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Wavelets in vector space without Cauchy-Schwartz inequality

Started by Edward Jensen September 4, 2009
Dear all.

I am studying wavelets and multi-resolution analysis (MRA) at the moment. 
One of the key problems with time-frequency analysis in general is the 
uncertainty principle with states that the localization in time and in 
frequency has a lower bound (called Heisenberg boxes in signal processing). 
My book presents MRA in a Hilbert space (L^2([0;1])) and proves the 
uncertainty relation by means of the Cauchy-Schwartz inequality. This made 
me think whether it is possible to (theoretically) develope a time-frequency 
analysis method (such as wavelets) in a vector space that doesn't support 
the Cauchy-Schwartz inequality in order to avoid this time-frequency 
resolution bound.

It should be noted that I'm an engineering student specializing in signal 
processing and I haven't had a rigorous education in functional analysis.

My train of though is as follows (and please correct me if I'm wrong):
*) The uncertainty relation is based on the Cauchy-Schwartz inequality
*) Without the Cauchy-Schwartz inequality the uncertainty relation does not 
hold
*) The Cauchy-Schwartz inequality requires an inner product.space

So my question is whether it is possible to develope time-frequency analysis 
(theoretically) in a vector space which it not an inner product space, for 
example in a Banach space where there the norm is not complete with respect 
to the metric.

Does this make sense at all?

Thanks in advance,
Edward

Ps. Cross-posted to sci.math and followup to comp.dsp


On 5 Sep, 00:17, "Edward Jensen" <edw...@jensen.invalid> wrote:
> Dear all. > > I am studying wavelets and multi-resolution analysis (MRA) at the moment. > One of the key problems with time-frequency analysis in general is the > uncertainty principle with states that the localization in time and in > frequency has a lower bound (called Heisenberg boxes in signal processing). > My book presents MRA in a Hilbert space (L^2([0;1])) and proves the > uncertainty relation by means of the Cauchy-Schwartz inequality. This made > me think whether it is possible to (theoretically) develope a time-frequency > analysis method (such as wavelets) in a vector space that doesn't support > the Cauchy-Schwartz inequality in order to avoid this time-frequency > resolution bound.
Nope. The Heissenberg Inequality is a measure for how 'far' away two vectors need to be, before the become orthogonal. With the complex exponentials in the N-pt DFT, vectors with normalized frequencies that are 2*pi*k/N apart are orthogonal. With wavelets, the intuition isn't quite as clear, but the mathemathics is the same.
> It should be noted that I'm an engineering student specializing in signal > processing and I haven't had a rigorous education in functional analysis. > > My train of though is as follows (and please correct me if I'm wrong): > *) The uncertainty relation is based on the Cauchy-Schwartz inequality > *) Without the Cauchy-Schwartz inequality the uncertainty relation does not > hold > *) The Cauchy-Schwartz inequality requires an inner product.space > > So my question is whether it is possible to develope time-frequency analysis > (theoretically) in a vector space which it not an inner product space, for > example in a Banach space where there the norm is not complete with respect > to the metric.
What DSP is concerned, it's the inner product that is the reason why one wants to do the transform in the first place. The transforms, be it wavelets or Fourier transforms, are done because one for some reason or another finds that the transformed *representation* of the signal is somehow easier to deal with than the original *representation* of the same signal. Since the original and transformed *representations* of the signals are formally equal, the key is that one wants a 1:1 relation between the transformed signal and the signal in original domain. The inner product provides that 1:1 relation. Rune
On Sat, 5 Sep 2009 00:17:15 +0200, "Edward Jensen"
<edward@jensen.invalid> wrote:

>Dear all. > >I am studying wavelets and multi-resolution analysis (MRA) at the moment. >One of the key problems with time-frequency analysis in general is the >uncertainty principle with states that the localization in time and in >frequency has a lower bound (called Heisenberg boxes in signal processing). >My book presents MRA in a Hilbert space (L^2([0;1])) and proves the >uncertainty relation by means of the Cauchy-Schwartz inequality. This made >me think whether it is possible to (theoretically) develope a time-frequency >analysis method (such as wavelets) in a vector space that doesn't support >the Cauchy-Schwartz inequality in order to avoid this time-frequency >resolution bound. > >It should be noted that I'm an engineering student specializing in signal >processing and I haven't had a rigorous education in functional analysis. > >My train of though is as follows (and please correct me if I'm wrong): >*) The uncertainty relation is based on the Cauchy-Schwartz inequality >*) Without the Cauchy-Schwartz inequality the uncertainty relation does not >hold >*) The Cauchy-Schwartz inequality requires an inner product.space > >So my question is whether it is possible to develope time-frequency analysis >(theoretically) in a vector space which it not an inner product space, for >example in a Banach space where there the norm is not complete with respect >to the metric.
Detail: Huh? First, there's no such thing as an incomplete Banach space. But I know what you meant: A normed vector space that's not complete. But again, huh? Completeness has more or less nothing to do with being an inner-product space. An "incomplete Banach space" can certainly be an inner-product space. Never mind, on to your main question:
>Does this make sense at all?
Not much that I can see. The uncertainty principle is a _fact_. Avoiding something that's needed in the proof might allow a person to remain ignorant of this fact, but it can't make the fact false - the uncertainty principle remains true whether you're able to prove it or not.
>Thanks in advance, >Edward > >Ps. Cross-posted to sci.math and followup to comp.dsp >
David C. Ullrich "Understanding Godel isn't about following his formal proof. That would make a mockery of everything Godel was up to." (John Jones, "My talk about Godel to the post-grads." in sci.logic.)