If you want a copy of the graphics files mentioned in this document please tell me by sending a message to BOTH of the following addresses: mcwikel@math.technion.ac.il babamc@gmail.com When you send your request please also tell me if you are willing to help me if necessary with each of the following things: 1. Forwarding these files to other students who may need them. 2. Sending invitations to open gmail accounts to other students who may need them. I certainly do not expect such help from those of you who do not have convenient access to email from a protected place, and so far I am managing fine without such help. In response to your request you should receive 9 very big messages, each of about 3 or 4 megabytes, containing the files, respectively, for lectures 1,4,5,6,7,8,10,11 and 12. If you only receive some of these 9 messages, please wait for one or two hours and then inform me. The material for lectures 13 and 14 is accessible from my private Fourier webpage, and the material for lectures 2,3 and 9 is summarized in the document below. If a non trivial number of students request it, I will prepare more detailed descriptions of the material of lectures 2,3 and 9. ----------------------------------------------------------------- Version 3 (Updated on 25.7.06. Since version 2, includes lecture 5 which was somehow forgotten in version 1.) For a more "reader friendly" version of this document, but without details about graphics files see http://www.math.technion.ac.il/~mcwikel/fourier/Summary706.txt ---------------------------------- This document is designed to list all, or most of the topics which were covered in the lectures this semester for the Fourier course. You are expected to know all these topics for the exam, except part of the very last topic of the very last lecture, which uses material from the course "Funktsiyot Merukavot". Most of the topics appear as titles of graphics files of some of the pictures which I took of the board during my lectures. You can also see a complete list of the picture files here. If any of you need to view these pictures, I will try to find some way to send them to you or put them on a website. Altogether there are 228 files and their combined size is 109,622,621 bytes. MEANWHILE I HAVE REPLACED THEM BY MUCH SMALLER FILES. THEY HAVE THE SAME NAMES, EXCEPT THAT "FS" HAS BEEN CHANGED TO "A-fs1" FOR THE FIRST LECTURE, AND TO "b-fs" FOR ALL THE OTHER LECTURES. Some of the graphics files were edited after the lecture, so their dates/times may be different to what you expect. In general a file name which begins with the four letters/digits FSAB where AB is a number from 01 to 12 corresponds to a picture for lecture number AB. DATE TIME SIZE FILE NAME SUBJECT 20/03/2006 10:55 457,121 FS010157Introduction.JPG 20/03/2006 10:55 452,974 FS010158.JPG 20/03/2006 10:55 463,915 FS010159ComplexExponentials.JPG 20/03/2006 10:55 467,876 FS010161DerivativesAndIntegralsOfComplexValuedFunctions.JPG 20/03/2006 10:55 471,659 FS010162.JPG 20/03/2006 11:09 498,716 FS010163.JPG 20/03/2006 11:09 466,398 FS010164.JPG 20/03/2006 11:09 465,419 FS010165.JPG 20/03/2006 11:09 468,877 FS010166LinearSpaces.JPG 