Orthogonal Basis of Periodic Functions: Beyond Sines and Cosines

[QuantumCuriosity42]

TL;DR Summary

Investigating if sines and cosines are the only orthogonal basis for periodic functions that generate the entire function space.

Hello everyone,

I've been delving deep into the realm of periodic functions and their properties. One of the fundamental concepts I've come across is the use of sines and cosines as an orthogonal basis for representing any functions. This is evident in Fourier series expansions, where any function can be represented as a sum of sines and cosines of different frequencies.

However, this leads me to a question: Are sines and cosines the only orthogonal basis for periodic functions that can generate the entire function space? Specifically, can we consider other periodic functions, like square pulses or triangular waves of various frequencies, as potential candidates for an orthogonal basis?

Would love to hear insights and if there are any relevant literature or studies that have explored this topic further.

Thank you in advance!

 

[anuttarasammyak]

You can expand a periodic function as a series of any complete orthogonal set of base functions. Sinusoidal base function is an example often used.

 

[QuantumCuriosity42]

You can expand a periodic function as a series of any complete orthogonal set of base functions. Sinusoidal base function is an example often used.

To clarify, I was referring to expanding any function (whether periodic or not) in terms of an orthogonal basis of periodic functions.

 

[WWGD]

Note these are Schauder bases, not Hamel bases, in that we deal with convergence, given these are infinite sums. Those not only Linear Algebra but Topology is also involved.

 

[anuttarasammyak]

To clarify, I was referring to expanding any function (whether periodic or not) in terms of an orthogonal basis of periodic functions.

If periodic, the function is expresed as series, e.g. Fourier series.
If not periodic, the function is expressed as integral. e.g. Fourier integral.

 

[QuantumCuriosity42]

If periodic, the function is expresed as series, e.g. Fourier series.
If not periodic, the function is expressed as integral. e.g. Fourier integral.

Yes, but my question is what other basis are apart from the one Fourier transform uses (sine and cosine).

 

[DaveE]

Laplace and Wavelet come to mind, but I think there are many.

 

[QuantumCuriosity42]

Laplace and Wavelet come to mind, but I think there are many.

Those don't use a basis of periodic functions.

 

[WWGD]

Wouldn't Parsevals Theorem apply?

 

[QuantumCuriosity42]

Wouldn't Parsevals Theorem apply?

What do you mean?

 

[anuttarasammyak]

Yes, but my question is what other basis are apart from the one Fourier transform uses (sine and cosine).

How about this set as an exasmple ?

 

[QuantumCuriosity42]

How about this set as an exasmple ?
View attachment 334783

Is that set ortogonal and generates the space of all possible functions? I don't know. Is there any proof like there is for the Fourier transform basis?
Moreover, I would prefer to find a basis of continous functions like sin and cos.

 

[anuttarasammyak]

Is that set ortogonal and generates the space of all possible functions? I don't know. Is there any proof like there is for the Fourier transform basis?
Moreover, I would prefer to find a basis of continous functions like sin and cos.

I just give you a hint or suggestion which might be of your interest (I hope so). It is your problem and you can do it.

For smooth functions you like, instead of sin x and cos x bases, using their combination
$$\sin x + \cos x$$
$$\sin x - \cos x$$
is an easy way to demonstrate.

 

[QuantumCuriosity42]

I just give you a hint or suggestion which might be of your interest (I hope so). It is your problem and you can do it.

For smooth functions, instead of sin x and cos x, using their combination
$$\sin x + \cos x$$
$$\sin x - \cos x$$
is an easy way to demonstrate.

Thanks, but I don't have a level of math good enough to prove that.
By the way, it is not a problem, just that my question originated because I wonder why in physics we always decompose waves in sinusoids, when maybe there are other valid basis of periodic functions. That is, there is no "fundamental frequency", the frequency depends in the basis of periodic functions used.

 

[anuttarasammyak]

when maybe there are other valid basis of periodic functions.

To clarify your queations what other canditates are in your mind ?

 

[QuantumCuriosity42]

To clarify your queations what is an example in your mind ?

