Discrete Operators associated with Linking Operators

We associate to an integral operator a discrete one which is conceptually simpler, and study the relations between them.


Introduction and notation
It is well-known that integral operators are very useful in Approximation Theory. However, sometimes they are expressed in terms of complicated integrals with respect to complicated measures. In this paper we associate to an integral operator a discrete one which is conceptually simpler, and study the relations between them. Some results in this direction can be found also in [11].
After introducing the necessary definitions we present some general results. Then we recall the definitions of Baskakov type operators, genuine Baskakov-Durrmeyer type operators, and their Kantorovich modifications. We construct the discrete operators associated with these integral operators and apply our general results in this context.

Preliminaries
Let I ⊂ R be an interval and H a subspace of C(I) containing e 0 , e 1 and e 2 . Let L : H −→ C(I) be a positive linear operator such that Le 0 = e 0 . The second moment of L is defined by For a fixed x ∈ I consider the functional H ∋ f −→ Lf (x) and define Var x L := Le 2 (x) − (Le 1 (x)) 2 ; then, roughly speaking, Var x L shows how far is the functional from being a pointwise evaluation.
It is easy to verify that Now let L be of the form where A j : H −→ R are positive linear functionals, Then, generally speaking, Var A j shows how far is A j from the point evaluation at b j . The discrete operator associated with L is defined by The point evaluation functional at b j is conceptually simpler than A j ; from this point of view, D is simpler than L. We shall investigate the relations between L and D.
It is easy to verify that Moreover, according to (1) and (3) Combined with (6), this shows that Now define We have With (4) and (6) this leads to Combined with (7), this yields Finally, let f ∈ H ∩ C 2 (I) and suppose that f ′′ ∞ < ∞. Then by Taylor's formula,

This entails
Moreover, according to (2) and (5), and so We conclude that Using (12) we see that E(L)(x) shows how far is L from D.

Linking operators and discrete operators
In [9,10] Pȃltȃnea introduced operators depending on a parameter ρ ∈ R + , which constitute a non-trivial link between the Bernstein and Szász-Mirakjan operators, respectively, and their genuine Durrmeyer modifications. In [5] this definition was extended to a non-trivial link between Baskakov type operators and genuine Baskakov-Durrmeyer type operators and their k-th order Kantorovich modification. For k = 1 this means a link between the Kantorovich modification of Baskakov type and Baskakov-Durrmeyer type operators.
In what follows for c ∈ R we use the notations In the following definitions of the operators we omit the parameter c in the notations in order to reduce the necessary sub and superscripts. Let Then the basis functions are given by In the following definitions we assume that the function f is given in such a way that the corresponding integrals and series are convergent. The operators of Baskakov-type are defined by and the genuine Baskakov-Durrmeyer type operators are denoted by for c < 0 and by Depending on a parameter ρ ∈ R + the linking operators are given by x, y > 0, we denote Euler's Beta function. Note that in case c < 0 the sums in (13) and (15) are finite, as p n,j (x) = 0 for j > −n/c. The k-th order Kantorovich modification of the operators B n,ρ are defined by where D k denotes the k-th order ordinary differential operator and For k = 0 we omit the superscript (k) as indicated by the definition above. We recall some results concerning lim ρ→∞ B Let ε > 0 be arbitrary. As the space of polynomials P is dense in L p [0, 1], · p , 1 ≤ p < ∞ and C[0, 1], · ∞ , p = ∞, we can choose a polynomial q, such that f − q p < ε. Then As the operators B c ≥ 0: Let n, k ∈ N, n − k ≥ 1, ρ ∈ N and f ∈ W ρ n . Then we have the representation We consider the operators for which we have V (k) n,ρ e 0 = e 0 . They are of the form (2); more precisely, where (see (17), (18) and (20)) If nρ > kc, we can consider the barycenter of A (k) n,ρ,j . As tp m+c,l−1 (t) = l m p m,l (t) and The discrete operators (5) associated with V (k) n,ρ are n,ρ,j )p n+kc,j .
Under the form n,ρ is a Stancu-type modification of the operator (13) A direct calculation similar to (23) shows that
As explained in Section 2, Var x V . This assertion, and other similar ones, are made more precise in the following Theorem. Consequently, Since x n,k,j ∈ j n , j+k n , it follows that |x n,k,j − 2j+k 2n | ≤ k 2n , and so This proves (37).
Let us return to Var A and this shows that Var A (k) n,ρ,j is decreasing w.r.t. ρ. A similar transformation of (28) (or an inspection of (8)) reveals that E (k) n,ρ is also decreasing w.r.t. ρ.
Finally (33) can be put under the form which shows that Var x V (k) n,ρ is decreasing w.r.t. ρ.
Moreover (see [3,12]), |B n,∞ (f g)(x) − B n,∞ f (x)B n,∞ g(x)| ≤ 1 2 (1 − S n,c (x)) osc n (f ) osc n (g), where osc n (f ) := sup {|f ( j n ) − f ( i n )| : i, j ∈ N 0 }. Let n and x be fixed; consider the functional f → B n,∞ f (x). Then, roughly speaking 1 − S n,c (x) shows how far is the functional from being a multiplicative functional, and ultimately how far is it from being a point evaluation.
Consider (p n,0 (x), p n,1 (x), . . . ) as a probability distribution parameterized by x. Then 1 − S n,c (x) is the associated Tsallis entropy. Moreover, S n (x) is viewed as one of the indices measuring the inequality and diversity, i. e., the degree of uniformness of the distribution (see [8, pp. 556-559]).