For questions related to wavelets and wavelet theory

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Boundedness of the Fourier transform of a Battle-Lemarie scaling function

Could anyone please give a short and simple proof of the following proposition: the Fourier transform $\hat\phi$ of Battle-Lemarie scaling function (of arbitrary order) is bounded on $\mathbb{R}$, ...
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58 views

A basis for $\text{span}_{L^2(0,1)} \left\{ \eta_\alpha,0<\alpha<1 \right\}$, and how compute coordinates

Let $$\text{span}_{L^2(0,1)} \left\{ \eta_\alpha,0<\alpha<1 \right\},$$ where we take $$\eta_\alpha(t)= \left\{ \frac{\alpha}{t} \right\} -\alpha \left\{ \frac{1}{t} \right\},$$ and $ \left\{ x ...
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2answers
47 views

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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19 views

Daubechies wavelet for structural dynamic analysis

I am working on the structural dynamic response with Daubechies wavelet. The Equation of Motion of dynamic system with single degree of freedom can be described as $$ m\ddot{x}(t) +c\dot{x}(t)+k{x}(t)...
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8 views

Can you find shearlets in CNNs?

I read that you can find wavelets in the first layers of a CNN. Can you also find shearlets, and if not, why? Edit (further explanations): I read that, when you train a CNN the filters in the first ...
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1answer
72 views

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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29 views

Non-uniform convolution with discrete wavelet transform

I understand that if you have a circular N-dimensional convolution matrix, it can be diagonalized with the Fourier transform of the convolution operator. This makes it easy to calculate the density of ...
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1answer
18 views

Admissibility condition for wavelets

The admissibility condition for a wavelet $\psi$ is: $\int \frac{|\hat\psi(x)|^2}{|x|} dx < \infty$, with $\hat\psi$ the Fourier transform of $\psi$. A necessary and sufficient condition should ...
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1answer
25 views

How can I analyse signal with discrete wavelet transform?

With CWT it's clear enough. We have function of two variables which are scale and translation. But what about DWT? Here is Matlab code: ...
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16 views

Smoothness in Haar basis

The rate of decay of fourier co-efficients of a function $f$ determines the order of differentiability of $f$. Is there an equivalent result for the case when Fourier basis is replaced by Haar wavelet ...
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1answer
39 views

What is wavelet tranform in simple words?

I have read wiki and other sources and have still problem understanding the wavelet transform. What is the basic idea in simple words? Does the Fourier uncertainty hold for wavelet transform?
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1answer
77 views

How the second and third equalities can be achieved?

I am reading this paper. In the Proof of Lemma 3.3, How the second(*) equality can be achieved? How can i use Parseval's identity in third(**) equality?
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21 views

How to obtain an analytic approximation of this wavelet-looking function?

I would like the approximating function to be infinitely differentiable. I have tried fitting (1) a polynomial and (2) a combination of sin, cos, and exp functions to approximate the function shown ...
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12 views

Orthogonal basis for scaling function

I computed a scaling function of a wavelet [phi, ~, ~] = wavefun('dB4',6); . Because I want to calculate the coefficients to annhilate polynomials, I need a dual ...
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18 views

Choose a Daubechies wavelet to approximate a polynomial function

Assume I have a polynomial function $s(x) = \sum\limits_{i=0}^d a_i x^i$, and $\boldsymbol{s}_{d}$ is a discretized sample from $s(x)$, i.e. $\boldsymbol{s}_{d}=(s(0), s(\Delta x), \ldots, s((n-1)\...
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2answers
33 views

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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1answer
63 views

where can I find the proof of this theorem of wavelet frames?

Let $\mathbb{K}$ be a local field. Definition of local field: Let $\mathbb{K}$ be a field and a topological space. Then $\mathbb{K}$ is called a locally compact field if both $\mathbb{K}^+$ ...
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39 views

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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62 views

Using wavelets to capture the $L^2$ norm of $f''$

The goal I have in mind is to express the $L^2$ norm of the second derivative of a quite regular function as a sum of some coefficients. My idea was that such coefficients involved some system of ...
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33 views

Biorthogonal (discrete wavelet) noise bases?