20/03/2006 11:21 514,003 FS010167.JPG 20/03/2006 11:21 478,363 FS010168.JPG 20/03/2006 11:21 459,339 FS010169.JPG 20/03/2006 11:40 448,202 FS010170.JPG 20/03/2006 11:40 517,563 FS010171.JPG 20/03/2006 11:59 478,669 FS010172TheSpaceE[a,b].JPG There is also a brief summary of the topics of the lecture on 20/3/06 at http://www.math.technion.ac.il/~mcwikel/fourier/1stLect306.ps DATE TIME SIZE FILE NAME SUBJECT 23/03/2006 08:26 89,413 FS010173.JPG 23/03/2006 08:31 511,108 FS010174.JPG 20/03/2006 11:59 467,182 FS010175Norms.JPG 20/03/2006 11:59 478,596 FS010176.JPG 20/03/2006 12:16 517,492 FS010177.JPG 20/03/2006 12:16 453,113 FS010178.JPG 20/03/2006 12:16 468,314 FS010179.JPG 20/03/2006 12:16 463,921 FS010180.JPG 20/03/2006 12:16 447,547 FS010181InnerProducts.JPG 20/03/2006 12:17 510,609 FS010182.JPG 20/03/2006 12:25 510,648 FS010183.JPG 20/03/2006 12:25 477,586 FS010184.JPG 27/3/06 Lecture given by Prof. Daoud Bshouty 4/4/06 Lecture given by Prof. Baruch Solel Here is an approximate description of the topics of those lectures: Cauchy-Schwartz Inequality in a general inner product space. Examples in particular spaces. Orthogonal elements in an inner product space. Orthogonal systems, orthonormal systems, "Pythagoras' theorem" or some other similar result, in inner product spaces. If x and y are orthogonal then ||x+y||^2 = ||x||^2 + ||y||^2 . Orthogonal projections onto finite dimensional spaces. Proof about "best approximation" or "nearest point" property of orthogonal projection. Examples, Gram Schmidt procedure for construction of an orthonormal system. Generalized Fourier coefficients, Bessel's inequality, and the Riemann-Lebesgue lemma. DATE TIME SIZE FILE NAME SUBJECT 13/04/2006 05:36 66,696 FS040698OrthogonalAndOrthonormalSystems.JPG 13/04/2006 05:38 82,260 FS040699.JPG 13/04/2006 05:38 666,068 FS040700.JPG 13/04/2006 05:38 505,214 FS040701.JPG 13/04/2006 05:39 493,702 FS040702.JPG 13/04/2006 05:39 671,873 FS040703.JPG 13/04/2006 05:42 98,352 FS040704.JPG 13/04/2006 05:39 602,689 FS040705.JPG 13/04/2006 05:40 580,490 FS040706.JPG 13/04/2006 05:44 73,407 FS040707TheBigQuestion.JPG 13/04/2006 05:45 514,983 FS040712ConvergenceInNorm.JPG 13/04/2006 05:45 577,517 FS040713.JPG 13/04/2006 05:45 656,547 FS040714.JPG 13/04/2006 05:45 611,262 FS040715.JPG 13/04/2006 05:45 610,174 FS040716.JPG 13/04/2006 05:48 89,518 FS040717.JPG 13/04/2006 05:48 595,812 FS040718ClosedAndCompleteOrthonormalSystems-ParsevalIdentity.JPG 13/04/2006 05:49 509,783 FS040719.JPG 13/04/2006 05:50 66,336 FS040720.JPG 13/04/2006 05:51 91,086 FS040721.JPG 13/04/2006 05:51 492,551 FS040722.JPG 13/04/2006 05:52 507,766 FS040723.JPG 13/04/2006 05:52 507,567 FS040724.JPG 24/04/2006 10:04 529,494 FS050105FourierSeriesOnInterval-PiToPi.JPG 24/04/2006 10:04 502,280 FS050106.JPG 24/04/2006 10:05 457,334 FS050107.JPG 24/04/2006 10:05 454,912 FS050108.JPG 24/04/2006 10:05 510,982 FS050109MeaningOfSymbol~.JPG 24/04/2006 10:25 494,721 FS050110.JPG 24/04/2006 10:25 494,027 FS050111.JPG 24/04/2006 10:25 452,712 FS050112.JPG 24/04/2006 10:25 445,628 FS050113.JPG 24/04/2006 10:26 517,410 FS050114.JPG 24/04/2006 11:00 530,842 FS050115ConvergenceOfFourierSeries.JPG 