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For questions related to wavelets and wavelet theory.

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Equivalent definitions of a wavelet

Im new in the field of wavelet theory. I see two different notations in the literature about the definition of wavelets: First one A wavelet is a function $\psi \in L^2(\mathbb{R})$ which satisfies ...
Mathstudent's user avatar
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Riez sequences and frames of subspaces, provided an orthonormal basis

Given a Hilbert space $\mathcal H$ and a sequence $\lbrace f_n\rbrace\subset\mathcal H$, we say that it is a frame for $\mathcal H$ if there are constants $A,B>0$ such that $A\|f\|^2\leq \sum |\...
confusedTurtle's user avatar
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"Elementary" $L^2$ inequality for combination of compactly supported functions

I am reading the paper Ondelettes et poids du Muckenhoupt by Lemarié and, at some point, he needs to prove a certain operator is bounded in $L^2(\mathbb R)$ to apply weighted inequalities theory. The $...
confusedTurtle's user avatar
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How to prove a multi-resolution analysis $V_j: j\in \mathbb Z$ intersects to zero

Suppose that $V_j: j\in \mathbb Z$ is an increasing sequence of subspaces of $L^2(\mathbb R),$ such that $f\in V_0$ iff $f(2^j \cdot)\in V_j.$ Assume also that $\{\phi(\cdot -n): n\in \mathbb Z\}$ is ...
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Wavelets - Upsampling and downsampling relation.

Suppose $M \in \mathbb{N}, N=2 M, z \in \ell^2\left(\mathbb{Z}_N\right)$, and $w \in \ell^2\left(\mathbb{Z}_M\right)$. I need to prove that $$ \langle D(z), w\rangle=\langle z, U(w)\rangle . $$ Note ...
THIRUMAL 5688's user avatar
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Wavelet admissibility and orthogonality

I have been studying the continuous wavelet transform and came across the following result on the Wikipedia: A wavelet $\psi(t) \in L^1(\mathbb{R})\cap L^2(\mathbb{R})$ is admissible if it has a ...
Isaac Mortiboy's user avatar
3 votes
1 answer
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Why does the Haar orthonormal system span the whole $L^2$?

I am reading "Real Analysis with an Introduction to Wavelets and Applications" because I want to understand wavelets better for my work. I got stuck on a detail about the Haar Basis in ...
Matteo Aldovardi's user avatar
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1 answer
34 views

Proof that a specific exponential integral converges (Admissibility of complex Morlet wavelet)

As part of a proof of the admissibility of the complex Morlet wavelet, I am trying to show that the following integral is positive and finite $$ 0<\int_0^\infty{\frac{(e^{\sigma \omega}-1)^2e^{-\...
Isaac Mortiboy's user avatar
2 votes
0 answers
53 views

An unanswered question from Daubechies' Ten Lectures on Wavelets

In I. Daubechies' "Ten Lectures on Wavelets" it is stated in note 2 of chapter 2 (p. 51) that A. Grossmann, J. Morlet, and T. Paul's article "Transforms associated to square Integrable ...
user920957's user avatar
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Theoretical derivation of discrete-time multiresolution decomposition

I started three weeks ago to study wavelets with the intention of later applying them to some deep learning architectures. As a Ph.D. student in mathematics my interest lies mainly in the theoretical ...
Pietro Cestola's user avatar
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Wavelet Denoising of Random Walk

I have a time series of log prices that looks like a random walk. I want to denoise this series using wavelet denoising. I care about predicting future returns (so predicting the difference in the ...
James Pinkerton's user avatar
1 vote
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46 views

Exploring the Potential Wavelet Frame Properties of a Set

I am reading the definition of wavelet frame given from Ole Christensen's book "An Introduction to Frames and Riesz Bases." The definition reads: "A frame for $L^2(\mathbb{R})$ of the ...
Mark's user avatar
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Will inverse DWT of approx. coefficients results in an approximate signal in original space?

