## "Babes-Bolyai" University of Cluj-Napoca Faculty of Mathematics and Computer Science

 Wavelet Transforms
 Code Semes-ter Hours: C+S+L Credits Type Section MC267 2 2+2+0 9 compulsory Matematică Aplicată MC267 2 2+2+0 9 compulsory Modele matematice in Mecanica si Astronomie
 Teaching Staff in Charge
 Prof. AGRATINI Octavian, Ph.D., agratini@math.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.
 Assessment Oral examination.