Power series as Fourier Series

Two types of series expansion ubiquitous in mathematics are the power series of an analytic function, and the Fourier series of an integrable function on the circle. In the complex domain, well-known theorems of elementary complex analysis guarantee the existence of locally uniformly convergent power series expansions of holomorphic functions in domains with ample symmetry such as disks and annuli, where “holomorphic” can be taken in the sense of Goursat, i.e., the function is complex-differentiable at each point. On the other hand, the convergence of a Fourier series is a subtle matter, and its study has led to many developments in analysis (see [Zyg02]). The close connection between these two types of outwardly different series expansions is a recurring theme in many areas of classical analysis, e.g., in the theory of Hardy spaces.

It is however not difficult to see that at a certain level of abstraction, Fourier series and power series are in fact two examples of the same phenomenon, the representation theory of the circle group. The aim of this article is to take this idea seriously, and use it to recapture some basic results of complex analysis. The take-home message is that many properties of holomorphic functions, such as the almost-supernatural regularity phenomena are profitably thought of as expressions of symmetry, more precisely the invariance of certain locally convex spaces under the Reinhardt action of the torus group on ${\mathbb{C}}^{n}$. It is hoped that the point of view taken here has pedagogical as well as conceptual value, and will be of interest to students of complex analysis.

We begin in Section 2 with an account of Fourier series associated to a continuous representation of the $n$-dimensional torus $\mathbb{T}^{n}$ on locally convex topological vector spaces, using some ideas of [Joh76]. This provides the unifying language of sufficient generality to encompass both classical Fourier expansions and the power series representations of complex analysis. Even at this very general and “soft” level, one can establish a version of Fejér’s theorem on the summability in the Cesàro sense (Theorem  2.1), which can be thought of as a “completeness” statement for the holomorphic monomials.

Using the abstract framework of Section 2, we recapture in Section 3, in a novel way, some of the basic classical facts about holomorphic functions. We show that on a Reinhardt domain, a distribution which satisfies the Cauchy-Riemann equations (which we call a holomorphic distribution) has a complex power series representation that converges uniformly along with derivatives of all orders on compact subsets of a “relatively complete” log-convex Reinhardt domain (see Theorem 3.1 below), and thus a holomorphic distribution is a function, in fact an analytic function. Traditionally, to prove such an assertion, one would start by showing that a holomorphic distribution is actually a (smooth) function. This can be done either by ad hoc arguments for the Laplacian going back to Weyl (see [Wey40]), or in a more general way, by noticing that fundamental solution $\frac{1}{\pi z}$ of the Cauchy-Riemann operator $\frac{\partial}{\partial\overline{z}}$ is smooth, in fact real-analytic, away from the singularity at the origin. It then follows by standard arguments about convolutions (see [Hör03, Theorem 4.4.3]), that any distributional solution of $\frac{\partial}{\partial\overline{z}}u=0$ is real-analytic. Therefore, a holomorphic distribution is a holomorphic function in the sense of Goursat, and classical results of elementary complex analysis give the power series expansion via the Cauchy integral formula. The extension of the domain of convergence to the envelope of holomorphy can be obtained by convexity arguments (see [Ran86]).

While this classical argument has the admirable advantage of placing holomorphic functions in the context of solutions of hypoelliptic equations, it also has the shortcoming that the crucial property of complex-analyticity (and the associated Hartogs phenomenon in several variables) of holomorphic distributions is proved not from an analysis of the action of a differential operator on distributions, but by falling back on the Cauchy integral formula. After using the real-analytic hypoellipticity of $\frac{\partial}{\partial\overline{z}}$ to conclude that holomorphic distributions are real-analytic, we discard all of this information, except that holomorphic functions are $\mathcal{C}^{1}$ and satisfy the Cauchy-Riemann equations in the classical sense. In our approach here, however, complex analyticity of holomorphic distributions is proved directly in a conceptually straightforward way, by expanding a holomorphic distribution on a Reinhardt domain in a Fourier series, and then showing that the resulting series (the Laurent series of the holomorphic function) converges in the $\mathcal{C}^{\infty}$-topology, not only on the original set where the distribution was defined, but possibly on a larger domain, thus underlining the fact that Hartogs phenomenon can be thought of as a regularity property of the solutions of the same nature as smoothness. Our proof also clearly locates the origin of the remarkable regularity of holomorphic distributions in

