# Tagged Questions

For questions related to wavelets and wavelet theory

20k views

### Difference between Fourier transform and Wavelets

While understanding difference between wavelets and Fourier transform I came across this point in Wikipedia. The main difference is that wavelets are localized in both time and frequency whereas ...
11k views

### What is a “vanishing moment”?

In this paper, Sweldens says about desireable properties of wavelets: To analyze and represent such signals we need wavelets that are local in space and frequency. Typically this is achieved by ...
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### What is the precise mathematical definition of what a wavelet is and what is its relation to linear algebra?

I was reading on wavelets and it seems that its hard to find a precise mathematical definition of what this concept is. My confusion first arose due to Gilbert Stang's linear algebra book. In ...
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### Decomposing a discrete signal into a sum of rectangle functions

Hello math@stackexchange 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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### Absolute Continuity of Finite Borel Measure Characterized by Orthonormal Basis

I've been working through Fundamentals of Stochastic Filtering (Bain, Crisan) and am a little perplexed by the following (initially) seemingly straightforward exercise and its given solution. We are ...
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### An introduction to wavelets, and the wavelet transform

I am looking for a good introduction to the wavelet transform, particularly in the context of image processing. I am very comfortable with the Fourier transforms, and I've got a good background in ...
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### Which time-frequency coefficients does the Wavelet transform compute?

(I asked this on Stack Overflow a while ago and didn't get a satisfying answer, so I'm trying again here.) The Fast Fourier Transform takes O(N log N) operations, while the Fast Wavelet Transform ...
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### expansion for $1-|t|$

Let $f$ be a continuous function on $\mathbb{R}$ with compact support with exactly one maximum. Form the functions $$f_{m,k}(x)=f^m\left(x-\frac{k}{2^m}\right)$$ I am wondering if one can expand ...
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### Wavelet Theory — where do I start?

I am in the process of implementing a Fixed-Point Fast Fourier Transform. The Fixed-Point FFT requires mathematical background in the area of wavelets and lifting schemes. What are good resources/...
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### definition of Bessel sequence

A sequence $\{f_{k}\}_{k=1}^{\infty}$ is called a Bessel sequence in a Hilbert space $H$, if there exists $B>0$ such that $$\sum_{k=1}^{\infty}|\langle f,f_{k}\rangle|^{2}\leq B\|f\|^{2}$$ for all ...
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### A theorem about Lipschitz regularity and Fourier transform

How to prove that: A function $f$ is uniformly Lipschitz $\alpha$ over $\mathbb R$ if $$\int_{-\infty}^{+\infty}|\hat f(\omega)|(1+|\omega|^\alpha)d\omega<+\infty$$ A function $f$ is uniformly ...
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### Definition of “uniformly regular” signals (as used in the book “Wavelet Tour of Signal Processing”)

The author uses the term "uniformly regular" and I get the idea of it's meaning through the context, yet the phrase is used as if could also have a precise mathematical meaning. Is there a definition ...
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### Calderon's admissibility condition for wavelets explained

Calderon's admissibility condition is a central argument in a number of recent wavelet-like constructs, like curvelets, shearlets, to name a few. It states that if $\psi$'s Fourier transform conforms ...
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### How to intuitively interpret Gabor lambda param?

I have troubles understanding in an intuitive way (not by writing complicated math formulas) what is the meaning of the lambda parameter in the Gabor functions. (I have basic math understanding, grad ...
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### Multi dimensional multiresolution analysis, designing biorthogonal wavelets.

With inspiration from this question, I'm wondering if biorthogonal bases for arbitrary dimensions are possible to construct with the same mechanism. I am thinking a subsampling of a factor of $N$ in ...
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### Wavelet through the lens of Harmonic analysis

I need some references for starting to study about wavelet. I have enough information about abstract Harmonic analysis. Thanks!
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### Regularity of Daubechies wavelet

I am reading the book Wavelets: Theory and applications by A. K. Louis, D. Maass, A. Rieder (http://books.google.ca/books?id=58hpQgAACAAJ&dq=wavelets:+theory+and+application&hl=en&sa=X&...
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### Error bounds in representing a vector using a truncated Moore-Penrose biorthogonal basis

I was reading and trying to reproduce the results in the arXiv preprint of Periodic Gabor Functions with Biorthogonal Exchange: A Highly Accurate and Efficient Method for Signal Compression by Asaf ...
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### Find an function that oscillates between a given upper and lower envelope

Suppose I'm given two real, continuous functions $f(x)$ and $g(x)$ such that $f(x)\ge g(x)$ for all real $x$. I'd like to determine an oscillating function $h(x)$ that has $f(x)$ as its upper-envelope ...
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### Morlet's wavelet reconstruction formula

The CWT (continuous wavelet transform) of a signal $x(t)$ is $$X_w(a,b)=\frac{1}{\sqrt{|a|}} \int_{-\infty}^{\infty} x(t)\psi^{\ast}\left(\frac{t-b}{a}\right)\, dt$$ In order to reconstruct the ...
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### What does $\psi \in \mathscr{L}^2(\mathbb{R}^2)^7$ mean?

I'm aware that $\mathscr{L}^2(S)$ usually means the $L^p$-norm over some set in the context of metric spaces. So I guess that the first part "$\mathscr{L}^2(\mathbb{R}^2)$" means a set of $L^2$-norms ...
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### Function Approximation by Wavelet method

If $f(x)\in{L^{2}(R)}$ , then $f(x)$ can be projected into the subspace $V_{j}$ as, $$P_{j}f(x) = \sum\limits_{k\in{Z}}c_{j,k}\phi_{j,k}(x)$$ where $k\in{Z}$. The projection equation is ...
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### Need to learn wavelet, suggest steps and resources

I am looking for a good introduction to wavelets and wavelet transforms. that covers the following: Basics Vector Spaces – Properties– Dot Product – Basis – Dimension, Orthogonality and ...
In the book "An Introduction To Wavelet Analysis" by David F. Walnut, there is, Exercise 7.45. Show that if $\psi(x)$ is a wavelet, then $\sum\limits_{j}{\left|\hat{\psi}(2^j\gamma)\right|^2} = 1$ ...