5 ms·
You get to do projections as in the Pythagorean theorem. The coefficients you need in the projections are just the values of some inner products. With random
by graycat 11y ago
You get to do projections
as in the Pythagorean theorem.
The coefficients you need in the
projections are just the
values of some inner products.
With random variables, those
coefficients are covariances,
that is, much the same as
correlations, that commonly
can estimate from data.
In the multivariate Gaussian
case, uncorrelated implies
independence.
Fourier theory is easier in
L2 than in L1. E.g., in
classic Fourier series, the
error in the approximation
is in L2 and is from the
L2 orthogonality of the
harmonics.
Yes, L-infinity can
also be nice: The
uniform limit of a sequence
of continuous functions
is continuous.
Or, with L2, often get a
Hilbert space but with
L1 or L-infinity usually
get at best just a Banach space --
that is, a complete, normed
vector space. Then, yes, can
get the Hahn-Banach theorem,
but the same thing in Hilbert
space is easier.
There is a sense in which L1 and
L-infinity are duals of
each other, but L2 is self-dual
which is nicer.
Filling in all these details and
more is part of functional analysis
101. There tough to miss at least
three books of W. Rudin:
Principles of Mathematical Analysis,
Real and Complex Analysis,
and Functional Analysis.
There's more, but I've
got some bugs to get out
of the software of my
Web pages!
I like the question -- asked
it myself at the NIST early in my career.
The answer I gave here is better
than what people told me then.
I've indicated likely most of the
main points, but my answer here is
rough and ready (I typed too fast),
and a quite polished answer is
also possible -- I just don't have
time today
to dig out my grad school
course notes, scan through Rudin,
Dunford and Schwartz,
Kolmogorov and Fomin, much of
digital filtering, much of
multi-variate statistics, etc.
- sukilot 11y agoThia sounds like lampposting -- closing a model because the math is nice, not because it is an accurate model.
- graycat 11y agoNo: If what you want is the L-infinity norm, then go for it. A standard place for that is numerical approximations of special functions -- want guarantees on the worst case error. And there is some math to help achieve that. It's sometimes called Chebyshev approximation. But, in practice, the usual situation, e.g., signal processing, multi-variate statistics, there's no good reason not to use L2 and many biggie reasons to use it. E.g., for a given box of data, commonly the better tools in L2 just let you do better. Or to the customer: "If you will go for a good L2 approximation, then we are in good shape. If you insist on L1 or L-infinity, then we will need a lot more data and still won't do as well.". Again, a biggie example is just classic Fourier series. Sure, if you are really concerned about the Gibbs phenomenon, then maybe work on that. Otherwise, L2 is the place to be. E.g., L1 and L-infinity can commonly take you into linear programming. Generally you will be much happier with the tools available to you in L2.
- graycat 11y agoAgain, just now I just don't have time for a more full, complete, and polished explanation. A really good explanation would require much of a good ugrad and Master's in math, with concentration on analysis and a wide range of applications. I've been there, done that but just don't have time to write out even a good summary of all that material here.
- selimthegrim 11y agoputting a shout in for Bollobas here, surely all roads to functional analysis don't lead through Rudin?
- graycat 11y agoRight. Also, say, Kolmogorov and Fomin. And Dunford and Schwartz. No doubt the full literature is enormous -- I don't know all of it! But Rudin is a good author, and as a writer got better, less severe in style and, thus, easier to read, over time in his career.
- tomrod 11y agoI intuit that you're getting at the real answer with the self-dual. That makes a lot of sense. Also, from a practical perspective L2 is very nice because it causes the problem of error reduction to be quadratic, so it scales well. Time to dig out Rudin.