I have a signal $y$ in real-time space $V_0$. Hence assume $y = y^0$, the approximation coefficients of the signal at level $V^0$ (as per Mallat's pyramidal algorithm). I applied DWT and obtained ...
Kiran Kumar's user avatar
1 vote
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29 views

What is the error of reconstruction of a smooth function observed only on a fixed grid by projection on a wavelet basis?

Context I'm a PhD student in Statistics and I have evaluations of a $L_2([0,1])$ function $f$, that is $m$ times derivable, on a regular grid of $[0,1]$ $$f\left(\frac{k}{p-1}\right), 0\leq k \leq p-1....
Rocinante's user avatar
5 votes
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I know that $ \int_0^\infty \frac{|f(x)|^2}{x} < \infty $, which condition should I have to prove $ \int_0^\infty \frac{|f(x)|^2}{x^2} < \infty $?

I'm working with the wavelet transform, and I'm facing a problem proving that $$ \int_0^\infty \frac{|f(x)|^2}{x^2}\mathrm{d}x $$ converges. The only thing I know is that $f(x)$ is the Fourier ...
Jzsb's user avatar
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Compactly supported orthonormal basis of $L^2(\mathbf R)$ with certain properties

Take a positive integer $\alpha$. I am looking for an orthonormal basis $(\phi_n)$ for $L^2(\mathbf R)$ with roughly speaking the following properties: each $\phi_n$ is compactly supported and $C^\...
George C's user avatar
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1 vote
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Orthonormal Basis in $L^2[0,5/8]$

It is well-known that the Haar wavelet system forms an orthonormal basis for $L^2[0,1]$. I am interested in forming a similar orthonormal basis for $L^2[0,\frac58]$. We write $[0, \frac58] = [0,\...
AinvAchor's user avatar
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3D wavelet transform in the form of a matrix?

I was wondering if anyone may know of any method for the construction of a 3D wavelet transform in matrix form? I've been able to build matrices to perform 1D & 2D transforms. Yet, am finding very ...
hubble's user avatar
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Is a sequence in a Hilbert space that forms biorthogonal with a Schauder basis also a Schauder basis?

Suppose $H$ is a Hilbert space and has a Schauder basis $\{e_{j}\}_{j=1}^{\infty} $. $\{\hat{e_{j}}\}_{j=1}^{\infty}$ is a sequence biorthogonal with $\{e_{j}\}_{j=1}^{\infty}$; that is, $<e_{j},\...
Ksw's user avatar
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6 votes
1 answer
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What is the functional space of (general) Wavelet transform of squared-integrable functions?

Let $\psi(x) = \exp(-x^2)$. The (general) Wavelet transform $W: L^2(\mathbb R) \rightarrow X$ is defined by $$Wf(a, b) = b^{-1/2}\int_{x\in \mathbb R} f(x) \psi\left( \frac{x-a}{b}\right)dx$$ for $(a,...
Leonard Neon's user avatar
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1 answer
68 views

Quadratic function and the Gaussian integral

I'm trying to find the CWT (Continuous Wavelet Transform) for the function $$f(t) = \alpha t^2 + \beta t + c$$ and the wavelet $$\psi(t) = (1-\frac{t^2}{\sigma^2})\exp(-\frac{t^2}{2\sigma^2})$$Note ...
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3 votes
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Haar Representation of Brownian Motion Understanding Proof of Continuity

I have begun reading about the Haar function representation of Brownian motion and am trying to understand the proof of why the representation defines a Brownian motion. Specifically, I am looking at ...
chris7347's user avatar
1 vote
1 answer
156 views

Two different formulations of Haar basis in $L^2(\mathbb R)$

I found two different formulations of Haar basis in $L^2(\mathbb R)$. For example, in Heil, Christopher. A basis theory primer: expanded edition. Springer Science & Business Media, 2010. we found: ...
Mark's user avatar
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35 views

How to orthogonalize the Fourier-Iagolnitzer transform in the same direction of a wave train?

