The precise relationship between Fourier series and Fourier transform

[docnet]

Would someone be able to explain like I am five years old, what is the precise relationship between Fourier series and Fourier transform?

Could someone maybe offer a concrete example that clearly illustrates the relationship between the two?

I found an old thread that discusses this, but I feel that it was not so clear.

Thanks again,

 

[jedishrfu]

While we ponder how best to answer your question, this diversion may help:

https://betterexplained.com/articles/an-interactive-guide-to-the-fourier-transform/

and a cool video diversion:

 

[Svein]

The short version: Fourier series is for repetitive functions (as in f(x+A)=f(x) for some constant A). Fourier transform is for non-repetitive functions.

 

[onatirec]

Fourier series is for repetitive functions (as in f(x+A)=f(x) for some constant A). Fourier transform is for non-repetitive functions.

I would not agree with this. Besides not really addressing the relationship between the series and transform... One can take the FT of a repetitive function.

I think of it like this:

Just as a point in R3 can be described by a string (or sum) of cartesian coordinates, so can a wave be described by a sum of pure sinusoids. The Fourier series is that sum (coordinates, if you will, in a sinusoidal basis). The transform is a method of finding the coefficients for this new sinusoidal basis. Loosely speaking, it's like taking the projection of a signal onto a different basis of unit vectors (similar to the dot product), except this basis is pure sinusoidal terms instead of the 3 orthogonal vectors (i, j, k).

 

[mathwonk]

as i recall, it is analysis on a group, and these involve different groups. so the series is a special case (on the "circle group" R/Z ≈ S^1) of the transform, which applies to a large variety of groups. sorry for a tantalizing but non explicit "answer", or merely comment rather.

ah yes, a little more is coming back: the transform relates analysis on a group with analysis on its "dual group". (The dual of a locally compact abelian group is the group of continuous homomorphisms from the group into the circle group.)
https://link.springer.com/chapter/10.1007/0-387-27561-4_7

And it seems the integers are the dual of the circle group, (a periodic function is a function on the circle, and a series is a function on the integers), while the real line group R is dual to itself. Thus the transform of a periodic function, i.e. a function on the circle, is a function on the integers, i.e. a (fourier) series, and the two functions, if nice enough, determine each other. The transform of a (perhaps rapidly vanishing at infinity) function on the line, is another (rapidly vanishing) function on the line, and the two determine each other. Since rapidly vanishing functions are dense in the lebesgue integrable functions this correspondence extends to integrable functions. [See Lang, Analysis II, chapter XIV.]

the point is the transform sets up a one - one correspondence between functions on a group and functions on its dual group, and the group properties cause certain operations on one space of functions to correspond to different operations on the functions in the other space. In some classical examples differentiation corresponds to multiplication, so solving differential equations translates into solving algebraic equations. In other variations on this theme, solutions of one more difficult differential equation transform into solutions of an easier differential equation, e.g. in the case of "Paley Wiener" type transforms. (One such transform I recall essentially changes ∂/∂zbar into ∂/∂t, changing the job of finding holomorphic solutions to a certain equation, into the easier job of finding solutions of another equation, that are constant in t.) There is also an exotic version in algebraic geometry, the "Fourier-Mukai" transform, relating derived categories of sheaves on abelian varieties (compact complex groups), and their duals, that apparently has a relevance to string theory.

for a brief intro to duality, try this article, at least the first few paragraphs.
https://en.wikipedia.org/wiki/Pontryagin_duality

and for the very diligent reader, here is a historical survey of the origins of the whole subject (harmonic analysis) in number theory, probability and mathematical physics, especially section 8 beginning on page 565.

https://www.ams.org/journals/bull/1980-03-01/S0273-0979-1980-14783-7/S0273-0979-1980-14783-7.pdf

and for the derived category stuff: (not suitable for 5 year olds [nor me])

https://en.wikipedia.org/wiki/Fourier–Mukai_transform

Although very abstract, this theory has useful consequences: a classical theorem, due to R. Torelli, of great interest in algebraic geometry says that an algebraic curve is entirely determined by its "jacobian variety", a compact complex group with a certain subvariety called the theta divisor (defined as the zeroes of a certain holomorphic, almost periodic, Fourier like theta-series of exponentials
https://mathworld.wolfram.com/RiemannThetaFunction.html).

