All Questions
15 questions
0
votes
1
answer
71
views
Asymptotic expansion inverse discrete Fourier transform
Let $\ell^1(\mathbb{Z})$ be the space of biinfinite sequences $f = (f(n))_{n \in \mathbb{Z}} \subset \mathbb{C}$ such that it is absolutely summable. The discrete Fourier transform or Fourier series ...
3
votes
1
answer
157
views
How can discrete Fourier transform approximation prove the completeness of complex exponentials in $L^2(T)$?
I have a question about the completeness of complex exponentials in function spaces.
For the discrete set $ S = \{1, 2, \ldots, n\} $, it is clear and intuitive that $ e^{2\pi ikx/n} $ for $ k = 0, 1, ...
- 807
2
votes
0
answers
80
views
Prove uniqueness of Radon transform without using Fourier transform
The uniqueness of Radon transform can be expressed by the following claim (I assumed that the function has compact support for simplicity):
If a continuous function with compact support has zero ...
- 807
1
vote
0
answers
86
views
Fourier transform relation for spherical convolution
Let $f$ and $g$ be two functions defined over the 2d sphere $\mathbb{S}^2$.
The convolution between $f$ and $g$ is defined as a function $f * g$ over the space $SO(3)$ of 3d rotations as
$$(f*g)(R) = \...
- 2,306
8
votes
1
answer
1k
views
Who introduced the discrete Fourier transform?
I am trying to find the original reference which introduced the definition of discrete Fourier transform as used today. When did this modern formulation (which includes the indexing from n to N-1) of ...
- 879
0
votes
1
answer
273
views
Fourier transform of measures on $\mathbb{T}$
I'm currently working with Fourier transforms of measures on the $\mathbb{T}^n$ (more specifically in dimension two), i.e.
$$
\hat{\mu}(k) = \int_{\mathbb{T}^n} e^{i k \cdot x} d\mu(x)
$$
or something ...
- 595
3
votes
1
answer
305
views
What corresponds to the operation of taking traces in of the Fourier transformation on a finite group?
I have a question about the Fourier transfomation on a finite non-comutative group. I hope that it is a known fact in the Representation Theory but I cannot find it written explicitly in textbooks.
...
- 41.9k
1
vote
0
answers
151
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Fourier transforms exhibiting symmetries about their critical points
Upon looking at the graphs of various Fourier sine and cosine transforms (ones without Dirac deltas in their domain) I've noticed a pattern that is probably already known, but that I thought would be ...
- 113
4
votes
0
answers
595
views
On smoothness of a function and decay of its Fourier transform
I am not sure that this question is research level, but it was not answered at MSE for several days, so I place it here.
I am interested in a quantitative version of the principle that smoothness of ...
- 41
1
vote
0
answers
50
views
Comparison of (square) of a function and its Fourier transform in an integral
I am completely stuck on a comparison between $f(t)^2$ and $\hat{f}(t)^2$ in an integral.
Considering $f(t)$ of rapid decrease at infinity such that near zero: $f(t) \sim_0 t^{-\frac{1}{2}- \alpha}+o(...
- 1,199
11
votes
1
answer
691
views
Reference request: Fourier transform on the multiplicative group of real numbers
Let us consider the three groups $(\mathbb{R},+)$, $(\mathbb{Z}/2\mathbb{Z},+)$ and $(\mathbb{R}^\times,\cdot)$ (where $\mathbb{R}^\times := \mathbb{R} \setminus \{0\}$). We endow $\mathbb{R}$ with ...
- 12.6k
3
votes
2
answers
487
views
Where to find a table of fair Fourier transforms? [closed]
I am greatly dissatisfied with those tables of Fourier transforms, available online. I simply have no guess what method they use to derive their tables, but it seems completely off to me.
For ...
- 10.1k
2
votes
0
answers
120
views
request for any expository works in pointwise convergence of double Fourier series and especially a paper by Hardy
Quart. J. Math. Volume 37, Issue 1, Pages 53-79
On double Fourier series, and especially those which represent the double zeta-function with real and incommensurable parameters.
Hardy, G.H.
I am not ...
- 698
1
vote
1
answer
1k
views
Fourier approximation error in L^2 for piecewise continuous functions
Let $u:[0,2\pi)\to \mathbb{R}$ be the step function
$$u(x) = \begin{cases}
1 & \text{if } x \in [0,\pi), \\
0 & \text{if } x \in [\pi,2\pi)
\end{cases}$$
By a direct computation, one ...
- 837
1
vote
1
answer
229
views
Result of Beurling concerning absolute convergence of Fourier series of |f|
Let $f\in L^{1}(\mathbb T)$ and define the Fourier coefficient of $f$ : $\hat{f}(n)=\frac{1}{2\pi} \int _{-\pi}^{\pi} f(t) e^{-int} dt; (n\in \mathbb Z)$ and we put,
$$A(\mathbb T):= \{f\in L^{1}(\...
- 1,051