Polar-Analytic Functions: Old and New Results, Applications

Here we review the notion of polar analyticity introduced in a previous paper and successfully applied in Mellin analysis and for quadrature formulae over the positive real axis. This approach provides a simple way of describing functions which are analytic on a part of the Riemann surface of the logarithm. New results are also obtained.


Introduction and Motivation
One of the main results of Fourier analysis is the Paley-Wiener theorem (see e.g. [17]). It characterizes the so-called bandlimited functions by entire functions of exponential type. More precisely, it states that a function f ∈ L 2 (R) has a Fourier transform f that vanishes outside a compact interval [−σ, σ], where σ > 0, if and only if f has an extension to the whole complex plane as an entire function of exponential type σ, that is, with a constant C > 0. This basic theorem has many applications, in particular in Shannon's sampling theory, the reproducing kernel theory, the investigation of quadrature formulae over the real line and various other topics.
During the nineties, P.L. Butzer and S. Jansche started a systematic study of the Mellin transform with the aim to establish a Mellin analysis as a fully independent counterpart of Fourier analysis (see [10,11,13]). Later on, many results were obtained in the framework of Mellin theory, especially contributions to approximation theory. One of the main motivations for this development relies on the exponential sampling theory, introduced in a formal way by optical physicists and engineers, due to its applications in light scattering, Fraunhofer diffraction, radio-astronomy and so on (see, e.g., [9]). Using Mellin analysis, Butzer and Jansche gave a rigorous version of the exponential sampling formula for functions defined over the positive real axis (see [12]). Later on, interconnections with other basic formulae were studied (see [1]). The exponential sampling formula in its classic form can be obtained under the assumption that the involved function f is bandlimited in the Mellin transform sense. Therefore it was quite natural to investigate the structure of the space containing the Mellin bandlimited functions. This was done in several papers (see, e.g., [25]), but using a "real" approach involving certain differential operators, called transmutation operators. A fully complex version was given by Z. Szmydt and her school in Cracow (see, e.g., [23,24]) in which the authors characterize the Mellin transform of functions (distributions) with compact support. But, unlike what happens in Fourier analysis, the inverse of a Mellin transform is not a Mellin transform (in L 2 setting). Therefore the study of the structure of Mellin bandlimited functions requires a quite different approach.
A first complex result in this direction was obtained in [2] involving functions defined over the Riemann surface of the complex logarithm S log and their analytic branches on it. Indeed, it was observed that a non-trivial, Mellin bandlimited function cannot be extended to the whole complex plane as an entire function, but it may be extended to S log in such a way that the analytic branches f k are of exponential type in the sense that However, the handling of analytic branches is quite inconvenient for computational purposes. Therefore, in [3], we introduced a different approach, based on a simple geometric representation of the Riemann surface of the logarithm as a helicoidal surface in R 3 . Following this interpretation, we finally arrived at a new definition of analyticity, called polar analyticity, which uses the polar plane as a representation of S log . Here we note that this kind of representation of the Riemann surface was also considered, in different studies, in [19]. Further developments were presented in [6]. Our notion of polar analyticity is motivated by considering the complex variable z in polar coordinates z = re iθ and treating (r, θ) as Cartesian coordinates. This modified notion of analyticity gives a simple description of analytic function on parts of the Riemann surface of the logarithm without employing branches. It also reveals a direct and simple applicability to several topics in Mellin analysis, from exponential sampling theory to quadrature formulae over the positive real axis (in this respect see the papers [5,7,8]). Moreover, it is useful in order to define function spaces like Hardy-type spaces in Mellin setting.
The present article gives a survey on the development of the theory of polar-analytic functions, including also some new results, obtaining counterparts of the Cauchy integral theorems, series expansions, conformality, residuetype theorems, identity principles, and so on. The new results involve polaranalytic versions of the maximum modulus principle, Liouville's theorem with an application to a version of the fundamental theorem of algebra for the socalled polar polynomials, and Rouché's theorem. The final sections are devoted to various applications in the framework of Mellin analysis. In particular, we describe applications to exponential sampling and to quadrature formulae. These applications testify the usefulness of this new theory.

