Generalization of JakimovskiLeviatan type Szasz operators
Introduction
In 1950, Szasz [1] introduced the following operators where x ≥ 0 and f ∈ C[0, ∞) whenever the right-hand side of (1.1) exists. In this paper, Szasz investigated the detailed approximation properties of operators (1.1).
Later in 1969, Jakimovski and Leviatan [2] gave a generalization of Szasz operators by using the Appell polynomials. Let be an analytic function in the disc |z| < R, (R > 1) and assume g(1) ≠ 0. According to Chihara, the Appell polynomials pk(x) are defined by the generating functions Jakimovski and Leviatan constructed the operators Pn(f; x) by The authors studied the approximation properties of these operators in view of Szasz’s method [1]. For the special case we obtain the Appell polynomials and from (1.3) we meet again Szasz operators given by (1.1).
Wood [3] proved that the operators given by (1.3) are positive in [0, ∞) if and only if for Taking into account of this fact, Ciupa [4], [5] studied the order of approximation of the function f by means of the linear positive operators, introduced another variants of operators (1.3) and examined the approximation properties by virtue of Korovkin’s theorem [6]. Several important contributions for Jakimovski–Leviatan type operators can be found in [7], [8], [9].
Ismail [10] presented another generalization of Szasz operators (1.1) by using Sheffer polynomials which also include the operators (1.3). Let (a0 ≠ 0) and (h1 ≠ 0) be analytic functions in the disc |z| < R, (R > 1) where ak and hk are real. A polynomial set {pk(x)}k ≥ 0 is said to be Sheffer polynomial set if and only if it has a generating function of the form
Under the assumptions Ismail investigated the approximation properties of the following
For one can easily obtain that the operators given by (1.6) reduce to the operators (1.3). On one hand, by choosing and we get therefore (1.6) leads to the well-known Szasz operators (1.1).
In this paper, under the assumptions (1.5), we define a Stancu type generalization of the operators (1.6) as below: where α, β ≥ 0. Convergence of the operators (1.7) is examined with the help of the well-known Korovkin theorem. The degree of convergence is established by using classical and the second modulus of continuity. Operators including Meixner polynomials which form a sequence of discrete orthogonal polynomials and the 2-orthogonal polynomials of Laguerre type are given as examples. In addition, numerical examples are presented to illustrate the theoretical results. It is worthy to note that when we find the operators (1.6).
Section snippets
Approximation properties of operators
We give the following lemmas and definitions which are used in the sequel.
Definition 1 Let and δ > 0. The modulus of continuity ω(f; δ) of the function f is defined by
where is the space of uniformly continuous functions on [0, ∞). Definition 2 The second modulus of continuity of the function f ∈ C[a, b] is defined by
where . Lemma 1 (Gavrea and Rasa [11]) Let z ∈ C2[0, a] and (Ln)n ≥ 0 be a sequence of linear
Examples
Example 1 Meixner polynomials [14] which form a sequence of discrete orthogonal polynomials have the generating functions of the form
and the explicit representations
where 0 < c < 1, γ > 0 and (α)k is the Pochhammer’s symbol given by
Note that from (3.2), it is understood that when x ≤ 0, 0 < c < 1 and γ > 0, Meixner polynomials are positive. On the other hand, Meixner polynomials are also
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