Questions tagged [wavelets]
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54 questions
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What's the problem in using spanning Bessel sequences that are not frames to decompose vectors?
This is related to a question I recently asked on math.SE.
Consider a subset $G\equiv \{g_k\}_{k\in\mathbb{N} }\subseteq\mathcal H$ in a separable Hilbert space $\mathcal H$, and suppose $G$ spans the ...
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How linearly independent are orthonormal wavelet systems (Daubechies wavelets)?
Disclaimer: Of course, any orthonormal system of functions is linearly independent in the sense of linear algebra, but I am interested in infinite linear combinations with potentially "ugly" ...
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Reference request: injectivity of CWT, density of dilations and translations in $L^p$
Recently, I encountered the notion of Continuous Wavelet Transform (CWT), and I find it very intriguing (for a reference, see the wiki). I believe it offers a different and more general perspective on ...
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Multiplication with dilations of nonzero measurable function is injective
Denote $f_s(x):=f(sx)$ as the dilation of a function $f$. I want to know whether the following statement is true:
Suppose $f$ and $g$ are measurable functions on $\mathbb{R}$, and $f$ is not almost ...
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Correlation Matrix Problem of Three Decomposition Level of DWT
I'm trying to apply a DWT with 3 composition levels and the following question arose when calculating the composition matrix.
The step I'm trying to follow is:
The DWT coefficientes are obtained from ...
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122
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Admissibility condition of wavelet functions
After a badly formulated question, I decided to make a new post searching for help.
The basic problem is the follows: I have a wavelet function $\psi(t)$ (real or complex) and would like to compute (a)...
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Why for $\psi$ square-integrable function the zero mean condition is equivalent to $\hat{\psi}(0) = 0$?
I am studying the classical book "Ten Lectures on Wavelets" written by Ingrid Daubechies and I do not understand a specific point. I would appreciate it if someone could help me with ...
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Morlet wavelet transform of binary dataset in R
I want to perform a Morlet Wavelet transform analysis (WTA) on a sequence of binary data (0, 1), length about 19000 observations. The result seems reasonable, but I have my doubts whether WTA can be ...
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549
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A proof of Bernstein's inequality
I'm studying the Meyer's book, "Wavelets and operators", and I'm confused about a proof of Bernstein's inequality at page 47, which is stated below:
"The function $\frac{\xi^\beta}{|\xi|...
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104
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Wavelet decomposition of $C^{k}$-functions on smooth manifolds
Background (compactly supported wavelet decomposition of $\mathbb{R}^n$):
Fix compactly supported “mother and father wavelets” $\phi,\psi^{\epsilon}:\mathbb{R}^n\rightarrow \mathbb{R}$ where $\epsilon$...
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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 ...
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Deriving periodical processes from a finite time series
Suppose we have a finite time series of real-world events measured at $(t_k), k \in \mathbb{N}$ with $(t_{k-1} < t_k)$. The content of the actual events is irrelevant.
I would like an automated ...
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Wavelet momentum identity
I am reading an article on wavelet connection coefficients (G. Beylkin, "On the representation of operators in bases of compactly supported wavelets", 1992 (MSN)) and I came across Equation (3.31):
\...
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Advantage of fractional Fourier transform over multiscale wavelet
What is the best argument of fractional Fourier transform over multiscale wavelet in data analysis purpose.
Optimization of the good time-frequency domain parameter ? "Good" will be, find the %time-%...
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Relationship between wavelet shape and filter points
MATLAB has a library of wavelet functions, showing their "continuous forms" as well as the the decomposition and reconstruction filters.
In decimated wavelet transform the filter size remains the ...
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Is there a transform similar to the shearlet transform that uses a rotation matrix rather than shearing?
I Have been working in wavelet and shearlet analysis for the past couple of months. However I am working in the analysis side rather than the numerics side. In my work I have been considering the ...
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$f \in L^p(\mathbb{R}^2)$ for all $p \geq 1$, and $f$ has zero integral. What can we say about this function's fourier Transform?
