On time-frequency analysis and some operators

Time-frequency analysis is a modern branch of harmonic analysis which comprises of theory and its applications of mathematical techniques using the translations and modulations to analyze functions and operators. Time-frequency analysis is a form of local Fourier analysis that threats time and frequency simultaneously and symmetrically. In classical Fourier Analysis, the signals on the time space are studied by transforming them in to the frequency space. However, in the frequency space, some informations related to time is lost which makes analysis of non-stationary signals difficult. To overcome this deficiency, a variety of methods are proposed, one of which is a local form of Fourier analysis, called the short-time Fourier transform. In this thesis, techniques of Fourier analysis and time-frequency analysis are examined and compared. The thesis is concentrated mainly on time-frequency space. Today the theory time-frequency analysis has important technological and scientific applications extending the known results on Fourier analysis. In the first part of the thesis classical Fourier analysis and its properties are given. In particular Fourier transform is introduced and uncertainty principles are given. In the second and main part of the thesis, time-frequency analysis is introduced and basic operators togerther with their properties are given. Next frame theory, in particular Gabor frames are studied in connection with some key operators related to frames. Finally, Zak transform is introduced and its properties are given. In addition, the relationship of Zak transform to Fourier transform and Gabor frames is studied.