Fourier Question?

  • Thread starter ghotra
  • Start date
  • #1
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Main Question or Discussion Point

Actually this might not be a fourier question, but it certainly reminds of Fourier series.

Suppose,

[tex]
\sum_{n=0}^\infty a_n \, g_n(x) = 0
[/tex]

Does it necessarily follow that [itex]a_n = 0 \: \forall n[/itex]? If so, please provide a proof. If not, a counterexample would be helpful. If not, can I deduce anything about the the coefficients?

A similar formula,
[tex]
f(x) g(y) = 0
[/tex]

only implies that each function must be a constant.
[tex]
\sum_n f_n(x) \, g_n(y) = 0
[/tex]
Under a sum, my guess is that we can't say anything about each of the functions.
 

Answers and Replies

  • #2
Tide
Science Advisor
Homework Helper
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The [itex]a_n[/itex] will necessarily be zero only if the basis functions [itex]g_n[/itex] are mutually orthogonal (linearly independent).
 
  • #3
53
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Haha! I knew I had seen that sum before! Independence!

lol...thanks.
 

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