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

Code	Subject	
MMC1004	Linear Approximation Processes	

Section	Sem.	Hours: C+S+L	Category	Type	
Applied Mathematics	3	2+1+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 operators.
3. Knowing basic elements of Fourier Analysis and Wavelets Analysis.	
Content	1. Modulus of continuity. M-order moduli of smoothness. K-functional. Applications.
2. Operators and functionals. Properties. Basic theorems of approximation by linear positive operators. Korovkin closures and Korovkin subspaces for the identity operator. 
3.  Discrete operators for functions defined on unbounded intervals. Kantorovich and Durrmeyer type integral extensions. Summation methods. Shift invariant operators.
4. Fourier Transforms. Properties. Gabor formulas for Windowed Fourier Transforms.
5. Wavelet Transforms. Multiresolution Analysis. Wavelet decompositions and reconstructions.	
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. Debnath, L., Wavelet Transforms and their Applications, Birkhauser, Boston, Basel, Berlin, 2002.
5. Gasquet, C., Witomski, P., Analyse de Fourier et applications, Masson, Paris, 1990.
6. 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.