The wavelets tag has no usage guidance.

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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 ...

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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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### Gaussian gabor frame

It is widely known that $\phi(x)=e^{-\frac{x^2}{2}}$ does not define a Gabor frame if we consider translations by units of $1$ and multiplication by $e^{2 \pi inx}$for $n \in \mathbb{N}.$ A way to ...

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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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### 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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### 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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### 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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### 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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### 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 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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200 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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### 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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### 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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### [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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### 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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### 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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### Introduction to wavelets?

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