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3
votes
2answers
830 views

Need a formula for a quadratic spline

I'm trying to reproduce some results from a paper and I need an explicit formula for a specific quadratic spline to do so. The problem is, I've only got a plot of it. The quadratic spline is from ...
6
votes
0answers
250 views

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 ...
2
votes
0answers
66 views

Lower bound for the eigenvalue

For a given real number $c>0$ define functions $\left(\psi_{k,c}(\cdot)\right)_{k\ge0}$, as an eigenfunctions of the Sturm-Liouville operators $L_c$ defined $$ ...
2
votes
0answers
165 views

Hermite functions and integral

Let $$ h_n(x)=(-1)^n\gamma_ne^{x^2/2} \frac{d^n}{dx^n}e^{-x^2}, $$ where $\gamma_n=\pi^{-1/4}2^{-n/2}(n!)^{-1/2}$, be Hermite function. Consider $$ ...
1
vote
0answers
77 views

$n$-th derivative of the prolate spheroidal function

For a given real number $c>0$ define functions $\left(\psi_{k,c}(\cdot)\right)_{k\ge0}$, as an eigenfunctions of the Sturm-Liouville operators $L_c$ defined $$ ...
2
votes
0answers
222 views

$L_2$-norm representation of the function

Let $$ f^{\alpha}_+(x)=\frac{1}{\Gamma(\alpha+1)}\sum_{k\ge 0}(-1)^k{\alpha+1 \choose k}(x-k)^{\alpha}_+, $$ where $\alpha > -\frac 12$(see for reference ...
0
votes
2answers
3k views

DFT and DWT difference?

what is the basic difference between the Discrete Fourier Transform and the Wavelet Transform ? and why does JPEG2000 preferred DWT over DCT or DFT ?
12
votes
2answers
7k 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 ...
2
votes
1answer
153 views

How do I divide Laurent polynomials?

I have an example from a paper (listed below) that I cannot figure out. I can divide normal polynomials, but the alternative ways to divide Laurent polynomials is beyond me at the moment. The paper ...
1
vote
1answer
511 views

Extrapolating signals using wavelets

I am an absolute beginner to wavelets, and I've read a few articles on how wavelets are used for predicting future points of a dataset, notably Wavelet prediction for Oil Prices and 1D Signal ...
0
votes
1answer
238 views

Why is the dot product of 2 wavelet domain functions a real value?

I'm working on some code here, and here is what I have done. It is based on the work by Ng et al. An example of what this looks like is here. Background: Here, a "lighting cubemap" is a bunch of ...
1
vote
1answer
133 views

What other wavelets (besides the Haar system) form a basis of $L^2(0,1)$?

The Haar system of wavelets forms a basis of $L^2[0,1]$. What other wavelets are there that also form bases of $L^2[0,1]$ (or $L^2[0,a]$ in general)?. Thanks
1
vote
0answers
109 views

Is there a wavelet frame for $L^2[0,\infty)$?

What systems of wavelets provide a frame for $L^2[0,\infty)$. For example, the Haar system of wavelets provides a basis for $L^2[0,1]$, and the harmonic wavelets provide a basis for all of ...
2
votes
1answer
155 views

Best way to find magnitude and phase of a specific frequency in an empirical time series…

I've a discrete, univariate time series, and I'm interested in to investigate a specific frequency component. Assume I'm interested in a frequency with a cycle-time of $f$ samples - and I need to get ...
2
votes
0answers
163 views

Singular Value Decomposition

I want to decompose an image $A$ using the Discrete Wavelet Transform and then find the singular values, $S$, such that $A=USV$. I will then do the same to another image such that $B=USV$. I will ...
1
vote
1answer
266 views

What is the “fourth” Haar basis function of the 4×4 Haar basis?

I know the three Haar basis functions //mother wavelet: [1 1 -1 -1] //dilation: [1 -1 0 0] //translation: [0 0 1 -1] However, the Haar basis consists of four ...
5
votes
1answer
569 views

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 ...
4
votes
1answer
165 views

Question in solving $\phi(t)=\phi(2t)+\phi(2t-1)$, $\phi\ne0$

Actually one can resort to the two-scale equation in multiresolution analysis. Perform Fourier transformation on both side of $\phi(t)=\phi(2t)+\phi(2t-1)$, it turns out that ...
0
votes
1answer
100 views

How to calculate dual frames under constraints?

Denote orthonormal basis in $\mathbb{R}^2$: $(\epsilon_1,\epsilon_2)=\begin{pmatrix}1&0\\0&1\end{pmatrix}$ and ...
1
vote
2answers
561 views

How to implement the Daubechies wavelet?

http://en.wikipedia.org/wiki/Daubechies_wavelet#Transform.2C_D4 I find it is difficult to understand the pseudo-code on this Wiki page. ...
1
vote
0answers
545 views

Mathematica: How to convert scales to frequencies?

According to the transform $$w(u,s)=\frac{1}{\sqrt{s}}\int _{-\infty }^{\infty }x(t) \psi ^*\left(\frac{t-u}{s}\right)dt,$$ the frequency should be $f=\omega/(2\pi)=1/(2\pi s)$ (is it right?), where ...
7
votes
3answers
731 views

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 ...
6
votes
3answers
572 views

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 ...
5
votes
2answers
556 views

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 ...