Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Master
SUBJECT

Code	Subject	
MMC1004	Linear Approximation Processes	

Section	Semester	Hours: C+S+L	Category	Type	
Applied Mathematics	3	2+2+0	speciality	optional	

Teaching Staff in Charge	Prof. AGRATINI Octavian, Ph.D., agratini@math.ubbcluj.ro	
Aims	1. Approaching different construction methods of linear positive operators.
2. Knowing discrete and continuous classes of approximation opearators.
3. Using these sequences for approximation of functions belonging to various spaces and  investigation of shape preserving properties.	
Content	1. Modulus of continuity. M-order moduli of smoothness. Ditzian - Totik moduli of smoothness. Weighted moduli of smoothness. Applications.
2. Operators and functionals. Properties. Basic theorems of approximation by linear positive operators. Korovkin closures and Korovkin subspaces for the identity operator. Finite Korovkin sets.
3. Bernstein type operators. Discrete operators for functions defined on unbounded intervals. Kantorovich and Durrmeyer type integral extensions. Convolution operators. Shift invariant operators.
4. Variation diminishing property. Preservation of Lipschitz classes.Convergence of iterates. A probabilistic approach - Feller operators. Statistical convergence.
5. Tensor products. Boolean sums. Korovkin type theorems. Estimates with moduli of continuiy.	
Bibliography	1. Agratini, O., Aproximare prin operatori liniari, Presa Universitară Clujeană, 2000.
2. Altomare, F., Campiti, M., Korovkin-type Approximation Theory and its Applications, Walter de Gruyter, Vol. 17, Berlin - New York, 1994.
3. Anastassiou, G. A., Gal, S. G., Approximation Theory. Moduli of Continuity and Global Smoothness Preservation, Birkhauser, Boston, 2000.
4. Gavrea, I., Aproximarea funcţiilor prin operatori liniari, Editura Mediamira, Cluj Napoca, 2001.
5. Stancu, D. D., Coman, Gh., Agratini, O., Trîmbiţaş, R., Analiză numerică şi teoria aproximării, Vol I, Presa Universitară Clujeană, 2001.	
Assessment	During the semester: a Control Paper.
In the session: written exam.
The final grade: arithmetic mean of the above two marks having the weights 1/3 and 2/3, respectively.