Szemerédi regularity lemma

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A multiresolution analysis (MRA) or multiscale approximation (MSA) is the design method of most of the practically relevant discrete wavelet transforms (DWT) and the justification for the algorithm of the fast wavelet transform (FWT). It was introduced in this context in 1988/89 by Stephane Mallat and Yves Meyer and has predecessors in the microlocal analysis in the theory of differential equations (the ironing method) and the pyramid methods of image processing as introduced in 1981/83 by Peter J. Burt, Edward H. Adelson and James Crowley.

Definition

A multiresolution analysis of the space L2() consists of a sequence of nested subspaces

{0}V0V1VnVn+1L2()

that satisfies certain self-similarity relations in time/space and scale/frequency, as well as completeness and regularity relations.

  • Completeness demands that those nested subspaces fill the whole space, i.e., their union should be dense in L2(), and that they are not too redundant, i.e., their intersection should only contain the zero element.

Important conclusions

In the case of one continuous (or at least with bounded variation) compactly supported scaling function with orthogonal shifts, one may make a number of deductions. The proof of existence of this class of functions is due to Ingrid Daubechies.

Assuming the scaling function has compact support, then V0V1 implies that there is a finite sequence of coefficients ak=2ϕ(x),ϕ(2xk) for |k|N, and ak=0 for |k|>N, such that

ϕ(x)=k=NNakϕ(2xk).

Defining another function, known as mother wavelet or just the wavelet

ψ(x):=k=NN(1)ka1kϕ(2xk).

One can show that the space W0V1, which is defined as the (closed) linear hull of the mother wavelet's integer shifts, is the orthogonal complement to V0 inside V1.Template:Cn Or put differently, V1 is the orthogonal sum (denoted by ) of W0 and V0. By self-similarity, there are scaled versions Wk of W0 and by completeness one hasTemplate:Cn

L2()=closure of kWk,

thus the set

{ψk,n(x)=2kψ(2kxn):k,n}

is a countable complete orthonormal wavelet basis in L2().

See also

Template:Inline

References

  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

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  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

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