Summary of Topics in the Fourier course, WITHOUT list of graphics files. Version 1. 25/7/06 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". Lecture 1: Complex valued functions of a real variable. Example: The exponential function e to the power c t, where c is a constant complex number. Derivatives And Integrals Of Complex Valued Functions Linear Spaces The Space E[a,b] of piecewise continuous complex valued functions on the interval [a,b]. 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 Lecture 2: Norms and Inner Products on linear spaces, examples. Lecture 3: given by Prof. Daoud Bshouty Lecture 4: given by Prof. Baruch Solel Here is an approximate description of the topics of these two 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. Lectures 4-8: Orthogonal And Orthonormal Systems in general inner product spaces. (These were already mentioned in the previous lecture.) The "Big Question": Suppose that for n=1,2,.... the elements u_1, u_2,.... form an orthonormal system in some inner product space V Let f be an element in V and let a_n = for each n. Let S_N be the sum of the elements u_n for n from 1 to N. Does S_N converge to f in some sort of way when N tends to infinity? This is the big question. This course will give some positive answers to this question in some special but very important cases. (In finite dimensional spaces an orthonormal system u_1, u_2,... can only have finitely many elements. So the sum of the vectors u_n has only finitely many terms. This sum from 1 to N will equal f for every f in the space V, if and only the vectors u_1, u_2,....,u_N span V.) Convergence In Norm of a sequence or a series in an inner product space. Closed And Complete Orthonormal Systems - Parseval's Identity Fourier Series On the Interval [-Pi, Pi] . The Meaning Of The Symbol ~ , when applied to Fourier series. Pointwise Convergence Of Fourier Series. PV-sums (The "P.V." or "principal value" sum of a series "Sigma" a_n where the numbers a_n are defined for all positive and non positive integers n, is the limit as N tends to +infinity of the sum as n ranges from -N to N of the numbers a_n.) Periodic Extensions of functions initially defined on [-Pi, Pi]. Various preliminary results which will be needed for the proof of Dirichlet's Theorem. The Dirichlet kernel, Proof of Dirichlet's Theorem Different Types Of Convergence. Uniform Convergence Of Fourier Series. More about Two Sided Series. The Weierstrass M-Test for uniform convergence. Term By Term Integration Of Fourier Series. The Fourier Orthonormal System of complex exponential functions e to the power n t, Is a Closed Orthonoormal system. Similarly for the system of functions cos nx, sin nx , 1/(square root of 2). Parseval's Identity For Fourier Series. Smoothness Of Functions And Decay Of Their Fourier Coefficients Fourier Series On Other Intervals The Fourier Transform Piecwise Continuity of functions define On Unbounded Intervals The Space G(R) of piecewise continuous and absolutely integrable functions. Lecture 9: 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. [Remember: 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 inclued the formula for the transform of f(ax+b) if the transform of f is known. Lectures 10-12: The Lebesgue Dominated convergence theorem, and Leibniz Theorem For differentiating an integral depending on a parameter, in the case of integrals on an unbounded interval. (without proofs!) Derivatives Of Fourier Transforms. The Fourier Transform Of the function exp{-x^2} The Riemann Lebesgue Lemma For Fourier Transforms The Integral Of (sin x)/x on [0, infinity) Fubini's Theorem for Change Of Order Of Integration. (without proof!) The Fourier Inverse Theorem. (Sometimes called Dirichlet's theorem for Fourier transforms.) Fourier Transforms Of Derivatives. Remark: Functions In G(R) Do Not Always Tend To Zero at infinity and - infinity. Smooth Functions Have Fast Decaying Fourier Transforms Plancherel's Formula Convolutions The Solution Of the Heat Equation on the real line, using the Fourier transform. Lectures 13-14: 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. If there is a demand for it, future versions of this document will include a list of the main topics mentioned in lectures 13 and 14: Hopefully you can easily get the two files which summarize these lectures from my site.