Here are some links to play with sums of trigonometric polynomials:

- Sum of three sine functions with different frequencies .
- Approximation of the Sawtooth function .
- The Fejer Kernel .

A few more links to study the Fourier transformation. (Thanks to David for the good ideas!)

- We have studied the Fourier transformation of continuous functions, but our definitions only require an integrable function. The so called Square Wave is a noncontinuous integable function, and it's Fourier series is demonstrated here . (Square wawe in red, approximations to Fourier series in blue.) The approximations start oscillating wildly. Where does the approximating function have the most fluctuation? The fluctuation does not go away even for large N - this is known as Gibb's phenomenon.
- Study the sum of smooth functions given here . How does the smoothness of the function change when you vary \alpha, \beta, and N?

We studied the sine and cosine functions defined via the complex exponential map. Then we studied some motivations for the Fourier theory we start to approach.

We proved the Weierstrass M-test, together with a simple Lemma, stating that if a sequence of continuous mappings is Cauchy w.r.t. the supremum norm and converges pointwise to a map, then the sequence actually converges uniformly to said map. Furthermore we proved (mostly) Theorem 3.7.1. stating that if a sequence of maps has continuous derivatives converging uniformly to a limit map, then the limit map itself converges (with a few extra assumptions).

You might find this worksheet by David to be useful to understand the nuances of changing limits w.r.t. integrals and derivatives.

We studied continuity, especially we showed that a mapping is continuous if and only if the pre-image of any open set is open.

We showed that compact implies sequantially compact, the reverse direction can be found from Tao's book. From now on we will use mostly sequential compactness! We also went through the Heine-Borel theorem: a set in the euclidean space is compact if and only if it is closed and bounded. Furthermore we defined continuity of a mapping between metric spaces, and looked at a few examples.

We defined the concept of relative topology, and talked about how certain concepts, like opennes and closedness, are dependant on the ambient space. We are approaching the concept of completeness, which will not depend on the ambient structure. On the way there we defined what is a Cauchy sequence, and showed that converging sequences are Cauchy and that a Cauchy sequence is always bounded.

As a complete side remark, I mentioned the Kempner series, see also this.

We defined what is a complete metric space, and went through several examples. Any space with a discrete metric is complete, as are the space of real numbers with the euclidean metric and the space of continuous functions of the unit interval equipped with the supremum metric. On the other hand, equipped with the standard metric the open unit interval and the set of rational numbers are not complete.

We defined the concept of compactness and looked at a few examples. We will mostly use the definition of sequential compactness in the course, but we also looked at the definition via open covers.

We began by showing that a norm always induces a metric, and noted that the discrete metric is not induced by a norm. After this we defined a ball in a metric space and the convergence of a sequence.

We defined the interior, boundary and exterior points of a set, as well as the closure of a set. (Rougly section 1.2. in Tao's book. We will prove some more properties on friday. The definition we use in the book (and used in the lectures) is equivalent to another defintion which is more handy in proofs. I mentioned in the lectures that these two definitions equal, but here is a proof written out formally.

We talked about adherent points and their connection to closure. This means that we have now gone through the theory on 1.1.1. (Definitions an Examples) and 1.1.2. (Some point set topology of metric spaces) from Tao's book.

The course 131B has roughly three main themes. For the first 3.5 weeks we will mostly study the theory of Metric Spaces. During this time we will discover that many of the notions given on earlier courses, e.g. continuity of functions, convergence of sequences, can be meaningfully defined outside the setting of the real line - all that we need is a concept of distance. Such a distance, or a metric, is our main focus for the first few weeks.

The second part of the course, weeks 4-6, focuses on power series and their theory. Much like series of numbers, which can heuristically be thought as "infinite sums of numbers", we can in a similar way take "infinte sums of functions". This gives rise to a surprising amount of useful techniques.

The final part of the course aims to the definition of the Fourier transformation and the Fourier series. Together with differential equations Fourier analysis is not only a beautiful mathematical theory, but also immensely applicable in physics and engineering.

- The definition of a normed space.
- The epsilon-delta definition of the continuity of a function.
- The basic properties of (Riemann) integration of functions. (E.g. you should remind yourself why a continuous function is integrable, and why it holds true that if the integral of a non-negative continuous function is zero, then the function itself must be constant zero.)

We defined what is a metric and a metric space, and proved that the max-metric in the plane and the sup-metric in the space of continuous functions are metrics. The sup-metric in the space of continuous functions was a bit abstract (and somewhat technical) proof, so here is the proof written out explicitly. I recommend looking at Tao's book, pages 1-5 for more interesting examples of metrics!