Teacher: Mihalis Kolountzakis

▶ **Announcements**

- The midterm exam will be held in class on Wednesday, April 6, 2022. Please be there no later than 15:00.
- Notice Monday's session moved to 13:00-15:00 instead of 15:00-17:00.
- The session of Monday the 14th of Feb. 2022 will not be held. Class will be held as scheduled on Wednesday the 16th.
- Το μάθημα θα διδαχθεί στα Αγγλικά μια και στο ακροατήριο θα υπάρχουν φοιτητές από το πρόγραμμα Erasmus. Θα γίνει με τέτοιο τρόπο ώστε με μια στοιχειώδη γνώση Αγγλικών να μην έχει κανείς πρόβλημα να το παρακολουθήσει. Όλες οι ασκήσεις και τα διαγωνίσματα θα είναι και στις δύο γλώσσες.

▶ **Schedule**

Room A208. Mon 13:00-15:00 and Wed, 15:00-17:00.

Teacher's office hours: Wed 12:00-13:00.

▶ **Course description**

**Goal:**
Introduction to harmonic analysis, mostly Fourier Series.

**Contents:**
Quick introduction to Lebesgue measure on the real line.
Periodicity, trigonometric polynomials, orthogonality.
Fourier coefficients and Fourier series of periodic functions.
Absolute convergence and size of Fourier coefficients with respect to function smoothness.
Uniqueness, convolution, kernels, Cesaro means, Fejer's theorem and applications.
The $L^2$ theory.
Pointwise convergence of Fourier series.
Norm convergence.
Localization.
Bernstein's inequality.

▶ **Books and lecture notes**

- [K]: Y. Katznelson, An introduction to Harmonic Analysis, 3rd corrected edition, Cambridge Univ. Press, 2004.
- [KP]: M. Kolountzakis and Ch. Papachristodoulos, Fourier Analysis, 2015. In Greek.

Book last updated on:**January 09 2022 21:50:11.** - [SS]: E. Stein and R. Shakarchi, Fourier Analysis: an introduction, Princeton Univ. Press, 2003.

▶ **Student evaluation**

Intermediate exam 40%, final exam 60%. This remains the same for all further examination periods.

▶ **Class diary**

No class today.

We talked about Lebesgue measure today and stopped short from introducing the Lebesgue integral. We said how Lebesgue measure is defined, through coverings of the set by a sequence of open intervals, and mentioned many properties of Lebesgue measure without proving many of them. We went through the construction of the ternary Cantor set, proved that it has measure zero and that it is a uncountable set, which gave us the opportunity to revisit the Cantor diagonal argument for disproving that the real numbers (or, the set of all sequences on two symbols) is countable.

__Problems:__ No 1.
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*Turn in solutions in class on Wednesday of the following week.*

We introduced the Lebesgue integral. First we defined it for simple functions (functions which take finitely many values) and then we extended the definition to nonnegative functions via a natural limiting process. Finally we extended the definition to signed or complex-valued functions by decomposing them to a sum of nonnegative functions (with coefficients). We proved some properties of the integral, solved some of the problems in the notes and introduced the spaces $L^1(\RR)$ and $L^1(A)$.

Today we covered the main convergence theorems for the Lebesgue integral, the Monotone Convergence Theorem and the
Dominated Convergence Theorem. We saw several implications of these. Finally we talked about Lebesgue integration
in $\RR^d$ and stated Fubini's theorem for multiple integration. We used Fubini's theorem in order to prove that the
*convolution* of two functions $f, g \in L^1(\RR)$
$$
f*g(x) = \int f(y) g(x-y) \,dy
$$
is well defined for almost all $x \in \RR$ and that it satisfies the inequality
$$
\int\Abs{f*g} \le \int \Abs{f} \cdot \int \Abs{g}.
$$

__Problems:__ No 2.
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*Turn in solutions in class on Wednesday of the following week.*

We defined the $L^p(A)$ spaces along with the $L^p(A)$ norm, for $1\le p \lt \infty$ initially and then we defined the essential supremum of a function. This we called the $L^\infty(A)$ norm. We talked about the properties of norms on vector spaces and mentioned (without proof) Minkowski's inequality $$ \Norm{f+g}_p \le \Norm{f}_p + \Norm{g}_p, $$ which is valid for $1 \le p \le \infty$. We also mentioned Holder's inequality (and its special case, the Cauchy-Schwarz inequality). We mentioned, without proof, the fact that all $L^p(A)$ spaces are complete (such complete normed vector spaces are called Banach spaces). We then assumed that $m(A) \lt \infty$ and proved that the $L^p(A)$ are nested: $p_1 \le p_2 \Longrightarrow L^{p_2}(A) \subseteq L^{p_1}(A)$. We also showed that this inclusion is always proper: there is a function in $L^{p_1}(A)$ not in $L^{p_2}(A)$. Under the extra assumption that $m(A)=1$ we proved first that $\Norm{f}_{p_1} \le \Norm{f}_{p_2}$ and then that $\lim_{p \to \infty} \Norm{f}_p = \Norm{f}_{\infty}$.