Basic Notions of Mellin Analysis
In what follows, we denote by N, N 0 , Z, the sets of positive integers, nonnegative integers and integers, respectively, by R and R + , the sets of real and positive real numbers, respectively, and by C the set of complex numbers.
Let C(R + ) be the space of all continuous functions defined on R + , and C (r) (R + ) be the space of all functions in C(R + ) having a derivative of order r that belongs to C(R + ). Analogously, by C ∞ (R + ) we denote the space of all infinitely differentiable functions. By L 1 loc (R + ), we denote the space of all measurable functions which are integrable on every bounded interval in R + .
For 1 ≤ p < +∞, let L p (R + ) be the space of all Lebesgue measurable and p-integrable complex-valued functions defined on R + endowed with the usual norm f p . Analogous notations hold for functions defined on R.
For p = 1 and c ∈ R, we introduce the space (see [10]) endowed with the norm More generally, let X p c denote the space of all functions f : R + → C such that f (·)(·) c−1/p ∈ L p (R + ) with 1 < p < ∞. Finally, for p = ∞, we define X ∞ c as the space comprising all measurable functions f : Setting In the Mellin frame, the natural concept of a pointwise derivative of a function f is given by the limit of the difference quotient involving the Mellin translation; thus if f exists, then This gives the motivation for the following definition (see [10]): The pointwise Mellin differential operator Θ c and the pointwise Mellin derivative Θ c f of a function f : R + → C and c ∈ R are defined by provided that f exists a.e. on R + . The Mellin differential operator of order r ∈ N is defined recursively by For convenience, we set Θ r := Θ r 0 and Θ 0 c := I with I denoting the identity operator.
The Mellin transform of a function f ∈ X c is the linear and bounded operator defined by (see, e.g., [10]) where in general L p (c + iR), for p ≥ 1, will mean the space of all functions More generally, for 1 < p ≤ 2, the Mellin transform M p c of f ∈ X p c is given by (see [13] for p = 2) where p is the conjugate exponent of p, that is, 1/p + 1/p = 1. Analogously, for example for p = 2, we define the inverse Mellin transform of a function g ∈ L 2 (c + iR) by where the limit is taken in see [13].

Polar-Analytic Functions
In this section, we will describe a new concept of analyticity which turns out to be very useful in the development of Mellin analysis. We begin with some preliminary facts concerning complex functions in polar coordinates. Let us first recall that in classical complex analysis the Cauchy-Riemann equations in polar coordinates are obtained by setting z = re iθ , i.e., given an analytic function f = u + iv on a domain in the complex plane, one considers the function g(r, θ) := f (re iθ ) and uses the chain rule for deriving partial derivatives of u, v with respect to the variables r, θ (see, e.g., [18]). Due to the 2π-periodicity of the exponential function e iθ , this implies a periodicity with respect to θ of the function g. As we shall see, in our definition of polar analyticity, we again derive the same Cauchy-Riemann equations in polar form, but in general this periodicity does not appear. As a consequence, this approach gives a simple description of analytic functions defined on a part of the Riemann surface of the complex logarithm without involving branches.

Definition and Elementary Properties
Let H := {(r, θ) ∈ R + × R} be the right half-plane. By a domain in H, we mean a non-empty, open and connected subset of H. Let D be a domain in H.
Polar analyticity of two functions is inherited by their arithmetic combinations and the familiar rules known for classical differentiation hold for polar derivatives as well. For details see [6].
Let f : (r, θ) → u(r, θ) + iv(r, θ) be polar-analytic on D, where u and v are real-valued. Identifying C with R 2 , we may interpret f as a mapping from a subset of the half-plane H into R 2 . Then we have (see [6]):

