Fourier Analysis (201-1-0041)

## Fourier Analysis for Electrical Engineering Students (201-1-0041)

Administrative Matters

PREREQUISITES: Differential and Integral Calculus (Hedva) II for Electrical Engineering Students (201-1-9821), Linear Algebra for Physics and Engineering Students (201-1-9641), Ordinary Differential Equations for Electrical Engineering Students (201-1-9841), Foundations of Complex Function Theory (201-1-0071, in parallel).

SYLLABUS:
Normed vector spaces and inner product spaces, best approximation and orthogonal projections, orthonormal systems, Bessel inequality and Parseval equality, closed orthonormal systems, Haar's system.
Fourier series, Fejer's Theorem, L^2 convergence of Fourier series, uniform convergence, Gibbs' phenomenon, termwise integration and differentiation, multidimensional Fourier series.
Fourier transform, Plancherel's Theorem, Hermite functions, L^2 Fourier transform, Fourier inversion formula, convolution, Shannon-Kotelnikov sampling theorem.
Laplace transform, relation to the Fourier transform, the inversion formula.
An introduction to distributions.

The FINAL GRADE in the course will be calculated as follows:
2 Best Mid Term Tests out of Three: 25% Each,
Final Exam: 50%.

The midterm tests will take place on Friday 25.11.2011, Friday 23.12.2011, and Friday 13.01.

If you cannot or could not attend one of the three midterms due to a justified reason (reserve service, hospitalization, time conflict with another test), please notify your lecturer at once. The grade of the missing midterm will be replaced by a simple average between the grades of the two other midterms and the grade of the final exam.

Office Hours

Lecturers:
Dr. Orr Shalit (phone: 08-6477815)
Tuesday 16-17 and Wednesday 10-11 (Room 310, Building 58).
Dr. Yossi Strauss
Wednesday 16-18 (Room -109, Building 58).
Prof. Victor Vinnikov (phone: 08-6461618)
Monday 15-16 and Wednesday 17-18 (Room 103, Building 58).

Teaching Assistants:
Yoav Bar-Sinai
Monday 11-13 (Room 121, Building 58).
Nir Shreiber
Wednesday 14-15 and 18-19 (Room -127, Building 58).
Yonatan Yehezkeally
Sunday 12-13 and Wednesday 17-18 (Room 227, Building 51).

Practice Homework

Practice Homework No. 1 (vector spaces, normed vector spaces, inner product spaces)

Practice Homework No. 2 (best approximation and orthogonal projections, convergence in norm, orthonormal systems)

Practice Homework No. 3 (Fourier series, revised)

Practice Homework No. 4 (Fourier transform)

Practice Homework No. 5 (more on Fourier transform; Laplace transform; distributions)

Mid Term Tests

The first midterm test will take place on Friday 25.11.2011 at 09:00. Location: Building 28 (Rooms 202,203,204,205) and Building 32 (Rooms 206,207,208,209,210).
Material: vector spaces, normed vector spaces, inner product spaces, best approximation and orthogonal projections, finite orthonormal systems, Bessel's inequality. (Notice that Bessel's inequality is the only material on infinite orthonormal systems included in this test.)
Relevant Practice Homeworks: No. 1 (all problems), No. 2 (problems 1,2,3,4a,7a,9a,9c,11).

The second midterm test will take place on Friday 23.12.2011 at 11:00. Location: Building 28 (Rooms 202,203,204,205), Building 32 (Rooms 206,207,208,209,309), and Building 34 (Room 214).
Material: convergence in norm, infinite orthonormal systems, the Haar system, Fourier series, Fejer's Theorem, L^2 convergence of Fourier series; the material of the first midterm test is not specifically targeted, but it may appear as well; notice that pointwise and uniform convergence of Fourier series, and termwise differentiation and integration, are not included.
Relevant Practice Homeworks: No. 2 (all problems, but especially those that were not relevant for the first midterm test), No. 3 (problems 1-3 and 8-11 in the revised version), as well as No. 1.

The third midterm test will take place on Friday 13.01.2012 at 09:00. Location: Building 90 (Rooms 123,125,127,134,135,136,137,138,139,140,141,144,145,225,226).
Material: Fourier series - pointwise and uniform convergence, differentiation and integration, Fourier series on arbitrary intervals and multidimensional Fourier series; Fourier transform - definition, basic properties, Placherel's Theorem, L^2 Fourier transform, the inversion formula. Notice that convolution and Laplace transform are not included. Notice also that while the material of the first and the second midterm test is not specifically targeted, it may appear as well.
There will be a formulae sheet available at the test.
Relevant Practice Homeworks: No. 3 (all problems, but especially those that were not relevant for the second midterm test), No. 4 (except for problems 4-5 and 8-9; in problem 7, you can ignore the words ``Shannon's sampling theorem''), as well as No. 1 and No. 2.

Midterm Test No. 3 - Questionnaire
Midterm Test No. 3 - Solutions (pp. 1-4, pp. 5-7)

Final Exam

The final exam will cover the entire course. You will have to choose three problems out of four, with each problem worth 35 credit points.
There will be a formulae sheet available at the exam.

Miscallenous

A PostScript Source of the Book (Hebrew) Fourier Series and Integral Transforms by Allan Pinkus and Samy Zafrany.

Tutorials by Yonatan Yehezkeally and by Yoav Bar Sinai (the students are encouraged to print the tutorial and bring it with them to the exercise session).

Midterm Test No. 1, Fall 2008/2009 - Questionnaire.
Midterm Test No. 1, Fall 2008/2009 - Solution (pp. 1-13 / problems 1, 2, 3).
Midterm Test No. 1, Fall 2008/2009 - Solution (pp. 14-18 / problems 4, 5).
Midterm Test No. 2, Fall 2008/2009 - Questionnaire.
Midterm Test No. 2, Fall 2008/2009 - Solution.
Midterm Test No. 3, Fall 2008/2009 - Questionnaire.
Midterm Test No. 3, Fall 2008/2009 - Solution.
Solution of Problem 5 - correction: the solution as written breaks down for omega = 0. Since f(x) is an odd function, it is immediate that hat f(0) = int_{-infinity}^{+infinity} f(x) dx = 0. (Alternatively, one can calculate hat f(0) as lim_{omega to 0} hat f(omega), using the fact that the Fourier transform of an integrable function is a continuous function.)
Midterm Test No. 4, Fall 2008/2009 - Questionnaire: page 1, page 2.
Midterm Test No. 4, Fall 2008/2009 - Solution: page 1, page 2, page 3, page 4, page 5, page 6, page 7, page 8.

Homework Assignment No. 3, Fall 2008/2009, with Partial Solution
(CORRECTIONS: Problem 5: in the definition of r_n(x) there should be x rather than t; the definition of w_n(x) should be r_{l+1}(x)^{epsilon_l} ... r_1(x)^{epsilon_0}. Problem 7.2: compute only the summation of 1/n^6 (ignore the summation of 1/n^5, and the hint). COMMENTS: Problem 3.3: a CLOSED vector subspace (or in general, a closed subset) is one which is equal to its closure; this has nothing to do with the notion of a closed orthonormal system. Problem 6.1: b is the IMAGINARY PART of the complex number w. FOR THE SOLUTIONS: Problems 9.3 and 10 are not included. In Problems 5.1 and 5.3 some details that have to be included are skipped. In Problems 6.2 and 7.2, the calculations are not carried through till the end.)