The Fourier-Bros-Iagolnitzer transform is given by the functional $$ \mathcal {F}\{f(t)\}=(2\pi )^{{-n/2}}\int _{{{{\mathbf R}}^{n}}}f(x)e^{{-a|x-y|^2/2}}e^{ix\cdot t},dx.$$ I applied this on a set of ...
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Duals In The Sense Of Riesz Representation Theorem

Can somebody help me understand this quote from Wikipedia: In mathematics, a dual wavelet is the dual to a wavelet. In general, the wavelet series generated by a square-integrable function will have ...
fweth's user avatar
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How to calculate the covariance between two signals using the wavelet transform?

I am currently in my last year of Master in Bioengineering, and am studying the application of the wavelet transform for various uses. For instance, wavelet analysis can be used to calculate the ...
Anais__'s user avatar
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0 answers
17 views

Maximal Overlap Wavelet Transform Energy Conservation Properties

When I perform the haar maximal overlap wavelet transform of a signal I get a series of coefficients and one approximation of the signal itself. ...
Emiliano Rosso's user avatar
2 votes
0 answers
36 views

Is $\left\{ \frac{1}{x-a\pm iw} \right\}_{a\in\mathbb R, w>0}$ dense in $L^2(\mathbb R)$?

For $f\in L^2(\mathbb R)$, one may want to expand $f$ in terms of other functions. Fourier transform is a prominent example. I want to consider another type of `basis functions'. Specifically, for $f\...
Laplacian's user avatar
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2 votes
1 answer
83 views

multiresolution analysis scaling identity

I'm reading the First Course in Wavelets with Fourier Analysis by Boggess & Narcowich. In chapter 5 section 1.2, it defines $$\newcommand{\scal}[1]{\langle{#1}\rangle}p_k = \sqrt2 \scal{ \phi, \...
YAC's user avatar
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0 votes
1 answer
62 views

Cascade algorithm for wavelet and scaling functions

It has been described to me that the wavelet and scaling functions can be computed using the Cascade Algorithm applied to the low and high pass filter coefficients. Where can I find some proofs of how ...
Veak's user avatar
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0 votes
1 answer
73 views

Computing db2 to db20 coefficients

I want to implement db2 to db20. Is there a specific formula for the wavelet function coefficients, which determine its shape and properties? How about the associated scaling function?
Veak's user avatar
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1 vote
0 answers
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low pass and high pass filter defined as a pair

In Strang's Wavelets and Filter Banks, Problem 1.3.6 asks, If $H_0$ is the response of a lowpass filter, what is the response $H_1$ of a corresponding highpass filter? If $h(0), \dots, h(N)$ are ...
YAC's user avatar
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0 votes
1 answer
48 views

Which methods are best to analyse wave-packets?

I would like to find several methods to analyse wave-packets. My primary aim is to find a way to decompose wave-packets into "sub-wave packets" that, when subjected to some operation give ...
Superunknown's user avatar
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Which methods apply best in the analysis of non-linear phenomena?

I am going to set up my Bachelors project in analysis of non-linear wave phenomena, and I am planning to study plots and graphs of nonlinear spectra. In this context, I am quite familiar with the ...
Superunknown's user avatar
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1 vote
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Haar wavelet decomposition, show that: $\psi_{j,k}=\sum_{n\in\mathbb{N}}b_{n-2k}\phi_{j-1,k}$

We have a $L^2(\mathbb{R})$ multiresolution $(V_j)_j$ through the family of scaling functions $\phi_j$ and the orthogonal wavelets $\psi_j$ such that $$\phi(x)=\sum_{n\in\mathbb{N}}a_n\phi(2x-n)$$ and ...
tareqath's user avatar
  • 107
1 vote
1 answer
37 views

What is a waveform dictionary?

If I google "waveform dictionary" or "waveform dictionaries" I find some papers that use these mathematical objects e.g. The Analysis of Foreign Exchange Data Using Waveform ...
Mark's user avatar
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2 votes
0 answers
46 views

Extracting frequency derivative from STFT

I have a signal that oscillates on short timescales and changes frequency of that oscillation on larger scales. If I do a Short Time Fourier Transform(STFT) or a constant Q transform, I get a feature ...
niklasrb's user avatar
1 vote
0 answers
60 views

When are LU factors of sparse matrices surely sparse?

