Analysis Books for learning Fourier series expansion

[Adesh]

Hello Everyone!

I want to learn about Fourier series (not Fourier transform), that is approximating a continuous periodic function with something like this $a_0 \sum_{n=1}^{\infty} (a_n \cos nt + b_n \sin nt)$. I tried some videos and lecture notes that I could find with a google search but those materials were not very helpful. I want to know how we came up with something like that and its applications.My current knowledge is that I know mathematics up to Calculus I (that is single variable calculus) and just a beginner in Multivariable Calculus and Real Analysis. Please suggest me some books that explains Fourier series approximation for any function in a detailed manner. I think a complete separate book on Fourier Analysis (which google search is giving me) requires some high level knowledge of analytical tools, I need something introductory.

Thank you.

 

[etotheipi]

Hey Adesh, I don't know if it's exactly what you're after but this is quite a good reference, there are also some problems at the end. Maybe it is useful

 

[Mondayman]

The Dover book by Tolstov looks pretty good for a beginner, I have it and plan on getting to it once I complete my current courses.

 

[jasonRF]

If you want an actual book, I would recommend looking at books on differential equations that also cover boundary value problems. These should provide a basic treatment of Fourier series, but also show them in context of solving problems such as Laplace’s equation and the heat equation. Used copies of old editions would be fine. One example that I like is
https://www.amazon.com/dp/B010WFCGAO/?tag=pfamazon01-20

Another option might be a a book on math methods for physics, such as
http://www.physics.miami.edu/~nearing/mathmethods/

Jason

 

[BvU]

I really liked Nearing's advice in the intro

How do you learn intuition?
When you've finished a problem and your answer agrees with the back of the book or with your friends or even a teacher, you're not done. The way to get an intuitive understanding of the mathematics and of the physics is to analyze your solution thoroughly. Does it make sense? There are almost always several parameters that enter the problem, so what happens to your solution when you push these parameters to their limits? In a mechanics problem, what if one mass is much larger than another? Does your solution do the right thing? In electromagnetism, if you make a couple of parameters equal to each other does it reduce everything to a simple, special case? When you're doing a surface integral should the answer be positive or negative and does your answer agree?
When you address these questions to every problem you ever solve, you do several things. First,
you'll find your own mistakes before someone else does. Second, you acquire an intuition about how the equations ought to behave and how the world that they describe ought to behave. Third, It makes all your later efforts easier because you will then have some clue about why the equations work the way they do. It reifies the algebra.
Does it take extra time? Of course. It will however be some of the most valuable extra time you
can spend.

Is it only the students in my classes, or is it a widespread phenomenon that no one is willing to
sketch a graph? ("Pulling teeth" is the cliche that comes to mind.) Maybe you've never been taught that there are a few basic methods that work, so look at section 1.8. And keep referring to it. This is one of those basic tools that is far more important than you've ever been told. It is astounding how many problems become simpler after you've sketched a graph. Also, until you've sketched some graphs of functions you really don't know how they behave.

 

[Adesh]

I really liked Nearing's advice in the intro

I have two questions for you:

1. I would like to know Nearing, who is he?
2. Did he really mean to sketch a graph or can we use computer software to make graphs? Believe me or not but making graph on a paper by a pen/pencil does really make you familiar with the behavior of the function.

 

[BvU]

I don't know more than what I can google

Copyright 2003, James Nearing
Permission to copy for individual or classroom use is granted.
QA 37.2 Rev. May, 2010

Do hope PF use of his good advice is allowed ...