Composition of functions and operator algebra.

[tony42]

Hello.

I would like to turn an equation (function) like 4x²+2 into an operator, L, acting on x. Lx.

I want to get into operator algebra so that I can manipulate equations like √(4x²+2) using operators (e.g. the last equation could be RLx).

I'm finding difficulties:-

1. Many common functions/equations (like the above) add an integer at the end. I don't really know what I'm talking about but aren't these some kind of affine operation, making them nonlinear. And how would you go about turning functions into operators anyway?

2. Everytime I try to look up operator algebra on the web I am met with very advanced double dutch on banach spaces, operator spectra and such like.

3. No-one seems to be talking about manipulating collections of operators using commutativeness or other techniques and how to set those operators up. I think you can understand what I'm trying to get at. If I did succeed in turning function f and function g into two operators F and G. I could look at FG and, say, find that they commuted to make GF-x, then use that identity to simplify function compositions. This I what I've heard about operators; you can manipulate them without even worrying about the x or whatever they are operating upon.

I really hope someone out there can shed some light on this or point me in the right direction or even just make some suggestions for further researches.

Thanks for your time reading this. Cheers.

 

[Deveno]

there is a natural algebraic structure for functions f:S-->S, which is monoidal (one speaks of the monoid of transformations of S). one common way to think of this monoid is that S is a collection of "states" and the individual operators represent "processes", so that a product represents, in some sense, an "evolution" of the states.

this is done (for example) in computer programming, where the S is some data structure, and the f,g (etc.) represent individual programs that manipulate the data structure, so that fg means: run g, then run f.

the subcollection of "reversible processes" form a group, and this essentially is what permutation groups are (and permutation groups arise quite generally, in any kind of system that possesses symmetry).

for a collection of functions to possesses "more" properties, you usually desire a second operation (like +) that is "compatible" with composition. a common compatiblity condition is that left- and right-compostion be "+morphisms":

f(g+h) = fg + fh
(g+h)f = gf + hf.

here, you can see why affine functions and non-linear functions run into trouble: suppose S has an additive structure, and that f(x) = Ax + b (whatever "Ax" might mean).

then f(g+h) = A(g+h) + b, whereas:

fg + fh = Ag + b + Ah + b = Ag + Ah + b+b.

equating the two means that we have to have A(g+h) = Ag + Ah, and b = b+b.

so "A" has to be an additive homomorphism, and b = 0, which pretty much kills the hope of creating an affine function composition structure compatible with +.

simliarly suppose f(x) = x².

then f(g+h) = (g+h)² = (g+h)(g+h)

whereas fg + fh = g²+h² = gg + hh.

even if we have distributivity, so that (g+h)(g+h) = gg + gh + hg + hh.

the two still won't be equal unless gh + hg = 0, which rather limits the kinds of functions we can consider.

linearity "fixes" these problems, we don't have to worry about "cross-terms". nevertheless, thare are still quite broad classes of functions that can be manipulated algebraically without reference to the "bound variable" x (whatever kind of thing "x" might be). polynomials are one example- where, for example, the polynomial can be represented purely in terms of its coefficients, instead of writing p(x) = x²+x-2, we consider the vector (1,1,-2) which "captures" everything we need to know about p.

another example is rational functions in x, where we can consider expressions like:

(f²g - 3fh)/(2f - 3g)² and manipulate these like "ordinary fractions".

a lot of trigonometric function manipulation falls into this same category, the variable θ is often "just along for the ride", we are just interested in 2 functions s and c where s²+c² is the constant function 1 (although it "looks strange" to think of sin(2θ) as s°(2_) ).

if you do find some simple rules, for even limited sets of functions, that make calculations simpler, and easier to understand, by considering functions as "operators", by all means, knock yourself out.

 

[tony42]

That's sort of interesting, but I really need how to find out about this at my level. Any suggestions for books or websites?

 

[Deveno]

what is "your level"?

have you studied linear algebra yet?

 

[tony42]

My level? Well I have A-level Maths and A-level Further Maths. I'm an interested party really. I've done introductory stuff on complex numbers and quantum mechanics. I really hope to use the operators as a way of finding short-cuts in maths.