I don't know an example basis different from cos(kx) with k being a real number. Which is the one the Fourier transform uses (ignoring the imaginary part).
But my question was precisely the example you are asking me to provide, or I am not understanding you correctly. But for example, maybe all triangular (or pulse) shaped functions of different frequencies also form a basis, I don't know.

 

[anuttarasammyak]

Other than Trigonometric Functions, Some example of periodic base functions in physics are Bessel Functions, Legendre Polynomials, Chebyshev Polynomials, Hermite Polynomials and Walsh Functions. You can see what they are and how they are used in websites.

I think among these functions, trigonometric fundtion is most easy and most universal due to its simple mathematics of
$$\frac{d}{dx}e^{ikx}=ik e^{ikx}$$
$$[\frac{d^2}{dx^2}+k^2]e^{ikx}=0$$
what is solution of equation of motion of harmonic oscillator.

[EDIT]My bad, Bessel functions and others are not periodic itself.

 

[QuantumCuriosity42]

Other than Trigonometric Functions, Some example of periodic base functions in physics are Bessel Functions, Legendre Polynomials, Chebyshev Polynomials, Hermite Polynomials and Walsh Functions. You can see what they are and how they are used in websites.

I think among these functions, trigonometric fundtion is most easy and most universal due to its simple mathematics of
$$\frac{d}{dx}e^{ikx}=ik e^{ikx}$$
$$[\frac{d^2}{dx^2}+k^2]e^{ikx}=0$$
what is solution of equation of motion of harmonic oscillator.

I think all the functions you listed are not periodic ?
And even if they were, why some things in nature work in terms of the frequencies of sines (like E=h*f), even if trigonometric function are easy to work with. It does not make sense that nature chose the same thing that is easy or convenient for us.

 

[anuttarasammyak]

In a map North-South axis and East-West axis is given and we appoint number coordinates to a place.
We can use another orothogonal axis, e.g. NW-SE axis and NE-SW axis and appoint another number coorinate to the same place. The both maps work equally. Convenience or whch we are familiar with is another issue.

We have a same situation in expansion of functions. sinusoidal bases are common in use but any orther bases, #11 as an example, work equally.

 

[anuttarasammyak]

With two different base of $\{\phi_n\},\{\psi_n\}$, F is expanded as
$$F(x)=\sum_n p_n\phi_n(x)$$
$$F(x)=\sum_n q_n\psi_n(x)$$
and base function is also expressed by the other base,
$$\phi_n(x)=\sum_m r_{nm}\psi_m(x)$$
So
$$F(x)=\sum_n p_n\sum_m r_{nm}\psi_m(x)=\sum_m \sum_n p_n r_{nm}\psi_m(x)=\sum_n(\sum_m p_m r_{mn} )\psi_n(x)$$
Thus
$$q_n=\sum_m r_{mn} p_m$$

We can go from one expansion to the other in this way. As an example $\{\phi_n\}$ is sinusoids ,$\{\psi_n\}$ is base in #11.

 

[DaveE]

For smooth functions you like, instead of sin x and cos x bases, using their combination
sin⁡x+cos⁡x
sin⁡x−cos⁡x
is an easy way to demonstrate.

Thanks, but I don't have a level of math good enough to prove that.

This is the Hartley transform which only uses real numbered coefficients for real valued functions.

 

[DaveE]

I think among these functions, trigonometric fundtion is most easy and most universal due to its simple mathematics

Yes. The sinusoids are eigenfunctions for linear systems, which is VERY convenient. This allows you to do spectral analysis at single frequencies.

 

[QuantumCuriosity42]

Yes. The sinusoids are eigenfunctions for linear systems, which is VERY convenient. This allows you to do spectral analysis at single frequencies.

Could you explain that more deeply? Why the basis of #11 doesn't allow you to do spectral analysis at single frequencies?

 

[QuantumCuriosity42]

How about this set as an exasmple ?
View attachment 334783

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

 

[DaveE]

Could you explain that more deeply? Why the basis of #11 doesn't allow you to do spectral analysis at single frequencies?

