Some approximation results for generalized Kantorovich-type operators
© Mursaleen et al.; licensee Springer. 2013
Received: 25 September 2013
Accepted: 18 November 2013
Published: 17 December 2013
In this paper, we construct a new family of operators, prove some approximation results in A-statistical sense and establish some direct theorems for Kantorovich-type integral operators.
MSC: Primary 41A10, 41A25, 41A36.
KeywordsLagrange polynomial Korovkin approximation theorems A-statistical convergence Kantorovich-type operators positive linear operators modulus of continuity Peetre’s K-functional Lipschitz class
1 Introduction and preliminaries
Statistical convergence  and its variants, extensions and generalizations have been proved to be an active area of recent research in summability theory, e.g., lacunary statistical convergence , λ-statistical convergence , A-statistical convergence , statistical A-summability , statistical summability , statistical summability , statistical summability  and statistical σ-summability etc. Following the work of Gadjiv and Orhan , these statistical summability methods have been used in establishing many approximation theorems (e.g., [5, 11–20] and ). Recently, the statistical approximation properties have also been investigated for several operators. For instance, in  Butzer and Hahn operators; in  and q-analogue of Stancu-Beta operators; in  Bleimann, Butzer and Hahn operators; in  Baskakov-Kantorovich operators; in  Szász-Mirakjan operators; in  analogues of Bernstein-Kantorovich operators; and in q-Lagrange polynomials were defined and their statistical approximation properties were investigated. Most recently, the statistical summability of Walsh-Fourier series has been discussed in . In this paper, we construct a new family of operators with the help of Erkuş-Srivastava polynomials, establish some A-statistical approximation properties and direct theorems.
Let us recall the following definitions.
In this case, we write . Note that every convergent sequence is statistically convergent but not conversely.
Let , , be an infinite matrix. For a given sequence , the A-transform of x is defined by , where , provided the series converges for each n. We say that A is regular if . Let A be a regular matrix.
In this case, we denote this limit by .
Note that for , the Cesàro matrix of order 1, A-statistical convergence reduces to the statistical convergence.
2 Construction of a new operator and its properties
The case of the polynomials given by (2.5) corresponds to the familiar (two-variable) Lagrange-Hermite polynomials considered by Dattoli et al. .
corresponds to (2.3).
3 A-statistical approximation
For typographical convenience, we will write in place of if no confusion arises.
which gives (3.7). Now, combining (3.5)-(3.7), and using the statistical version of the Korovkin approximation theorem (see Gadjiv and Orhan , Theorem 1), we get the desired result.
This completes the proof of the theorem. □
In a similar manner, we can extend Theorem 3.1 to the -dimensional case for the operators given by (2.9) as follows.
Remark 3.1 If in Theorem 3.2 we replace by the identity matrix, we immediately get the following theorem which is a classical case of Theorem 3.2.
is uniformly convergent to f on .
Finally, we display an example which satisfies all the hypotheses of Theorem 3.2, but not of Theorem 3.3. Therefore, this indicates that our A-statistical approximation in Theorem 3.2 is stronger than its classical case.
However, since the sequence defined by (3.10) is non-convergent, Theorem 3.3 does not hold in this case.
4 Direct theorems
This completes the proof of the theorem. □
uniformly on .
uniformly with respect to .
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