Articles of wavelets

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 $f\in H$. But my question is: is this an equivalent definition for if $\sum_{k=1}^{\infty}|\langle f,f_{k}\rangle|^{2}<\infty$ for all $f\in H$, then $\{f_{k}\}_{k=1}^{\infty}$ is a Bessel sequence. If yes, how to show that […]

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 given a certain finite Borel measure $\mu$ on $\mathbb{R}^d$ and we wish to determine whether or not it is absolutely continuous with respect to Lebesgue measure. The […]

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 (rectangular) function $r(x)$ $$r(x)=\begin{cases} 1 & \mbox{if }0\leq x \leq 1; \\ 0 & \mbox{elsewhere} \end{cases}$$ I would like to find the coefficients […]

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 building wavelets which have compact support (localization in space), which are smooth (decay towards high frequencies), and which have _vanishing moments_ (decay towards low […]

Construction of Brownian Motion using Haar wavelets

I want to construct brownian motion using Haar Wavelets where $\varphi_n, n \in \mathbb{N}$ is an orthonormal basis of $L^2[0,1]$. I take the inner product: $$\langle f,g \rangle = \int_0^1 f(t)g(t)dt$$ Now I take $$\Phi_n(t) = \int_0^t \varphi_n(s) ds$$ This is obviously well defined since $\varphi_n \in L^2[0,1] \subset L[0,1]$. Now I take a series […]

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 the standard Fourier transform is only localized in frequency. I did not understand what is meant here by “localized in time and frequency.” Can someone please […]

Potential uses for viewing discrete wavelets constructed by filter banks as hierarchical random walks.

I have some weak memory that some sources I have encountered a long time ago make some connection between random walks and wavelets, but I am quite sure it is not in the same sense. What I was thinking is to in the matrices representing filtering operations replace every $a\in \mathbb{R}$: $$\cases {\phantom{-}a\to \begin{bmatrix}a&0\\0&a\end{bmatrix}\\-a\to \begin{bmatrix}0&a\\a&0\end{bmatrix}}$$ […]