Harmonic Analysis, Spring 2019-20

Mon, 3 Feb. 2020:

Today we talked about Lebesgue measure on the real line, how it is defined, saw some examples (for instance, we proved that all countable setss have measure 0) and listed some basic properties of the set function $m(E)$ (the Lebesgue measure of the set $E \subseteq \RR$). We proved some (but not all) of these properties. We also defined the ternary Cantor set and saw that it has measure 0 without being a countable set.

Problems: No 1 ( / )

Thu, 6 Feb. 2020:

We used the concept of Lebesgue measure to define the integral of arbitrary functions. First we defined the integral of simple functions (functions that take finitely many values only) and then we defined, using them, the integral of arbitrary nonnegative functions. The integral of signed (or complex valued) functions is obtained by linearity. We also defined the function space $L^1(A)$, which consists of those functions defined on $A \subseteq \RR$ for which $\int_A \Abs{f} \lt \infty$. Using these definitions we proved several statements regarding the integral (exercises 1.9, 1.10, 1.12, 1.13 (Markov’s inequality)). We stated (without proof) the monotone convergence theorem and, using it, we did exercises 1.15, 1.16.

Problems: No 2 ( / )

Mon, 10 Feb. 2020:

Today we talked about the Dominated Convergence Theorem and its applications. Then we talked about Lebesgue measure in $\RR^2$ and also saw that the Lebesgue integral in $\RR^2$ is defined in exactly the same way as in $\RR$. We stated Fubini’s theorem on repeated integration and saw how this is applied in order to show that the convolution of two functions in $L^1(\RR)$ is defined almost everywhere and its integral is bounded by the product of the integrals of the absolute values of the factors: $\int_\RR \Abs{f*g} \le \int_\RR \Abs{f} \int_\RR \Abs{g}$. Finally we talked about norms in vector spaces and saw their axioms and that the $L^1(A)$ norm $$ \Norm{f}_1 = \int_A \Abs{f} $$ for functions in $L^1(A)$ has these properties provided we do not distinguish functions that differ only on a set of measure 0.

Problems: No 3 ( / )

Thu, 13 Feb. 2020:

Today we completed our crash-course on Lebesgue measure and integration (Chapter 1 of [KP]). We studied $L^p$ spaces and the basic inequalities for $L^p$ norms (Hölder and Cauchy’s inequalities) and saw that $L^p(A) \subseteq L^q(A)$ if $m(A) \lt \infty$ and $p \gt q$. We also saw that $C_0(\RR)$ is dense in $L^p(\RR)$ (for $p\lt\infty$) and saw how to use this to prove (a) that translation is continuous in $L^p(\RR)$ ($p\lt\infty$) and also to prove the Riemann-Lebesgue lemma for the Fourier Transform on $L^1(\RR)$.

Problems: No 4 ( / )

Mon, 17 Feb. 2020:

We talked about periodic functions and then about trigonometric polynomials. We proved (uniqueness theorem) that a trigonometric polynomial of degree $\le N$ which is 0 at $2n+1$ different points in $[0, 2\pi)$ is necessarily the zero polynomial (has all its coefficients equal to 0). Then we defined the inner product on $[0, 2\pi)$ and saw some of its elementary properties. Please read the entire Chapter 2 of [KP].

Problems: No 5( / )

Thu, 20 Feb. 2020:

We covered § 3.1 and 3.2 of [KP]. We defined the Fourier coefficients and the Fourier series of a function in $L^1(\TT)$ and saw some elementary properties of it, such as the bound $$ \Abs{\ft{f}(n)} \le \Lone{f} = \frac{1}{2\pi}\int_0^{2\pi}\Abs{f}, $$ which implies that, for any $k \in \ZZ$, the linear map $f \to \ft{f}(k)$ is a bounded and, therefore, continuous map from $L^1(\TT) \to \CC$. We computed the Fourier series of some simple functions. We spoke about general trigonometric series (as opposed to Fourier series) and saw that an absolutely convergent trigonometric series converges uniformly to a continuous function and it is the Fourier series of that function. Finally we defined the so-called Poisson kernel via its Fourier series and found a formula for it, from which we derived some of the properties of this function as the parameter $r \to 1 - $.

Problems: No 6( / )

Mon, 24 Feb. 2020:

Today we saw how some simple changes in the function (translation, modulation, etc) translate in the Fourier side, i.e., what the Fourier coefficients of the changed function are compared to those of the unchanged function. Then we introduced the $O(\cdot)$ and $(o(\cdot)$ asymptotic notations and proved that if $f \in C^j(\TT)$ then $\ft{f}(n) = O(n^{-j})$. In particular, if $f \in C^2(\TT)$ it follows that the Fourier series of $f$ is absolutely and uniformly convergent. We finished Chapter 3 of [KP].

Problems: No 7( / )

Thu, 27 Feb. 2020:

We proved our first interesting uniqueness theorem today. A nice corollary of it is that for $f \in C^2(\TT)$ its Fouier series converges uniformly to it on $\TT$. Then we reviewed some basic facts about convolutions of functions (that we had already seen earlier in the course) and talked about convolution on $\TT$ and its properties. Then we saw that the partial sums of a Fourier series of a function are given by convolving the function with the so-called Dirichlet kernel $D_N(x)$. Read [KP, § 4.1-4.4].

Problems: No 8( / )

Thu, 5 Mar. 2020:

We started by proving the lower bound $\Lone{D_N} \ge C \log N$ for the $L^1$ norm of the Dirichlet kernel. This is an inequality that is at the heart of the difficulties in many problems of convergence of the partial sums of a Fourier series to the corresponding function. We then defined the so-called Cesaro means $$ \sigma_N f = \frac{S_0f + S_1f + S_2 f+ \cdots + S_N f}{N+1}, $$ and we saw that $\sigma_N f = f*K_N$ where $$ K_N(x) = \sum_{k=-N}^N \left(1-\frac{\Abs{k}}{N+1}\right) e^{ikx} = \frac{1}{N+1}\frac{\sin^2 \frac{(N+1)x}{2}}{\sin^2\frac{x}{2}}. $$ This is the so-called Fejer kernel and we saw that it has several good properties. We stated and proved Fejer's theorem (that $\sigma_n f \to f$ uniformly for $f \in C(\TT)$) and saw that it implies the uniqueness theorem for all of $L^1(\TT)$. Read [KP, § 4.5-4.7].

Problems: No 9 ( / )

Mon, 9 Mar. 2020:

We did several problems from previous homework sets. Then we started talking about equidistribution of sequences. We defined the concept and saw an example of a sequence which is equidistributed. Then we stated Weyl's equidistribution criterion but did not finish the proof of it. Read part of [KP, § 4.9].