This next article shows that the Fourier mukai transform of the jacobian essentially changes the theta divisor into the original curve, thus proving the theorem.

https://arxiv.org/abs/math/9811136

oops, sorry. I guess this is both more (and less) than you wanted to know. But thanks for the fun question. And maybe someone more expert (like an analyst) will give a better, more focussed, answer. But I do hope the idea of groups and dual groups helps relate the two concepts you asked about. what do you think @docnet?

 

[mathwonk]

trying again:

Fourier analysis

In the classical setting, there are two types of representations of functions using exponential functions, Fourier series for periodic functions, and Fourier integrals for more general functions on the real line, e.g. compactly supported ones. These are special cases of the same general construction. Both take advantage of the additive group structure on the real line and its quotient groups.

The general construction transforms a function on a (locally compact abelian) group G into a function on its dual group G*, (where G* is the set of continuous group homomorphisms from G to the multiplicative group S^1 of complex numbers of absolute value 1, the circle group), using integration (or summation in the case of finite or discrete group).

Example: if G = Z the integers, then for every point e^it of S^1, there is a unique continuous group homomorphism Z—>S^1 taking 1 to e^it, hence taking n to e^int. Thus the dual of Z is S^1 itself. Conversely, a continuous homomorphism of S^1 to S^1, must wrap the circle around itself a finite number n of times, and is determined by this number n, hence has form e^it—>e^int, for some n in Z. Hence the dual of S^1 is Z. In general, as in this example, G** = G.

Example: If G = R, the real line with additive group structure, then a continuous homomorphism R—>S^1 is determined, up to taking inverses, by its kernel, or by the smallest positive number t sent to 1, thus has form

t—>e^2πixt for some unique real x. Hence the dual of R is again R.

The construction is the following: given a function f:G—>C from the group G to the complex numbers C, define its transform as the function
f*:G*—>C, where for y in G*,
f*(y) = the integral (or sum) over all x in G, of f(x).y(x).

Hence if f is a function on S^1, we consider it as a periodic function on R, so that its value at e^it is considered as its value at t. A typical element y of the dual group Z, is considered as an integer n, or as the equivalent homomorphism from S^1 to S^1, or the equivalent periodic map R—>S^1, defined by e^int. Thus the function on the dual group Z has value at n, given by the integral over S^1, or over the appropriate parametrizing t-interval , f*(n) = integral of f(t).e^int. These are the Fourier coefficients of f.

Conversely, given a function on Z, namely a sequence a(n), the transform is the following function a*(t) on S^1: given an element t thought of as an element e^it of S^1, mapping n to e^int, we have a*(t) = sum over all n, of a(n).e^int. This is the Fourier series determined by the coefficients a(n).
The theorem is that with certain restrictions, these two operations invert each other.

Now consider the other basic example, where G = R, and we have a function
f:R—>C. Then the transform f*:R—>C has value at y in R, (thought of as the homomorphism R—>S^1 taking 1 to e^iy), equal to the integral over R, of f(x).y(x) = f(x).e^ixy. Again this operation is self inverting on suitable spaces of functions.

So I tried to show how both Fourier series and Fourier integrals (or transforms), are special cases of the same abstract construction, but I have given no idea why the constructions work! And I am not an expert, so this could be wrong in details, or more globally. and you may well laugh, but I meant this to be the 5 year old version.

 

[vela]

Would someone be able to explain like I am five years old, what is the precise relationship between Fourier series and Fourier transform?

Could someone maybe offer a concrete example that clearly illustrates the relationship between the two?

I found an old thread that discusses this, but I feel that it was not so clear.

The link is broken.

