# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Equi-statistical convergence of positive linear operators. (English) Zbl 1131.41008
The authors study a Korovkin-type approximation theorem using the equi-statistical convergence which is stronger than the statistical uniform convergence. By an example it is shown that the new approximation result works while its classical and statistical cases do not work. Furthermore, the rate of equi-statistical convergence of a sequence of positive linear operators is computed, and a Voronovskaya-type theorem in the equi-statistical sense for a sequence of positive linear operators constructed by means of the Bernstein polynomials is given.
##### MSC:
 41A36 Approximation by positive operators 40A30 Convergence and divergence of series and sequences of functions 41A30 Approximation by other special function classes 41A25 Rate of convergence, degree of approximation
##### References:
 [1] Altomare, F.; Campiti, M.: Korovkin-type approximation theory and its applications, (1994) [2] Balcerzak, M.; Dems, K.; Komisarski, A.: Statistical convergence and ideal convergence for sequences of functions, J. math. Anal. appl. 328, 715-729 (2007) · Zbl 1119.40002 · doi:10.1016/j.jmaa.2006.05.040 [3] Bojanic, R.; Cheng, F.: Estimates for the rate of approximation of functions of bounded variation by Hermite – Fejér polynomials, Proc. conference canadian math. Soc. 3, 5-17 (1983) · Zbl 0554.41012 [4] Bojanic, R.; Khan, M. K.: Summability of Hermite – Fejér interpolation for functions of bounded variation, J. natur. Sci. math. 32, 5-10 (1992) · Zbl 0762.41002 [5] Devore, R. A.: The approximation of continuous functions by positive linear operators, Lecture notes in math. 293 (1972) · Zbl 0276.41011 [6] Duman, O.; Orhan, C.: $\mu$-statistically convergent function sequences, Czechoslovak math. J. 54, 413-422 (2004) · Zbl 1080.40501 · doi:10.1023/B:CMAJ.0000042380.31622.39 [7] Duman, O.; Khan, M. K.; Orhan, C.: A-statistical convergence of approximating operators, Math. inequal. Appl. 6, 689-699 (2003) · Zbl 1086.41008 [8] Erkuş, E.; Duman, O.: A Korovkin type approximation theorem in statistical sense, Studia sci. Math. hungar. 43, 285-294 (2006) · Zbl 1108.41012 · doi:10.1556/SScMath.43.2006.3.2 [9] Erkuş, E.; Duman, O.; Srivastava, H. M.: Statistical approximation of certain positive linear operators constructed by means of the chan – chyan – Srivastava polynomials, Appl. math. Comput. 182, 213-222 (2006) · Zbl 1103.41024 · doi:10.1016/j.amc.2006.01.090 [10] Fast, H.: Sur la convergence statistique, Colloq. math. 2, 241-244 (1951) · Zbl 0044.33605 [11] Gadjiev, A. D.; Orhan, C.: Some approximation theorems via statistical convergence, Rocky mountain J. Math. 32, 129-138 (2002) · Zbl 1039.41018 · doi:10.1216/rmjm/1030539612 · doi:http://math.la.asu.edu/~rmmc/rmj/vol32-1/CONT32-1/CONT32-1.html [12] Korovkin, P. P.: Linear operators and approximation theory, (1960)