I am slightly interested in discrete wavelet transforms (DWT), but so far I have mostly used already-derived and existing well-known wavelets, such as Daubechies, Cohen-Daubechies-Faveau, Symlets and ...
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14 views

Interpretation of diagonal detail in Haar Wavelet Transforms

I am a statistics grad student, and I have just begun exploring the topic of wavelet regression (specifically, Haar wavelets for discrete functions). I understand the generalization from a one ...
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0answers
30 views

Wavelet zero mean condition

I'm trying to understand more about wavelets so I went and read the wikipedia article and some other papers on the topic. I have learned that wavelet functions are compact supported and belong to the ...
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66 views

Scale resolution and frequency resolution in continuous wavelet transform

I'm reading the well known wavelets tutorial by Robi Polikar here. In part 3, about figure 3.7 and 3.8, it says "lower scales (higher frequencies) have better scale resolution (narrower in scale, ...
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11 views

What are the different type of Daubechies Wavelet transform?

Like Daub4 are there others named as Daub2, daub3 or we only have daub4 , daub8, daub16? What is the order of a transform(represented usually by N)? Does this order have any resemblance with the ...
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22 views

Invariance of stationary wavelet transform

Suppose we are given 64 points $x_1,\ldots ,x_{64}$ and divide them into two groups $x_1,\ldots, x_{32}$ and $x_{33},\ldots , x_{64}$. Then we apply stationary wavelet transform to both these groups ...
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1answer
160 views

Signal differentiation by wavelet transform

I have a noisy signal x(t) that I want to differentiate in time, in order to obtain x'(t). Though numerical finite difference approximantion does not work well on noisy signal, I would like to ...
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36 views

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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32 views

Build polynomial orthogonal to set of other pre-defined polynomials

Basically. the question is simple. Is there any algorithm so I can build a polynomial, orthogonal for the set of pre-defined polynomials? I need the algorithm(like Gram-Schmidt) that would be ...
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29 views

How to orthogonalize piecewise polynomial with respect to set of piecewise polynomials?

Currently I'm working with multiwavelet basis(need it for solving the stochastic ODE system as described here) and can't understand how to build it. Basically, I have a set of functions $p_{i} = x^{i}$...
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1answer
37 views

Absolute convergence related to Fourier analysis

If the Fourier transform of a function $f \in L^2$, whose frequency $\xi$ satisfies $|\xi| \leq \pi$, has compact support, it is famous that \begin{align*} f(x) = \sum_n f(n)\frac{\sin\pi(x-n)}{\pi(...
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1answer
68 views

Wavelet admissibility

This question is about Daubechies' book "Ten Lectures on Wavelets", section 2.4. This author says, if $\psi \in L^2(\mathbf{R})$, $\int_{\mathbf{R}}dx\ \psi(x) = 0$, and \begin{align*} \exists \alpha&...
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22 views

Why is the scaling and shifting variable for wavelet transform is chosen to be multiple of 2?

The continuous wavelet transform is $f \mapsto \gamma(s,\tau)=\int f(t)\psi(\frac{t-\tau}{s})dt$ where we choose the variables s and $ \tau $ to be discrete integers from -$\infty$ to $+\infty$. We ...
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16 views

Why does multiplying by a wavelet equal to bandpass filtering?

The wavelet transform is given as $ \gamma(s,\tau) = \int f(t) \psi (\frac{t - \tau}{s})dt $ However when it comes to carrying out this transform it turns out that it has a "band pass" effect on ...
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1answer
73 views

What is time-frequency plane and when do we use it?

I have seen frequency-amplitude plane for fourier analysis however this time-frequency plane makes no sense to be. It lacks the amplitude information. What is the way to understand and interpret this ...
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1answer
89 views

What does it mean to “simultaneously localize a function in time and frequency domain”?

