Fourier transform of a holomorphic function

Question

Question: Is there a simple method for calculating the Fourier transform of a holomorphic complex function ${f{{\left({z}\right)}}}:\Omega\to{\mathbb{{{C}}}}$?

In order for my question to be well-posed I define a holomorphic function ${f}:\Omega\to{\mathbb{{{C}}}}$ to posses continuous first partial derivatives and satisfy the Cauchy-Riemann equations in a simple connected domain ${\Omega}\subseteq{\mathbb{{{C}}}}$ without any singularities. I am quite familiar with a Fourier transform for a real, periodic function ${f}:{\mathbb{{{R}}}}\to{\mathbb{{{R}}}}$ that uses complex exponentials as a basis of eigenfunctions to generate an expansion ${f{{\left({x}\right)}}}={\sum_{{{n}=-\infty}}^{{\infty}}}{A}_{{{n}}}{e}^{{in{x}}}$.

Given that all functions satisfying the Cauchy-Riemann equations are harmonic, I wondered if the Laplace PDE with homogeneous Dirichlet boundary conditions ${\Delta}{f{{\left({z}\right)}}}={0}\forall{z}\in{\Omega}$ and ${f{{\left({z}\right)}}}{\mid}_{{\partial{\Omega}}}={0}$ could be used to generate a class of harmonic functions in ${\mathbb{{{R}}}}^{{{2}}}$. Admittedly, none would be guaranteed to correspond with analytic functions let alone approximate a desired function ${f{{\left({z}\right)}}}$ within a sufficiently small error bound.

Next, I considered the viability of taking a Fourier decomposition of the real and imaginary components separately, which could be superposed to recover the original function. While this approach merits consideration for sufficiently simple functions, I noticed that it would fail for cases where separability is more enigmatic. For an example, I turn to the Schwarz-Christoffel transform.

$${f{{\left({z}\right)}}}={\int_{{{z}_{{{0}}}}}^{{{z}}}}\frac{{{A}{\left.{d}{z}\right.}}}{{{\prod_{{{j}={1}}}^{{{n}}}}{\left({z}-{x}_{{{j}}}\right)}^{{{k}_{{{j}}}}}}}+{B}$$

In the above, ${A},{B}\in{\mathbb{{{C}}}}$ are both taken to be constants. Given the integral representation of the formula, I find that it would present a particular challenge to separate the components for an arbitrary choice of ${x}_{{{j}}}$.

Answer 1

It isn't clear what you mean by the Fourier transform of a holomorphic function. You would need to supply a definition of this concept before you can get coherent answers. Perhaps your question is about an attempt to define such a transform?

It seems like you are interested in expressing functions in some orthonormal basis. For this, you might be interested in "Bergman spaces". One example of a Bergman space is the set of all holomorphic and square integrable functions on an open set $\Omega \subset \mathbb{C}$. One can prove that these are complete Hilbert spaces, and possess an orthonormal basis. One could view this as a kind of "Fourier transform" like situation. In fact, these spaces are even nicer since they are "reproducing kernel Hilbert spaces", and have a kernel ("the Bergman Kernel") such that one can recover point values of $f$ by integrating against the kernel.

Most information about Bergman Kernels is in the Several Complex Variables literature. Krantz's SVC textbook has a good treatment.

Another direction you might be interested in exploring are all the various flavors of "Paley-Wiener" theorems. One which might be of interest to you is the following:

Let $\mathbb{H}$ be the lower half plane. Let $f: \mathbb{H} \to \mathbb{C}$ be a holomorphic function. Define $f_y: \mathbb{R} \to \mathbb{C}$ by $f_y(x) = f(x+iy)$. If the integral

$$ \int_{-\infty}^{\infty} |f_y(x)|^2 \textrm{d}x $$

is bounded above by a constant independent of $y$ then there is a function $g \in L^2(0,\infty)$ so that the Fourier transform of $g$

$$ \mathcal{F}(g)(z) = \int_0^\infty g(t)e^{-2\pi itz} \textrm{d}t $$

is equal to $f$. This gives you an inverse Fourier transform for holomorphic functions of this type defined on the lower halfplane.

For a quick reference

https://www.ndsu.edu/pubweb/~littmann/Topics15/class10-7.pdf

(beware several typos involving negative signs and factors of $2\pi$).

More in depth treatments can be found in the ``Logarithmic Integral" tomes of Koosis.

Note: We need $z$ in the lower half plane since if $z = a+bi$, we have $|e^{-2\pi itz}| = e^{2\pi tb}$. We want this to decay to $0$ as $t \to \infty$ so that the integral is well defined, so we want $b < 0$.

Answer 2

The Fourier transform for entire functions has been with us from the birth of modern distribution theory. (There are many reasons for regarding the latter as the natural framework for considering FT‘s in this context). They were considered by Schwartz in his original treatise. The key phrase is „Paley-Wiener theorem for distributions“. This gives precise estimates on an entire function which characterises when it is the FT of a distribution with compact support. A more modern discussion in English can be found in the monograph „A guide to distribution theory and Fourier transforms“ by Strichartz where this subject is treated in chapter 7–the relevant PW theorem is 7.2.4.

The subject is usually formulated in the form: the FT of a distribution is an entire function of certain type but could, of course, be regarded as stating that the FT of certain entire functions are distributions, given the symmetry between the FT and its inverse.

The simplest (and probably most useful) example is that of polynomials, whose FT’s are linear combinations of higher derivatives of the delta function— one of the reasons why it is natural to use distributions in this context.