The other week I revisited an old classic factorization from my bachelors studies, the LU-factorization. The LU factorization of a (square) matrix M finds lower and upper triangular matrices (L and U ...
mathreadler's user avatar
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0 votes
0 answers
34 views

Relationship between primal and dual wavelet filter coefficients

I've been working with wavelets for a project, more specifically implementing the Fast Wavelet Transform, which requires the use of the dual high pass and low pass filters (for the forward FWT). In ...
voyseto's user avatar
  • 11
1 vote
0 answers
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Looking to Learn more about compressed sensing. Looking for guidances (on algorithms).

I am looking to learn more into compressed sensing algorithms (especially parallelizable ones). I am familiar with some formulations of the problem (minimize L2 norm of recovered data against original ...
Manuel Jenkin's user avatar
2 votes
1 answer
176 views

Applying Riesz Representation Theorem to show existence of dual basis

Given a square-integrable function $\psi\in L^2(\mathbb{R})$, define the series $\{\psi_{jk}\}$ by $$\psi_{jk}(x) = 2^{j/2}\psi(2^jx-k)$$ for integers $j,k\in \mathbb{Z}$. Such a function is called an ...
FreeMind's user avatar
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1 vote
0 answers
61 views

Inner product of a function with a dual function

I have been reading a textbook on wavelets. I faced the following statement, Let us relax the condition that $A_s$ and $W_s$ be orthogonal to each other and assume that the wavelet $\psi_{k,s}\in W_s$ ...
FreeMind's user avatar
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0 votes
1 answer
135 views

Show that $f_n$ converges uniformly to $f$ on $[0, 1]$

Show that $f_n$ converges uniformly to $f$ on $[0, 1]$. Please help me. I have no idea.
user avatar
1 vote
1 answer
69 views

Multiresolution analysis, change of integral and summation and convergence a.e.

Let $(V_n, \phi)$ be multiresolution analysis with scaling function $\phi$ and $V_n \subset L_2(\mathbb{R})$. I need an explanation why $\phi$ can be written in following way (taken from Bachman's ...
User154's user avatar
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1 vote
0 answers
55 views

Inverse Fourier transform of equation to define a wavelet

First time asking the internet a question... I am trying to learn wavelet analyses using the filter bank method described in this article (https://www.sciencedirect.com/science/article/pii/...
pjxs's user avatar
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0 answers
127 views

How to Calculate a Wave Travelling Along the Circumference of a Circle?

I'm wondering how you might calculate a wave that traverses the circumference of a circle. This image is a terribly drawn attempt at showing what I mean, Imagine that the black wavy pattern is a ...
Thor Ether's user avatar
3 votes
0 answers
39 views

Intersection of wavelet spaces is trivial

Let $\Phi\in C_c(\mathbb R)$ be a continuous and compactly supported function, satisfying $\int\Phi(x-m)\Phi(x-n)=\delta_{m,n}$. Set $V_j=\mathrm{span}\{\Phi(2^jx-k)\}_{k\in\mathbb Z}$ and consider $\...
Václav Mordvinov's user avatar
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0 answers
45 views

How space of wavelets is created from {0}?

I have learned about wavelets and there is a space $$... V_{-1} \subset V_0 \subset V_1 $$ and they create as a closure of sum of $V_j$ all space $L^2(\Bbb{R})$. There are some other points, but one ...
koralgooll's user avatar
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0 answers
21 views

Information-based function approximation. Series of bits instead of series of real/complex coefficients

There are multiple series and functional bases (e.g. Taylor, Fourier) that allow approximating a function. In some sense, such approximations associate a function with an infinite sequence of real or ...
Ark-kun's user avatar
  • 256
1 vote
0 answers
164 views

How to calculate Gaussian wavelets

What is the formula for the $n$th order derivative Gaussian wavelet? A Gaussian wavelet is given by $$\psi(t) = C \exp(-\frac{(t-\mu)^2}{2\sigma})$$ with $\mu$ and $\sigma$ the mean and standard ...
suitendaal's user avatar

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