Fourier Analysis - Theory and Applications : 18.103 (Spring 2003)

Lectures

Textbooks

Graders

Examinations and Homework

• HW #1, due Thursday, February 13: (Chapter 2) 2, 4(abc), 6, 7, 12, 15, 16, 18, 21 ,23.
• HW #2, due Tuesday, February 25: (Chapter 2) 24, 25, 26, 29, 30, 31, 32, 34, 35. Extra credit: 47.
• HW #3, due Tuesday, March 4: (Chapter 5) 13, 14, 15, 17, 19, 20, 21, 22, 23.
• HW #4, due Tuesday, March 11: (Chapter 6) 1, 2, 5, 9, 10, 11, 12, 14, 16, 17.
• HW #5, due Tuesday, March 18: (Chapter 6) 19, 21, (Chapter 7) 6, (Chapter 8) 1, 2, 3, 4, 5, 6.
• HW #6, due Tuesday, April 1: (Chapter 8) 7, 8, 9, 10, 11, 12, 16, 17, 18.
• HW #7, due Tuesday, April 8: (Chapter 10) 1, 2, 3, 4, 5, 6, 7, 9, 10.
• HW #8, due Tuesday, April 15: (Chapter 10) 12, 13, 18, 20, (Chapter 12) 2, 8, 12, 13.
• HW #9, due Thursday, April 24: (Chapter 13) 2, 3, 4, 5, 6, 7.
• HW #10, due Tuesday, May 6: (Chapter 13) 26, 27, 39, 40, 43, 45, 46.
• HW #11, due Thursday, May 15: (Chapter 14) 4, 5, 8, 14, 15, 16, 30.

Brief Lecture notes

• Lecture 1: February 4
  
  1. Measure of regtangles, polygons, open sets, compact sets.
• Lecture 2: February 6
  
  1. Properties of measure for compact and open sets.
  2. Remarks about Lebesgue integral and Riemann integral.
• Lecture 3: February 11
  
  1. Cantor ternary set.
  2. Outer measure and inner measure.
  3. Lebesgue measurability for finite outer measure.
  4. Main approximation theorem (measurable <-> can approx. by G open and K closed such that G \ K has arbitrarily small measure).
• Lecture 4: February 13
  
  1. Lebesgue measurability.
  2. Unions and complements.
  3. Countable subadditivity.
• Lecture 5: February 20
  
  1. Properties of Lebesgue measure.
  2. Approximation theorem.
  3. Caratheodory criterion.
  4. Axiom of choice -> existence of a non-measurable set.
• Lecture 6: February 26
  
  1. Sigma-algebras.
  2. Borel sets = sigma-algebra generated by the open sets.
  3. A Lebesgue measurable set which is not Borel.
  4. The extended real number system.
• Lecture 7: February 28
  
  1. Measurable functions.
  2. Sums, products, and compositions of measurable functions.
  3. Simple functions.
• Lecture 8: March 4
  
  1. Approximation of measurable functions by simple functions.
  2. Integral of simple function.
  3. Integral of measurable functions.
  4. Lebesgue's monotone convergence theorem.
• Lecture 9: March 6
  
  1. Fatou's Lemma.
  2. General measurable functions.
  3. Lebesgue's dominated convergence theorem.
  4. Almost everywhere.
• Lecture 10: March 11
  
  1. Integration on subsets of R^n.
  2. Riemann integral.
  3. Comparison of Lebesgue and Riemann integral.
• Lecture 11: March 13
  
  1. Fubini's Theorem for nonnegative functions.
• Lecture 12: March 18
  
  1. Fubini's Theorem for L^1 functions.
  2. Exam review.
• March 20
  
  1. Exam I.
• Lecture 13: April 1
  
  1. L^p spaces.
  2. Holder inequality.
  3. Minkowski inequality.
• Lecture 14: April 3
  
  1. Metric spaces and completeness.
  2. Normed spaces, Banach spaces.
  3. Riesz-Fischer Theorem (completeness of L^p).
  4. Convergence in L^p.
• Lecture 15: April 8
  
  1. Essentially bounded functions.
  2. L^{\infty} is a Banach space.
  3. The limit of L^p norms as p -> \infty.
  4. Definition of L^p(X), for X a measurable subset of R^n.
  5. If \lambda(X) < \infty, then L^q(X) \subset L^p(X) for p < q.
• Lecture 16: April 10
  
  1. Local L^p spaces.
  2. Convexity relations between L^p norms.
  3. Convolutions.
  4. Young's Inequality.
• Lecture 17: April 15
  
  1. Fourier Transform
  2. FT of L^1 function is bounded and continuous.
  3. FT of convolutions.
  4. Density of C^0 functions in L^1.
• Lecture 18: April 17
  
  1. Continuity of translation in L^1.
  2. Riemann-Lebesgue Lemma.
  3. Examples of Fourier Transforms.
• Lecture 19: April 24
  
  1. Fourier inversion Theorem in L^1.
  2. Schwartz functions.
  3. Parseval's Identity for Schwartz functions.
• Lecture 20: April 29
  
  1. Approximation of L^p function by smooth functions.
  2. Fourier-Plancherel transform on L^2.
• Lecture 21: May 1
  
  1. Applications of Fourier transform.
  2. Hilbert spaces.
• Lecture 22: May 6
  
  1. Fourier Series of periodic functions.
  2. Geometry of Hilbert spaces.
  3. Exam review.
• May 8
  
  1. Exam II
• Lecture 23: May 13
  
  1. Examples of Fourier series.
  2. Convergence of Fourier series.
• Lecture 24: May 15
  
  1. Fourier inversion theorem.
  2. Parseval's Identity.