# Analysis IV (Fourier Theory and Hilbert Spaces) Spring 2023

Lecturer
Mikaela Iacobelli
Coordinator
Federico Franceschini

## Time and place

Lectures take place in HG F3 on Wednesdays form 9 to 10 and on Fridays from 10 to 12. The first lecture will be on February 22nd, 2023.

Exercise classes take place on Wednesday in different locations and different groups according to the table below. Please enroll to the exercise classes in MyStudies. The first exercise class will be on March 1st, 2023.

WhenWhereWhoLanguage
Wednesday 10-12HG G 26.1J. ChenEnglish
Wednesday 10-12LEE D 105A. DawidEnglish or German
Wednesday 10-12ML F 40L. MarslandEnglish
Wednesday 10-12ML H 43D. SchlagenhaufEnglish
Wednesday 10-12ML J 34.1D. UrechGerman
Wednesday 12-14ML F 40J. WarnettEnglish

For further information please refer to the course catalog.

## Content

This course will cover the following topics

• Real and complex Hilbert spaces, Hilbert bases and Riesz representation Theorem. See Chapter 4 of [AmDaMe] and Chapter I and III section 1-3 of [Co].
• Fourier series of a function in L^2([-π, π]; C), relationship between the regularity of a function and the asymptotic behaviour of the Fourier coefficients. Application to the resolution of linear partial differential equations with various boundary conditions in [-π, π].
• Fourier Transform in R^d and its elementary properties, relationship between the regularity of the function and the asymptotic behaviour of its Fourier transform, relationship between the summability of the function and the regularity of it Fourier transform. Application to the resolution of linear partial differential equations with various decay conditions in R^d.
• Compact operators on Hilbert spaces, Self-adjoint operators, the spectral theorem and applications to eigenvalue problems.

## Diary

Here you can find the diary of the Lectures. Unfortunately, no recording/streaming of the lectures will be available.

