I should have noticed one year earlier that the “convolution” in probability and the “convolution” in Fourier analysis are the same “convolution”.
To prepare for the graduate analysis course next quarter, I planned to read some materials in the winter break. Other than this reason, I have not learned Fourier Analysis throughout before. While it is a very interesting and useful subject, I decided to learn as a beginner.
I picked the book Fourier Analysis and Its Applications by Gerald B. Folland as the beginning. I read the basics of Fourier Series this week (The first two Chapters). Beginning with some examples, the book introduces the Fourier series of a 2π-periodic function. It then talks about the convergence of Fourier series, the behavior of the Fourier series in derivatives and integrals, and finally using the periodic extension to get the Fourier series of a function on a single interval. During the rest of the break, I planned to learn the Fourier Transform later and jumped to the topic of generalized function and Green’s function.
Another book that I found as an option is a book Fourier Analysis by T. W. Kòrner. It is actually a fun book to read. More specifically, it reads like a storybook rather than a textbook. However, since the content is relatively deeper, I may try to pick it up later.
After I finished chapter 1 of the basics of the Ordinary Differential Equation, professor Stingo asked me to read chapter four of the book, topics about the contraction mapping and the proofs of some other main theorems. I now finished the proof of contraction mapping and the introduction of Picard approximations with the relation of contraction mapping. The concepts are very well connected in the book, and I am looking forward to the next topic.
Since the last time I learned Algebra is one year earlier, and I have to continue my undergraduate Algebra course in the next quarter, I will review the materials from the book Algebra by Michael Artin. I just completed the definition of homomorphism between groups, and the book discusses the topic in detail.
Approximation Algorithm and Semidefinite Programming:
After finishing the introduction of semidefinite programming and the topic of Shannon Capacity and Lovasz Theta, the next chapter of the book talks about the Cone Programming. In this chapter, the book first works on the more general setting of the cone programming, and it talks bout the Separation theorem, Farkas lemma, and further into the weak and strong duality theorem. One thing interests me is that the book introduces the concept of an adjoint linear operator, which is a concept introduced in function analysis as well. Moreover, the wording “dual cone” reminds me of the wording “dual space” in Analysis. Is there any relation between the two concepts?
This post ends with an interesting quote of T. W. Kòrner in his book Fourier Analysis after he introduced the Cesàro limit and Fejér‘s discovery on finding a continuous function with the given Fourier coefficients:
Any reader discouraged by Fejér’s precocity should note that a few years earlier his school considered him so weak in mathematics as to require extra tuition.
——T. W. Kòrner