• (a)

  the invariance properties of the space of holomorphic distributions under Reinhardt rotations and translations,

• (b)

  symmetry and convexity properties of the Laurent monomial functions $z\mapsto z_{1}^{\alpha_{1}}\dots z_{n}^{\alpha_{n}},$
  $\alpha_{j}\in\mathbb{Z}$ and

• (c)

  the fact that radially symmetric holomorphic distributions are constants.

It is also interesting that the fact that $\frac{1}{\pi z}$ is a fundamental solution of the Cauchy-Riemann operator, which is key to many results of complex analysis including the Cauchy integral formula, does not play any role in our approach.

The method of proving analyticity via Fourier expansion can be used in other contexts. For example, replacing the representation theory of $\mathbb{T}^{n}$ by that of the special orthogonal group $SO(n)$, the method can be used to show that a harmonic distribution in a ball of ${\mathbb{R}}^{n}$ is in fact a real analytic function and admits an expansion in solid harmonics (see, e.g., [CH53, pp. 316-317]), which converges uniformly along with all derivatives on compact subsets of the ball. Similarly, one can obtain the Taylor/Laurent expansion of a monogenic function of a Clifford-algebra variable in “spherical monogenics”, the analogs for functions of a Clifford-algebra variable of the monomial functions $z\mapsto z^{n},n\in\mathbb{Z}$ (see [BDS82]).

While the theory of generalized functions forms the natural context for solution of linear partial differential equations such as the Cauchy-Riemann equations, for aesthetic and pedagogical reasons it is natural to ask whether it is possible to develop the Fourier approach to power series, as outlined in Section 3, using only classical notions of functions as continuous mappings, and derivatives as limits of difference quotients, and without anachronistically invoking distributions. We accomplish this in the last Section 5 of this paper. We start from the assumption that a continuous function on the complex complex plane satisfies the hypothesis of the Morera theorem, and show directly that it is complex-analytic without any recourse to the Cauchy integral representation formula (which, being a case of Stoke’s theorem, requires the differentiability of the function, at least at each point). Not unexpectedly, one of the steps of the proof uses the classical triangle-division used in the standard proof of the Cauchy theorem for triangles.

The first author would like to thank Luke Edholm and Jeff McNeal for many interesting conversations about the topic of power series. He would also like to thank the students of MTH 636 and MTH 637 at Central Michigan University over the years for their many questions, which led him to think about the true significance of power series expansions.

Sections 2 and 4 of this paper are based on part of the Ph.D. thesis of the second author under the supervision of the first author. The second author would also like to thank his other committee members, Dmitry Zakharov and Sonmez Sahutoglu for their support. Other results from the thesis have appeared in the paper [Daw21].

We begin by recalling some notions and facts about functional analysis in topological vector spaces. See the textbooks [Trè67, Rud91, Bou04] for more details on these matters.

Let $X$ be a locally convex Hausdorff topological vector space (we use the standard abbreviation LCTVS). Recall that the topology of $X$ can also be described by prescribing the family of continuous seminorms on $X$: a net $\{x_{j}\}$ in $X$ converges to $x$, if and only if for each continuous seminorm $p$ on $X$, we have $p(x-x_{j})\to 0$ as a net of real numbers. In practice, we describe the topology of an LCTVS by specifying a generating family of seminorms (analogous to describing a topology by a subbasis): a collection of continuous seminorms $\{p_{k}:k\in K\}$ on $X$ is said to generate the topology of $X$ if for every continuous seminorm $q$ on $X$, there exists a finite subset $\{k_{1},\dots,k_{n}\}\subset K$ and a $C>0$ such that

and further, for every nonzero $x\in X$ there exists at least one $k\in K$ such that $p_{k}(x)\neq 0$ (this separating property ensures that the topology of $X$ is Hausdorff). Then clearly a net $\{x_{j}\}$ converges in $X$ if and only if $p_{k}(x_{j}-x)\rightarrow 0$ for each $k\in K$.

