# Questions tagged [wavelets]

For questions related to wavelets and wavelet theory.

342 questions
Filter by
Sorted by
Tagged with
43 views

### 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 ...
1 vote
71 views

8 views

### 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 ...
• 7,534
29 views

### 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 ...
29 views

### 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 ...
94 views

### 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 ...
34 views

136 views

### 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 ...
• 53
151 views

29 views

### 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 ...
• 619
57 views

• 1,364
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 ...
• 4,359
32 views

### 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 ...
1 vote
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: ...
• 7,880
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 ...
• 2,973
43 views

### 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 ...
• 3,574
79 views

### 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 ...