Proof or Counterexample on Fourier Series for a_n=0

[ghotra]

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

Suppose,

$$\sum_{n=0}^\infty a_n \, g_n(x) = 0$$

Does it necessarily follow that $a_n = 0 \: \forall n$? 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,
$$f(x) g(y) = 0$$

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

 

[Tide]

The $a_n$ will necessarily be zero only if the basis functions $g_n$ are mutually orthogonal (linearly independent).

 

[ghotra]

Haha! I knew I had seen that sum before! Independence!

lol...thanks.