Which function space do square waves span?

[Kontilera]

Hello!
As the topic suggests I´m interested which functions space square waves span?
Lets say we define them as https://wikimedia.org/api/rest_v1/media/math/render/svg/5b8953debf86627276f45bf8822140ff2bbaee56 .
Do they span the same space as the sines and cosines in Fourier analysis? :/

Thanks!

 

[scottdave]

The Wikipedia page, where these formula resides, talks about the Fourier Series for a square wave. A square wave consists of odd harmonics of the fundamental frequency. https://en.wikipedia.org/wiki/Square_wave

 

[Kontilera]

The Wikipedia page, where these formula resides, talks about the Fourier Series for a square wave. A square wave consists of odd harmonics of the fundamental frequency. https://en.wikipedia.org/wiki/Square_wave

Maybe I am missing something now but does that necessarily mean that an infinite set of square waves span the same space as the sinusoidal waves do? And in that case, why?

 

[scottdave]

Maybe I am missing something now but does that necessarily mean that an infinite set of square waves span the same space as the sinusoidal waves do? And in that case, why?

It's the other way around. A square wave can be represented by summing up an infinite series of sine waves. Or it could be cosine waves, or you may need both sine and cosine, depending on when the square wave "starts" . in relation to t=0.

 

[Kontilera]

It's the other way around. A square wave can be represented by summing up an infinite series of sine waves. Or it could be cosine waves, or you may need both sine and cosine, depending on when the square wave "starts" . in relation to t=0.

Yes, but my question is which function space the square waves span.

 

[Klystron]

A square wave is not a continuous function. By nature, ideal square waves exhibit discontinuities transitioning from minimum to maximum value.

 

[Kontilera]

A square wave is not a continuous function. By nature, ideal square waves exhibit discontinuities transitioning from minimum to maximum value.

A sinus wave is a periodic function bounded by its amplitude in both positive and negative y-direction..

 

[FactChecker]

Do [even and odd square waves] span the same space as the sines and cosines in Fourier analysis?

This is a good question. Due to the discontinuities, they contain components of all sine and cosine frequencies. Can those frequencies be recovered as the limit of linear combinations of the square waves? I do not immediately see the answer, although I suspect they can be.

 

[Kontilera]

This is a good question. Due to the discontinuities, they contain components of all sine and cosine frequencies. Can those frequencies be recovered as the limit of linear combinations of the square waves? I do not immediately see the answer, although I suspect they can be.

Although I believe this is a dangerous argument since not much is intuitive when it comes to summing up an infinite number of terms, but...
Lets say that we want to express the quarter of the unit circle that is positioned in the first quadrant as a sum of square waves. The arc length of this curve is one half pi. Although by just adding square waves it seems that the arc length for the generated curve always will give 2.
This means that although we expect our generated curve to converge to the original curve, some properties won't converge.
Could this be an argument that our square waves doesn't span the function space?

 

[Mark44]

Lets say that we want to express the quarter of the unit circle that is positioned in the first quadrant as a sum of square waves. The arc length of this curve is one half pi. Although by just adding square waves it seems that the arc length for the generated curve always will give 2.

How do you get 2 for the arc length? For a quarter circle of radius 1, if I divide the interval [0, 1] into n equal subintervals, the total length of the horizontal tops of the waves is $\frac {n - 1} n$, which approaches 1 as n grows large.

This is the same idea of why you can't approximate the length along a curve by a sum of horizontal pieces -- $\int_a^b \sqrt{1 + (f'(x))^2} dx$ gives the correct arc length, but $\int_a^b 1 dx$ just gives the length of the interval [a, b]. If you want to approximate the area within the quarter circle, rectanular subareas work just fine, but they don't work at all if you want to approximate the length along the quarter circle.

This means that although we expect our generated curve to converge to the original curve, some properties won't converge.

I don't think this is important. The point of Fourier series is to be able to approximate discontinuous functions such as square waves, sawtooth waves, and so on, with a series of continuous functions that are sums of sine and/or cosine functions of varying periods and amplitudes. All that is required is that the Fourier series function values agree with the function values of the discontinuous square wave, sawtooth wave, etc.

Could this be an argument that our square waves doesn't span the function space?

What function space? If you're talking about the function space of all continuous functions, then square waves aren't in that space, because they aren't continuous.

 

[Svein]

In this context, the "square waves" are called "Walsh functions". Please check out https://en.wikipedia.org/wiki/Walsh_function.

 

[FactChecker]

In this context, the "square waves" are called "Walsh functions". Please check out https://en.wikipedia.org/wiki/Walsh_function.

But these are very restricted regarding where the step changes are.

 

[WWGD]

@Kontilera It seems we are talking Schauder basis here, so we are talking convergence and not equality , as in Hamel bases. This means the underlying space where these functions has a norm , which gives rise to a topology , or a given topology of other sort. What is the underlying space

 

[Kontilera]

Thank you for the answers!