Fourier analysis, PDEs: Fourier analysis

Fourier analysis



Fourier series

• periodic function
• a function
  
   is called a periodic function if
  
   is defined for all real x (perhaps except at some points, like
  
   and if there is some positive number p, called a period of
  
  , such that

     for all x.

• first few case, we will study various functions
  
   of period
  
  .

 

• given function of period
  
  , Fourier series
• given function of period
  
   can be represented by a trigonometric series; Fourier series,

• coefficients of Fourier series are the so-called Fourier coefficients of
  
  , given by the Euler formulas

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Fourier series expansion and function approximation

lecture note. Partial fraction expansion and Fourier series expansion

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Partial fraction expansion
Fourier series expansion
     periodic function (definition and example)
     Fourier series representation of periodic function, f(x)
     Fourier series coefficient: Euler's formula when period is 2*pi and p

lecture note. function approximation and the square error of function, frequency and amplitude

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fourier series expansion of f(x)
approximation of f(x), f_N(x)
the square error of f_N(x), E_N: definition, derivation formula which in practrical using
limitation of the square error of f_N(x), E*: definition, parseval's equality
frequency and amplitude spectrum, derivation other form of fourier series
     frequency spectrum of f(x)
     nth harmonic frequency of f(x)
     amplitude spectrum of f(x)
     nth harmonic of f(x)