Wavelet Transforms 
ter 

Teaching Staff in Charge 

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, LittlewoodPaleyStein, Franklin and Stromberg, respectively. Discret and continuous wavelet transforms. Examples. Basic properties. Definition of multiresolution analysis. Properties of scaling functions and orthonormal wavelet bases. Timefrequency 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. 