All Questions
Tagged with fa.functional-analysis signal-analysis
19 questions
0
votes
0
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47
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Fractal dimension using wavelets [closed]
I'm trying to estimate the fractal dimension of a function.
I created the log(energies) Vs log(scales) plot and I'm computing the Fractal Dimension (D) from the slope using the relation
$$
\alpha = -...
1
vote
1
answer
281
views
Continuous wavelet transform of a periodic function
I have a question regarding the Continuous Wavelet Transform (CWT) of non finite energy functions, such as $g(t) = a\exp(i\omega_0t)$. We know that the CWT is defined for functions in the Hilbert ...
CommunityBot
- 1
1
vote
1
answer
86
views
Discrete uniqueness sets for the two-sided Laplace transform?
Let $f : \mathbb R_+ \to \mathbb C$ be a measurable and integrable function where $\mathbb R_+ = [0,\infty)$. The Laplace transform of $f$ is given by
$$
Lf(s) = \int_0^\infty f(x)e^{-sx} \, dx.
$$
A ...
- 91.8k
2
votes
0
answers
43
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Selecting some linearly independent columns of a particular matrix
Let us consider the matrix $C=A_1+A_2$ where :
$A_1=(a_{k,l})_{k,l=0}^{n-1}$ is the $n$ by $n$ matrix given by $a_{k,l}=\frac{2}{\sqrt{n}}(\cos\frac{2kl\pi}{n})$
$A_2$ is the the $n$ by $n$ block ...
- 4,058
7
votes
1
answer
290
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Square-root lattices: where do they appear?
As an experimental physicist working on crystallography I'm often dealing with the reconstruction of an object from intensity data that emerge from an imaging device. In mathematics the problem is ...
CommunityBot
- 1
6
votes
1
answer
491
views
Harmonic analysis for a beginner
I am currently dealing with discrete Fourier transform and correlation technique to construct the spectrum of a broad band signal. It's already known that if I have enough observations of the signal, ...
- 188k
1
vote
1
answer
322
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A particular commutator of the discrete Fourier matrix
For $N$ be a fixed natural number, define $w=e^{\frac{2\pi i}{N}}$ and $z=e^{\frac{\pi i}{N}}$, so that $z^2=w$. Let $D$ be the diagonal matrix $D=\operatorname{diag}(1,z,z^2,\ldots,z^{N-1})$ and $F$ ...
- 52.4k
2
votes
0
answers
122
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Consistent approximation of weighted Radon transform of smooth probability density, using kernel density estimation
Let $X$ be a random vector in $\mathbb R^d$, with "sufficiently smooth" probability density function on $\rho$. For unit-vectors $w$ and $u$ in $\mathbb R^d$, and a scalar $b \in \mathbb R$, ...
- 6,853
6
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2
answers
333
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Is there a way to reconstruct the convolution $(f * g)(x)$ of $f$ with a Gaussian $g$ from sampled values, $(f*g)(a), a \in A$?
Suppose that $f: \mathbb{R} \to \mathbb{C}$ is a function which has support in $[-1,1]$. Let $g = g_\sigma$ be a centered Gaussian with variance $\sigma^2$. Is there a way to reconstruct the ...
- 437
9
votes
2
answers
907
views
When is a mapping the proximity operator of some convex function?
Is there a characterization of mappings $p : \mathbb R^n \rightarrow \mathbb R^n$ which are proximity operators (in the sense of Moreau) of l.s.c (extended) real-valued functions ?
That is, given $p : ...
- 56
2
votes
0
answers
57
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The significant role of dual frames in the progress of Frame theory
For a given frame $\{\zeta_i\}_{i=1}^\infty$, any Bessel sequence $\{\eta_i\}_{i=1}^\infty$ satisfying in the following identity for every $\xi\in H$
$$\xi=\sum_{i=1}^\infty \langle \xi, \eta_i\...
- 21
2
votes
1
answer
127
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Are the Prolate Spheroidal Wave Functions absolutely integrable?
I would like to know if the Prolate Spheroidal Wavefunctions (PSWFs, defined below) are in $L^1(\mathbb{R})$. I know that they are square integrable, but cannot decide about absolute integrability.
...
- 91.8k
1
vote
0
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123
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Discrete Wavelet Transform and Gaussian decay
I have a question regarding the possibility of constructing a Discrete Wavelet Transform based on a scaling function having Gaussian decay (and no more decay than that). More specifically, I am ...
- 11
3
votes
2
answers
340
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How far can the domain of definition of multiplier operators be extended?
Given any $g \in L^\infty(\mathbb{R})$, we define the associated multiplier operator $T_g \colon L^2(\mathbb{R}) \to L^2(\mathbb{R})$ by
$$ \mathcal{F}(T_g f) \ = \ g.\mathcal{F}f $$
where $\mathcal{F}...
- 16.2k
2
votes
0
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132
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Wiener-Ikehara Theorem and Signal Processing
I am trying to understand the Wiener-Ikehara Tauberian theorem which can be a step to understanding the prime number theorem. Let
$$ \hat{a}(s) = \int_0^\infty e^{-us}\, da(u) $$
with $a(u)$ some ...
- 13.9k
0
votes
0
answers
35
views
Approximate rank of the set formed by all delayed replicas of a bandlimited signals between 0 and T
Given a complex-valued signal with a certain delay $s(t-\tau)$ for which we sample $N$ instants
$$
\mathbf{s(\tau)}=\left[s(0-\tau),\ldots,s\left(\frac{N-1}{f_s}-\tau\right)\right]^T
$$
at Nyquist ...
- 345
1
vote
0
answers
146
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In which sense Daubechies wavelets converge to the Shannon wavelet?
My question is about wavelets theory. Consider $\psi_n$ the Daubechies wavelet of order $n \geq 1$; that is, the Daubechies wavelet with $n$ vanishing moments. We also define the Shannon wavelet in ...
- 2,306
2
votes
1
answer
508
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What kind of role has Functional Analysis played in Signal Processing? [closed]
Does it serve mainly as a narration or is there any substantive consequence which might not be derived without tools of functional analysis?
- 25.3k
10
votes
5
answers
1k
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What is a rigorous statement for "linear time-invariant systems can be represented as convolutions"?
In Signal Processing books, a fundamental theorem is that linear time invariant systems can be represented as a convolution with a distribution. Could you give a mathematically rigorous statement of ...
- 2,745