Math 144: Introduction to Fourier Analysis, Fall 2019

Instructor: Ioan Bejenaru

Office: AP&M 6230
E-mail: ibejenaru@ucsd.edu; Include Math 144 in the subject line.
Office Hours: Mo 3-4pm, Fri 1-2pm
I can also schedule appointments upon request. E-mail at the address above.

Teaching Assistant

Nandagopal Ramachandran

E-mail: naramach@ucsd.edu

Office: APM 5748

Office hours: Th 5-6pm, Fr 9:15-10:15am

Announcements:

• Final information. Material covered: all material covered by the midterm plus Chapter 3 (except 2.2), Chapter 4 (sections 2 and 4 but not 1 and 3), Chapter 5 (except section 4).
  OH during final week: Nandagopal Mo 3-6pm, Ioan Tu 2:30-4pm.
• Midterm information. Material covered: Chapters 1: 1.2 (only the superposition of standing waves), 2.2 (steady state heat equation in the disc), Chapter 2: sections 1,2,3,4,5, Chapter 3: 1 and 2.1 (note 2.2 is not covered)
  
• Quiz information. Material covered: Chapters 1: 1.2 (only the superposition of standing waves), 2.2 (steady state heat equation in the disc), Chapter 2: sections 1,2,3
  
• All course information will be updated on this site. Canvas will be used only for recording scores for HW, Midterm, Final.

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  Prerequisites: Math 140B or Math 142B, or consent of instructor.

  Course description: Rigorous introduction to the theory of Fourier series and Fourier transforms. Topics include basic properties of Fourier series, mean square and pointwise convergence, Hilbert spaces, applications of Fourier series, the Fourier transform on the real line, inversion formula, Plancherel formula, Poisson summation formula, Heisenberg uncertainty principle, applications of the Fourier transform.

  Textbook: Fourier Analysis by E.M. Stein and R. Shakarchi. We will cover most of chapters 1-5.

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  Homework: Homework will be assigned weekly. Homework is due every Friday by 4pm and should be placed in the drop-box in the basement of APM. No late homework will be accepted. If, for any reason, you cannot turn in a homework assignment, keep in mind that the lowest two scores will be dropped.
  

  Presentation: Every student is expected to give a (roughly) half hour presentation in front of their peers. This can be: teach new material in class, lead a discussion session for half hour, present some interesting topic they discovered that is somehow related to the course etc.

  Quiz: There will be one in-class quiz on October 22nd . There will be no make-up exams.

  Midterm Exam: There will be one in-class midterms on November 7th . There will be no make-up exams.

  Final exam: The final exam is scheduled for Wednesday, December 11, 11:30-2:30pm.

  Exam policy (applies to all midterms and final): Cheating will be taken very seriously and automatically reported to the Academic Integrity Office. The use of any electronic devices (cell-phone, smart watch, earbud, calculator, etc) is prohibited during exams and it will be considered cheating, thus it will be reported to the Academic Integrity Office.

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  Grading: Your final score will be calculated as follows
  15% Homework + 5% Presentation+ 10% Quiz + 25% Midterm + 45% Final Exam

  In addition you must earn a passing grade on the final examination in order to pass the course.

  Regrades: Homework and midterm exams will be returned in the discussion sections. If you wish to have your homework or exam regraded, you must return it immediately to your TA. Regrade requests will not be considered once the homework or exam leaves the room. If you do not retrieve your homework or exam during discussion section, you must arrange to pick it up from your TA within one week after it was returned in order for any regrade request to be considered.

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  Schedule of lectures:
  • Lecture 1 (9/26): Historical background, Simple harmonic motion (Chapter 1), Solution to the wave equation (1.2)
  • Lecture 2 (10/1): Derivation of the wave equation 1.1, superposition of standing waves (1.2), the plucked string (1.3), derivation of the heat equation (2.1)
  • Lecture 3 (10/3): Steady-state heat equation in the disk (Chapter 1, 2.2), Basic properties of Fourier series (2), main definitions (2.1.1)
  • Lecture 4 (10/8): Examples 1-5 (Chapter 2, 1.1), Uniqueness of Fourier series (Chapter 2, 2)
  • Lecture 5 (10/10): Convolution(Chapter 2, section 3)
  • Lecture 6 (10/15): Good kernels (Chapter 2, section 4), Cesaro and Abel summability (Chapter 2, section 5)
  • Lecture 7 (10/17): Cesaro and Abel summability (Chapter 2, section 5),

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    List of homework assignments:

    • Homework 1, due Oct 11: Chapter 1, exercises 5,9,10,11 and Problem 1
    • Homework 2, due Oct 18: Chapter 2, exercises 1,2,3,6,9,10,11 and Problem 1     Solutions
    • Homework 3, due Oct 25: Chapter 2, exercises 12,15,16,17,19 and Problem 2     Solutions
    • Homework 4, due Nov 1: Chapter 3, exercises 2,6,8,9,11,15,16,17
    • Homework 5, due Nov 12: Chapter 3, exercises 19, Problem 1; Chapter 4: exercise 1,11
    • Homework 6, due Nov 15: Chapter 4: exercise 5,7,8,9,10; Problem 2
    • Homework 7, due Nov 22: Chapter 5: exercises 1,2,3,4,5
    • Homework 8, due Dec 2: Chapter 5: exercises 7,8,9,11,12 Problem 1; optional Exercise 20
    • Homework 9, not to be handed in: Chapter 5: 14,15,20,21; Optional 17,18