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Representation theorem for wavelets

Started by Rune Allnor February 7, 2006
Hi all.

In Real Analysis there is a theorem (Weierstrass'?) that any
"reasonable"
function can be approximated to arbitrary presicion by means of any
linear basis for the function space, or

|f(x) - sum_n=0^\inf g_n(x)| < eps

where g_n(x) is the set of basis functions. This is "easy" to
see/derive
for basis functions like complex sinusiodals that have a finite
magnitude
for all x.

Is there a similar theorem for wavelets, that is, functions that have
vanising
magnitudes on large intervals of x?

Alternatively, how is this theorem formulated in terms of wavelets?
Any help and comments are welcome.

Rune

Rune Allnor wrote:
> Hi all. > > In Real Analysis there is a theorem (Weierstrass'?) that any > "reasonable" > function can be approximated to arbitrary presicion by means of any > linear basis for the function space, or > > |f(x) - sum_n=0^\inf g_n(x)| < eps > > where g_n(x) is the set of basis functions. This is "easy" to > see/derive > for basis functions like complex sinusiodals that have a finite > magnitude > for all x. > > Is there a similar theorem for wavelets, that is, functions that have > vanising > magnitudes on large intervals of x? > > Alternatively, how is this theorem formulated in terms of wavelets? > Any help and comments are welcome. > > Rune
Hmm. Well, I'd come at it differently. If wavelets (regardless of their shape) can be shown to be a basis for some space and if the function you're approximating is in that space, then precisely the same theorems that apply to wavelets as to other bases. But perhaps I've missed your question altogether. Cheers! --M
Sorry if this appears twice, the server disappeared while sending...

Rune Allnor wrote:

> In Real Analysis there is a theorem (Weierstrass'?) that any > "reasonable" function can be approximated to arbitrary presicion by > means of any linear basis for the function space, or > > |f(x) - sum_n=0^\inf g_n(x)| < eps > > where g_n(x) is the set of basis functions. This is "easy" to > see/derive for basis functions like complex sinusiodals that have > a finite magnitude for all x. > > Is there a similar theorem for wavelets, that is, functions that > have vanising magnitudes on large intervals of x? > > Alternatively, how is this theorem formulated in terms of > wavelets? Any help and comments are welcome.
I believe you want the term "multiresolution analysis". In short, this is a sequence of nested subspaces of L2(R) each of which is spanned by the translates of the scaling function, dilated to a particular scale, and whose union is L2(R) itself. So projecting a given function f onto successive subspaces in the multiresolution analysis gives successive approximations to f, and we recover f in the limit. For the proper definition, look at one of the top hits from: http://www.google.com/search?q=multiresolution+analysis+approximation+space Martin -- Quidquid latine scriptum sit, altum viditur.
Martin Eisenberg wrote:
> Sorry if this appears twice, the server disappeared while sending... > > Rune Allnor wrote: > > > In Real Analysis there is a theorem (Weierstrass'?) that any > > "reasonable" function can be approximated to arbitrary presicion by > > means of any linear basis for the function space, or > > > > |f(x) - sum_n=0^\inf g_n(x)| < eps > > > > where g_n(x) is the set of basis functions. This is "easy" to > > see/derive for basis functions like complex sinusiodals that have > > a finite magnitude for all x. > > > > Is there a similar theorem for wavelets, that is, functions that > > have vanising magnitudes on large intervals of x? > > > > Alternatively, how is this theorem formulated in terms of > > wavelets? Any help and comments are welcome. > > I believe you want the term "multiresolution analysis". In short, > this is a sequence of nested subspaces of L2(R) each of which is > spanned by the translates of the scaling function, dilated to a > particular scale, and whose union is L2(R) itself. So projecting a > given function f onto successive subspaces in the multiresolution > analysis gives successive approximations to f, and we recover f in > the limit. > > For the proper definition, look at one of the top hits from: > http://www.google.com/search?q=multiresolution+analysis+approximation+space
Thanks. This seems to be what I was after. Rune
Please send me representation theorem for wavelets
mlimber wrote:
> Rune Allnor wrote: > > Hi all. > > > > In Real Analysis there is a theorem (Weierstrass'?) that any > > "reasonable" > > function can be approximated to arbitrary presicion by means of any > > linear basis for the function space, or > > > > |f(x) - sum_n=0^\inf g_n(x)| < eps > > > > where g_n(x) is the set of basis functions. This is "easy" to > > see/derive > > for basis functions like complex sinusiodals that have a finite > > magnitude > > for all x. > > > > Is there a similar theorem for wavelets, that is, functions that have > > vanising > > magnitudes on large intervals of x? > > > > Alternatively, how is this theorem formulated in terms of wavelets? > > Any help and comments are welcome. > > > > Rune > > Hmm. Well, I'd come at it differently. If wavelets (regardless of their > shape) can be shown to be a basis for some space and if the function > you're approximating is in that space, then precisely the same theorems > that apply to wavelets as to other bases. But perhaps I've missed your > question altogether. > > Cheers! --M
Please send me representation theorem for wavelets
mlimber wrote:
> Rune Allnor wrote: > > Hi all. > > > > In Real Analysis there is a theorem (Weierstrass'?) that any > > "reasonable" > > function can be approximated to arbitrary presicion by means of any > > linear basis for the function space, or > > > > |f(x) - sum_n=0^\inf g_n(x)| < eps > > > > where g_n(x) is the set of basis functions. This is "easy" to > > see/derive > > for basis functions like complex sinusiodals that have a finite > > magnitude > > for all x. > > > > Is there a similar theorem for wavelets, that is, functions that have > > vanising > > magnitudes on large intervals of x? > > > > Alternatively, how is this theorem formulated in terms of wavelets? > > Any help and comments are welcome. > > > > Rune > > Hmm. Well, I'd come at it differently. If wavelets (regardless of their > shape) can be shown to be a basis for some space and if the function > you're approximating is in that space, then precisely the same theorems > that apply to wavelets as to other bases. But perhaps I've missed your > question altogether. > > Cheers! --M
alka_gudadhe@yahoo.com wrote:
> mlimber wrote: > > Rune Allnor wrote: > > > Hi all. > > > > > > In Real Analysis there is a theorem (Weierstrass'?) that any > > > "reasonable" > > > function can be approximated to arbitrary presicion by means of any > > > linear basis for the function space, or > > > > > > |f(x) - sum_n=0^\inf g_n(x)| < eps > > > > > > where g_n(x) is the set of basis functions. This is "easy" to > > > see/derive > > > for basis functions like complex sinusiodals that have a finite > > > magnitude > > > for all x. > > > > > > Is there a similar theorem for wavelets, that is, functions that have > > > vanising > > > magnitudes on large intervals of x? > > > > > > Alternatively, how is this theorem formulated in terms of wavelets? > > > Any help and comments are welcome. > > > > > > Rune > > > > Hmm. Well, I'd come at it differently. If wavelets (regardless of their > > shape) can be shown to be a basis for some space and if the function > > you're approximating is in that space, then precisely the same theorems > > that apply to wavelets as to other bases. But perhaps I've missed your > > question altogether. > > > > Cheers! --M > > Please send me representation theorem for wavelets
Please see the rest of the thread. If you don't understand something about it, ask a specific question. Cheers! --M