Proposition 1. A polar-analytic function is differentiable in the classical sense.
It can be verified that f = u + iv with u, v : D → R is polar-analytic on D if and only if u and v have continuous partial derivatives on D that satisfy the differential equations Note that these equations coincide with the Cauchy-Riemann equations of an analytic function g defined by g(z) := u(r, θ) + iv(r, θ) for z = re iθ . For the derivative D pol , we easily find that Since f = u + iv, equations (1) can be written in a more compact way as and then formula (2) takes the form Also note that D pol is the ordinary differentiation on R + . More precisely, if ϕ(·) := f (·, 0), then (D pol f )(r, 0) = ϕ (r).
When g is an entire function, then f : (r, θ) → g(re iθ ) defines a function f on H that is polar-analytic and 2π-periodic with respect to θ. Moreover, by (2) one has (D pol f )(r, θ) = g (z) with z = re iθ . A converse statement is not true in general. If f is polar-analytic on H and 2π-periodic with respect to the second variable, there may not exist an entire function h such that As examples, let us consider the function g(z) = z a , a > 0. Defining f (r, θ) := g(re iθ ) = r a e iaθ , we have f (r, θ) = u(r, θ) + iv(r, θ) := r a cos(aθ) + ir a sin(aθ) and so Analogously, taking g(z) = sin z and f (r, θ) := sin(re iθ ), we have (D pol f )(r, θ) = cos(re iθ ) = g (z).
The main novelty of the definition of polar-analytic function is that, using this approach, we avoid periodicity with respect to the argument θ, and in this way we can avoid the use of Riemann surfaces.
A simple example of a polar-analytic function that is not 2π-periodic is the function L(r, θ) := log r + iθ, which is easily seen to satisfy the differential equations (1). In this approach, we consider the logarithm as a single-valued function on H, without the use of the Riemann surface S log . Employing (2) and setting z = re iθ , we find that In order to state a connection with analytic functions on S log , for α, β ∈ R with α < β, we consider the set If f : H → C is polar-analytic but not 2π-periodic with respect to θ, then we can associate with f a function g that is analytic on the Riemann surface S log of the logarithm. The restriction of f to a strip H α+2kπ,α+2(k+1)π , where k ∈ Z, defines an analytic function g k in the slit complex plane C\{re iα : r > 0} by setting g k (re iθ ) := f (r, θ). The functions g k , for k ∈ Z, are the analytic branches of g.
As to the geometric properties of polar-analytic functions, we recall that a remarkable geometric property of an analytic function g in the classical sense is that at any point z 0 with g (z 0 ) = 0 it preserves angles and orientation. More precisely, this means the following. Let γ 1 and γ 2 be smooth arcs that intersect at z 0 where they have tangents t 1 and t 2 , respectively. Suppose that we have to rotate t 1 in the mathematically positive sense by an angle α around z 0 in order that it coincides with t 2 . Then the same is true for the arcs g • γ 1 and g • γ 2 , that is, if τ 1 and τ 2 , respectively, are their tangents in g(z 0 ), then we have to rotate τ 1 in the mathematically positive sense by the angle α around g(z 0 ) in order that it coincides with τ 2 . We cannot expect that this property extends to polar-analytic functions.
The following propositions inform us about the different geometric behavior (for the details see [6]). Examples are discussed in [6]. According to the notion of Mellin derivatives, we give the following definition (see [6]).

Definition 2.
For a fixed real number c ∈ R, we define the Mellin polar derivative as provided that the polar derivative D pol f exists at the point (r, θ) ∈ H.

Analysis of Polar-Analytic Functions
In this section, we give counterparts of the classical Cauchy integral theorems and formulae, and a Taylor-like expansion. Concerning the Cauchy integral theorems, the key result is given in the subsequent Theorem 1. It was proved in [3] for convex domains only since this sufficed for the envisaged applications. However, that approach extends to simply connected domains as may be seen by employing [16,Theorem 8.4, page 372].

Cauchy and Morera Type Theorems
The following theorem on line integrals for polar-analytic functions includes an analogue of Cauchy's fundamental theorem of complex function theory. Here a piecewise continuously differentiable curve will be called a regular curve.