This concludes our introduction to the use of Lebesgue measure and integration. From the next meeting on we start with harmonic analysis proper.

We quickly reviewd some basic facts about complex numbers and the function $e^z$ in connection with the functions $\cos{x}, \sin{x}$. We then talked about periodic functions. We observed that the sets of periods of a function forms an additive group. We then proved that the period group of a continuous, non-constant function on $\RR$ is either trivial (no periods but 0) or of the form $T\ZZ$, for some $T>0$, the minimum period of the function. We defined the $L$-periodization of a function $f \in L^1(\RR)$ by $$ F(x) = \sum_{n \in \ZZ} f(x+nL) $$ and proved that this series converges for almost all $x \in \RR$ to a $L$-periodic function $F$ and that $\int_0^L F = \int_{\RR} f$. We then defined the trigonometric polynomials of degree $N$ $$ p(x) = \sum_{n=-N}^N p_n e^{i n x}.\ \ \ \ (TP) $$ We then proved that the exponentials functions are linearly independent (which implies that a trigonometric polynomial function cannot be written in more that one way as in (TP). We even proved that the collection of functions $$ e^{\lambda x},\ \ \ \lambda \in \CC, $$ is linearly independent. Then we defined the torus $\TT = \RR/(2\pi \ZZ)$ and the corresponding function spaces $L^p(\TT)$ and $C(\TT)$, and defined the inner product $$ \inner{f}{g} = \nint f(x) \overline{g(x)}\,dx $$ for $f \in L^p(\TT)$, $g \in L^q(\TT)$, with $\frac{1}{p}+\frac{1}{q}=1$. We showed that the integer frequency exponentials for an orthonormal system with respect to this inner product $$ \inner{e^{imx}}{e^{inx}} = \One{m = n}. $$

__Problems:__ No 3.
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*Turn in solutions in class on Wednesday of the following week.*

We defined the Fourier coefficients of a function $f \in L^1(\TT)$ and the Fourier series of such a function. We computed the Fourier coefficients of a few simple functions. We proved the Riemann-Lebesgue lemma (the Fourier coefficients tend to 0). We spoke about general trigonometric series (of which Fourier series are a subset) and saw that if the coefficients of such a series are absolutely summable then the series converges uniformly to a continuous $(2\pi)$-periodic function. Finally we proved that if the Fourier series of an integrable function is absulutely summable then it converges uniformly to a continuous (hence $L^1$) function which has the same Fourier coefficients. Later we will see that these two functions are identical.

__Problems:__ No 4.
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*Turn in solutions in class on Wednesday of the following week.*

We started by computing the trigonometric series $\sum_{n=-\infty}^\infty r^{\Abs{n}} e^{i n x}$, for $0 \le r \lt 1$, which
turns to be the function $\Ds \frac{1-r^2}{1-2r \cos{x}+r^2}$ (the *Poisson kernel*).
Then we saw how some elementary operations affect the Fourie coefficients of a function. The operations are translation of the function,
multiplication by an exponential, reflection of the function and conjugation.
We then reminded the definition of the big $O(\cdot)$ and little $o(\cdot)$ notation.
Then we proved that smoothness of a function implies decay of the Fourier coefficients: if $f \in C^j(\TT)$ then $\Abs{\ft{f}(n)} = O(n^{-j})$.
For this we first proved that $\ft{f\prime }(n) = i n \ft{f}(n)$ if $f \in C^1(\TT)$.

We explained why one does not need a continuous derivative for $f$ in order to have $\ft{f\prime}(n) = in \ft{f}(n)$. Then we stated a proved a uniqueness theorem for Fourier series: if $f \in L^1(\TT)$ has all its Fourie coefficients equal to 0 then it is equal to 0 at any point of continuity. We then reminded ourselves a few basic properties of convolution on the real line and saw how the definition is amended in order to convolve functions on $\TT$.

__Problems:__ No 5.
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*Turn in solutions in class on Wednesday of the following week.*

We spoke a bit more about convolutions on $\TT$ and their properties (specifically how smoothness of a convolution factor implies smoothness of the convolution product). Then we defined the Dirichlet kernel $D_N(x)$ and we saw the basic identity $(S_Nf)(x) = f*D_n(x)$. We computed a formula for $D_N(x)$ and, using it, we saw that $\Lone{D_N} \ge C \log N$. We defined the Cesaro means of the partial sums of the Fourier series, $\sigma_Nf$ and also the Fejer kernel $K_N$ defined via the identity $\sigma_N f = f*K_N$.

We first computed a formula for $K_N(x)$, the Fejer kernel of order $N$. Then we defined what it means for a sequence $k_n(x) \in L^1(\TT)$ to be
a *good kernel* and proved that the Fejer kernel is indeed a good kernel. We then showed that if $f \in C(\TT)$ and $k_n$ is a good kernel then
$f * k_n \to f$ uniformly. Specializing to the Fejer kernel we deduce Fejer's theorem, that $\sigma_Nf \to f$ uniformly for $f \in C(\TT)$.
We then showed that the density of continuous functions in the integral $L^p$ spaces implies that this theorem also holds for $L^p$ functions.
As an application of the density of trigonometric exponentials that this implies in $C(\TT)$ we proved Weyl's criterion for uniform distribution
and used it to prove that $n \alpha$ is uniformly distributed mod 1 when $\alpha$ is irrational.