24/04/2006 11:01 471,684 FS050116PV-sums.JPG 24/04/2006 11:01 462,664 FS050117.JPG 24/04/2006 11:01 448,754 FS050118.JPG 24/04/2006 11:01 511,230 FS050119PeriodicExtensions.JPG 24/04/2006 11:25 520,061 FS050120.JPG 24/04/2006 11:25 451,813 FS050121Dirichlet'sTheorem.JPG 24/04/2006 11:25 482,645 FS050122.JPG 24/04/2006 11:25 465,750 FS050123.JPG 24/04/2006 11:25 496,410 FS050124.JPG 01/05/2006 10:46 455,544 FS060006FourierSeries.JPG 01/05/2006 10:46 494,083 FS060007PeriodicFunctions.JPG 01/05/2006 10:46 473,104 FS060008.JPG 01/05/2006 10:46 476,316 FS060009.JPG 01/05/2006 10:46 440,094 FS060010.JPG 01/05/2006 10:46 495,188 FS060011.JPG 01/05/2006 11:07 471,874 FS060012.JPG 01/05/2006 11:07 479,988 FS060013.JPG 01/05/2006 11:07 533,813 FS060014.JPG 01/05/2006 11:07 474,246 FS060015DirichletKernel.JPG 01/05/2006 11:07 475,498 FS060016.JPG 01/05/2006 11:08 518,009 FS060017.JPG 01/05/2006 11:20 518,606 FS060018DirichletTheorem.JPG 01/05/2006 11:20 464,128 FS060019.JPG 01/05/2006 11:20 498,089 FS060020.JPG 01/05/2006 11:20 456,100 FS060021.JPG 01/05/2006 11:20 526,928 FS060022.JPG 01/05/2006 11:38 464,617 FS060023.JPG 01/05/2006 11:38 481,025 FS060024.JPG 01/05/2006 11:38 489,450 FS060025.JPG 01/05/2006 11:58 499,824 FS060026.JPG 01/05/2006 11:58 482,928 FS060027.JPG 01/05/2006 11:58 484,129 FS060028.JPG 01/05/2006 11:58 449,277 FS060029TermByTermDifferentiationOfFourierSeries.JPG 01/05/2006 11:58 495,857 FS060030.JPG 01/05/2006 11:59 494,891 FS060031.JPG 01/05/2006 12:22 462,656 FS060032.JPG 01/05/2006 12:22 482,925 FS060033.JPG 01/05/2006 12:22 467,147 FS060034.JPG 01/05/2006 12:22 463,447 FS060035.JPG 01/05/2006 12:22 459,248 FS060036.JPG 01/05/2006 12:23 505,367 FS060037.JPG 01/05/2006 12:25 469,757 FS060038.JPG 08/05/2006 10:51 496,292 FS070119DifferentTypesOfConvergence.JPG 08/05/2006 10:51 502,616 FS070120.JPG 08/05/2006 10:51 470,930 FS070121.JPG 08/05/2006 10:51 451,417 FS070122.JPG 08/05/2006 10:51 461,837 FS070123UniformConvergenceOfFourierSeries.JPG 08/05/2006 10:51 492,928 FS070124.JPG 08/05/2006 11:14 508,493 FS070125.JPG 08/05/2006 11:14 472,417 FS070126.JPG 08/05/2006 11:14 454,767 FS070127.JPG 08/05/2006 11:14 449,023 FS070128TwoSidedSeries.JPG 17/05/2006 08:20 355,197 FS070130.JPG 08/05/2006 11:40 535,210 FS070131.JPG 08/05/2006 11:41 463,416 FS070132WeierstrassM-Test.JPG 08/05/2006 11:41 459,945 FS070133.JPG 08/05/2006 11:41 488,142 FS070134TermByTermIntegrationOfFourierSeries.JPG 08/05/2006 11:41 468,774 FS070135.JPG 08/05/2006 12:10 493,773 FS070136.JPG 08/05/2006 12:10 478,260 FS070137.JPG 08/05/2006 12:10 477,168 FS070138.JPG 08/05/2006 12:10 447,658 FS070139.JPG 08/05/2006 12:10 506,968 FS070140.JPG 08/05/2006 12:20 494,672 FS070141TheFourierOrthonormalSystemIsClosed.JPG 08/05/2006 12:20 486,413 FS070142.JPG 08/05/2006 12:20 454,525 FS070143.JPG 08/05/2006 12:21 441,095 FS070144.JPG 08/05/2006 12:21 490,861 FS070145.JPG 15/05/2006 10:48 539,293 FS080254.JPG 15/05/2006 10:48 460,731 FS080255Parseval'sIdentityForFourierSeries.JPG 15/05/2006 10:48 460,549 FS080256.JPG 15/05/2006 10:48 453,608 FS080257.JPG 15/05/2006 10:48 504,515 FS080258.JPG 15/05/2006 11:03 482,313 FS080259.JPG 15/05/2006 11:03 487,632 