Week 2: 1. Motivation of the study of Fourier series; 2. Solution of the one-dimensional wave equation; 3. Traveling waves and standing waves; 4. The Dirichlet problem for the unit disk; 5. Convergence of the Fourier series; 6. The N-th Dirichlet kernel; 7. The Poisson kernel; 8. Uniqueness of Fourier series.
Week 3: 1. Convolution; 2. A family of good kernels; 3. Pointwise convergence for continuous functions; 4. The family of Dirichlet kernels are not good kernels.
Week 4: 1. Cesaro means and summability; 2. Fejer’s kernel; 3. Fejer’s theorem; 4. Approximation of a continuous function by trigonometric polynomials; 5. Poisson’s kernel; 6. Abel means and summability; 7. Solution of the Dirichlet problem; 8. Mean convergence; 9. Bessel’s inequality; 10. Parseval’s identity.
Week 5: 1. A local result of the pointwise convergence; 2. Riemann localization theorem; 3. Dini’s test; 4. Jordan’s test; 5. Gibb’s phenomenon.
HW#4(due 10/18): Chap 3, Exercises 15,16; Problems 1, 3.
Week 6: 1. Gibb’s phenomenon; 2. Some applications of Fourier series; 3. The isoperimetric inequality; 4. Hurwitz’s proof; 5. Weyl’s distribution theorem; 6. Equidistributed sequences.
Week 7: 1. Proof of Weyl’s equidistribution theorem; 2. Connection to the Fourier series; 3. Weyl’s criterion; 4. Ergodicity; 5. Examples of continuous nowhere differentiable functions.
HW#5(due 10/25): two problems assigned in class, Chap 4, Exercise 4, 5, 6, 8.
Week 8: 1. Fourier transform of an integrable function; 2. Properties of Fourier transform; 3. Convolution; 4. Young’s theorem; 5. Fourier inversion formula.
Week 9: 1. Fourier inversion formula for an integrable function; 2. Fourier transforms of square integrable functions; 3. Fourier-Plancherel formula; 4. Tempered distributions; 5. Fourier transform of a tempered distribution; 6. Poisson summation formula; 7. Periodization of a function; 8. The heat equation and the heat kernel.
HW#6(due 11/8): one problem assigned in class, Chap 5, Exercise 3, 14, 15, 16.
Week 10: 1. The heat equation on the real line and on the circle; 2. Poisson summation formula; 3. Poisson kernels for the upper half-plane and the disc; 4. The uncertainty principle; 5. The X-ray transform; 6. The Radon transform; 7. The uniqueness question.
HW#7(due 11/17): one problem assigned in class, Chap 5, Exercise 21, 23, Problem 1, 4.
Week 11: 1. The inversion formula of Radon transform; 2. The backprojection operator.
Week 12: 1. Some properties of the backprojection operator; 2. The Riesz potentials; 3. The Hilbert transform; 4. The inversion formula of Radon transform; 5. Local reconstruction formula and nonlocal reconstruction formula; 6. Symmetry properties of Fourier transform; 7. Decomposition of L^1(R).
HW#8(due 11/29): Chap 6, Exercise 12, 13, 14, Problem 6.
Week 13: 1. Spherical harmonics; 2. Harmonic polynomials; 3. Dimension of the space of spherical harmonics of degree k; 4. Denseness of spherical harmonics; 5. Expansion of a function defined on the sphere in terms of spherical harmonics.
Week 14: 1. Discrete Fourier transform; 2. Fast Fourier transform; 3. The sampling theorem; 4. Shannon’s sampling theorem.
Week 15: 1. Shannon’s sampling theorem; 2. Nyquist condition; 3. Over-sampling and under-sampling; 4. Aliasing; 5. Filtering; 6. The wave-front set.
Week 16: 1. Distributions; 2. Space of test functions; 3. Tempered distributions; 4. Fourier transform of a tempered distributions; 5. Distributions with compact supports; 6. Frequency set; 7. The wave front set.