On the stability of some positive linear operators from approximation theory

Recently, Popa and Raşa have shown the stability/ instability of some classical operators defined on $$[0,1]$$[0,1] and obtained the best constant when the positive linear operators are stable in the sense of Hyers–Ulam. In this paper we show that the Kantorovich–Stancu type operators, King’s operator, Bernstein–Stancu type operators, and Kantorovich–Bernstein–Stancu type operators with shifted knots are Hyers–Ulam stable. Further we find the best Hyers–Ulam stability constants for some of these operators. We also prove that Szász–Mirakjan and Kantorovich–Szász–Mirakjan type operators are unstable in the sense of Hyers and Ulam.

in 1940: "Given a metric group (G, ., ρ), a number ε > 0 and a mapping f : G → G which satisfies the inequality ρ( f (x y), f (x) f (y)) < ε for all x, y ∈ G, does there exist a homomorphism h of G and a constant k > 0, depending only on G, such that ρ( f (x), h(x)) ≤ kε for all x ∈ G?" If the answer is affirmative the equation f (x y) = f (x) f (y) of the homomorphism is called stable; see [2,3]. The first answer to Ulam's problem was given by Hyers [4] in 1941 for the Cauchy functional equation in Banach spaces, more precisely he proved: "Let X, Y be Banach spaces, ε a nonnegative number, f : X → Y a function satisfying f (x + y) − f (x) − f (y) ≤ ε for all x, y ∈ X , then there exists a unique additive mapping h : X → Y with the property f (x)−h(x) ≤ ε for all x ∈ X ." Due to the question of Ulam and the result of Hyers this type of stability is called today Hyers-Ulam stability of functional equations. A similar problem was formulated and solved earlier by Pólya and Szegö in [5] for functions defined on the set of positive integers. After Hyers result a large amount of literature was devoted to study the Hyers-Ulam stability for various equations. A new type of stability for functional equations was introduced by Aoki [6] and Rassias [7] by replacing ε in the Hyers theorem with a function depending on x and y, such that the Cauchy difference can be unbounded. For other results on the Hyers-Ulam stability of functional equations one can refer to [2,8,9]. The Hyers-Ulam stability of linear operators was considered for the first time in the papers by Miura, Takahasi et al. (see [10][11][12]). Similar type of results are obtained in [13] for weighted composition operators on C(X ), where X is a compact Hausdorff space. A result on the stability of a linear composition operator of the second order was given by Brzdek and Jung in [14].
Recently, Popa and Raşa obtained [15] a result on Hyers-Ulam stability of the Bernstein-Schnabl operators using a new approach to the Fréchet functional equation, and in [16,17], they have shown the (in)stability of some classical operators defined on [0, 1] and find best constant when the positive linear operators are stable in the sense of Hyers-Ulam.
The aim of this paper is to show that Kantorovich-Stancu type operators, an operator introduced by J. P. King and Bernstein-Stancu type and Kantorovich-Bernstein-Stancu type operators with shifted knots are Hyers-Ulam stable. Further we find the best Hyers-Ulam stability constants for some of these operators. We also prove that Szász-Mirakjan and Kantorovich-Szász-Mirakjan type operators are unstable in the sense of Hyers and Ulam.