DateTopic discussedNotes
Wednesday, February 22 Introduction to the course, definition of (real and complex) vector spaces and inner product spaces. Examples: R^d, C^d, l^2_C.
Friday, February 24 Examples of inner product spaces L^2, C([0,1]) (examples 1.8 and 1.11 LN). Norms, Cauchy-Schwarz inequality and triangular inequality with proof, Parallelogram identity, Pythagora’s theorem, Polarisation formulas, Ptolemy inequality. Lemma 1.22 without proof (continuity of the inner product), exercise 1.2, example 1.25 (some norms on finite dimensional ), example 1.26 (C([0,1])) is an infinite dimensional v.s. and some classical norms.
Wednesday, March 1 L^p spaces are normed spaces, some topological definitions: open ball, interior point, open set, closed set, convex set, topological vector space, Cauchy sequences, completeness. Comparison between norms on the same space, definition of Hilbert spaces, examples of Hilbert spaces (C^d, l^2, L^2).
Friday, March 3 A subspace of a Hilbert space is Hilbert if and only if it is closed; example of inner product space that is not complete: C([-1,1], L^2 norm), notion of density and closure of a set, separability, example of separable spaces, example of a non separable Hilbert space, basis of a Hilbert space, definition of orthonormal set for a Hilbert space. Statement of Bessel inequality and Parseval identity.
Wednesday, March 8 Proof of the theorem about orthonormal sets of a Hilbert space (Bessel inequality, Parseval identity), Hilbert basis, Completeness criterion for separable Hilbert Spaces (proof of i) implies ii)).
Friday, March 10 End of proof completeness criterion for separable Hilbert Spaces, Theorem of existence of an Hilbert basis, example of l^2, and example of compactly supported sequences. Every separable Hilbert space of infinite dimension is isometric to l^2, Hilbert bases are not algebraic bases, theorem about the existence of the projection on closed subspaces and characterisation of the orthogonal projection. Statement of the theorem about the projection on closed convex set.
Wednesday, March 15 Characterization of the projection onto a convex set, with proof. Projection onto a subspace in terms of a Hilbert basis of the subspace. Every closed subspace has an orthogonal complement and H = V \oplus V^\perp.
Friday, March 17 Ending of the proof of the splitting H = V \oplus V^\perp. Linear operators between normed vector spaces are continuous if and only if they are bounded, with proof. Definition of the norm of a linear operator and proof that it is equal to the Lipschitz constant. Continuity of linear operators if the dimension is finite. Example: the identity is not necessarily continuous if we change the topology.
Wednesday, March 22 Riesz' representation Theorem of continuous linear functionals on an Hilbert space, with proof. Corollary: canonical isometric isomorphism between H and H' (the dual space). Application: the Radon-Nikodym theorem (proven, but not examinable).
Friday, March 24 Motivation of the Fourier series, definiton of Fourier coefficients and Fourier partial sum. The Complex Stone Weierstrass Theorem, (only statement). Theorem: the Fourier basis forms an Hilbert basis of L^2((-\pi,\pi);\C) (proof using the Complex Stone Weierstrass).
Wednesday, March 29 Consequences of the fact that the Fourier basis is an Hilbert basis: L^2 convergence of partial sums, Parseval's identity, scalar product in terms of the Fourier coefficients. Remark: every continuous function can be approximated in L^\infty with trigonometric polynomials, but some continuous functions are not even pointwise limit of their Fourier partial sums. Example: Fourier series of sines and cosines. Proposition: characterization of real valued functions trough their Fourier coefficients.
Friday, March 31 Almost everywhere convergence of the Fourier partial sums, Carleson's Theorem (only statement). Examples: Foureir sereis of trigonometric polynomial, Fourier series of x and computation of the sum of inverse squares of the integers. Theorem: absolutely convergent series and square summable sequences of orthogonal vectors, converge in Hilbert spaces. Example: given a sequence of C^k functions if the sup-norm of the derivatives of order \le k is summable, then the limit is differentiable k times and the convergence of the function and its first k derivatives is uniform. Proposition: if a periodic function is piecewise C^1 then the Fourier coefficients of f and f' are related, and have certain better summability properties (more than \ell^2). Proposition: generalisation of the previous proposition to C^k functions, with periodic first k-1 derivatives.
Wednesday, April 5 Proposition: If the Fourier coefficients {c_k} of an L^2 function f are such that \sum_{k}|k|^\alpha |c_k| is finite, then the function f is fact of class C^h_{per}, where h is the largest integer less or equal than \alpha (only statement). Theorem (pointwise convergence): if f is Holder continuous at a point then the Fourier series converges at that point. Beginning of the proof: S_N(f) = f *D_N, where D_N is the Dirichelt kernel.
Friday, April 7 No class, Easter Holidays
Wednesday, April 12 No class, Easter Holidays
Friday, April 14 No class, Easter Holidays
Wednesday, April 20 Proof of the pointwise convergence Theorem using the Riemann-Lebesgue Lemma.
Friday, April 21 Proof of the Riemann-Lebesgue Lemma. Remark: the pointwise convergence Theorem implies the completeness of the trigonometric system without the use of the Ston-Weierstrass Theorem. The Heat equation in a one-dimensional periodic spatial domain, formal solution via Fourier series. Theorem with proof: the formal solution is a smooth solution if the Fourier coefficients of the initial datum are summable. Remarks on the solution of the heat equation: 1) solutions becomes smooth for positive times, 2) u(t,\cdot) converges to the initial average temperature as t goes to \infty, 3) localized initial data spread out instantaneously (unphysical behaviour) 4) Time arrow: the equation is very hard to solve for negative times.
Wednesday, April 26 General recap of the heat equation. Theorem: uniqueness of continuous solutions of the heat equation that -for positive times - are C^2 and 2pi periodic in the space variable and C^1 in the time variable.
Friday, April 28 Informal introduction to the Fourier transform as a continuous version of the Fourier series. Definition of the Fourier transform for functions in L^1(\R^d). Proposition: Their F.T. is a continuous function which vanishes at \infty (with proof). Lemma: Behaviour of the F.T. under dilation, translation, multiplication by imaginary exponentials. Reminder of convolution and the Young inequality (seen in Analysis III).
Wednesday, May 3The Fourier transform of the convolution is the product of the Fourier transforms. Fourier transform of even/odd/real functions. Example: computation of the Fourier transform of \exp{-|x|}.
Friday, May 5 Computation: the Fourier transform of the Gaussian in d variables. Theorems: the Fourier transform transforms directional derivatives in multiplications by the respective coordinate function, and viceversa. Application: alternative computation of the Fourier transform of the gaussian via ODE. Definition of the Schwartz class of functions.
Wednesday, May 10 Theorem: the Fourier transform of a Schwartz function is itself a Schwartz function, and the formulas realting directional derivatives and multiplication by coordinate functions holds.
Friday, May 12Theorem: the inversion formula for Schwartz functions (with proof). Generalisations to functions such that both f and \hat f are in L^1(\R^d). Lemma: the Fourier transform is formally self-adjoint with respect to the L2 scalar product.
Wednesday, May 16 Remark: the Fourier transform is injective in L^1. Examples: explicit computation of some Fourier Transforms. Extension of the Fourier transform to to an isometry in L^2(\R^d), Plancherel identity. Extension of the formula for the F.T> of the derivative for L^2 functions.
Friday, May 19 Example: the Fourier transform of x/(1+x^2). Theorem: the Heisenberg inequality, characterization of Gaussians equality cases. Quantum mechanical meaning of the Heisenberg inequality. The heat equation in R^d, formal representation with the heat kernel.
Wednesday, May 24 The heat equation in \R^2, existence and smoothness of the solution defined trough the heat kernel. The harmonic oscillator. Definition of Hermite functions and Hermite polynomials. Proposition: the Hermite functions are eigenfunctions of the Fourier transform.
Friday, May 26 The creation and annihilation operators A and A^* and their commutation rules and their behaviour on Hermite functions. Theorem: Hermite function are an Hilbert basis of L^2(\R).
Wednesday, May 31 End of the proof of completeness of the Hermite function. Application of Hermite functions to the solution of the time-dependent Schr\"odinger equation for the quantum harmonic oscillator. (NOT EXAMINABLE FROM HERE) Evolving for a certain time a function trough the harmonic oscillator Schrodinger equation one find the Fourier transform of the initial datum. The Hardy's uncertainty principle. Statement of Paley-Wiener theorem."
Friday, June 2 (NOT EXAMINABLE) Proof of the Paley - Wiener theorem. Discussion about uncertainty principles (Heisenberg, Hardy, Paley Wiener), relationship with the solutions of the sphere packing problem in dimension 8 and 22.

