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Results tagged with fourier-analysis
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user 9652
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.
3
votes
What are the known conditions for the log of the Fourier transform of a 2D real discrete sig...
This turns out to be too long for a comment.
I am not sure if one can say much in the discrete two-dimensional case, but something is known in one-dimension for continuous signals:
There is a theore …
- 12.7k
5
votes
A problem on real valued functions in $\mathbb{R}^2$ with least variation
You want to solve this problem
$$
\min \{V(f,\Omega)\ : f\in BV(\Omega),\ f|_{\partial \Omega}\equiv J \}
$$
and there is quite some background required to analyze this problem.
First, one should cla …
- 12.7k
1
vote
Minimizing the L1 norm of odd-term trigonometric polynomial
Not a full answer but too long for a comment.
We rewrite the problem as follows: Define $A:\mathbb{R}^n\to L^1(0,1)$ by $Aa(t) = \sum_k a_k\sin(2\pi kt)$. Then the problem is
$$
\min_a \|f-Aa\|_1
$$
…
- 12.7k
5
votes
How to do DFT for irregular sampling period ?
If you are looking for software: Have a look at NFFT by Potts, Kreiner and Kunis.
- 12.7k
3
votes
General procedure for inverse of an integral transform
Probably a "general procedure" would be as follows: Find appropriate Hilbert spaces $X$ and $Y$ such that the operator $Lf(\xi) \int_a^b f(x) g(x,\xi) dx$ maps boundedly from $X$ to $Y$. Often the spa …
- 12.7k
3
votes
A metric on the set of BV functions, is it mentioned/studied in literature?
This is closely related to the so-called metric of strict convergence which is
$$
d(u,v) = \|u-v\|_{L^1} + |TV(u)-TV(v)|
$$
where $TV(u)$ denoted the total variation of $u$. This is indeed a metric on …
- 12.7k
3
votes
Range of the Radon Transform
Probably you refer to some theorem in "Mathematics of Computerized Tomography" by Frank Natterer (e.g. Theorem 4.2)? Then you are assuming that the domain in $\mathcal{S}$ and if I remember correctly, …
- 12.7k
3
votes
2
answers
1k
views
Do you know this form of an uncertainty principle?
I hope this question is focused enough – it's not about real problem I have, but to find out if anyone knows about a similar thing.
You probably know the Heisenberg uncertainty principle: For any func …
- 12.7k
11
votes
Function and Fourier transform vanish on an interval
A distribution for which both the distribution and its Fourier transform vanish on some interval is the Shah function aka Dirac comb
$$
Ш = \sum_{k=-\infty}^\infty \delta_k
$$
which is its own Fourier …
- 12.7k
12
votes
Origin of the term "sinc" function
The Wikipedia page for the Shannon-Whittaker reconstruction formula states that Whittaker used the term "cardinal series" for the reconstruction formula
$$
f(t) = \sum_{n\in\mathbb{N}} f(n)\mathrm{sin …
- 12.7k
5
votes
Emergence of the discrete from the continuum
(Answer related to the one of Dustin Mixon.)
This appears as relaxation in discrete optimization: if you have an optimization problem $\min f(x)$ with a vector $x$ and the constraint that $x_i\in\{0,1 …
- 12.7k