Invisible wrote:
> Can somebody who knows what they're talking about confirm this?

	That probably shouldn't include me, but I'll try.

> If I'm understanding this right, if I can find a complete set of
> orthogonal functions, I should be able to construct any possible
> function as a linear combination of them.

	By definition, you can construct any possible piecewise continuous
function with them: In the domain that they are complete (which may only
be an interval on the real line).

See http://mathworld.wolfram.com/CompleteOrthogonalSystem.html

	Now when they say that the limit of the error goes to 0, they mean in a
sort of "average" sense (see the Lebesgue integral?).

	More concretely, this means that if you have your (usually infinite)
set of complete orthogonal functions, and a given function f, you can
construct something that's "almost" identical (if not identical) to f
using a linear combination of the orthogonal functions - BUT it may
disagree with f on a number of points (which constitutes a set of
measure 0).

	For a concrete example, take your sines and cosines that you mentioned.
It can (easily) be shown that any linear combination of them (even an
infinite one), must be continuous. Hence, you cannot use them to
construct discontinuous functions (like a square wave) *exactly*.
However, you can construct something that looks a lot like it but
differs at only a "few" points. The square wave and your constructed
function will disagree at the points of discontinuity.

	See Gibb's Phemonenon:

http://en.wikipedia.org/wiki/Gibbs_phenomenon

	The "overshoot" at the point of discontinuity for a square wave is 9%
even when you take the infinite limit (the Wikipedia article, I believe,
is incorrect in claiming it goes away in that limit).

	The reason sin and cos are labeled as "complete" even though they can't
replicate a simple square wave is that in the infinite limit, the sum is
identical to the square wave everywhere except at a finite number of
points.

> If I'm not mistaken, "orthogonal" means that one function can't be
> constructed from the others (so there's no duplication of
	
	Not quite. You can have a function that cannot be obtained from the
others, but not be orthogonal to them.

	In function spaces, a scalar (or inner) product is defined, akin to the
one that you get with vectors in space. You take two functions,
calculate their scalar product, and get a scalar. Two functions are
orthogonal if their scalar product is 0.

	The analogy with vectors is a good one. Think in 3-D space: You can
easily get unit vectors from which you can construct any 3 dimensional
vector, but with none of these basis vectors being orthogonal. An
example would be:

A=[1 1 0], B=[0 1 1], C = [1 0 1]

(all divided by sqrt(2) for normalization). One wants orthogonality in
functions for the same reason you do with vectors: Computational
convenience.

	There are a number of ways to define a scalar product of 2 functions,
but they usually have some properties:

http://mathworld.wolfram.com/InnerProduct.html

(For function spaces, things are often complex (vs real), and so you use
the "alternate" rule given further down the page in place of rule 3).
For systems I occasionally deal with, the inner product of 2 functions
is given by:

(f, g) = Integral(f*g') where g' is complex conjugation. Occasionally a
weighting function is included as well.

> So, like, how do you tell if two functions are orthogonal? And how do
> you tell when a set of them is complete?

	To test for orthogonality, compute the scalar product. OR decompose it
into its components with respect to a known basis (e.g. sines and
cosines) and then take the inner product of those (hard to describe
here, but if you can do the decomposition, the inner product is trivial
as you know that the sines and cosines are orthogonal).

	Forgot how to test for completeness...

	I may not have described things well here, but once you "get" it, it's
quite illuminating. Basically an extension of how you dealt with vectors
in space. All concepts there have analogs in function spaces.

-- 
ASCII stupid question... get a stupid ANSI!


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