"Harmonic Analysis". Faculty of Engineering. (Ya. Yakubov)

Background: calculus, ordinary differential equations, functions of complex variables (a parallel course).

1. Fourier coefficients with respect to an orthonormal system, trigonometric bases in the interval [-pi,pi].
2. Partial sums and Dirichlet integral. Riemann-Lebesgue theorem, Riemann principle.
3. Sin and Cos series (real Fourier series), complex representation of Fourier series.
4. Dini and Dirichlet convergence criteria, and convergence at a discontinuity. Pointwise convergence and uniform convergence.
5. Integral and derivative of Fourier series, inner product vector spaces, orthonormal bases, the best approximation, Bessel inequality, Parseval identity. Completeness of a trigonometric system. Gibbs example.
6. Fourier integral in L_1(R), inverse formula. Fourier transform.
7. Properties of Fourier transform: continuity and decay at infinity, dependance of the decay on the smoothness. Fourier transform of Sin and Cos.
8. Fourier transform in L_2(R), inverse Fourier transform. Difference between L_1(R) and L_2(R). Parseval-Plancherel formulas.
9. Laplace transform and its fundamental properties. Inverse Laplace transform.
10. Application to ordinary and partial differential equations.