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Continuous wavelets for piecewise polynomial functions

For continuous wavelets, as Haar is to piecewise constant functions, what is to piecewise linear functions and is there some wavelet basis that spans piecewise cubic spline functions?
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20 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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19 views

Does wavelet-based thresholding for denoising generally apply to details only?

I am trying to use the Discrete Wavelet Analysis for 1D signal denoising. The tool I am using is MATLAB's Wavelet Toolbox. The approach to signal denoising described in this tool's documentation is a ...
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53 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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50 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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107 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 $\Lambda_n\subset\mathcal{D}$ ...
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1answer
163 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, ...
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1answer
178 views

Scaling function

Why is it important for the scaling function to have unit area in wavelets?
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2answers
331 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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200 views

The Dunkl intertwining operator $V_k$ on $C(\mathbb{R}^d)$

The Dunkl intertwinig 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
194 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
596 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 ...
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649 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 ...
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1answer
312 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
299 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 ...
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1answer
619 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
275 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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8answers
1k views

Introduction to wavelets?

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