FS080260.JPG 15/05/2006 11:03 466,472 FS080261.JPG 15/05/2006 11:03 449,786 FS080262.JPG 15/05/2006 11:03 456,870 FS080263.JPG 15/05/2006 11:24 522,727 FS080264SmoothnessOfFunctionsAndDecayOfTheirFourierCoefficients.JPG 15/05/2006 11:24 453,001 FS080265.JPG 15/05/2006 11:24 472,554 FS080266FourierSeriesOnOtherIntervals.JPG 15/05/2006 11:25 450,033 FS080267.JPG 15/05/2006 11:25 453,598 FS080268.JPG 15/05/2006 11:25 494,826 FS080269.JPG 15/05/2006 12:04 505,047 FS080270.JPG 15/05/2006 12:04 488,497 FS080271.JPG 15/05/2006 12:04 474,829 FS080272TheFourierTransform.JPG 15/05/2006 12:04 471,803 FS080273.JPG 15/05/2006 12:04 501,386 FS080274.JPG 15/05/2006 12:21 479,162 FS080275.JPG 15/05/2006 12:21 516,042 FS080276.JPG 15/05/2006 12:21 465,529 FS080277PiecwiseContinuityOnUnboundedIntervals.JPG 17/05/2006 20:19 90,772 FS080278TheSpaceG(R).JPG 15/05/2006 12:21 473,878 FS080279.JPG 22/05/06 Lecture given by Prof. Baruch Solel. Subjects which were mentioned in that lecture would include the following. The "norm" on G(R). It is not quite a norm. Definition of generalized Riemann integral for functions on the interval [0, infinity) , on the interval (-infinity, 0] and on (-infinity, infinity) Definition of the P.V. integral (Cauchy principal value integral) on the interval (-infinity, infinity), connections with and differences from with the generalized Riemann integral on (-infinty,infinity). Absolutely integrable functions: if f is piecewise continuous,and |f| is integrable on an unbounded interval, then f itself is also integrable on the same interval. [[ f can take complex values .]] Application. If f is in G(R) then its Fourier transform is defined (and finite) for every point omega. It is also a bounded function. Fourier transform of the function which is 1 on [a,b] and zero for xb. Fourier transform of exp(-|x|) Riemann Lebesgue lemma for the Fourier transform. Proof in special cases. Theorem. If f in G(R) then its Fourier transform is continuous. (without proof.) Other elementary properties of Fourier transform. These included the formula for the transform of f(ax+b) if the transform of f is known. DATE TIME SIZE FILE NAME SUBJECT 05/06/2006 19:40 89,290 FS100155LebesgueAndLeibnizTheoremsForIntegralsOnUnboundedInterval.JPG 05/06/2006 19:41 125,392 FS100156DerivativeOfFourierTransform.JPG 05/06/2006 10:57 701,510 FS100158.JPG 05/06/2006 10:57 751,584 FS100159FourierTransformOf exp{-x^2}.JPG 05/06/2006 11:21 814,683 FS100160.JPG 05/06/2006 11:21 702,557 FS100161.JPG 05/06/2006 11:21 690,648 FS100162.JPG 05/06/2006 11:21 663,853 FS100163.JPG 05/06/2006 11:21 814,280 FS100164RiemannLebesgueLemmaForFourierTransforms.JPG 05/06/2006 11:56 705,067 FS100165IntegralOfSinX-DividedBy X.JPG 05/06/2006 11:56 738,813 FS100166.JPG 05/06/2006 11:56 757,892 FS100167.JPG 05/06/2006 11:56 733,833 FS100168.JPG 05/06/2006 11:56 819,172 FS100169.JPG 05/06/2006 12:22 704,968 FS100170Fubini'sTheoremChangeOfOrderOfIntegration.JPG 05/06/2006 12:23 720,313 FS100171.JPG 05/06/2006 12:23 710,162 FS100172.JPG 05/06/2006 12:23 703,242 FS100173FourierInverseTheorem.JPG 05/06/2006 12:23 703,284 FS100174.JPG 12/06/2006 10:45 687,249 FS110491.JPG 12/06/2006 