 133–165.

2. P. J. Davis, Interpolation and Approximation, Dover, New York, 1975.

3. H. Gonska, R. Păltănea, Simultaneous approximation by a class of Bernstein-Durrmeyer operators preserving linear functions, Czechoslovak Math. J. 60 (2010), 783–799.

4. H. Gonska, I. Raşa, E.D. Stănilă, The eigenstructure of operators linking the Bernstein and the genuine Bernstein-Durrmeyer operators, Mediterr. J. Math. (2013), DOI: 10.1007/s00009-013-0347-0

5. H. Gonska, X.-L. Zhou, Approximation theorems for the iterated Boolean sums of Bernstein operators, J. Comput. Appl. Math. 53 (1994), 21–31.

6. D. Kacsó, E. D. Stănilă, On the class of operators U_{n}^{ϱ} linking the Bernstein and the genuine Bernstein-Durrmeyer operators, J. Appl. Funct. Anal. 9 (2014), 335–348.

7. A. Lupaş, Die Folge der Betaoperatoren, Ph.D. Thesis, Stuttgart: Universität Stuttgart 1972.

8. G. Mastroianni, G. V. Milovanović, Interpolation Processes. Basic Theory and Applications, Springer, 2008.

9. G. Mastroianni, M. R. Occorsio, Una generalizzazione dell’operatore di Bernstein, Rend. Accad. Sci. Mat. Fis. Nat. Napoli, Serie IV, 44 (1977), 151–169.

10. , Una generalizzazione dell’operatore di Stancu, Rend. Accad. Sci. Mat. Fis. Mat. Napoli, Serie IV, 45 (1978), 495–511.

11. G. Mühlbach, Rekursionsformeln für die zentralen Momente der Pólya und der Beta-Verteilung, Metrika 19 (1972), 171–177.

12. R. Păltănea, A class of Durrmeyer type operators preserving linear functions, Ann. Tiberiu Popoviciu Sem. Funct. Equat. Approxim. Convex. (Cluj-Napoca) 5 (2007), 109–117.

13. E. Popoviciu, Teoreme de Medie din Analiza Matematică şi Legatura lor cu Teoria Interpolării, Editura Dacia, Cluj, 1972.

14. I. Raşa, T. Vladislav, Analiza Numerică. Aproximare, problema lui Cauchy abstractă, proiectori Altomare, Ed. Tehnica, Bucureşti 1999.

15. D. D. Stancu, O. Agratini, Gh. Coman, R. Trâmbiţaş, Analiză Numerică şi Teoria Aproximării, vol. I, Cluj-Napoca: Presa Universitară Clujeană 2001.

16. H. J. Wenz, On the limits of (linear combinations of) iterates of linear operators, J. Approx. Theory 89 (1997), 219–237.

University of Duisburg-Essen Faculty of Mathematics Forsthausweg 2 47057 Duisburg Germany

 heiner.gonska@uni-due.de
 elena.stanila@stud.uni-due.de

Technical University of Cluj-Napoca Department of Mathematics Str. Memorandumului nr. 28 RO-400114 Cluj-Napoca Romania

 Ioan.Rasa@math.utcluj.ro