Instructor: Laszlo Erdös
Teaching Assistant: Johannes Alt
Fourier transform is a fundamental tool of many areas of mathematics, including analysis, partial differential equations, probability, analytic number theory, as well as many applications (engineering, data analysis etc.).
This course offers an introduction to the mathematical side of Fourier analysis where certain results will be rigorously proven, others will only be mentioned and explained in semi-rigorous fashion. It is mainly recommended for students who have encountered Fourier transform in one way or another but never had a chance to learn it somewhat properly.
There will be a written exam during the last lecture on June 23.
The final grade is determined by the final exam.
|May 3, 2016:||Motivations for Fourier series: I) Expansion, II) Solution to partial differential equations. We discussed the first point: concept and difficulties with Taylor expansion (requires smoothness, problem with convergence and sometimes even if it converges, it does not represent the “right” function). Moreover, Taylor expansion is not well suited for periodic functions. In contrast, for periodic functions it is better to use trigonometric polynomials (or standard exponential functions) as basis. Conventions about the form of the Fourier series (trigonometric or exponential; 2 pi). Advertisement: Fourier series behave much better than Taylor series.|
|May 10, 2016:||Today we discussed the second motivation for Fourier transform: solution of partial differential equations (PDE). Example: wave equation in 1 spatial (and 1 time) dimension. We derived the wave equation for the vibrating motion of an elastic string starting from the Newton’s equation (after discretizing the string). We first looked for solutions of product form, u(x,t)= f(x) g(t), which led to ordinary differential equations. Then we realized that linearity of the wave equation allows one to construct more complicated solutions by linear combinations of factorized solutions. We discussed boundary conditions and initial conditions. The method is very general, applies to PDE’s other than the wave equation, e.g. we introduced heat equation and Laplace equation. However, there are two important conditions for the applicability of the Fourier method for PDE: (i) the equation must be linear and (ii) it must have constant coefficients.|
|May 12, 2016:||Fourier series with complex notation, we use complex exponential on the space of periodic functions on [0, 2pi]. L^2 theory is the simplest, so we start with that. We recalled the definition of the Hilbert space, completeness, orthogonal projection. Orthonormal systems (ONS) and orthonormal basis (ONB — defined as being complete ONS). Uniqueness of basis representation in arbitrary H-space (with countable dimension). By translating the general theory to the L^2[0,2pi] with complex exponentials as ONS, we got the complete L^2-theory of Fourier series, provided that the complex exponentials form a complete ONS, i.e., they are ONB. Next time we’ll prove their completeness.|
|May 17, 2016:||Fourier series of continuous functions, various types of convergence. For periodic continuous functions, the Fourier series is Cesaro summable in the uniform sense. The reason is that the Fejer kernel (characterizing the Cesaro sum) behaves better than the Dirichlet kernel (characterizing the partial sum alone). The Fejer kernel is nonnegative and decays faster, it behaves as an approximate delta function. For more details on the rigorous proof, see Section 1.2 of the enclosed notes .|
|May 19, 2016:||Detailed proof that continuous functions have uniformly Cesaro summable Fourier series, in particular we explained the good properties of the Fejer kernel standing behind the proof. Unitary isomorphism between periodic L^2 functions and their Fourier coefficients. Fourier series of a C^1 function is uniformly convergent. Decay of the Fourier coefficients of a C^k function; relations between differentiability and decay back and forth.|
|May 31, 2016:||Fourier transform diagonalizes derivative. It can be used to extend differentiation to certain non-differentiable functions. Sobolev spaces, generalized (weak) derivatives. Relations between classical C^k spaces and Sobolev spaces. Sobolev embedding (for the simplest one dimensional case). Leibniz rule and limit of difference quotients for H^1 functions. Fourier transform on R: L^1 to L^infinity boundedness. Extension to L^2 functions, Plancherel formula.|
|June 2, 2016:||Fourier transform on R^d. Proof of Plancherel formula and using it to extend Fourier transform to L^2 functions isometrically. Inverse Fourier transform can be defined for any L^2 functions and thus we established that Fourier transform is 1-1 isometry, i.e. it is unitary on L^2.|
|June 7, 2016:||Heuristic proof of the Fourier inversion formula. Informal introduction into the theory of distributions; delta function, its derivatives, regular distributions (=functions). Every distribution is differentiable any times. Relation with linear PDE’s. Distributions cannot be multiplied. Heaviside function and its derivative. Sobolev spaces in R^d. Definition of half derivative and its non-locality|
|June 9, 2016:||Quick introduction to basics of probability theory. Emphasis on the fact that probability theory is basically analysis on a measure space; expectation of a random variable is just integral of a function. Given this point of view, one can define the Fourier transform of a random variable (or rather: its distribution); in this context the Fourier transform is called characteristic function. We discussed basic properties of the characteristic functions, up to the Bochner theorem (continuous, positive definite functions, normalized to 1 at zero, are characteristic functions of some random variable (or probability measure))|
|June 14, 2016:||We discussed various types of convergence for a sequence of random variables, and introduced the concept of convergence in distribution. The latter is equivalent to the pointwise convergence of the characteristic function. We used this to prove the weak law of large numbers and the central limit theorem.|
|June 21, 2016:||Isoperimetric inequality via Fourier transform (for any closed curves of fixed length, the enclosed area is smaller or equal to that for the circle). I did not spell out, but the proof actually also gives that the largest area is achieved only for the circle (check it yourself). As a second application of Fourier transform, we showed that irrational rotation on the circle (i.e. the sequence of fractional parts of integer multiples of an irrational number) forms an equidistributed sequence. This holds only for irrational numbers and it clearly fails for rationals.|
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