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

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

- Fourier coefficients with respect to an orthonormal system, trigonometric bases in the
interval [-pi,pi].
- Partial sums and Dirichlet integral. Riemann-Lebesgue theorem, Riemann principle.
- Sin and Cos series (real Fourier series), complex representation of Fourier series.
- Dini and Dirichlet convergence criteria, and convergence at a discontinuity. Pointwise convergence
and uniform convergence.
- 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.
- Fourier integral in L_1(R), inverse formula. Fourier transform.
- Properties of Fourier transform: continuity and decay at infinity, dependance of the decay on
the smoothness. Fourier transform of Sin and Cos.
- Fourier transform in L_2(R), inverse Fourier transform. Difference between L_1(R) and
L_2(R). Parseval-Plancherel formulas.
- Laplace transform and its fundamental properties. Inverse Laplace transform.
- Application to ordinary and partial differential equations.

Books:
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).