## Exercises

The new exercises will be posted here on Wednesday afternoon, and will cover the topics that were covered the past week. Typically, the solutions will be discussed after a week.

Some points to keep in mind in order to have an healthy relationship with the problem sets:

• Handing in the solutions gives no points for the final exam, although working on the problem sets will greatly increase your chances to pass the exam.
• Problems are loosely ordered by increasing difficulty. You do not have to do all of them, some can be rather challenging. You can work in groups.
• The purpose of giving you problem sets is make you learn the material and develop your problem solving skills.
• The purpose of giving you the solutions and discuss them in exercise classes is to show you how to solve the problems you could not solve. Or to show you more efficient proofs.
• The purpose of letting you hand in the solutions and us correcting them is not assess your problem solving skills, but give you a feedback on how you write mathematics.

You can upload your solutions using the SAMUp tool or leave them in the coordinator postbox in HG F28.

Problem set 0 March 1st Submission Solutions
Problem set 1 March 8th Submission Solutions
Problem set 2 March 15th Submission Solutions
Problem set 3 March 22th Submission Solutions
Problem set 4 March 29th Submission Solutions
Problem set 5 April 5th Submission Solutions
Problem set 6 April 12th Submission Solutions
Problem set 7 April 19th Submission Solutions
Problem set 8 April 26th Submission Solutions
Problem set 9 May 3rd Submission Solutions
Problem set 10 May 10th Submission Solutions
Problem set 11 May 17th Submission Solutions
Problem set 12 May 24th Submission Solutions
Problem set 13 May 31th Submission Solutions

## Exam

The 3h written exam (joint with Analysis III) will take place Wednesday, 30 August in HIL G 61 at 13:00. Here you can find it (with solutions). The 20min oral exams (covering only Analysis IV) will take place on Tuesday, 29 August and Friday, 1 September in HG G 48.2.

The exam program (which is the same for the oral and written exam) is available here (updated with also problems from the problem set which is wise to revise!). Please notice that it is a subset of what we did in class, and only a part of the proofs are required. On the other hand, the ones that are required you should know how to reproduce without hints (this applies to both the oral and the written exam-takers).

Here you can find a 90min mock exam that resembles the part of the final exam that concerns Analysis IV. Here you can find the solutions of the mock exam. Please notice that it will be a closed-book exam, no notes of any kind will be allowed.

For every question concerning the program/exam write an email to federico.franceschini@math.ethz.ch and CC gerard.orriols@math.ethz.ch (who is in charge of the exam organisation).

## Literature

The script of the course (typesetted or handwritten) will be made available to students as it is created: here you can find the current version (password protected).

Here there are some textbooks that cover similar topics and that have been used to prepare lectures:
• [AmDaMe] "Introduction to measure Theory and Integration" by Ambrosio, Da Prato and Mennucci (Edizioni della Normale, 2011).
• [Bo] "Méthodes mathématiques pour les sciences physiques" by Bony (École polytechnique, 2000).
• [CaDA] "Introduction to Measure Theory and Functional Analysis" by Cannarsa and D'Aprile (Springer, 2008).
• [Co] "A Course in Functional Analysis" by Conway (Springer, 2007).
• [Ev] "Partial Differential Equations" by Evans (American Mathematical Society, 2010).
• [Kr] "Introductory Functional Analysis with Applications" by Kreyszig (John Wiley & Sons, 1978).
• [StSh] "Fourier Analysis: an Introduction" by Stein and Shakarchi (Princeton University Press, 2003)
• [Ci] "Exercises and Problems in Mathematical Methods of Physics" by Cicogna (Springer, 2020)