Let $\psi$ be an smooth admissible Shearlet with compact support, cand let $\mathcal{M}$ be a bounded region in $\mathbb{R}^2$ and let $m= \chi_{\mathcal{M}}$ be the characteristic function of $\...
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Example of (not necessarily compactly supported) Hölder continuous wavelet?
In Chapter six of “Ten lectures on wavelets” Daubechies presents a construction of compactly supported Hölder continuous wavelets. However, it seems that those wavelets cannot be represented by some ...
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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 $\xi\in H$
$$\xi=\sum_{i=1}^\infty \langle \xi, \eta_i\...
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Why do we consider some weakening frames like K-frames, frame sequences, and upper semi-frames?
I have found some applications of the Frame Theory in engineering sciences like signal processing, image processing, data compression, sampling theory, optics, filter-banks, signal detection.
As we ...
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Is it possible to find an atlas for the set: $\{F:FE = I, E \text{ is a frame for } \mathbb{R}^n\}$
Let $E$ be the matrix whos rows are $ \{e_i^{\top}\}_{i=1}^m$. Let $E$ also be a frame of $m$ elements for $\mathbb{R}^n$, $m \geq n$. This means there exist two constants $A, B > 0$ such that:
$$
...
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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 ...
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Can the wavelet bispectrum be normalised so that its integral "gives the right answer"?
Fix a rapidly decreasing function $\psi \in \mathcal{S}(\mathbb{R})$ with the properties that $\int_\mathbb{R} \psi = 0$, $\mathrm{Re}(\psi(\cdot))$ is an even function, and $\mathrm{Im}(\psi(\cdot))$ ...
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Power Spectral Density from a wavelet transform?
Is there anyway to obtain the Fourier Power Spectral Density from a [wavelet transform][1] of a time series?
I am particularly interested in this problem because I was wondering if there is any ...
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76
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Why is ideal wavelet selection a least-squares estimate?
In their classic paper "Ideal spatial adaptation by wavelet shrinkage" (http://biomet.oxfordjournals.org/content/81/3/425.short?rss=1&ssource=mfr), Donoho and Johnstone make the following ...
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What is the analogue of expansive matrix for automorphisms?
We say an invertible $n \times n$ matrix with entries in $\Bbb R^n$ is expansive if the absolute values of all of its eigenvalues exceed $1$. An easy calculation also shows that if we consider a ball ...
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275
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Computing 3-term connection coefficients for wavelets
I am trying to calculate the three-term connection coefficients
$$
Λ_{l,m}^{d_1,d_2,d_3} = \int_{-\infty}^\infty \varphi^{(d_1)}(x) \varphi^{(d_2)}_l(x) \varphi^{(d_3)}_m(x) dx
$$
for Daubechies ...
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Finite Parseval Frame
Assume that $G$ is a finite vector space over a finite field with order $|G|$. (For example, $G=Z_p^k$). Assume that $\{f_n\}_n$ is a Parseval frame for $l^2(G)$. Can we say that the sequence $\{f_n\}...
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Boundary behavior of harmonic function on the square
Is there a constant $C$ such that if $u:[0,1]^2\to \mathbb{R}$ is harmonic with $u\in L^\infty(\partial [0,1]^2)$ (if you prefer you can also assume $\|u\|_\infty = 1$ on the boundary and $u$ smooth ...
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Two questions on Elias Stein paper (1976)
I am working on some results related with a paper of Elias Stein (on the almost every where convergence of wavelet summation methods), and I have the following questions:
The maximal function ...
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Has anyone used the quincunx dilation matrix to form compactly supported wavelet functions?
Has anyone used the quincunx dilation matrix to form compactly supported wavelet functions? I know that it's possible, in fact a lot of references make the analog of the Harr wavelet basis, but I'm ...
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Using wavelets to capture the $L^2$ norm of $f''$
I posted this question on MSE a couple of days ago. Someone gave some hints, which, besides the fact that I struggle to understand them, go in a numerical analysis direction, which I am not interested ...
