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.,  agratinimath.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.
References
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.
Links: Syllabus for all subjects
Romanian version for this subject
Rtf format for this subject