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

 Code Subject MC267 Wavelet Transforms
 Section Semester Hours: C+S+L Category Type Applied Mathematics 1 2+2+0 compulsory Mathematical Models in Mechanics and Astronomy - in English 2 2+2+0 compulsory
 Teaching Staff in Charge
 Prof. AGRATINI Octavian, Ph.D.,  agratinimath.ubbcluj.ro
 Aims At first the course includes selected knowledge of Fourier analysis and windowed Fourier transforms. Further on, the course gives an introductory treatment of the basic ideas concerning wavelets, wavelet transforms, wavelet bases, multiresolution analysis and their applications.
 Content Fourier and inverse Fourier transforms. Fourier series. Signals and their classification. Windowed Fourier transform (the continuous Gabor transform). The radius and the width of a window function as well as discussions on the Uncertainty Principle. From Fourier to Haar. New directions of the 1930s. Atomic decomposition, from 1960 to 1980. Various definitions of a wavelet based on the work of Grossmann and Morlet, Littlewood-Paley-Stein, Franklin and Stromberg, respectively. Discret and continuous wavelet transforms. Examples. Basic properties. Definition of multiresolution analysis. Properties of scaling functions and orthonormal wavelet bases. Time-frequency algorithms. Wavelet expansions. Quantitative monotone and probabilistic wavelet type approximation. Estimates and sharpness. Applications of wavelets in functions estimation - wavelet versions of some types of statistical estimators.
 References [1] AGRATINI, O., CHIOREAN, I., COMAN, GH., TRIMBITAS, R., Analiza numerica si teoria aproximarii, Vol.III, Presa Universitara Clujeana, 2002. [2] CHUI,C.K.,An Introduction to Wavelets, Academic Press, Inc.Harcourt Brace Jovanovich, Publishers, New York, 1992. [3] DEBNATH, L., Wavelet Transform and Their Applications, Birkhauser, Boston Basel Berlin, 2002. [4] GASQUET, C., WITOMSKI, P., Analyse de Fourier et applications. Filtrage, Calcul numerique, Ondelettes, Masson, Paris, 1990. [5] MEYER, Y., Wavelets - Algorithms and Applications, SIAM, Philadelphia, PA, 1993. [6] OGDEN, R.T., Essential Wavelets for Statistical Applications and Data Analysis, Boston, Birkhauser, 1997. [7] Stanasila, O., Analiza matematica a semnalelor si undinelor, Matrix Rom, Bucuresti, 1997. [8] NEWTON, M., Wavelets - Utilization and Applications, Concord, NewHampshire, NH, 1999.
 Assessment During the semester: a Control Paper and short Disertation Paper. In the session: oral examination. The final mark: ponderate arithmatic mean of the above 3 marks.
 Links: Syllabus for all subjects Romanian version for this subject Rtf format for this subject