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My professor wanted me to master these topics from Fourier Analysis. I need a resource where these topics are discussed in brief. Although i know many of the topics in the list, i prefer a good resource to brush up my rusty knowledge and learn what i don't know. The topics are:

  • Fourier series: sin and cos as a basis set; calculating coefficients; complex basis; convergence, Gibbs phenomenon
  • Fourier transform: limiting process; uncertainty principle; application to Fraunhofer diffraction
  • Dirac delta function: Sifting property; Fourier representation
  • Convolution; Correlations; Parseval's theorem; power spectrum
  • Sampling; Nyquist theorem; data compression
  • Solving Ordinary Differential Equations with Fourier methods; driven damped oscillators
  • Green's functions for 2nd order ODEs; comparison with Fourier methods
  • Partial Differential Equations: wave equation; diffusion equation; Fourier solution
  • Partial Differential Equations: solution by separation of variables
  • PDEs and curvilinear coordinates; Bessel functions; Sturm-Liouville theory: complete basis set of functions
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  1. If your into solving a lot of examples and gathering some intuition i recommend Schaum's outline series. They have nice solved examples. (https://www.amazon.com/Schaums-Analysis-Applications-Boundary-Problems/dp/0070602190)

  2. If you are into more technical mathematical stuff, here is a textbook I used. (https://www.amazon.com/Introduction-Fourier-Analysis-Russell-Herman/dp/1498773702)

  3. A great way to learn about DFTs and Signal Processing in general, I recommend going through some coding problems and in such case, technical notes from NI and some coding textbooks helped a lot. (https://www.ni.com/ko-kr/innovations/white-papers/06/using-fast-fourier-transforms-and-power-spectra-in-labview.html)

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  • $\begingroup$ Hey, I know this is late, but could you please reccomend resources to learn multi-dimensional fourier transforms? Its for a Math Methods in Physics class I am taking now, we need to evaluate 2D/3D fourier transforms as well.... I need something that has a lot of worked examples and follows naturally from where the 1D version leaves off. Couldn't find a single book on this. $\endgroup$
    – F.N.
    Jun 4, 2022 at 15:58
  • $\begingroup$ Hey, did you try looking into Mathematical Methods for Physicists by Arfken, Hans J. Weber and Frank E. Harris? $\endgroup$ Jul 13, 2022 at 1:53
  • $\begingroup$ I did, but i needed something with more worked exaples and clearer explanations. Something with more exercises would be idea. Could you reccomend any? even if its a book for Math Majors I can manage. $\endgroup$
    – F.N.
    Jul 14, 2022 at 6:52

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