Fourier Analysis
Fourier Analysis
3-4 hours of lectures per week.
- Lecturer:
- Wojtek Slowikowski
- Content:
- Fourier analysis is a beautiful theory with far reaching implications also outside of mathematics. It is an indispensable tool in the theory of differential equations, quantum physics and probability theory but also e.g. in time series (statistics), signal processing and engineering. Though primary a part of analysis, it is also vital for algebra and geometry. We shall start with Fourier series, prove some basic results as e.g. the Fejèr theorem. Then we consider the
-setup and the eigenfunction expansion for derivation followed by some applications. We introduce the Fourier integral in 
and prove the Plancherel theorem and the inversion formula. The theory is extended to the multidimensional case and some important examples are considered. Then we discuss the harmonic oscillator, the creation and annihilation operators 
the Hermite functions expansion and the Segal-Bargmann entire function representation. - Prerequisites:
- Analysis 1
- Literature:
- H. Dym and H.P. McKean:,"Fourier series and integrals", Academic Press, 1972
-setup and the eigenfunction expansion for derivation followed by some applications. We introduce the Fourier integral in 
and prove the Plancherel theorem and the inversion formula. The theory is extended to the multidimensional case and some important examples are considered. Then we discuss the harmonic oscillator, the creation and annihilation operators 
the Hermite functions expansion and the Segal-Bargmann entire function representation.