# Questions tagged [wavelets]

The wavelets tag has no usage guidance.

54
questions

0
votes

0
answers

32
views

### 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 ...

0
votes

0
answers

60
views

### 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 ...

1
vote

0
answers

18
views

### 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 ...

3
votes

0
answers

64
views

### 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)...

0
votes

0
answers

116
views

### 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 ...

0
votes

0
answers

74
views

### Characterising wavelet frames using Fourier transform

As usual, for $f\in L^2(\mathbb R)$
$$
D_j(f)(x) = 2^{j/2} f(2^jx), T_k(f)(x) = f(x-k)
$$
and let $\{D_j T_k \varphi\}$ be a system of wavelets on $\mathbb R.$
A simple result is that, $T_k \varphi$ ...

0
votes

0
answers

57
views

### Multivariate continuous wavelet transform

I have used the continuous wavelet transform (CWT) and cross-wavelet transform (bivariate; XWT) several times while researching geophysical systems.
I'm trying to understand how the XWT can be ...

1
vote

1
answer

99
views

### 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

496
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

99
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 $\epsilon$...

1
vote

1
answer

241
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

29
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 ...

1
vote

1
answer

158
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

260
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
votes

1
answer

228
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 ...

2
votes

1
answer

57
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

231
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

60
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 ...

1
vote

0
answers

56
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\...

3
votes

1
answer

201
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 ...

2
votes

0
answers

98
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

108
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

214
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

724
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 ...

0
votes

0
answers

74
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 ...

1
vote

0
answers

126
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 ...

2
votes

1
answer

233
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 ...

1
vote

1
answer

88
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
votes

1
answer

325
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
votes

1
answer

929
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

43
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
votes

1
answer

245
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

221
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 ...

6
votes

1
answer

228
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(\...

1
vote

0
answers

143
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 ...

12
votes

1
answer

645
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

120
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^...

3
votes

1
answer

166
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

119
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

192
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 ...

1
vote

0
answers

79
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 ...

0
votes

0
answers

145
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

441
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 ...

1
vote

1
answer

206
views

### Scaling function

Why is it important for the scaling function to have unit area in wavelets?

0
votes

2
answers

421
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 ...

3
votes

1
answer

300
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

262
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 $\{\...

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)...

3
votes

0
answers

966
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

431
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 ...