*** WELCOME TO FOURIER SERIES & INTEGRAL TRANSFORMS ***
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*** ( F. S. I. T. ) ***
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We wish you a very pleasant, interesting and successful experience with this course.
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(THE MOST RECENT ANNOUNCEMENTS WILL BE INSERTED HERE. PLEASE GO FURTHER DOWN THIS PAGE TO SEE EARLIER ANNOUNCEMENTS.)
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13.10.06 Some explanations about the correction of the exam of 4.10.06.
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8.10.06 Khag Sameakh!
Here is a solution of the examination of 4.10.06 And here is one version of the examination itself.
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20.9.06 Here are some extra details about Question 5 of last week’s examination. Please look at this document, especially if you are thinking of submitting an “irur”.
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13.9.06 Here is a solution of today’s examination. (This has meanwhile been updated (to Version 3) on 6.10.06) And here is one version of the examination itself.
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8.9.06 Here, as requested by Liran, is a solution of Question 5 AND NOW ALSO QUESTION 10, from Homework No. 3. There are some associated pictures here.
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27.8.06. The “printer friendly” version of the file originally posted here on 2.7.06 (see below) has now been updated again. The changes are small. The main change is that the material in that document which is NOT required for the exams this/last semester now appears in blue and in a different font.
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10.8.06 We have decided to increase the weight of the bachan to 35% and the maximum possible weight of the homework magen to 15%. For more details see the email sent to all students today. It is also here.
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3.8.06 Here is a more detailed version of one of the calculations that I did in the video tutorial today.
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25.7.06 Here is a more “reader friendly” version of the document posted on 23.7.06 for those of you who do not need information about the graphics files.
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23.7.06. Here are details about the topics of the course which you are expected to know, and also about how to get the graphics files of pictures of the blackboard for most of the lectures this semester.
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12.7.06. The standard page (two sides) of formulae which you will be given for use during the exam is here. Do NOT bring your copy to the exam. Please be sure to read the email sent to all students today for further information.
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Here is a more precisely formulated version of the Laplace transform homework exercises which we published a few days ago.
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4.7.06 Here are two pictures relevant to the notes for last Monday’s lectures: the semicircle S_R and the curve Gamma_R. There is also a third picture, the graph of a certain function, related to an exercise in those lectures. I suggest that you try first to draw the graph yourself. Then you can see if you and I got the same answer, using the address (URL) given in the lecture notes.
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2.7.06 Here is a “printer friendly” version (Now version 3, updated on 27.8.06) of the file that I will use for tomorrow’s lecture. I suggest that you print a copy and bring it to the lecture.
And here are two homework exercises, an “unofficial” preliminary version of part of a set of homework exercises about the Laplace transform which will be available very soon.
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25.6.06 Here is a pdf file which displays the material which I will present in tomorrow’s lecture about Laplace transforms. (FILE UPDATED ON 26/6/06.)
This file does not include various additional remarks to be made during the lecture vocally and/or on the blackboard, for example, about integrals on [0,infinity) and convolutions. I will decide in consultation with you tomorrow to what extent I will present the lecture using a computer to display these pages, and to what extent I will lecture in the “traditional” way.
If you print these pages and bring them to the lecture you will be able to simply add comments to them instead of having to write all the details again.
Here is another (monochrome) version of this same file which is more suitable for printing. (THIS FILE WAS ALSO UPDATED ON 26/6/06.)
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19.6.06 Thanks Gilad (TODA GILAD!!) for noticing some misprints in Exercise 5 part b, on page 4 from the exercises posted here on 17.6.06. The link from 17.6.06 will now give you the corrected version. If you just want to see the single line on page 4 which has been changed, click here.
For those of you who want to think a bit more deeply about things related to today’s lecture, there is now an extra exercise (no. 11) in that set.
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17.6.06. A final version of those 8 exercises in Hebrew is now available here. I have also added exercises 9 and 10 (and now also 11). Exercises 5 and 10 from this document are part of the official homework assignment 3, about which Yoram has just sent you all a message.
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15.6.06 Here is a collection of 8 rather special exercises (in English). (The Hebrew version of them has meanwhile been posted on 17.6.06.)
CHANGE OF ORDER OF INTEGRATION FOR GENERALISED INTEGRALS ON AN UNBOUNDED INTERVAL.
