Fourier Analysis - Theory and Applications : 18.103 (Spring 2005)

Lectures

Textbooks

Graders

Examinations and Homework

• HW #1, due Thursday, February 10: (Chapter 2) #2, 4(abc), 6, 7, 12, 15, 16, 18, 21 ,23.
• HW #2, due Thursday, February 17: (Chapter 2) 24, 25, 26, 29, 30, 31, 32, 34, 35. Extra credit: 47.
• HW #3, due Tuesday, March 1: (Chapter 5) 13, 14, 15, 17, 19, 20, 21, 22, 23.
• HW #4, due Tuesday, March 8: (Chapter 6) 1, 2, 5, 9, 10, 11, 12, 14, 16, 17.
• HW #5, due Tuesday, March 15: (Chapter 6) 19, 21, (Chapter 7) 6, (Chapter 8) 1, 2, 3, 4, 5, 6.
• HW #6, due Tuesday, March 29: (Chapter 8) 7, 8, 9, 10, 11, 12, 16, 17, 18.
• HW #7, due Tuesday, April 5: (Chapter 10) 1 - 10.
• HW #8, due Thursday, April 15: (Chapter 10) 12, 13, 18, 20, (Chapter 12) 2, 8, 12, 13.
• HW #9, due Tuesday, April 26: (Chapter 14) 4, 5, 8, 14, 15, 16, 29, 30, 31, 32.
• HW #10, due Tuesday, May 3: (Chapter 13) 2, 3, 4, 5, 6, 7, (Chapter 14) 41, 42, 43, 44, 45.
• Extra Credit for Exam II, due Thursday, May 12: Write up complete and correct solutions to ALL problems on Exam II, to add 7 extra points to Exam II score.

Brief Lecture notes

• Lecture 1: February 1
  
  1. Measure of regtangles, polygons, open sets, compact sets.
• Lecture 2: February 3
  
  1. Properties of measure for compact and open sets.
  2. Remarks about Lebesgue integral and Riemann integral.
• Lecture 3: February 8
  
  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 10
  
  1. Lebesgue measurability.
  2. Unions and complements.
  3. Countable subadditivity.
• Lecture 5: February 15
  
  1. Properties of Lebesgue measure.
  2. Approximation theorem.
  3. Caratheodory criterion.
• Lecture 6: February 17
  
  1. Axiom of choice -> existence of a non-measurable set.
  2. Sigma-algebras.
  3. Borel sets = sigma-algebra generated by the open sets.
  4. A Lebesgue measurable set which is not Borel.
• Holiday, no class on February 22
  
• Lecture 7: February 24
  
  1. The extended real number system.
  2. Measurable functions.
  3. Sums, products, and compositions of measurable functions.
  4. Simple functions.
• Lecture 8: March 1
  
  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 3
  
  1. Fatou's Lemma.
  2. General measurable functions.
  3. Lebesgue's dominated convergence theorem.
  4. Almost everywhere.
• Lecture 10: March 8
  
  1. Integration on subsets of R^n.
  2. Riemann integral.
  3. Comparison of Lebesgue and Riemann integral.
• Lecture 11: March 10
  
  1. Fubini's Theorem for nonnegative functions.
• Lecture 12: March 15
  
  1. Fubini's Theorem for L^1 functions.
  2. Exam review.
• March 17
  
  1. Exam I.
• Lecture 13: March 29
  
  1. L^p spaces.
  2. Holder inequality.
  3. Minkowski inequality.
• Lecture 14: March 31
  
  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 5
  
  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 7
  
  1. Local L^p spaces.
  2. Convexity relations between L^p norms.
  3. Convolutions.
  4. Young's Inequality.
• Lecture 17: April 12
  
  1. Urysohn functions.
  2. C_c dense in L^1.
  3. C^{\infty} functions and mollifiers.
• Lecture 18: April 14
  
  1. Differentiating under the integral sign.
  2. C^{\infty}_c dense in L^p, 1 \leq p < \infty.
  3. Continuity of translation in L^p.
  4. Hilbert spaces
  5. Cauchy-Schwartz inequality.
  6. Periodic functions, f: S^1 -> C.
• Patriots Day, no class on April 19
  
• Lecture 19: April 21
  
  1. Absolutely convergent Fourier series.
  2. Complete orthonormal sequence in Hilbert spaces.
  3. Partial Fourier sums converge in norm.
  4. Parseval's equality.
  5. Real and complex forms of Fourier series.
  6. Properties of Fourier coefficients.
  7. Examples of Fourier series (square wave, sawtooth wave...).
• Lecture 20: April 26
  
  1. Fourier Transform
  2. FT of L^1 function is bounded and continuous.
  3. FT of convolutions.
  4. Riemann-Lebesgue Lemma.
  5. Examples of Fourier Transforms.
• Lecture 21: April 28
  
  1. Pointwise convergence of Fourier Series for Lipschitz functions.
  2. Uniform convergence for globally Lipschitz functions.
  3. Fourier Inversion Theorem for Fourier series in L^2.
• Lecture 22: May 3
  
  1. Fourier Inversion Theorem for Fourier transform.
  2. Schwartz class.
  3. Fourier-Plancheral Transform.
  4. Exam Review.
• May 5
  
  1. Exam II.
• Lecture 23: May 10
  
  1. Fejer kernel and Cesaro summation.
  2. Fourier inversion theorem for absolutely convergent Fourier Series.
  3. Weierstrass approximation theorem.
  4. Gibbs phenomenon.
  5. Application of Fourier transform to heat equation and Laplace's equation.
• Lecture 24: May 12, NOTE: Class to be held in 2-131.
  
  1. Shannon sampling theorem.
  2. Computer audio demonstration.