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Results tagged with fourier-analysis
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user 10583
The representation of functions (or objects which are in some generalize the notion of function) as constant linear combinations of sines and cosines at integer multiples of a given frequency, as Fourier transforms or as Fourier integrals.
4
votes
1
answer
2k
views
Characterizations of a linear subspace associated with Fourier series
Let $c_0$ be the Banach space of doubly infinite sequences $$\lbrace
a_n: -\infty\lt n\lt \infty, \lim_{|n|\to \infty} a_n=0 \rbrace.$$ Let $T$ be the space of $2\pi$ periodic functions integrable …
- 744
3
votes
7
answers
1k
views
Continuous or analytic functions with this property of sinc function
This question is motivated by my previous post in SE (math.stackexchange.com).
Prove or disprove that $\frac{\sin x}{x}$ is the only nonzero entire (i.e. analytic everywhere), or continuous,
function …
- 744
2
votes
Continuous or analytic functions with this property of sinc function
In fact, the identity is true for the function $f(x)=\frac{\sin ax}{ax}$ for each
$0\lt a\le \pi$.
For the integral part, just apply substitution.
For the series part, use the fact that $\sum_{n=1}^ …
- 744
2
votes
Accepted
Characterizations of a linear subspace associated with Fourier series
This is to summarize what were discussed in the comments, so the title will not be listed as unanswered.
The linear subspace $S$ of $c_0(\mathbb{Z})$ is equal to the convolution product of two copies …
- 744