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.
Grades (Midterm Tests / Final Exam Moed Alef / Final Grade in the Course)
Lecturers:
Teaching Assistants:
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)
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).
Midterm Test No. 1 - Solutions
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).
Midterm Test No. 2 - Solutions
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).
Midterm Test No. 3 - Questionnaire
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.
Final Exam Moed Alef - Questionnaire
A PostScript Source of the Book (Hebrew)
Fourier Series and Integral Transforms
by Allan Pinkus and Samy Zafrany.
Summary of the main convergence theorems for Fourier series.
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.
Final Exam Moed Alef, Fall 2009/2010, with sketches of solutions.
Homework Assignment No. 3, Fall 2008/2009, with
Partial Solution
Office Hours
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).
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
Mid Term Tests
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).
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.
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 -
Solutions
(pp. 1-4,
pp. 5-7)
Final Exam
There will be a
formulae sheet available at the exam.
Final Exam Moed Alef - Solution
Miscallenous
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.
(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.)
