"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.
  • G. B. Folland, "Fourier Analysis and its Applications", Wadsworth and Brooks/Cole, 1992.
  • G. P. Tolstov, "Fourier Series", Prentice-Hall, 1965.
  • M. H. Protter, C. B. Morrey, "A First Course in Real Analysis", UTM Series, Springer-Verlag, 1991.
  • S. Zafrani, A. Pinkus, "Turei Fourier ve hatmarot integralijot", The Technion, 1997 (in Hebrew).