Administrative Matters
The final grade in the course will be calculated as follows:
4 Mid Term Tests, 20% Each: 80%,
Homework Assignments: 20%.
NOTICE THAT THERE WILL BE NO FINAL EXAM!
Please submit the homework assignments at the course mailboxes
on the ground floor of Math Building (Building 58).
You have to submit the assignment into the mailbox bearing
the name and the number of the course.
FAILURE TO DO SO, OR
TO STAPLE THE PAGES IN YOUR ASSIGNMENT,
OR TO WRITE YOUR NAME AND ID NUMBER (mispar teudat zehut)
AT THE TOP OF THE FIRST PAGE,
MAY PREVENT YOU
FROM GETTING THE CREDITS.
All the assignments have to be submitted
by midnight of the due date.
No assignments will be accepted after that;
if you need an extension due to special circumstances
(reserve service, sickness, etc.),
please contact your lecturer for a permission.
Special Take Home Exam
(for students who missed two midterm tests and received a special approval;
to be submitted in person to Prof. Victor Vinnikov
on Thursday, April 16 at 11:00-11:30, Room 103, Building 58).
Lecturers:
Prof. Michael Gil
(phone: 08-6477805)
Sunday 18:00-20:00 (Room 306, Building 58).
Prof. Victor Vinnikov
(phone: 08-6461618)
Monday 16:00-17:00, Thursday 13:00-14:00 (Room 103, Building 58).
Dr. Yossi Strauss
Tuesday 14:00-15:00, Thursday 13:00-14:00 (Room -109, Building 58).
(Please use English rather than Hebrew when emailing us;
we do not necessarily have Hebrew fonts at hand.)
Teaching Assistants:
Dr. David Kats
Thursday 12:00-14:00 (Room -109, Building 58).
Dr. Irina Lerman
Tuesday 16:00-18:00 (Room 318, Building 58;
NOTICE THAT THE OFFICE HOURS WERE CHANGED).
Homework Assignment No. 1(to be submitted on Sunday, November 30).
SUBMISSION EXTENDED TILL SUNDAY, DECEMBER 7.
Homework Assignment No. 2(to be submitted on Thursday, December 11).
SUBMISSION EXTENDED TILL FRIDAY, DECEMBER 12 AT 09:00 (before the midterm test).
NOTICE: you may submit the assignment without Problem 8 (that deals with
infinite orthonormal systems) and submit Problem 8 together with
Homework Assignment No. 3.
Homework Assignment No. 3(to be submitted on Thursday, January 22).
SUBMISSION EXTENDED TILL SUNDAY, JANUARY 25.
NOTICE: if you submitted Homework Assignment No. 2 without Problem 8,
you have to submit that problem together with this homework assignment.
NOTICE: you may submit the assignment without Problem 5 and submit
Problem 5 together with Homework Assignment No. 4 (this is relevant
for the students of Prof. Michael Gil (Section 2) who still did not
study the Haar system; while the knowledge of the Haar system is not
necessary to solve Problem 5, it certainly helps).
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.
Homework Assignment No. 4(to be submitted on Thursday, February 19).
SUBMISSION EXTENDED TILL SUNDAY, FEBRUARY 22.
CORRECTIONS:
Problem 3:
the function to be used is f(x) = cos ax (NOT f(x) = sin ax); also, in
the sum on the right hand side, one of the two terms should be
1/(a pi + n pi).
Problem 4:
it should be 2x/pi rather than 2pi/x.
Problem 8:
it should be sin bx, cos bx (the variable is x, not t).
Homework Assignment No. 5(to be submitted on Sunday, March 1).
NOTICE:
if you did not submit yet Problem 5, Homework Assignment No. 3 and / or
Problem 1, Homework Assignment No. 4, you have to submit them with this
homework assignment.
Homework Assignment No. 6:
page 1,
page 2,
page 3,
page 4
(to be submitted on Tuesday, March 31).
Each midterm test will last for 2.5 hours. You will have to solve 4 problems
out of 5, each problem being worth 25 credit points.
Midterm Test No. 1: Friday, Dec. 12 at 09:00.
Material to be covered: vector spaces, normed vector spaces, inner product
spaces, best approximation and orthogonal projections,
finite orthonormal systems, Gramm-Schmidt orthogonalization process,
convergence in normed vector spaces.
NOTICE: infinite orthonormal systems, Bessel's inequality, Parseval's equality
and closed orthonormal systems will not be covered.
Relevant homework assignments: assignment no. 1 and assignment no. 2 (except
for Problem 8).
No accessory material (khomer ezer) will be allowed at the test.
Midterm Test No. 2: Thursday, Feb. 12 at 18:00.
Material to be covered: infinite orthonormal systems, closure of a set
in a normed vector space, completeness in normed vector spaces and
in inner product spaces, Fourier series (both real and complex, including
Parseval's equality, not including Fejer's Theorem).
Notice: Haar system is not included.
Notice: the material covered during the previous test(s) can appear on this
test as well.
Relevant homework assignments: Problem 8 of Assignment No. 2 and Assignment
No. 3 (except Problem 5).
No accessory material (khomer ezer) will be allowed at the test.
Midterm Test No. 3:
Thursday, Feb. 19 at 18:00 (NOTICE THE CORRECTION).
Material to be covered: Fourier series - Dirichlet kernel, Fejer kernel,
Fejer Theorem, pointwise and uniform convergence, differentiation and
integration of Fourier series; Fourier transform - definition and basic
properties.
Notice: more general kernels satisfying the same key properties as the
Fejer kernel (compare Problem 1, Assignment 4) AND the properties of
the Fourier transform having to do with shift, dilation, rotation, and
differentiation (compare Problems 8 and 9, Assignment 4) are not
included; all of these, as well as Haar systems (compare Problem 5,
Assignment 3) will be included on Midterm Test No. 4.
Notice: the material covered during the previous test(s) can appear on this
test as well.
Relevant homework assignments: Assignment No. 4 (except Problems 1, 8,
9).
Accessory material (khomer ezer): one standard sheet (A4), two sided,
with a material of your choice.
Midterm Test No. 4: Friday, Feb. 27 at 09:00.
Material to be covered: Fourier transform (the entire topic: properties,
Plancherel Theorem, inverse Fourier transform, Fourier transform in the
L^2 setting, convolution theorem); also, Haar system and
more general kernels satisfying the same key properties as the
Fejer kernel (compare Problem 1, Assignment 4).
Notice: the material covered during the previous test(s) can appear on this
test as well.
Notice: Hermite functions are not included.
Relevant homework assignments:
Problem 5, Assignment No. 3, Problems 1, 8, 9, Assignment No. 4,
and Assignment No. 5.
Accessory material (khomer ezer): one standard sheet (A4), two sided,
with a material of your choice.
If you are on reserve service on (or immediately before)
one of the midterm tests, please notify your lecturer as soon as possible.
Your final grade will be calculated based on the other 3 midterm tests.
Summary of the main convergence theorems for Fourier series.
Midterm Test No. 1 - Questionnaire.
Midterm Test No. 1 - Solution (pp. 1-13 / problems 1, 2, 3).
Midterm Test No. 1 - Solution (pp. 14-18 / problems 4, 5).
Midterm Test No. 2 - Questionnaire.
Midterm Test No. 2 - Solution.
Midterm Test No. 3 - Questionnaire.
Midterm Test No. 3 - 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 - Questionnaire:
page 1,
page 2.
Midterm Test No. 4 - Solution:
page 1,
page 2,
page 3,
page 4,
page 5,
page 6,
page 7,
page 8.
Homework Assignment No. 3 - Partial Solution.
Notice: 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.
