I'm currently in my summer break and when I begin again next year I'm taking a course which directly translated is called "Functions and series". Now I'd like to start with studying for the subject and look for some good books on it, but I wouldn't know where to start. The course description is as follows (Wherever I'm unsure of the translation, I've placed the word in quotes.):
After a short recap of the definitions and basic facts of the differentiation of functions of multiple variables, "switching theorems" will be treated. These exist of switching limits and order of differentiation, limit taking and differentiation under the integral sign and switching the order of integration.
In the second part of the course, series of functions are studied. Especially power and fourier-series. Both of which are very important in a plethora of applications. We'll later treat the convergence properties of fourier series, where the "Dirchlet kernel" plays an important role. The "Gibbsphenomenon" with fourier series of functions with gaps will also be describes.
To close, the concept of Hilbert space will be introduced and the fourier theory on $L^2(\mathbb{R})$ treated.
Knowledge and insicht: After finishing this course, the student is capable of:
- Coming up with functions on $\mathbb{R}^n$ with certain differentiability properties.
- Determining properties of the limit function of a fourier series.
- Can determine whether "switching theorems" can or can not be applied.
- Knows that they need to argument more carefully with calculations which are not differentiable.
Skills: The student is capable of
Proving the "Switching theorems".
Determining the power series expansion of a function.
Calculating the radius of convergence of a power series.
Calculating fourier series.
Some of these topics seem to come back in Rudin's Mathematical analysis (Which I find great), but I don't know to what extent (and have only worked through the first 6 or 7 chapters so far). The prerequisite knowledge includes an introductory course in real analysis and linear algebra, together with a solid understanding of the differentiation and integration operations in one variable.
I also really enjoyed Hoffman and Kunze's Linear algebra. Both in the way the book was set up, as well as the way the exercises were treated (The increasing difficulty and the closure of every paragraph with exercises.)
I'm hoping that this will be enough information to help me out. Thanks in advance!