Orthogonality

[marble sedge]

I have had a class about orthogonality of functions ( inner product of two functions and stuff) and I haven’t really understood the meaning of the following definition:

f(t) and g(t) are orthogonal in [a,b] with respect to the weight function w(t) <=> <f,g>w = 0 and <f,f>≠0 and <g,g>≠0

Could someone help me understand this definition?

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[mild pier]

Think of functions as elements of a vector space. So, vectors.
Two vectors are orthogonal when their dot (or, more generally, inner) product is 0.

The latter conditions in the definition just ensure that the function isn't 0 almost anywhere (so, not a "zero" vector).

[marble sedge]

But how can functions be orthogonal?

Or isn’t it anymore the same as perpendicular?

[mild pier]

Don't think of it in terms of geometry.
Orthogonality is defined by two non-zero vectors (as in, generalized vectors - members of an arbitrary vector space) having a zero inner product.

[mild pier]

Yeah, no geometric meaning there.

[marble sedge]

Okay, so now the meaning of orthogonality of functions is basically if the dot product of the two is 0, which you can compute with the integral

[mild pier]

Yeah.

[marble sedge]

And what does the weight function specifically mean

[mild pier]

You know how when dealing with usual vectors, you can define inner products differently using a matrix?
As in, if A is a positive definite matrix, then <u, v> = u^T Av is a valid dot product.

It's kiiinda the same idea.

Though, in case of functions, a weight function is not just something you can add, but something necessary, rather.

Can't really explain why exactly there is a distiction like that, sorry. I haven't gone deep enough into the subject, as I'm not a math student.

[boreal pike]

The best question I can ask you here is, how is your inner product defined?

[marble sedge]

Same here though, I was just wondering because our prof just instantly gave the definition with the weight function without really explaining what it is or what it does

[boreal pike]

I would assume something along the lines of:
$$\forall f, g \in \mathcal{C}([a, b]), \langle f, g \rangle = \int_{a}^{b} f(t)g(t)dt$$

[thorny pewter]

Rion

[marble sedge]

For the inner product of two functions?

[boreal pike]

Yeah

[marble sedge]

This was the definition he gave:
given two function g(t) and f(t) in [a,b] that is an element of all real numbers. The scalar product of f and g with respect to the weight function w(t) is :

$$<f,g>=\int_{a}^b f(t) g(t) w(t) dt$$

[thorny pewter]

Reyess

[marble sedge]

but g(t) is conjugate complex

I didn’t know how to do the little bar over g(t)

[boreal pike]

Ah, so they're continuous functions from [a, b] to C

Gotcha

So that's a hilbert space

$$\forall f, g \in \mathcal{C}([a, b], \bC), \langle f, g \rangle_w = \int_{a}^{b} f(t) \overline{g(t)} w(t) dt$$
Where $w : [a, b] \to \bR_+$ is nonzero

[thorny pewter]

Rion

[boreal pike]

The weighting function here just says that you'll define a different notion of orthogonality with every weighting function

[marble sedge]

yeah that one

[boreal pike]

basically, when you have a C-vector space, you don't necessarily have notions of geometry like a norm or right angles

but when you endow your C-vector space with an inner product, you get this missing part

so functions are vectors, and this inner product will basically tell you how large your vectors are and how they are positioned relatively to each other

but the inner product depends on w, and you'll get a different one for different w's

so two w's will define two different notions of orthogonality and distances

it's similar to what Darp was talking about earlier, but with R^n

[marble sedge]

Ah okay yeah that makes a bit more sense

[boreal pike]

as in, <x, y>_A = <x, Ay> when A is psd

so here you just define a different notion of orthogonality

than the standard one on R^n

You can note that the map $A : f \mapsto wf$ is also psd

[marble sedge]

Ah I see, ty. Sometimes it’s just hard for me to understand these things without them telling what it does

[thorny pewter]

Rion

[marble sedge]

Also sorry but what did you mean with psd?

I am not familiar with all the terms in english

[boreal pike]

positive semi definite

it's also symmetric I guess

$\langle f, Ag \rangle = \int_{a}^{b} f\overline{wg} = \int_{a}^{b} fw\overline{g} = \langle Af, g \rangle$

(since $w$ is real-valued)

[thorny pewter]

Rion

[marble sedge]

What does this tell me? That it is the same if I take the dot product of Af with g and f with Ag?

[boreal pike]

What it tells you is that this behaves similarly to symmetric matrices in R^n

just to make the link with what Darp was explaining earlier

But more importantly: $\langle f, Af \rangle \in \bR_+$

[thorny pewter]

Rion

[marble sedge]

Ah I see, yeah

Also have you heard about this formula, because I don’t really get what it is saying

well mean quadratic error or something

what is the difference between f and f_i?

because we defined ci as <f,fi>/(norm(f))^2

[boreal pike]

No idea, I can check

$\lVert f - f_i \rVert^2 = \int_{a}^{b} \left( f(x) - f_i(x)\right) \overline{\left( f(x) - f_i(x)\right)}dx$

[thorny pewter]

Rion

[boreal pike]

So if you expand that

you will get something like $\lVert f \rVert^2 + \lVert f_i \rVert^2 - \langle f, f_i \rangle - \langle f_i, f \rangle$

[thorny pewter]

Rion

[boreal pike]

but I don't think this is what you are looking for

[marble sedge]

No I don’t think so

I can show you what they did to prove it

[boreal pike]

I mean first of all, what are you computing?

[marble sedge]

well to be honest, I am just kind of trying to understand what they were explaining in the class. He proved the formula but didn’t give a lot more info about it

so it’s more or less some error you’re trying to find ig

but I am not sure myself

[boreal pike]

probably would be good to make sure

[marble sedge]

Yeah that’s why I was wondering what the terms here mean

but yeah I have to maybe look somewhere

[midnight zenith]

@marble sedge

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[midnight zenith]

@boreal pike @mild pier The user still needs help with this help request.

[marble sedge]

I had the following question still, let’s say they give me some sequence and tell me that those are orthogonal functions in the interval 0,2pi. And I have to show that

How do you know which dot product you have to take?

[boreal pike]

you tell me man

which product did you endow C([0, 2pi], K) with?