Because your basis functions contain different frequencies which LTI systems respond differently to. Each response wouldn't have just a single gain and phase, but an infinite set of them for each harmonic frequency too. I suppose you could do it with a bunch of math. Since the output is no longer a simple modification of the input, then you'll have to deconvolve the whole mess. Then you're back to using sinusoids. I guess the phrase "single frequency" kind of requires sinusoids by definition.

But, honestly, I haven't thought much about this. Maybe in a square wave world the engineers understand the square wave domain like we understand the frequency domain. We are trained with delta functions, step functions and sinusoids because the math is easy. Each illuminates system behaviors in ways we understand, maybe just because that's what we are used to.

 

[marcusl]

To expand on Dave’s post, sines and cosines are the eigenfunctions of linear time-invariant systems, and, since this class of system is ubiquitous, Fourier analysis is everywhere.

There are other classes of periodic function for more complex systems, like the elliptic functions that are doubly periodic.

 

[anuttarasammyak]

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

You are right. My bad. Thanks.
Insted of my failure attempt, a good example of bases for periodic function other than sinusoids is Legendre polynomials which is illustrated in https://upload.wikimedia.org/wikipe...mials6.svg/360px-Legendrepolynomials6.svg.png

I think all the functions you listed are not periodic ?

[EDIT]My bad, Bessel functions and others are not periodic itself.

To make it periodic is not difficult. For an example new Pn(x) would be
$$P_n(2(\frac{x+1}{2}-[\frac{x+1}{2}])-1)$$
so that it is periodic outside of [-1,1], where [ ] is floor function. Normalization is easily done also. Completeness is proved though it is complicated to me.

And even if they were, why some things in nature work in terms of the frequencies of sines (like E=h*f), even if trigonometric function are easy to work with. It does not make sense that nature chose the same thing that is easy or convenient for us.

That discussion on nature seems beyond OP which is a question on mathematics: Investigating if sines and cosines are the only orthogonal basis for periodic functions that generate the entire function space. The answer is we have plural bases other than sinusoids.

 

[QuantumCuriosity42]

You are right. My bad. Thanks.
Insted of my failure attempt, a good example of bases for periodic function other than sinusoids is Legendre polynomials which is illustrated in https://upload.wikimedia.org/wikipe...mials6.svg/360px-Legendrepolynomials6.svg.pngTo make it periodic is not difficult. For an example new Pn(x) would be
$$P_n(2(\frac{x+1}{2}-[\frac{x+1}{2}])-1)$$
so that it is periodic outside of [-1,1], where [ ] is floor function. Normalization is easily done also. Completeness is proved though it is complicated to me.
That discussion on nature seems beyond OP which is a question on mathematics: Investigating if sines and cosines are the only orthogonal basis for periodic functions that generate the entire function space. The answer is we have plural bases other than sinusoids.

So if we have more possible basis of periodic functions that generate the entire function space, why is it that in nature for example, the colors of light depend on the frequencies of the armonic decomposition (in sines) of the wave? It is too much of a coincidence and it does not make any sense to me.
Also, that basis of periodic legendre polynomials to decompose functions has a well known name like Fourier transform? Or can you reference the proof of completnees you talked about?

 

[QuantumCuriosity42]

Because your basis functions contain different frequencies which LTI systems respond differently to. Each response wouldn't have just a single gain and phase, but an infinite set of them for each harmonic frequency too. I suppose you could do it with a bunch of math. Since the output is no longer a simple modification of the input, then you'll have to deconvolve the whole mess. Then you're back to using sinusoids. I guess the phrase "single frequency" kind of requires sinusoids by definition.

But, honestly, I haven't thought much about this. Maybe in a square wave world the engineers understand the square wave domain like we understand the frequency domain. We are trained with delta functions, step functions and sinusoids because the math is easy. Each illuminates system behaviors in ways we understand, maybe just because that's what we are used to.

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

 

[anuttarasammyak]

Also, that basis of periodic legendre polynomials to decompose functions has a well known name like Fourier transform? Or can you reference the proof of completnees you talked about?

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

 

[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.