Here is a formulation of Fubini’s theorem which is useful for changing order of integration in connection with Fourier transforms.
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12.6.06 Here is an exercise (in Hebrew) which will help you check to what extent you understood today’s lecture.
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5.6.06 Here are some explanations about (1) the Lebesgue dominated convergence theorem and (2) the formula of Leibniz for differentiating under the integral sign. We consider both of these results in versions which are suitable for use with the Fourier transform
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16.5.06 One version of the midterm test is here and a solution for all versions is here.
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9.5.06 Three things today:
(a) Here is a summary (Version 4.6, revised on 9.5.06). It deals with TWO topics:
(1) Differentiation and integration of Fourier series, term by term, including a connection with uniform convergence, and
(2) Fourier series on other intervals instead of [-pi,pi].
We are now discussing or will soon be discussing Topic (1). Topic (2) is simpler and may be left for you to study privately.
(b) Topic (1) includes a mention of uniform convergence. (hitkansut b’mida-shava). This is something which you are supposed to remember from Hedva 1M (or Hedva 2M). Although, in general, uniform convergence is very important, it only plays a limited role in this particular course. Here is a quick summary, a “Uniform Convergence Survival Kit” containing the main things you need to know about uniform convergence for this course.
(c) Here are some notes about two sided series. (A new version with small changes was posted here on 17/4/06.)
2.5.06 Since the deadline for submitting Question 5 of the first set of homework exercises has now passed, we are now publishing a solution of this question. (As of 3.5.06 this solution has been slightly modified with a comment added at the end.)
If you have questions before the test you are invited to come and ask them between 17:30 and 19:30 on Sunday 7/5/2006 in Amado 233. Yoram or I will be there during that time.
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15.4.06 Chag Sameakh, vKhofesh Nayim.
The first set of homework exercises is available here. Some small (mostly obvious) clarifications about the formulation of these exercises can be found in a email message sent today to all students.
There is also another set of five exercises about orthonormal systems here. Two of these are the same as two of the homework exercises. Most of these five exercises contain information which will be useful later on in the course.
Even though we have not yet proved that the standard trigonometric orthonormal systems are closed in C[-pi,pi] or E[-pi,pi] you can assume this already in exercises where you may need it, before we give the proof.
Some of my old notes are available here. (English, 4 pages). They give the easy proof of the equivalence of Parseval’s identity to closedness of an orthonormal system, and describe the Gram-Schmidt procedure. (Both of these topics were also discussed fully in the lectures and in the book of Pinkus-Zafrany.) The last two pages are a discussion of some more exotic questions, not compulsory for this course, about the connection between closed orthonormal systems and complete orthonormal systems.
For other perhaps simper versions of exotic counterexamples showing that an orthonormal sequence may sometimes be complete but not closed, see some notes prepared in Hebrew by Dr. Alla Shmukler and also my notes here.
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25.03.06 Some of the topics to be covered soon in this course are the theorem about orthogonal projections, Bessel’s inequality and the Riemann-Lebesgue lemma. Here are some notes about these three topics which I wrote in 2002. The approach in those notes is slightly different from the approach used in the course textbook. The approach which we will use in lectures this semester may also be slightly different.
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23.3.06 Here is a summary of the lecture that I gave on Monday. It also corrects three small misprints in some formulae I wrote on the board, when I was defining piecewise continuous functions. Thanks Yahel for noticing two of these. If you want to see just the corrected versions of those formulae you can click here.
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22.3.06 To go to the OFFICIAL website for this course click here. That site also contains ESSENTIAL information, exercises, text book, previous examinations, etc. The documents there include two pages of general information and instructions and this semester’s syllabus. There is also a link from there, and from here to the page of the Metargel Akhrayi where you can see office hours for your teachers, dates of examinations etc.
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To go directly to download or read the text book for this course, written by Allan Pinkus and Samy Zafrany, click here, or here,
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BELOW THIS LINE ARE OLD ANNOUNCEMENTS FROM EARLIER SEMESTERS.
(Some of these can still be quite useful.)6.6.2005. Here is a pdf file which displays the “slides” which I used in today’s lecture about Laplace transforms for the classes of Avi Levy and Gershon Wolansky. (Some misprints, hopefully all misprints have been corrected.) This file does not include various additional remarks made during the lecture vocally and on the blackboard, for example, about integrals on [0,infinity) and convolutions.
