Undergrad Orthogonal Basis of Periodic Functions: Beyond Sines and Cosines

[QuantumCuriosity42]

https://en.wikipedia.org/wiki/Legendre_polynomials would give you an indroductory ideas.

But that completeness is only proved on [-1, 1] interval, I don't know if it is valid to extend it to (-inf, +inf), and I don't think it is valid to repeat that polinomial more times to fill the (-inf, +inf) interval. And even if we did that, then how do you change the frequency of the extended Legendre polynomial to the full interval (-inf, +inf)?
To decompose a function in terms of fundamental frequencies you need to prove ortogonality between all versions of the same function but each time with a different frequency. For example, cos(mx) is ortogonal with cos(nx), for all values of n. That is why it forms a basis that can be used in the Fourier transform.

 

[anuttarasammyak]

For any priodic function of
f(x)=f(x+L)
we can concentrate our investigation on [0, L] and forget outside because they behave same with inside. The period [-1,1] for Legendre polynomials is easily transformed to cover [0, L].

Why you say$(-\infty,+\infty )$? Function of $L=\infty$ is not periodic.

 

[QuantumCuriosity42]

For any priodic function of
f(x)=f(x+L)
we can concentrate our investigation on [0, L] and forget outside because they behave the same with inside.
The period [-1,1] for Legendre polynomials to to cover [0, L].
Why you say (-\infty,+infty )? function of $L=\infty$ is no more periodic.

Ah I see, I didn't know that.
I didn't mean to say L=inf, I meant, a Legendre polynomial only is defined on the interval [-1, 1], while cosine function is defined in (-inf, +inf), and we can use any frequency like cos(kx), k being the frequency.
To do the equivalent thing with Legendre poynomials, we should choose only one Legendre polynomial, extend it to (-inf, +inf), and having a way of changing its frequency, how could we do that? Is it even possible?

All of this with the intention of finding another infinite basis of ortogonal and periodic functions, each of a different frequency, like cos(nx) is ortogonal with cos(mx). To see if there are more or the only one is the one formed by infinites cosines of different frequencies (or sines, but they are the same). These are called the "fundamental frequencies". I want to know if they are called fundamental because they are convenient to work with, or really because they are the only basis possible that is ortogonal for all possible versions of the function with different frequencies.

 

[QuantumCuriosity42]

In fact, now that I think about it, Legendre polynomial ortogonality is proved between all Legendre polynomials. But I don't think it could be proved in that way I say after choosing one, and proving its ortogonality with different versions of itself, each with a different frequency. In order to have another suitable basis of fundamental frequencies to have a similar "Fourier transform", but with other basis of fundamental frequencies which aren't those of sines.

 

[anuttarasammyak]

To do the equivalent thing with Legendre poynomials, we should choose only one Legendre polynomial, extend it to (-inf, +inf), and having a way of changing its frequency, how could we do that? Is it even possible?

I have already shown

To make it periodic is not difficult. For an example new Pn(x) would be
Pn(2(x+12−[x+12])−1)
so that it is periodic outside of [-1,1], where [ ] is floor function.

which is periodic with period L=2 and is defined in $(-\infty,+\infty)$.
I have no idea of "frequency" you say.

 

[QuantumCuriosity42]

I have already shown

which is periodic with period L=2 and is defined in $(-\infty,+\infty)$.

What is n, and what is x there? Can you explain what that expression shows?
I don't see a way to change its frequency.

 

[anuttarasammyak]

What is n, and what is x there? Can you explain what that expression shows?
I don't see a way to change its frequency.

n is number attached to Legendre polunomials

Graph of the variable of polynomials

So the graph of Legendre Polynomial for n=7 is

You see it periodic in all the region.

 

[QuantumCuriosity42]

n is number attached to Legendre polunomials

Graph of the variable of polynomials
View attachment 334829

So the graph of Legendre Polynomial for n=7 is

View attachment 334830

You see it periodic in all the region.

Ah I see now, thanks. But I still can't see how you could change its frequency with an argument of the function, like cos(kx), being k the frequency.

 

[anuttarasammyak]

Ah I see now, thanks. But I still can't see how you could change its frequency

I have no idea of "frequency" you say.

 

[QuantumCuriosity42]

@anuttarasammyak What I mean is that, we can change the frequency of a cosine ( cos(k*x) ) just by changing its parameter k (it is its frequency), cos(m*x) is ortogonal with cos(n*x), for all m not equal to n.
That is why we can use them as a basis formed by infinite cosines each with a different frequency in the Fourier transform.
What I asked was how can we decompose all possible functions with another basis of periodic functions which are not cosines. And you proposed to use a Legendre polynomial, with that extension to (-inf, +inf), but we lack the ability to change its frequency, like we do for cos(k*x). And for them to form a basis it would also be needed to prove the ortogonality between that extendend polynomial and all possible versions of it with different frequencies.
My question is that I think there is no way to change the frequency of a Legendre polynomial (or the extended one you proposed), like we do for cos(k*x), where k is the frequency and we can change it.