The second answer in this stackexchange thread gives a pretty straightforward explanation. It may be too advanced for a five-year-old, but I don't know of any five-year-olds doing Fourier analysis.

 

[mathwonk]

Thanks for that link. Reading that one and this related one is enlightening.

https://math.stackexchange.com/ques...ourier-series-and-fourier-transformation?rq=1

It seems that one is trying to generalize the spectral theorem from linear algebra, that tells how to diagonalize a linear transformation on a finite dimensional vector space, by finding a basis of eigenvectors for the underlying vector space. recall that an eigenvector for T is a vector v such that T(v) = av, for some scalar a. Having such a basis makes it easy to solve problems about that linear transformation. In Fourier analysis it seems one is trying to diagonalize the derivative operator, acting on certain spaces of functions, hence helping solve problems involving derivatives, i.e. differential equations. The new aspect seems to relate to the function spaces having large, infinite dimensions.

Just as in a finite dimensional space in linear algebra, we have a finite basis and try to write a vector as a finite linear combination of eigenvectors, for the space of functions on the (compact) circle group, we have a countably infinite "eigenbasis" (a set whose finite linear combinations are dense in the space), and hence we represent a function as the limit of an infinite sum of such "eigenfunctions".

(I.e. note crucially that the derivative d/dx acts on the functions e^ax, by multiplication, d/dx (e^ax) = a.e^ax, so e^ax is an eigenfunction for d/dx.) With Fourier series, we use as our eigenbasis the sequence of eigenfunctions e^nx for integers n, and try to write a function defined on the circle, as an infinite sum of multiples of these functions.

In the case of functions on the (non compact) real line, it seems one needs a further generalization of linear combinations, and must use not just the countable collection of eigenfunctions e^nx, for integers n, but the full uncountable collection of eigenfunctions e^ax, for all real numbers a. Then one must also integrate over this collection of functions, since not even an infinite sum will suffice.

So the difference between Fourier series and Fourier integrals seems due in some sense to the larger (infinite) "dimension" of the space of functions on the non compact group R, as opposed to the (also infinite dimensional) space of functions on the compact circle group S^1. Here the notion of dimension apparently refers to the size of a complete set of eigenfunctions for that space, not the usual linear algebra concept of dimension involving finite linear combinations.

I admit to being one of those deprived individuals who was taught abstract functional analysis, including generalized spectral theorems involving integrals of projection operators taken over (paths in) uncountable spectral regions in the complex plane, without ever seeing or appreciating a Fourier series or Fourier integral, from which it seems to have all originated.

Riemann himself wrote a nice historical introduction to his paper on the topic if you have access to his work on representing functions as trigonometric series.

 

[onatirec]

ELI5...?

I'll have to go dust off my group theory textbook from kindergarten...

 

[AndreasC]

My 5 years old explanation: a Fourier series is a convenient way to write a function that repeats after an interval in terms of other functions we know how to manipulate better. That interval can have many different sizes, for instance you can have a function of time that repeats ever second, or every 10 seconds, or every million years. That's all fine, but you may want to deal with a function that does not repeat. Then the interval becomes infinite, and what you need there is a Fourier transform.

To be a little bit more precise, to write down the Fourier series, you need to find the Fourier coefficients. To do that you need an integral over the interval of repetition. The Fourier transform is kind of similar to finding these coefficients, except now the interval is infinite.

Additionally, to write down your function as a Fourier series you essentially have to sum a bunch of terms involving different "frequencies". How much different the next frequency is from the previous one depends inversely on the interval. So for a large interval you get a very dense sum of frequencies, whereas for a small one you get a pretty sparse one. In the limit where the interval becomes infinite, the sum of frequencies becomes infinitely "dense", which corresponds to an integral. So you can get back your old function from the transformed one by integrating the transformed one multiplied by the relevant frequencies. That's just the continuous analogue of a Fourier series.

Note that you can also Fourier transform a repeating function, nothing is stopping you.

Hope that helped. I'm not sure a 5 year old would understand this but I guess it depends on the 5 year old!