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If you want to see why we should not be robots and why I congratulate Yaacov Marko, you can visit my very old Fourier site here and look there at the entries for 17/9/01.
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ANNOUNCEMENTS FOR THE WINTER SEMESTER 2004/5
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19.3.05 Here is (one version of) the examination of 16.3.05. And here is a solution of it (updated on 24.3.05).
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1.3.05 Here is a probably not final version of some notes which explain an exact way of defining and using Fourier transforms of certain functions which are not absolutely integrable. In fact this definition works for “functions” which are not even really functions, like the Dirac delta “function”. Of course you are NOT expected to know any of this material for examinations in this course, but this material may help you have a deeper understanding of certain topics in later courses such as “Otot uMarakhot”. (A new version (1.9) of these notes was posted here on 2/2/2006, but it is quite similar to the previous version. Several small misprints have been corrected.)
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20.02.04 Here is a ‘compact’ version of the examination of 15/2/2005. And here is a solution of it.
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27.01.04 Here is a list of the topics that you will be expected to know for the examination.
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23.1.05 Here is Homework Exercises No. 4.
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17.1.05 The CD with pictures of the blackboards from my lectures has now been updated. For more details see here.
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11.1.05 Here is the third set of homework exercises.
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6.1.05 Here is the second set of homework exercises.
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27.12.04 You can see a solution of the mid term test on the official course website (page of the Metargel Akhrayi) or here. It can save a lot of time, both for you and for us if you look at this solution BEFORE you come to look at your makhberet from that test. Many thanks to Yoram Yihyie for writing most of the solution.
CHANGE OF ORDER OF INTEGRATION FOR GENERALISED INTEGRALS ON AN UNBOUNDED INTERVAL.
Here is a formulation of Fubini’s theorem which is useful for changing order of integration in connection with Fourier transforms.
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9.12.04 Since some students are still asking questions about what topics are included or not included in Sunday’s test, I have included further clarifications in the old announcement below. You can also see it directly here.
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1/12/04 The midterm test is on Sunday December 12 at 17:30. The rooms are shown in the usual place on the “undergraduate” website. You should be ready to to answer questions on all the subjects treated in our lectures and tirgulim up to and including Dirichlet’s theorem, including of course exercises which use Dirichlet’s theorem. If this does not define what you need to know sufficiently precisely for you, you can read this.
If you have questions before the test you can of course come to the sha’ot kabala of the lecturers and metargelim.
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27.11.04 Three things today:
(a) Here is a summary (Version 4.6, revised on 9.5.06). It deals with TWO topics:
(1) Differentiation and integration of Fourier series, term by term, including a connection with uniform convergence, and
(2) Fourier series on other intervals instead of [-pi,pi].
We are now discussing or will soon be discussing Topic (1). Topic (2) is simpler and may be left for you to study privately.
(b) Topic (1) includes a mention of uniform convergence. (hitkansut b’mida-shava). This is something which you are supposed to remember from Hedva 1M (or Hedva 2M). Although, in general, uniform convergence is very important, it only plays a limited role in this particular course. Here is a quick summary, a “Uniform Convergence Survival Kit” containing the main things you need to know about uniform convergence for this course.
(c) Viewing my lectures.
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22.11.04 A REVISED VERSION of the first set of homework exercises is here. (As announced earlier, there will not be any grade for homework and there will not be any correcting of homework. However on the mid term test and also on the final exam there will be at least one question taken from the homework exercises.)
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28.10.04 Please click here for VERY IMPORTANT information about (1) the format of the tests/exams, (2) the calculation of your tsiyun sofi, (3) homework exercises, (4) miluim during the semester or test or exam, (5) students with learning disabilities.
26.10.04 To go to the OFFICIAL website for this course click here. That site also contains ESSENTIAL information, exercises, text book, previous examinations, etc. There is also a link from there, and from here to the page of the Metargel Akhrayi where you can see office hours for your teachers, dates of examinations etc.
.To go directly to download or read the text book for this course, written by Allan Pinkus and Samy Zafrany, click here, or here,
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Click here for old announcements from 2003 and earlier.
(Some of this old information can be quite useful.)