Python Programming for Biology: Bioinformatics and Beyond

1

Fourier, J.B.J. (1822). Théorie Analytique de la Chaleur. Paris: Chez Firmin Didot, père

et fils.

2

 Titchmarsh,  E.  (1948).  Introduction  to  the  Theory  of  Fourier  Integrals  (2nd  edn.).

Oxford: Clarendon Press.

3

It has polarisation.

4

http://en.wikipedia.org/wiki/Discrete_Fourier_transform

.

5

since

for any integer k.

6

 Cooley,  J.W.,  and  Tukey,  J.W.  (1965).  An  algorithm  for  the  machine  calculation  of

complex Fourier series. Mathematics of Computation 19: 297–301.

7

 Press,  W.H.,  Teukolsky,  S.A.,  Vetterling,  W.T.,  and  Flannery,  B.P.  (2007).  Numerical

Recipes:  The  Art  of  Scientific  Computing  (3rd  edn.).  New  York:  Cambridge  University

Press.

8

 A  related  result  is  the  Nyquist-Shannon  sampling  theorem,  which  says  that  if  the

frequencies  are  bandlimited,  that  is,  |ω|  <  B  for  some  B,  then  the  entire  signal  can  be

exactly  reconstructed  from  an  infinite  sequence  of  samples  at  interval  Δt  as  long  as  B  <

1/(2Δt). Of course in practice an infinite sequence of samples is not taken.

9

We could set non-maximal points to some other value using origin, if zero has meaning.

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