Questions tagged [wavelets]

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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 ...
4 votes
2 answers
409 views

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|...
0 votes
0 answers
77 views

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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1 vote
1 answer
164 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 ...
1 vote
0 answers
25 views

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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1 vote
1 answer
147 views

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): \...
4 votes
1 answer
213 views

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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2 votes
1 answer
126 views

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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2 votes
1 answer
45 views

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 ...
3 votes
1 answer
220 views

$f \in L^p(\mathbb{R}^2)$ 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 $\...
1 vote
0 answers
46 views

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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1 vote
0 answers
51 views

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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3 votes
1 answer
193 views

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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2 votes
0 answers
94 views

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: $$ ...
1 vote
0 answers
90 views

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 ...
4 votes
0 answers
207 views

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 $\mathrm{Im}(\psi(\cdot))$ ...
2 votes
0 answers
644 views

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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0 votes
0 answers
73 views

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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1 vote
0 answers
124 views

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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2 votes
1 answer
192 views

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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1 vote
1 answer
84 views

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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5 votes
1 answer
269 views

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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4 votes
1 answer
915 views

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 ...
1 vote
0 answers
37 views

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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4 votes
1 answer
214 views

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 ...
3 votes
0 answers
197 views

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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5 votes
1 answer
214 views

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(\...
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1 vote
0 answers
126 views

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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12 votes
1 answer
595 views

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 ...
1 vote
0 answers
117 views

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^...
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3 votes
1 answer
127 views

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 ...
1 vote
1 answer
112 views

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 ...
2 votes
1 answer
156 views

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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1 vote
0 answers
76 views

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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0 votes
0 answers
140 views

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 ...
6 votes
1 answer
414 views

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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1 vote
1 answer
199 views

Scaling function

Why is it important for the scaling function to have unit area in wavelets?
0 votes
2 answers
413 views

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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3 votes
1 answer
294 views

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 ...
0 votes
1 answer
257 views

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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2 votes
3 answers
1k views

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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3 votes
0 answers
948 views

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 ...
3 votes
1 answer
418 views

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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1 vote
1 answer
343 views

[Numerical Mathemtics] How to solve hexagonal central differences

I want to simulate a 2d linear wave equation on a circle ($\displaystyle\frac{\partial^2 z(x,y,t)}{\partial t^2}=v^2\cdot\left(\displaystyle\frac{\partial^2 z(x,y,t)}{\partial x^2}+\displaystyle\frac{\...
3 votes
1 answer
1k views

Adjoint/transpose of wavelet transform

I'm using a wavelet transform in Matlab, so I think of it as a black-box. I'll represent it here as $W(x)$. There's a reconstruction function as well, which I'll write as $W^\dagger(y)$. I can ...
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1 vote
2 answers
480 views

Using Wavelet Transforms to Approximate Matrices

It's a long time since I worked on this kind of problem, so please bear with me. I have an approximate inverse matrix that I'm using as a preconditioner to solve the conjugate gradient method. ...
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17 votes
9 answers
4k views

Introduction to wavelets?

Are there any suggestions for introductory books on wavelets? I want a book, not online material or tutorials.