Questions tagged [wavelets]

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122 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): \...
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1answer
93 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-%...
2
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1answer
88 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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1answer
37 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
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1answer
200 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 $\...
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39 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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42 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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1answer
177 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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85 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: $$ ...
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52 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 ...
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0answers
203 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 $\mathrm{Im}(\psi(\cdot))$ ...
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433 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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66 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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116 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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1answer
98 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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1answer
76 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\}...
5
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1answer
199 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 ...
4
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1answer
896 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 ...
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0answers
33 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 ...
4
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1answer
190 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 ...
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180 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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1answer
181 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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106 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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1answer
494 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 ...
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0answers
111 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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1answer
112 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 ...
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1answer
100 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 ...
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85 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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0answers
73 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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81 views

phase prediction of wavelet coefficients for 1D signal [closed]

I was reading a paper 'A Flexible Framework for Local Phase Coherence Computation' (article URL) on predicting phases of wavelet coefficients across 3 consecutive scales in the 1D case, and I'm trying ...
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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 ...
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1answer
369 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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1answer
194 views

Scaling function

Why is it important for the scaling function to have unit area in wavelets?
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2answers
401 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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1answer
279 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 ...
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1answer
243 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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3answers
961 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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0answers
905 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
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1answer
398 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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1answer
334 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{\...
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1answer
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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2answers
426 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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9answers
3k views

Introduction to wavelets?

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