 

[anuttarasammyak]

My question is that I think there is no way to change the frequency of a Legendre polynomial (or the extended one you proposed), like we do for cos(k*x), where k is the frequency and we can change it.

Fourier series, in words of https://en.wikipedia.org/wiki/Fourier_series , has discrete set of frequencies or more precisely wave number of
\{2\pi n /P\}
Their numbers are countable infinite as well as n of Legendre polynomials. What' the difference you feel ?

 

[QuantumCuriosity42]

Fourier series, in words of https://en.wikipedia.org/wiki/Fourier_series , has discrete set of frequencies or more precisely wave number of
{2\pi n /P}
Their numbers are countable infinite as well as n of Legendre polynomials. What' the difference you feel ?

I am not talking about Fourier series, which are only valid to decompose periodic functions. I am refering to Fourier transform, which lets us decompose any function, wheter periodic or not.

With respect to Legendre polynomials, I am not talking about forming a basis with all Legendre polynomials, I mean to do it just with one of them. But with infinite versions of that one, and each one being the same original chosed polynomial but with different frequency. The same way that in Fourier Transform the basis is formed by infinite versions of the same cosine function, but each one with a different frequency.

 

[anuttarasammyak]

Now I know your interest is beyond OP topic of

Orthogonal Basis of Periodic Functions: Beyond Sines and Cosines​

 

[QuantumCuriosity42]

Now I know your interest is beyond OP topic of

Orthogonal Basis of Periodic Functions: Beyond Sines and Cosines​

Should I create a new thread?

 

[hutchphd]

One of the few useful pieces I remember from math classes:
Sturm–Liouville theory is the general study of Sturm–Liouville problems. In particular, for a "regular" Sturm–Liouville problem, it can be shown that there are an infinite number of eigenvalues each with a unique eigenfunction, and that these eigenfunctions form an orthonormal basis of a certain Hilbert space of functions.
Most useful functions are found within this rubric IMHO.

 

[DaveE]

LTI systems are more of an engineering rather than physics concept right?

Sorry, I don't really understand the difference; so sure, whatever... But I do see physicists using Fourier Transforms A LOT.

Could you provide some references about how LTI systems only behave linearly with sines?

Nope, that can't be done. Linear systems are linear for any input signal, not just sine waves. Consider that an input function can be expressed as a Fourier series (or transform), which is composed of sinusoids. If it's linear for sinusoids, it's also linear for linear combinations of sinusoids, which is nearly everything.

 

[marcusl]

That is interesting to know, but LTI systems are more of an engineering rather than physics concept right?
It looks like some nature properties like I said in my previous comment also depend on armonic frequencies.

LTI applies in physics as well. The propagation of light through any linear time-invariant medium (air, glass or colored filters) is an example. IWe can demonstrate that sines and cosines are the eigenfunctions (the natural modes or natural ways to characterize) an LTI system:
The output y(t) of an LTI system is given by the convolution of an input x(t) with the system's impulse response h(t) according to$$y(t)=x(t)*h(t)=\int^\infty_{-\infty}{x(t-\tau)h(\tau)d\tau}$$h on the right is independent of the time of day t, hence the characterization that the system is not time varying.
Now let the input be a wave of a single frequency $$x(t)=x(ω,t)=A(\omega)exp(i\omega t)$$where A is a complex number, so that x(t) consists of both a sine and cosine wave of the same frequency ω but of independent amplitudes. Then
$$y(t)=A(\omega)exp(i\omega t)\int^\infty_{-\infty}{exp(-i\omega \tau)h(\tau)d\tau}$$The integral is just the Fourier transform of the impulse response H(ω), which is called the frequency response of the system. Thus $$y(t)=H(ω)x(ω,t)$$We say that the complex exponential (sine/cosine, if you prefer) function x(ω,t) is a characteristic function of the LTI system and H(ω) is the corresponding characteristic value; these are also called eigenfunctions and eigenvalues. The frequency response or eigenvalue spectrum H(ω) completely characterizes the response of the system to any arbitrary input because every input can be decomposed into sines and cosines via Fourier transformation, and the system response to each frequency component is known from the frequency response. With more advanced math, it can be shown that this is the only set of eigenfunctions for an LTI system and that they are all orthogonal to each other.

The connection to light color is that light is a collection of electromagnetic waves, which are sines and cosines by definition. Propagation media such as air, glass or colored filters are LTI so decomposing light into its constituent frequencies (colors) and applying the frequency response function for the medium gives the output.

Another class of physical system is linear and spatially (rather than temporally) non-varying. They are treated mathematically in the same way but using spatially varying waves and spatial frequency responses.

 

[WWGD]

Well, now that I notice, those functions aren't orthogonal? For example the dot product between 1 and 3 is less than 0.