What does the statement "simultaneously localize a function in time and frequency domain" mean when it comes to signals and systems? What does it mean to localize?
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1answer
79 views

What does $L^2$(R) mean?

I am reading about wavelets and it mentions something about "a function in $L^2(\mathbb{R})$". What does that even mean?
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40 views

Monte Carlo (or other) Approximation to Infinite Summations

This question is sort of paired with a similar post I wrote in Stack Overflow, but not a repeat because I am asking about using another mathematical method. StackOverflow I am trying to approximate a ...
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1answer
31 views

How does a wavelet help in compressing data

I have an understanding of how we carry out image compression by using DCT along with Huffman encoding. The next subject is wavelets. I understand that wavelets are small waves and there are different ...
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32 views

Haar wavelet transformation of a binary vector data over $GF(2)$

I am trying to perform Haar wavelet transformation on the following vector which is defined over $GF(2)$. $[1, 0, 1, 0, 1, 0, 1, 0]$ I am doing it as follows. $[1, 0, 1, 0, 1, 0, 1, 0]$ $\implies ...
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1answer
44 views

Wavelet transform and taking out of frequencies

We use a scaled wavelet and move it across the signal taking out frequencies so that they need not to be processed with a differently scaled wavelet. How does this show up in the math behind wavelet ...
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2answers
104 views

A polynomial that satisfies $x^pf(1-x) + (1-x)^pf(x) = 1$

The context of this question is the construction of the Daubechies wavelet. $f$ is a polynomial of degree $p-1$ which satisfies the equation: $$ x^pf(1-x) + (1-x)^pf(x) = 1 \tag{1} $$ Since $$ f(x)...
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32 views

Directly compute image gradient from discrete wavelet or curvelet coefficients?

We know that (1) the discrete wavelet or curvelet transformation can be used in data compression, and (2) the Sobel operator can be used to compute the gradient of a 2D array of data. In the case of ...
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25 views

For wavelets, is a multiresolution analysis an example of a filtration? If so, is it a filtration on the algebra of L^2?

For wavelets, is a multiresolution analysis an example of a filtration? If so, is it a filtration on the algebra of $L^2$? Even elementary clarifications of basic concepts on the filtration side, in ...
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27 views

Rate of convergence of a Weyl-Heisenberg (Gabor) frame expansion

If $\{g_{m,n}\}$ is a Gabor frame for $L^2(R)$, with window function $g$, and $f \in L^2(R)$, is there a bound on the approximation error of $f$ using a finite subset of the frame? That is, is ...
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193 views

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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93 views

Compare between Short Time Fourier Transform and Wavelets

Fourier transform is localised in only frequency domain but Short time Fourier transform(STFT) is localised both in time and frequency domain same as in wavelets. I want to know How are STFT and ...
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1answer
27 views

About window function

In Charles.K.Chui's An introduction to wavelets, on Page54, the window function is a non-trivial function $w∈L^{2}(R)$ satisfying $tw(t)∈L^{2}(R)$. I want to ask how to understand the notion, and how ...
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0answers
24 views

Lipschitz property and wavelets

I'm trying to understand Nason et al(2000)'s proof of Proposition 3.3 in this paper: http://www.maths.bris.ac.uk/~guy/Research/papers/WavProcEWS1.pdf The expectation of the uncorrected periodogram is ...
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1answer
17 views

Filter output of a signal

So I have a filter $$H(z) = 0.5 + 0.5z^3 = (1/2, 0, 0, 1/2)$$ and need to find the output of it on a cyclical signal $$x = (..., 3, -1, 2, 1, 5, 2, 3,-1, 2, 1, 5, 2, 3,...) $$ Would the output be ...
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How to express a signal in terms of Riesz bases?

Fast discrete wavelet transform allows us to express any discrete signal in terms of wavelet bases by convolution with filter coefficients. How can one express a digital signal in terms of ...