Let $\Gamma$ be a directed set with order $\geq$. Recall that a net $\{x_{\alpha}\}_{\alpha\in\Gamma}$ in $X$ is said to be Cauchy if for every $\epsilon>0$ and every continuous seminorm $p$ on $X$, there exists $\gamma\in\Gamma$ such that whenever $\alpha,\beta\in\Gamma$ and $\alpha,\beta\geq\gamma$, we have $p(x_{\alpha}-x_{\beta})<\epsilon$. The space $X$ is said to be complete if every Cauchy net in $X$ converges. Observe that in the above definition we can use a generating family of seminorms rather than all continuous seminorms.

If $\{S_{k}\}_{k\in\mathbb{N}}$ is a sequence in a vector space, we can define the sequence of the corresponding Cesàro means $\{C_{n}\}_{n\in\mathbb{N}}$ by

The following is the analog for sequences in LCTVS of an elementary fact well-known for sequences of numbers:

For a formal series $\displaystyle{\sum_{j=0}^{\infty}x_{j}}$ in an LCTVS $X$, convergence is defined in the usual way, i.e. the sequence of partial sums converges in $X$. A formal sum ${\sum_{\alpha\in\mathfrak{A}}x_{\alpha}}$ over a countable index set $\mathfrak{A}$, where $x_{\alpha}$ are vectors in an LCTVS $X$, is said to be absolutely convergent if there exists a bijection $\tau:\mathbb{N}\rightarrow\mathfrak{A}$ such that for every continuous seminorm $p$ on $X$, the series of non-negative real numbers $\displaystyle{\sum_{j=0}^{\infty}p(x_{\tau(j)})}$ is convergent (see [KK97]). To check that a series is absolutely convergent, we only need to check the convergence of the above series for seminorms in a fixed generating family. If $X$ is a locally compact Hausdorff space, absolute convergence in the Fréchet space $\mathcal{C}(X)$ of continuous complex valued functions on $X$ is what is classically called normal convergence (see [Rem91, pp. 104 ff.]). Absolute convergence is typical for many spaces of holomorphic functions, e.g. in the space of holomorphic functions on a Reinhardt domain smooth up to the boundary (see [Daw21]).

The following result, whose proof mimics the corresponding result for numbers, shows that absolutely convergent series in LCTVS behave very much like absolutely convergent series of numbers:

The element $s\in X$ is naturally called the sum of the series, and we write $s=\sum_{\alpha\in\mathfrak{A}}x_{\alpha}$.

By a famous theorem of Dvoretzky and Rogers, the converse of the above result fails when $X$ is an infinite dimensional Banach space ([KK97, Chapter 4]).

We will also need the notion of the weak (or Pettis) integral of a LCTVS valued function which we will now recall (see [Bou04, p. INT III.32-39] for details). Let $K$ be a compact Hausdorff space and let $\mu$ be a Borel measure on $K$, and let $f$ be a continuous map from $K$ to an LCTVS $X$. An element $x\in X$ is called a Pettis integral of $f$ on $K$ with respect to $\mu$ if for all $\phi\in X^{\prime}$,

where $X^{\prime}$ denotes the dual space of $X$ (the space of continuous linear functionals on $X$), and the right hand side of (2.3) is an integral of a continuous function. If $X$ is complete, one can show that there exists a unique $x\in X$ such that (2.3) holds and we denote the Pettis integral of $f$ on $K$ with respect to $\mu$ by $\displaystyle{\int_{K}f\hskip 2.84526ptd\mu=x}$. In fact, the integral exists uniquely, as soon as the space $X$ is quasi-complete, i.e. if each bounded Cauchy net in $X$ converges, where a net $\{x_{\alpha}\}_{\alpha\in\Gamma}$ is bounded if for each continuous seminorm $p$, the net of real numbers $\{p(x_{\alpha})\}_{\alpha\in\Gamma}$ is bounded. While there are situations in which this more refined existence theorem for Pettis integrals is useful (e.g. when the space $X$ is a dual space with weak-* topology), in this paper, we only consider integrals in complete LCTVS. Since each Hausdorff TVS has a unique Hausdorff completion, we can define Pettis integrals in any LCTVS, provided we allow the integral to have a value in the completion.