Theorem 1. Let f be a polar-analytic function on a simply connected domain
D ⊂ H and let (r 1 , θ 1 ) and (r 2 , θ 2 ) be any two points in D. Then the line integral γ f (r, θ)e iθ (dr + irdθ) (4) has the same value for each regular curve γ in D that starts at (r 1 , θ 1 ) and ends (r 2 , θ 2 ). In particular, the integral vanishes for closed regular curves.
As a consequence, we obtain the following result (see [ We can also establish a converse of Theorem 1 in the spirit of Morera's theorem (see [3]). For sake of simplicity, we consider functions defined on the whole of H.

Theorem 3. Let f : H → C be a continuous function. If the integral (4) is zero on the boundary of every rectangle R ⊂ H with sides parallel to the axes, then
f is polar-analytic on H.

Cauchy Integral Formulae for Polar-Analytic Functions
In this section, we report on an analogue of the classical Cauchy integral formula. For details see [6]. In particular, if γ lies in a strip H a,b with 0 < b − a < 2π, then An interesting consequence of Theorem 4 is the following representation formula for polar derivatives:

A Series Expansion for Polar-Analytic Functions
In this section, we present a Taylor-like series expansions for a polar-analytic function. As we shall see, this expansion exists on strips of width less than 2π only. It amounts to taking an analytic branch on the Riemann surface of the logarithm and expanding it in the classical way.
In the next section, we will obtain another expansion involving the Mellin polar derivatives which has a global character, that is, it may exists on the whole of H.

Polar-Analytic Functions in Mellin Setting
In this section, we describe an analysis of polar-analytic functions in Mellin setting. A motivation comes from the fact that our analogue of Cauchy's integral formula of Theorem 4 is useful in horizontal strips of width less than 2π only since otherwise it may produce additional residues as undesired artifacts. Therefore one of the aims of this section is to establish a further analogue of Cauchy's integral formula which is always free of artifacts, and a version of the residue theorem for polar-analytic functions.

A Series Expansion with Mellin Polar Derivatives
We premise some notations and remarks. For z 0 = x 0 + iy 0 ∈ C and ρ > 0, consider the disk D(z 0 , ρ) := {z ∈ C : |z − z 0 | < ρ} in C. By x + iy → (log r, θ), where r = e x and y = θ, it is transformed into the region with one axis of symmetry in H, given by which we call a polar-disk with center (r 0 , θ 0 ) and radius ρ. Here r 0 = e x0 and θ 0 = y 0 . The boundary of this region is given by the graph of the functions Given a domain D ⊂ H, we define Now for z 0 = x 0 + iy 0 ∈ A, consider a disk D(z 0 , ρ) ⊂ A. Then, in order that the associated polar-disk E((r 0 , θ 0 ), ρ) is fully contained in D, the maximal admissible ρ > 0 is given by where ∂D denotes the boundary of D.
The following theorem gives a "global" version of the series expansion of the previous section and uses Mellin polar derivatives (see [6,7]).
A slight extension of the above expansion is given by the following theorem (see [7]).
converging on every polar-disk E (r 0 , θ 0 ), ρ ⊂ D, the convergence being uniform on compact subsets of these polar-disks.
A first remarkable consequence of Theorem 8 is the following identity theorem for polar-analytic functions (see [7]).

Theorem 9.
Let D be a domain in H and let f : D → C be polar-analytic. Suppose that (r 0 , θ 0 ) ∈ D is an accumulation point of distinct zeros of f . Then f is identically zero.

A General Cauchy Integral Formula for Polar-Analytic Functions in Mellin Setting
A second fundamental consequence of Theorem 8 is given by a general version of Cauchy's integral formula for Mellin polar derivatives (see [7]). Then, for (r 0 , θ 0 ) ∈ int(γ), c ∈ R and k ∈ N 0 , we have Other than the result of Theorem 5, the previous formula is also useful without a restriction to horizontal strips of width less than 2π since it does not produce additional residues as happens before.

A Residue Theorem for Polar-Analytic Functions in Mellin Setting
Here we introduce the notion of pole in its logarithmic representation and give a version of the residue theorem for polar-analytic functions which have poles as singularities; see [7]. We are now in a position to formulate a residue theorem for poles which will be suitable for many applications in Mellin analysis.