The Hyers-Ulam stability property of operators
In this section, we recall some basic definitions and results on Hyers-Ulam stability property which form the background of our main results. Definition 2.1 Let A and B be normed spaces and T a mapping from A into B. We say that T has the Hyers-Ulam stability property (briefly, T is HU-stable) [13] if there exists a constant K such that for any g ∈ T (A), ε > 0 and f ∈ A with T f − g ≤ ε, there exists an f 0 ∈ A such that T f 0 = g and f − f 0 ≤ K ε. The number K is called a HUS constant of T , and the infimum of all HUS constants of T is denoted by K T . Generally, K T is not a HUS constant of T (see [10,11]). Now let T be a bounded linear operator with the kernel denoted by N (T ) and the range denoted by R(T ). Consider the one-to-one operator T from the quotient space A/N (T ) into B: Theorem 2.2 [13] Let A and B be Banach spaces and T : A → B be a bounded linear operator. Then the following statements are equivalent: The main results used in our approach for obtaining, in some concrete cases, the explicit value of K T are the formula given above and a result by Lubinsky and Ziegler [18] concerning coefficient bounds in the Lorentz representation of a polynomial.
Let p ∈ Π n , where Π n is the set of all polynomials of degree at most n with real coefficients. Then p has a unique Lorentz representation of the form where c k ∈ R, k = 0, 1, . . . , n. Remark that, in fact, it is a representation in Bernstein-Bézier basis. Let T n denote the usual nth degree Chebyshev polynomial of the first kind. Then the following representation holds (see [18]): It is proved in [16] that d n,k = 2n 2k , k = 0, 1, . . . , n. Therefore Remark 2.3 [16] (1) Condition (i) expresses the Hyers-Ulam stability of the equation is given and f ∈ A is unknown.
So, in what follows, we shall study the HU-stability of a bounded linear operator T : A → B by checking the existence of a constant K for which (ii) is satisfied, or equivalently, by checking the boundedness of T −1 . (i) Bernstein operators [16] For each integer n ≥ 1, the classical Bernstein operators B n : C[0, 1] → C[0, 1] are defined by (see [8]) are stable in the Hyers-Ulam sense and the best Hyers-Ulam stability constant is given by , n ∈ N.
(iii) Beta operators [16] For each n ≥ 1, the Beta operator B n : is unstable in the sense of Hyers and Ulam.

(i) Kantorovich-Stancu type operators
The kernel of K n,m is given by which is a closed subspace of C[0, 1+m], and R(K n,m ) = Π n+m . The operator K n,m : is bounded since dimΠ n+m = n + m + 1, so according to Theorem 2.2 the operator K n,m is Hyers-Ulam stable. .
Proof Let p ∈ Π n+m , p ≤ 1, and its Lorentz representation Consider the piecewise constant function Then K n,m f p = p and K −1 n,m ( p) = f p + N (K n,m ). As usual, the norm of K −1 n,m : Π n+m → C[0, 1 + m]/N (K n,m ) is defined by On the other hand, let q(x) = T n (2x − 1), x ∈ [0, 1]. Then q = 1 and |c k (q)| = d n+m,k , 0 ≤ k ≤ n + m, according to Theorem 2.4. Consequently Then The inequality a k+1 a k ≥ 1 is satisfied if and only if k if n + m is an odd number, we conclude that .
This completes the proof of the theorem.
Remark For m = β = 0, the above operator reduces to the classical Kantorovich operator. Therefore the infimum of the HUS constant of the Kantorovich operator is .
This corrects the value provided by Theorem 3.3 in [17]. In [24], King defined the following interesting sequence of positive linear operators V n : C[0, 1] → C[0, 1] which generalizes the classical Bernstein operators B n defined by The kernel of V n is given by which is a closed subspace of C[0, 1], and R(V n ) = Π n . The operator K n : is bounded since dimΠ n = n + 1, so according to Theorem 2.2 the operator V n is Hyers-Ulam stable.
Proof The proof is similar as Theorem 3.3 for finding the best constant in the case of Bernstein operators in [16].

(iii) Szász-Mirakjan type operators
In [25], Lupaş proposed the linear positive operators with the help of the identity 1 Here we are taking an nth Szász-Mirakjan type operator L n : Proof Suppose that for a certain n ≥ 1, the operator L n is HU-stable. Then there exists a constant K such that for any f ∈ C b [0, +∞) with L n f ∞ ≤ 1 there exists According to Stirling's formula, lim k→∞ k k k!e k = 0, so that there exists a j ≥ 1 such that (K + 1) j j j!e j ≤ 1.
Let f ∈ C b [0, +∞) be the function defined by Then It is easy to check that L n f ∞ ≤ 1, so that there exists g ∈ N (L n ) with f − g ∞ ≤ K . But then g( j n ) = 0 and consequently K ≥ f − g ∞ ≥ | f ( j n ) − g( j n )| = K + 1 a contradiction. Thus the theorem is proved.  n , x ∈ [ j n , j+1 n ].
Of course f is linear on [ j−1 n , j n ] and [ j n , j+1 n ].