Fourier Analysis (201-1-0041)

## Fourier Analysis for Electrical Engineering Students (201-1-0041)  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 Mid Term Tests: 15% Each,
Final Exam: 70%.

The midterm tests will take place on Friday 30.11 and Friday 04.01.

If you cannot or could not attend one of the 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 the grade of the final exam. Office Hours Lecturers:
Monday 10-11 and Wednesday 10-11 (Room 110, Building 58).
Prof. Victor Vinnikov (phone: 08-6461618)
Monday 15:30-17:30 (Room 103, Building 58).

Teaching Assistants:
Yoav Bar-Sinai
Monday 14-16 (Room 121, Building 58).
Sunday 11-12 and Monday 11-12 (Room -123, Building 58; notice the minus sign - the room is on the floor -1 BELOW the ground floor). 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) Mid Term Tests The first midterm will take place on Friday, 30.11.2012.
Material for the test: vector spaces, normed vector spaces, inner product spaces, best approximation and orthogonal projections, finite orthonormal systems, convergence in norm, infinite orthonormal systems, orthogonal polynomials.
Notice: the Haar system and Weierstrass' theorem on uniform approximation by polynomials on a finite interval are not included.
Relevant Practice Homeworks: No. 1, No. 2 (except for Problem 10), and the Practice Homework on Orthogonal Polynomials. In the Practice Homework on Orthogonal Polynomials, you should concentrate especially on Problems 1.3, 2.5, 2.6, 3.4, 4.1, 4.2, 4.6, and 5; for your convenience, a summary of orthogonal polynomials has been posted below.
Duration 1.5 hours (1 hour and 30 minutes).
There will be no aids (khomer ezer) for the test. You should remember the basic formulae from calculus as well as the basic trigonometric formulae. You should also remember the basic formulae relating to orthogonal projections and orthonormal systems. There is no need to memorize the various formulae related to specific orthogonal polynomials (Legendre, Chebyshev, Hermite); if any of these will be required for the test, the relevant formulae will be reminded in the questionnaire.

The second midterm will take place on Friday, 11.01.2013.
Material for the test: the Haar system, Weierstrass' theorem on uniform approximation by polynomials on a finite interval and approximate identities, and all of the material on Fourier series; the material of the first midterm test is not specifically targeted, but it may appear as well.
Relevant Practice Homeworks: No. 2 (Problem 10) and the Practice Homework on Fourier Series, as well as No. 1, (all of) No. 2, and the Practice Homework on Orthogonal Polynomials. Final Exam Moed Bet Final Exam - Questionnaire.
Notice: there is a mistake in Problem 3; the correction is as follows: solve the equation - find the function or show that there is no solution. Miscallenous A PostScript Source of the Book (Hebrew) Fourier Series and Integral Transforms by Allan Pinkus and Samy Zafrany.

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.)