Connection between Fourier transform and Taylor series

Question

Both Fourier transform and Taylor series are means to represent functions in a different form.

What is the connection between these two? Is there a way to get from one to the other (and back again)? Is there an overall, connecting (geometric?) intuition?

Answer 1

Assume that the Taylor expansion $f(x)=\sum_{k=0}^\infty a_k x^k$ is convergent for some $|x|>1$. Then $f$ can be extended in a natural way into the complex domain by writing $f(z)=\sum_{k=0}^\infty a_k z^k$ with $z$ complex and $|z|≤1$. So we may look at $f$ on the unit circle $|z|=1$. Consider $f$ as a function of the polar angle $\phi$ there, i.e., look at the function $F(\phi):=f(e^{i\phi})$. This function $F$ is $2\pi$-periodic, and its Fourier expansion is nothing else but $F(\phi)=\sum_{k=0}^\infty a_k e^{ik\phi}$ where the $a_k$ are the Taylor coefficients of the "real" function $x\mapsto f(x)$ we started with.

Answer 2

A holomorphic function in an annulus containing the unit circle has a Laurent series about zero which generalizes the Taylor series of a holomorphic function in a neighborhood of zero. When restricted to the unit circle, this Laurent series gives a Fourier series of the corresponding periodic function. (This explains the connection between the Cauchy integral formula and the integral defining the coefficients of a Fourier series.)

But it's worth mentioning that the Fourier transform is much more general than this and applies in a broad range of contexts. I don't know that there's a short, simple answer to this question.

Edit: I guess it's also worth talking about intuition. One intuition for the Taylor series of a function $f(x)$ at at a point is that its coefficients describe the displacement, velocity, acceleration, jerk, and so forth of a particle which is at location $f(t)$ at time $t$. And one intuition for the Fourier series of a periodic function $f(x)$ is that it describes the decomposition of $f(x)$ into pure tones of various frequencies. In other words, a periodic function is like a chord, and its Fourier series describes the notes in that chord.

(The connection between the two provided by the Cauchy integral formula is therefore quite remarkable; one takes an integral of $f$ over the unit circle and it tells you information about the behavior of $f$ at the origin. But this is more a magic property of holomorphic functions than anything else. One intuition to have here is that a holomorphic function describes, for example, the flow of some ideal fluid, and integrating over the circle gives you information about "sources" and "sinks" of that flow within the circle.)

Answer 3

There is a big difference between the Taylor series and Fourier transform. The Taylor series is a local approximation, while the Fourier transform uses information over a range of the variable.

The theorem Qiaochu mentions is very important in complex analysis and is one indication of how restrictive having a derivative in the complex plane is on functions.

Answer 4

There is an analogy, more direct for fourier series. Both Fourier series and Taylor series are decompositions of a function $f(x)$, which is represented as a linear combination of a (countable) set of functions. The function is then fully specified by a sequence of coefficients, instead of by its values $f(x)$ for each $x$. In this sense, both can be called a transform $f(x) \leftrightarrow \{ a_0, a_1, ...\}$

For the Taylor series (around 0, for simplicity), the set of functions is $\{1, x , x^2, x^3...\}$. For the Fourier series is $\{1, \sin(\omega x), \cos(\omega x), \sin(2 \omega x), \cos(2 \omega x) ...\}$.

Actually the Fourier series is one the many transformations that uses an orthonomal basis of functions. It is shown that, in that case, the coefficients are obtained by "projecting" $f(x)$ onto each basis function, which amounts to an inner product, which (in the real scalar case) amounts to an integral. This implies that the coefficients depends on a global property of the function (over the full "period" of the function).

The Taylor series (which does not use a orthonormal basis) is conceptually very different, in that the coeffients depends only in local properties of the function, i.e., its behaviour in a neighbourhood (its derivatives).