Further Developments
Here we obtain some new results about polar-analytic functions in Mellin setting. In particular, we discuss polar Mellin versions of the classical maximum modulus principle, of Liouville's theorem and of the fundamental theorem of algebra for polar polynomials. Note that our proofs are independent of the classical complex analysis and use only tools from the theory of polar-analytic functions developed previously.

Theorem 12.
Under the assumptions of Theorem 10, we have Proof. Writing M := max (ρ,ϑ)∈γ |f (ρ, ϑ)|, we assume for a contradiction that there exists a point (r 0 , θ 0 ) ∈ int(γ) such that |f (r 0 , θ 0 )| > M. Since for any n ∈ N the function f n also satisfies the assumptions of Theorem 10, we have by (10) with k = c = 0 that Let γ(t) := ρ(t), ϑ(t) , t ∈ [0, 1], be a representation of γ, where ρ(·) and ϑ(·) are piecewise continuously differentiable functions. Then, by (11), we can write The integral on the right-hand side does not depend on n. Hence taking the n-th root on both sides and letting n → ∞, we obtain |f (r 0 , θ 0 )| ≤ M ; a contradiction. Now, using Theorem 12, we can derive an analogue of Liouville's theorem for polar-analytic functions. We shall say that f is polar-entire if it is polaranalytic on the whole of H. Proof. Let M > 0 be such that |f (r, θ)| ≤ M for every (r, θ) ∈ H. By Theorem 8, for c = 0, we have the expansion In view of (12), we obviously have The above series has the same convergence properties as the latter. Therefore g is a polar-entire function. From the definition of g we see that and so, by Theorem 12, we have Letting ρ → ∞, we find that g(r, θ) ≡ 0 on H. This implies that f (r, θ) = f (1, 0) on H, and so f is constant.
As a consequence, we obtain a polar version of the fundamental theorem of algebra for "polar polynomials", using only Mellin tools achieved before.
First we note that, in general, a polynomial P (r, θ) in the variables r and θ is not polar-analytic. Indeed, as an example, P (r, θ) = r + iθ is not polar-analytic.
In view of Theorem 7, the natural analogue of polynomials in Mellin analysis are functions of the form P (log r + iθ), where P (z) is an ordinary polynomial. Indeed, we define polar polynomials of degree n as functions of the form where a k are complex coefficients with a n = 0. We are now ready for the following result: Theorem 14 (Fundamental Theorem of Algebra). Every polar polynomial P n (r, θ) of degree n ≥ 1 has exactly n zeros in H when counted with their multiplicities.
Proof. First assume that P n (r, θ) has no zeros. Then the function f (r, θ) := 1 P n (r, θ) is polar-analytic on H. Next we prove the boundedness of f. Clearly, f is bounded on every polar disk with center at (1, 0). Moreover, there exists a sufficiently large R > 1 such that if | log r + iθ| > R then |P n (r, θ)| ≥ | log r + iθ| n |a n | − |a 0 | | log r + iθ| n − . . . − |a n−1 | | log r + iθ| > 1. Thus |f (r, θ)| < 1 outside the polar disk E (1, 0), R , and then f is bounded on H. Hence we can use Liouville's theorem to conclude that the function P n (r, θ) is a constant, which is a contradiction since a n = 0. So far we have shown the existence of a point (r 0 , θ 0 ) ∈ H such that P n (r 0 , θ 0 ) = 0. Therefore, by Theorem 7, we can write P n (r, θ) = log(r/r 0 ) + i(θ − θ 0 ) Q n−1 (r, θ), where Q n−1 (r, θ) can be shown to be a polar polynomial of degree n − 1. Now we can apply the first part of the proof to Q n−1 , and so the assertion follows by induction.
As an interesting application of the residue theorem for polar-analytic function, we derive the following integral formula.
where N is the number of zeros in int(γ) counted with their multiplicities and P is the number of poles in int(γ) counted with their orders.