__Problems:__ No 6.
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*Turn in solutions in class on Wednesday of the following week.*

We saw first how Fejer's theorem implies the Weierstrass Approximation Theorem. This is done by approximating a triginometric polynomial by an algebraic polynomial. One way to do this is to use the Taylor polynomial of cosines and sines. Next we introduced the de la Vallee Poussin kernel, which is a useful cross between the Dirichlet and Fejer kernel. We used it to prove that if a function $f \in C({\mathbb T})$ has non-zero Fourier coefficients only at frequencies $\pm 3^k$ then the partial sums of its Fourier series converge uniformly to $f$. We further intend to use the de la Vallee Poussin kernel in order to prove that the Weierstrass function $\sum_{n=0}^\infty 2^{-\alpha n} e^{i 2^n x}$, where $0 < \alpha < 1$, is a continuous function which is nowhere differentiable.

We finished the proof that the Weierstrass function $\Ds \sum_{n=0}^\infty 2^{-\alpha n} e^{i 2^n x}$, where $0 < \alpha < 1$, is a continuous function which is nowhere differentiable. Then we started talking about the $L^2(\TT)$ theory, mostly about general orthogonal systems and their properties.

We completed the basic $L^2$ theory. We defined what it means for an orthogonal system of functions in $L^2(\TT)$ to be complete (no function but 0 is orthogonal to all members of the system) and we showed that this is equivalent to all $\CC$-linear combinations of the members of the system being dense in $L^2(\TT)$. We showed Bessel's inequality for a general orthonormal system (ONS) and Parseval's formula for a complete ONS. We also saw a couple of applications (problems 5.10 and 5.11 of the notes).

We had our mideterm exam in class.

We saw how to prove the isoperimetric inequality using Fourier series. Then we started talking about the convergence of the partial sums of the Fourier series of a continuous function in preparation for showing that for some continuous function these may not converge.

We completed the proof of the existence of continuous functions whose Fourier series does not converge at 0. We proved the Banach-Stainhaus theorem for this. We also dealt with convergence of the partial sums in the $L^p$ norms. We saw that we cannot expect convergence in $L^1$ and $L^\infty$ but we do have convergence in $L^2$ as an easy consequence of Parseval. Though we did not prove this, we do have convergence in all $L^p$ with $1 < p < \infty$.

We proved that the convergence of the partial sums of the Fourier series of a function at a point $x$ depends only on the behaviour of the function in a neighborhood of $x$ (localization principle, Dini's criterion). We also proved a theorem of Hardy which says that if $\Abs{\ft{f}(n)} = O(1/n)$ then the partial sums of the Fourier series and the Cesaro means converge for the same $x$. We then started talking about the possible rate of decay of the Fourier coefficients and proved various results of the type that regularity of a function implies at least a certain rate of decay. Finally we proved a rate-of-decay theorem for functions that are increasing in $(-\pi, \pi)$ and $(2\pi)$-periodic.

We showed that whenever $a_n \ge 0$, $n \ge 0$, is a convex sequence tending to $0$ we can find an $f \in L^1(\TT)$ such that $\ft{f}(n) = a_{\Abs{n}}$, $n \in \ZZ$. This means that we can have a Fourier series whose coefficients decay to 0 at infinity arbitrarily slowly. We also showed however that if $\ft{f}(-n) = - \ft{f}(n) \ge 0$, for $n\ge 0$, then we must have $\Ds \sum_{n=1}^\infty \frac{\ft{f}(n)}{n} < \infty$. As a corollary we can conclude for some sequences that they cannot be the Fourier coefficients of an $L^1$ function, such as the sequence $\Ds \frac{1}{\log n}$.

__Problems:__ No 7.
(a) Solve problem 6.13 from [KP].
(b) Solve problem 4, p. 162, from [SS].
*Turn in solutions in class on Wednesday of the following week.*

We defined the Fourier Transform on $L^(\RR)$ and some of its elementary properties. We saw the space of Schwarz functions $\mathcal{S}(\RR)$ and saw how the FT is bijection from the space of Schwartz functions onto itself. Then we also saw that the FT preserves the inner product and $L^2$ norm on $\mathcal{S}(\RR)$.

We saw how we can extend the definition of the FT from the space of Schwartz functions to all of $L^2(\RR)$ using the density of the Schwartz functions and the preservation of the $L^2$ norm. Preparing to extend our definition of the FT to $L^p(\RR)$ for $1 \le p \le 2$ we stated the Riesz-Thorin interpolation theorem and saw its graphical representation.

We saw how one extends the definition of the FT to $L^2(\RR)$, using the Schwartz functions, their density and also the fact that the Fourier Transform preserves the $L^2$ norm, and is thus a bounded (continuous) operator on $\mathcal{S}(\RR) \cap L^2(\RR)$.

To extend the FT to the spaces $L^p(\RR)$, with $1 \le p \le 2$, we used the Riesz-Thorin interpolation theorem, which we eventually proved.