If $T:X\to Y$ is a continuous linear map of complete LCTVSs, and $f:K\to X$ is continuous then we have

since for any linear functional $\psi\in Y^{\prime}$,

using the fact that $\psi\circ T\in X^{\prime}$ and the definition (2.3) of the Pettis integral.

Let

be the $n$-dimensional unit torus. With the subspace topology inherited from ${\mathbb{C}}^{n}$ and binary operation defined as $(\lambda,\xi)\mapsto\lambda{\cdot}\xi=(\lambda_{1}\xi_{1},\cdots,\lambda_{n}\xi_{n})$, is a compact abelian topological group. For a LCTVS $X$, and a continuous function $f:\mathbb{T}^{n}\to X$, we denote the Pettis integral of $f$ with respect to the Haar measure of $\mathbb{T}^{n}$ (normalized to be a probability measure) by

Let $X$ be an LCTVS, and let $\lambda\mapsto\sigma_{\lambda}$ be a continuous representation of $\mathbb{T}^{n}$ on $X$. Recall that this means that for each $\lambda\in\mathbb{T}^{n}$, the map $\sigma_{\lambda}$ is an automorphism (i.e. linear self-homeomorphism) of $X$ as a topological vector space, the map $\lambda\mapsto\sigma_{\lambda}$ is a group homomorphism from $\mathbb{T}^{n}$ to $\mathrm{Aut}(X)$, and the associated map

is continuous.

Given a representation $\sigma$ of the group $\mathbb{T}^{n}$ on an LCTVS $X$, a continuous seminorm $p$ on $X$ is said to be invariant (with respect to $\sigma$) if $p(\sigma_{\lambda}(x))=p(x)$ for all $x\in X$ and $\lambda\in\mathbb{T}^{n}$.

Let $X$ be a complete LCTVS and let $\sigma$ be a continuous representation of $\mathbb{T}^{n}$ on $X$. For each $\alpha=(\alpha_{1},...,\alpha_{n})\in\mathbb{Z}^{n}$ and $x\in X$, define

the Pettis integral of the continuous function $\lambda\mapsto\lambda^{-\alpha}\sigma_{\lambda}(x)$ on $\mathbb{T}^{n}$ with respect to the Haar probability measure of $\mathbb{T}^{n}$. We will say that $\bm{\pi}^{\sigma}_{\alpha}(x)$ is the $\alpha$-th Fourier component of $x$ with respect to the representation $\sigma$. We use the standard convention with respect to multi-index powers, i.e., $\lambda^{\alpha}=\lambda_{1}^{\alpha_{1}}\dots\lambda_{n}^{\alpha_{n}}$. We will say that the subspace of $X$ defined as

is the $\alpha$-th Fourier mode of the space $X$, and we will call the map $\bm{\pi}^{\sigma}_{\alpha}$ the $\alpha$-th Fourier projection, both with respect to the representation $\sigma$. We note the following facts:

If $X$ is $L^{1}(\mathbb{T})$, the Banach space of integrable functions on $\mathbb{T}$, and if $\sigma$ is the continuous representation of $\mathbb{T}$ on $L^{1}(\mathbb{T})$ given by

an easy computation shows that for $\phi\in{\mathbb{R}}$, and $f\in L^{1}(\mathbb{T})$, we have

the $\alpha$-th term of the Fourier series of $f$.

It is therefore natural to define, for $x\in X$, the Fourier series of $x$ with respect to $\sigma$ to be the formal series

For an integer $k\geq 0$, define the $k$-th square partial sum of the Fourier series in (2.10) by

where $\left|\alpha\right|_{\infty}\coloneqq\max\big{\{}\left|\alpha_{j}\right|,1\leq j\leq n\big{\}}$. We are ready to state an abstract version of Fejér’s theorem.