- 398
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Wavelet characterization of Sobolev spaces
We know that there exist wavelets generating orthonormal bases in Sobolev spaces $W^{p,s}(\mathbb R^n)$, where $p$ is the index of integral and $s$ is the index of smoothness. Consider the orthonormal ...
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Discrete Wavelets
I am looking for research that has been done in Discrete wavelets. Let me be specific as Google doesn't give me what I want when I say "discrete wavelets". I don't want countable basis for $ L^2(\...
user66731
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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 ...
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Wavelet-like Schauder basis for standard spaces of test functions?
Edit: A more precise formulation of my question follows the separation line.
The Schwartz space of test functions $\mathcal{S}(\mathbb{R})$ is isomorphic to $\mathfrak{s}$ the space of sequences of ...
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Algorithm for finding eigenfunctions
I have an $ L^2(\mathbb{R}) $ operator that looks like
$$
\Omega = \int \partial\phi(a, b)\ \ |b, a\rangle \langle b, a |,
$$
where $ \langle x | a, b \rangle = f_a(x - b) e^{x^2/2} $ and $ f_a \in L^...
user66731
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1
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Why is it important to know if a frame is a Parseval frame?
I understand that a Parseval frame is one in which both upper and lower frame bounds equal 1. What's the main advantage to having this be the case? Or, more specifically, if I'm constructing a frame ...
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Normalized tight frame that is not orthonormal
Does anybody know an example of a normalized tight frame (wavelet frame) that is not an orthonormal frame of $L^2( \mathbb{R})$?
So in other words $\{\psi_{j,k}(x):=2^{j/2}\,\psi(2^j\,x-k)\}_{j,k \in ...
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Wavelet transform stability to deformations
I've come across the following claim in a paper of Mallat:
"High frequency instabilities [of a signal representation] to deformations can be avoided by grouping frequencies into dyadic packets in ...
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Wavelets in the spaces of harmonic functions
I plan to do something with the theory of wavelets but in harmonic function theory. My question is about this interconnection between wavelets and harmonic functions. Can you recommend me some paper ...
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How to bound Haar coefficients in terms of total variation?
I'm trying to get the basic idea behind the proof of Theorem 8.1 of this paper, but I'm having difficulty. Specifically, it says:
We shall show that there is a set $\Lambda_n\subset\mathcal{D}$ such ...
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Approximation power of wavelets
The Wikipedia article on Wavelet Transform states that:
Wavelet compression is not good for all kinds of data: transient signal characteristics mean good wavelet compression, while smooth, periodic ...
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209
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2
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436
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Discrete Wavelet Transform and L2 Basis
Using the mother wavlet $phi$ one obtains an orthonormal basis $\phi_{j,k}(x):=2^{j/2}\,\phi(2^j\,x-k)$of L^2 (on the unit interval say). Given a function $f$ on can calculate the coefficients using ...
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The Dunkl intertwining operator $V_k$ on $C(\mathbb{R}^d)$
The Dunkl intertwining operator $V_k$ on $C(\mathbb{R}^d)$ is defined by:
$$V_k f(x)=\int_{\mathbb{R}^d}f(y)d\mu_x(y),$$
where $d\mu_x$ is a probability measure on $\mathbb{R}^d$ with support in the ...
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Is there a wavelet frame for $L^2[0,\infty)$?
What systems of wavelets provide a discrete frame for $L^2[0,\infty)$?
Specifically, I need a mother wavelet $\psi(x)$ that has a continuous second derivative, such that the system of wavelets $\{\...
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Decomposing a discrete signal into a sum of rectangle functions
Hello mathoverflow community !
I have a simple question that seems to have a non trivial answer.
Given a discrete one dimensional signal $w(x)$ defined in a finite range, and the boxcar (rectangular)...
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Interpolating Wavelet Coefficients
Hi! I was instructed via reddit that this place would be the best place to post this question. Fingers cross you can help...
Ive been writing some code to get rid of noise "spikes" in a signal. I'm ...
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When does a mother wavelet generate a frame?
This question is about conditions on a mother wavelet that generates a countable familily of child wavelets via scaling and translation, that are both necessary and sufficient for the child wavelets ...
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