 

[mathwonk]

@onatirec: The group operations here are addition and multiplication of numbers, so they may occur in your source, (assuming of course your kindergarten covered the complex numbers).

 

[mathwonk]

Drawing on Riemann's historical survey, these topics arose when people like Bernoulli noticed that certain problems about functions, were easy to solve for the familiar trig functions sin and cos. They also noticed that taking linear combinations of solutions produced new solutions. Thus the question arose of whether arbitrary functions could be expressed as possibly infinite linear combinations of sines and cosines. This leads to Fourier series for periodic functions, and Fourier integrals for more general functions.

 

[docnet]

as i recall, it is analysis on a group, and these involve different groups. so the series is a special case (on the "circle group" R/Z ≈ S^1) of the transform, which applies to a large variety of groups. sorry for a tantalizing but non explicit "answer", or merely comment rather.

ah yes, a little more is coming back: the transform relates analysis on a group with analysis on its "dual group". (The dual of a locally compact abelian group is the group of continuous homomorphisms from the group into the circle group.)
https://link.springer.com/chapter/10.1007/0-387-27561-4_7

And it seems the integers are the dual of the circle group, (a periodic function is a function on the circle, and a series is a function on the integers), while the real line group R is dual to itself. Thus the transform of a periodic function, i.e. a function on the circle, is a function on the integers, i.e. a (fourier) series, and the two functions, if nice enough, determine each other. The transform of a (perhaps rapidly vanishing at infinity) function on the line, is another (rapidly vanishing) function on the line, and the two determine each other. Since rapidly vanishing functions are dense in the lebesgue integrable functions this correspondence extends to integrable functions. [See Lang, Analysis II, chapter XIV.]

the point is the transform sets up a one - one correspondence between functions on a group and functions on its dual group, and the group properties cause certain operations on one space of functions to correspond to different operations on the functions in the other space. In some classical examples differentiation corresponds to multiplication, so solving differential equations translates into solving algebraic equations. In other variations on this theme, solutions of one more difficult differential equation transform into solutions of an easier differential equation, e.g. in the case of "Paley Wiener" type transforms. (One such transform I recall essentially changes ∂/∂zbar into ∂/∂t, changing the job of finding holomorphic solutions to a certain equation, into the easier job of finding solutions of another equation, that are constant in t.) There is also an exotic version in algebraic geometry, the "Fourier-Mukai" transform, relating derived categories of sheaves on abelian varieties (compact complex groups), and their duals, that apparently has a relevance to string theory.

for a brief intro to duality, try this article, at least the first few paragraphs.
https://en.wikipedia.org/wiki/Pontryagin_duality

and for the very diligent reader, here is a historical survey of the origins of the whole subject (harmonic analysis) in number theory, probability and mathematical physics, especially section 8 beginning on page 565.

https://www.ams.org/journals/bull/1980-03-01/S0273-0979-1980-14783-7/S0273-0979-1980-14783-7.pdf

and for the derived category stuff: (not suitable for 5 year olds [nor me])

https://en.wikipedia.org/wiki/Fourier–Mukai_transform

Although very abstract, this theory has useful consequences: a classical theorem, due to R. Torelli, of great interest in algebraic geometry says that an algebraic curve is entirely determined by its "jacobian variety", a compact complex group with a certain subvariety called the theta divisor (defined as the zeroes of a certain holomorphic, almost periodic, Fourier like theta-series of exponentials
https://mathworld.wolfram.com/RiemannThetaFunction.html).

This next article shows that the Fourier mukai transform of the jacobian essentially changes the theta divisor into the original curve, thus proving the theorem.

https://arxiv.org/abs/math/9811136

oops, sorry. I guess this is both more (and less) than you wanted to know. But thanks for the fun question. And maybe someone more expert (like an analyst) will give a better, more focussed, answer. But I do hope the idea of groups and dual groups helps relate the two concepts you asked about. what do you think @docnet?

wow. i did not expect a post so informative and through in response my simple question. One would have to be a smart 5 year old to understand this, but this is amazing!