This is not really a problem, as Gram-Schmidt allows us to orthogonalize a basis, even an infinite one. In an inner-product space, of course.

 

[QuantumCuriosity42]

One of the few useful pieces I remember from math classes:
Sturm–Liouville theory is the general study of Sturm–Liouville problems. In particular, for a "regular" Sturm–Liouville problem, it can be shown that there are an infinite number of eigenvalues each with a unique eigenfunction, and that these eigenfunctions form an orthonormal basis of a certain Hilbert space of functions.
Most useful functions are found within this rubric IMHO.

I don't see what that has to do with my question about a basis formed by infinite versions of the same periodic function, each one with a different frequencies (like cos(kx)).

 

[hutchphd]

The point is that this is not unique, and should not therefore be expected a priori to be particularly special. You need to broaden your view.

 

[QuantumCuriosity42]

The point is that this is not unique, and should not therefore be expected a priori to be particularly special. You need to broaden your view.

But it is unique in terms of physics, for example, the energy of a photon E=h*f, depends on the frequency f, of the armonics that form the EM wave. Not on the frequency of other basis that are not armonics.

By the way, since you say it is not unique, could you provide a counterexample?

 

[hutchphd]

The vibration of a drumhead is described best using Bessel functions.
Diffraction of light by a circle by Airy functions. And so on.

 

[QuantumCuriosity42]

The vibration of a drumhead is described best using Bessel functions.
Diffraction of light by a circle by Airy functions. And so on.

And what about the De Broglie relation? https://en.wikipedia.org/wiki/Matter_wave
How is it that it relates the momentum with the frequency of a sinusoid, and not any other wave?
And what is more, how is it that not once in that Wikipedia article is mentioned to what kind of periodic wave (sinusoidal, triangular, square) the wavelength and frequency refer to. It is vital information to understand it, right?

Do you know an example of an infinite basis of periodic waves of different frequencies like the one Fourier transform uses? I really think it is the only one, and it explains why all of this coincidences happen.

 

[hutchphd]

The plane wave basis (sin cos or complex exponential) is most useful for square boxes (or crystals). The eigenvalues for any finite box are discrete because of boundary conditions.
The eigenbasis for a free atom does not use plane waves because they do not vanish with spherical distance as required. You can see that solution in any elementary quantum book, i will not describe it here. The allowed frequencies are not strictly "harmonic" nor distribution in space perfectly periodic but there is a whole lotta shakin' gong on......

 

[QuantumCuriosity42]

https://en.wikipedia.org/wiki/Black-body_radiation
In this article the same happens, wavelength and frequency are mentioned, while completely disregarding the type of periodic wave they refer to. I don't understand how is it that all people ignore that in all physics articles, it is crucial information, it seriously makes me go mad. I just don't get it.

 

[DaveE]

https://en.wikipedia.org/wiki/Black-body_radiation
In this article the same happens, wavelength and frequency are mentioned, while completely disregarding the type of periodic wave they refer to. I don't understand how is it that all people ignore that in all physics articles, it is crucial information, it seriously makes me go mad. I just don't get it.

Because sinusoids are so useful, common, and mathematically simple, I think most technical people just assume that either you know that's what they are talking about, or that it doesn't matter.

Plus, there is a semantic issue. For example, in my mind a triangle wave has one period but contains many frequencies. I know that doesn't make sense to everyone, but we don't always want to explain every detail.

 

[QuantumCuriosity42]

Because sinusoidals are so useful, common, and mathematically simple, I think most technical people just assume that either you know that's what they are talking about, or that it doesn't matter.

But like, why look at the radiation spectra of a black body in that basis of sines (Fourier transform)!?
We could use any other. Even more, I am starting to think that all that was derived after that black body radiation law, like E=hf is simply a suposition that the frequencies are of sinusoids, when they could be of any other periodic wave. What is going on?

 

[DaveE]

But like, why look at the radiation spectra of a black body in that basis of sines (Fourier transform)!?
We could use any other. Even more, I am starting to think that all that was derived after that black body radiation law, like E=hf is simply a suposition that the frequencies are of sinusoids, when they could be of any other periodic wave. What is going on?

You could use square waves, if you want. The rest of us like sine waves, we think they are easier.

 

[QuantumCuriosity42]

You could use square waves, if you want. The rest of us like sine waves, we think they are easier.

But if that is true, then all equations derived from that suposition are plainly wrong? We claim E=hf, <<after>> supossing that radiation is formed by sinusoidal waves.
I could suppose they are other kind of wave and derive E=h*sqrt(f), or something like that, what!?
Also how did people originally decompose black body radiation on its armonic spectra? They applied Fourier Transform?.

 

[marcusl]

The vibration of a drumhead is described best using Bessel functions.
Diffraction of light by a circle by Airy functions. And so on.

These are orthogonal basis functions but they are not periodic.