10:45 761,210 FS110492.JPG 12/06/2006 10:45 753,455 FS110493.JPG 12/06/2006 10:45 677,232 FS110494.JPG 12/06/2006 10:45 724,519 FS110495.JPG 12/06/2006 10:46 740,248 FS110496.JPG 12/06/2006 11:08 710,683 FS110497.JPG 12/06/2006 11:08 686,731 FS110498.JPG 12/06/2006 11:08 745,287 FS110499.JPG 12/06/2006 11:08 720,219 FS110500.JPG 12/06/2006 11:08 728,267 FS110501.JPG 12/06/2006 11:24 672,892 FS110502.JPG 12/06/2006 11:24 697,145 FS110503.JPG 12/06/2006 11:24 733,114 FS110504FourierTransformOfDerivative.JPG 12/06/2006 11:24 699,878 FS110505.JPG 12/06/2006 11:25 787,997 FS110506.JPG 12/06/2006 11:37 713,310 FS110507.JPG 12/06/2006 11:38 691,921 FS110508.JPG 12/06/2006 11:39 709,112 FS110509.JPG 12/06/2006 11:53 828,556 FS110510FunctionsInG(R)DoNotAlwaysTendToZero.JPG 12/06/2006 11:53 701,009 FS110511.JPG 12/06/2006 11:53 747,233 FS110512.JPG 12/06/2006 11:53 670,870 FS110513.JPG 12/06/2006 11:53 733,352 FS110514.JPG 12/06/2006 12:12 697,548 FS110515.JPG 12/06/2006 12:12 698,119 FS110516SmoothFunctionsHaveFastDecayingFourierTransforms.JPG 12/06/2006 12:12 751,098 FS110517.JPG 12/06/2006 12:12 702,065 FS110518.JPG 12/06/2006 12:12 814,055 FS110519Plancherel'sFormula.JPG For the first part of the lecture on 19/06/06 I used the following pictures from the corresponding lecture in Jan. 05. 10/01/2005 10:57 452,196 FS12A154.JPG 10/01/2005 10:57 486,589 FS12A155PlancherelFormula.JPG 10/01/2005 10:57 437,746 FS12A156.JPG 10/01/2005 10:57 445,531 FS12A157.JPG 10/01/2005 10:58 444,982 FS12A158Convolutions.JPG 10/01/2005 11:15 448,044 FS12A159.JPG 10/01/2005 11:15 459,350 FS12A160.JPG 10/01/2005 11:15 457,740 FS12A161.JPG 10/01/2005 11:15 471,324 FS12A162.JPG 10/01/2005 11:16 446,716 FS12A163.JPG 10/01/2005 11:22 462,754 FS12A164.JPG 10/01/2005 11:23 469,653 FS12A165.JPG 10/01/2005 11:46 445,711 FS12A166.JPG The second part of the lecture was given on the board. 19/06/2006 11:46 475,789 FS12B539Convolutions.JPG 19/06/2006 11:46 525,621 FS12B540.JPG 19/06/2006 11:46 528,484 FS12B541.JPG 19/06/2006 11:46 454,395 FS12B542.JPG 19/06/2006 12:06 491,527 FS12B543SolutionOfHeatEquation.JPG 19/06/2006 12:06 539,288 FS12B544.JPG 19/06/2006 12:06 529,292 FS12B545.JPG 19/06/2006 12:07 480,631 FS12B546.JPG 19/06/2006 12:26 466,971 FS12B547.JPG 19/06/2006 12:26 508,762 FS12B548.JPG 19/06/2006 12:26 530,694 FS12B549.JPG 19/06/2006 12:26 476,262 FS12B550.JPG The last two lectures (four hours) were about the Laplace transform. Summaries of these two lectures appear on my "private" Fourier website at http://www.math.technion.ac.il/~mcwikel/fourier/LAPLACE.pdf http://www.math.technion.ac.il/~mcwikel/fourier/heb2lap.ps In the last lecture we used the inverse Fourier transform to deduce a formula for the inverse Laplace transform under certain conditions. You are expected to know this material, and the material before it. But you are not expected to know the next thing: We explained quickly how to use residues (she'ariot) from funktsiyot merukavot to sometimes calculate that integral for the inverse Laplace transform. The reason you are not asked to know about this is that officially "funktsiyot merukavot" is not a prerequisite for the Fourier course.