1
vote
0answers
11 views

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 ...
0
votes
1answer
27 views

Confusion about space and multiresolution

I don't have a background in functional analysis so I find this hard to understand. How could it possible that each "dyadic" dilation of a function f(x) (where the dilation is f(2^i (x))) forms a ...
1
vote
1answer
71 views

What are unconditional bases and which wavelets have this property?

What are unconditional bases and which wavelets have this property? Haar wavelet seems to be one but how common is it? Unconditional bases are related to Riesz basis condition (ordering of the terms ...
0
votes
1answer
48 views

How are generalized frames related to biorthogonal bases?

How are generalized frames related to biorthogonal bases? It seems like frames are a possible solution if neither orthonormal nor biorthogonal bases are available. I thought the generalized frames ...
0
votes
1answer
35 views

Bessel Sequence proof check.

I have a similar question to definition of Bessel sequence, where it was solved using Banach-Steinhaus. A sequence $\{f_k\}_{k=1}^{\infty}$ is called a Bessel sequence in a Hilbert space ...
3
votes
0answers
75 views

Regularity of Daubechies wavelet

I am reading the book Wavelets: Theory and applications by A. K. Louis, D. Maass, A. Rieder ...
3
votes
1answer
60 views

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 ...
1
vote
3answers
70 views

Reducing an infinite dimensional expansion to a finite dimensional algorithm

The following is a big-picture question about the interplay between infinite dimensional function expansions and finite dimensional algorithms. I feel like I have a good understanding of these ideas ...
2
votes
0answers
219 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 ...