- #1
ghotra
- 53
- 0
Actually this might not be a Fourier question, but it certainly reminds of Fourier series.
Suppose,
<br /> \sum_{n=0}^\infty a_n \, g_n(x) = 0<br />
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,
<br /> f(x) g(y) = 0<br />
only implies that each function must be a constant.
<br /> \sum_n f_n(x) \, g_n(y) = 0<br />
Under a sum, my guess is that we can't say anything about each of the functions.
Suppose,
<br /> \sum_{n=0}^\infty a_n \, g_n(x) = 0<br />
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,
<br /> f(x) g(y) = 0<br />
only implies that each function must be a constant.
<br /> \sum_n f_n(x) \, g_n(y) = 0<br />
Under a sum, my guess is that we can't say anything about each of the functions.