For an open $\Omega\subset{\mathbb{R}}^{n}$, the space $\mathscr{D}(\Omega)$ of test functions is the $LF$-space of smooth compactly supported complex valued functions, topologized as the inductive limit of the Fréchet spaces $\mathscr{D}_{K}$ consisting, for a given compact $K\subset\Omega$, of those elements of $\mathscr{D}(\Omega)$ which have support in $K$. Recall that a subset $B\subset\mathscr{D}(\Omega)$ is bounded, if and only if there is a compact $K\subset\Omega$ such that $B\subset\mathscr{D}_{K}$, and for each nonnegative multi-index $\alpha\in\mathbb{N}^{n}$, we have

where, here and later, we will use standard multi-index conventions such as

The space of distributions $\mathscr{D}^{\prime}(\Omega)$ on $\Omega$ is the dual of $\mathscr{D}(\Omega)$, consisting of continuous linear forms on $\mathscr{D}(\Omega)$. We denote the value of a distribution $T\in\mathscr{D}^{\prime}(\Omega)$ at a test function $\phi\in\mathscr{D}(\Omega)$ by $\left\langle T,\phi\right\rangle.$ The space $\mathscr{D}^{\prime}(\Omega)$ is endowed with the usual strong dual topology. Recall that this topology is generated by the family of seminorms

In this topology, the space $\mathscr{D}^{\prime}(\Omega)$ is complete.

Given a locally integrable function $f\in L^{1}_{\mathrm{loc}}(\Omega)$, we can associate to $f$ a distribution $T_{f}$ defined by

where $dV$ denotes the Lebesgue measure of ${\mathbb{R}}^{n}$. Then the locally integrable distribution $T_{f}$ is said to be generated by $f$, and as usual we grant ourselves the the right to abuse language by identifying the distribution $T_{f}\in\mathscr{D}^{\prime}(\Omega)$ with the function $f\in L^{1}_{\mathrm{loc}}(\Omega)$.

We will use the abbreviations

for the basic constant coefficient differential operators of complex analysis, acting on functions or distributions on ${\mathbb{C}}^{n}$. If $\Omega\subset{\mathbb{C}}^{n}$ is an open set, define

the space of holomorphic distributions on $\Omega$. The subspace $\mathscr{O}(\Omega)$ is closed in the space $\mathscr{D}^{\prime}(\Omega)$ by the continuity of the operators $\bm{\overline{D}}_{j}$, and is therefore a complete LCTVS in the subspace topology.

We begin with some definitions and notational conventions. For $\lambda\in\mathbb{T}^{n}$ we denote by $\mathrm{R}_{\lambda}$ the Reinhardt rotation of ${\mathbb{C}}^{n}$ by the element $\lambda$, the linear automorphism of the vector space ${\mathbb{C}}^{n}$ given by

A domain $\Omega\subset{\mathbb{C}}^{n}$ is defined to be Reinhardt if and only if $\mathrm{R}_{\lambda}(\Omega)=\Omega$ for each $\lambda\in\mathbb{T}^{n}$. Let $Z$ denote the union of the coordinate hyperplanes of ${\mathbb{C}}^{n}$:

Recall that a Reinhardt domain $\Omega\subset{\mathbb{C}}^{n}$ is said to be log-convex, if whenever $z,w\in\Omega\setminus Z$, the point $\zeta\in{\mathbb{C}}^{n}$ belongs to $\Omega$ if there is a $t\in[0,1]$ such that for $1\leq j\leq n$

For $\alpha\in\mathbb{Z}^{n}$, let $e_{\alpha}$ be the monomial function given by

For a Reinhardt subset $\Omega\subset{\mathbb{C}}^{n}$ and $1\leq j\leq n$, let

where $\overline{\mathbb{D}}=\{\left|\zeta\right|\leq 1\}\subset{\mathbb{C}}$ is the closed disk. This can be thought of as the result of “completing” $\Omega$ in the $j$-th coordinate. Following [JP08], we say that the Reinhardt domain $\Omega\subset{\mathbb{C}}^{n}$ is relatively complete if for each $1\leq j\leq n$, whenever we have $\Omega\cap\{z_{j}=0\}\not=\emptyset$, we also have

We prove the following well-known structure theorem for holomorphic distributions on Reinhardt domains, as an application of the ideas of Section 2: