Integration of polar-analytic functions and applications to Boas’ differentiation formula and Bernstein’s inequality in Mellin setting

Abstract

We establish a general version of Cauchy’s integral formula and a residue theorem for polar-analytic functions, employing the new notion of logarithmic poles. As an application, a Boas-type differentiation formula in Mellin setting and a Bernstein-type inequality for polar Mellin derivatives are deduced.

Introduction

In [1], we introduced the notion of polar-analytic function as follows: Let \(\mathbb {H}\) be the half-plane \(\{(r,\theta ): r \in \mathbb {R}^+, \theta \in \mathbb {R}\}\), where \(\mathbb {R}^+\) the set of positive real numbers, and let f be a complex-valued function defined on a neighborhood of \((r_0, \theta _0)\in \mathbb {H}.\) Then the polar derivative of f at the point \((r_0, \theta _0)\), denoted by \((D_{\mathrm{pol}}f)(r_0, \theta _0),\) is given by the limit

$$\begin{aligned} \lim _{(r, \theta ) \rightarrow (r_0, \theta _0)}\frac{f(r, \theta ) - f(r_0, \theta _0)}{re^{i\theta } - r_0e^{i\theta _0}}. \end{aligned}$$

This definition leads naturally to the classical Cauchy–Riemann equations when written in their polar form, often considered in the literature; see e.g. [5, Sec. 23, p. 68]. Polar analyticity provides a simple alternative for functions which are analytic on a part of the Riemann surface of the logarithm and avoids the necessity of specifying analytic branches. The main applications of this concept are in Mellin analysis and in the realm of quadrature formulae on \(\mathbb {R}^+\); see [2]. It turns out to be very helpful for a foundation of Mellin analysis independent of Fourier analysis.

The corresponding notion of polar Mellin derivative has been introduced as (see [1])

$$\begin{aligned} \widetilde{\varTheta }_cf(r,\theta ):= re^{i\theta }(D_{\mathrm{pol}} f)(r, \theta ) + cf(r, \theta ). \end{aligned}$$

A first development of the theory of polar-analytic functions is given in the recent paper [3], in which extensions of Cauchy’s integral formula and Taylor-type series have been established among other results.

In Fourier analysis and signal processing, series expansions are often obtained by methods of complex analysis with the residue theorem as a decisive tool. Since in these applications only poles occur, the residue theorem can be replaced by skillful use of Cauchy’s integral formula and its extension to derivatives.

In the case of polar-analytic functions, our analogue of Cauchy’s integral formula in [3] is useful in horizontal strips of width less than \(2\pi \) only since otherwise it may produce additional residues as undesired artifacts; see [3, formula (10)]. Therefore the aim of this paper 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. As interesting applications, we establish an analogue of Boas’ differentiation formula for polar Mellin derivatives and, as a consequence, a Bernstein-type inequality. For the Fourier counterparts, see [7]. An interesting connection can be found in the papers [8, 9], in which Boas-type formulae were considered in the realm of general one-parameter groups of isometries in Banach spaces.

We wish to dedicate the present article to our close friend and fine mathematician Professor Domenico Candeloro (1951–2019). Among the wide range of interests of Professor Candeloro, we quote the paper [4], in which he gave an interesting connection between stochastic processes, approximation theory and Mellin analysis. This paper was rewarded with the JMAA Ames Award in 2014. We conclude this introduction with a personal remembrance of Paul Butzer:

Mimmo, Mimmo, please help me, I am stuck. Mimmo, Mimmo, I think that I have found a new theorem, another voice noted; Mimmo looked at the proof, with the words I feel it is an easy application of a classical result. It are these calls for assistance which drew my attention to the mathematician himself, Domenico Candeloro, “Mimmo” being the familiar name his close friends call him. It is his expertise in many areas, Calculus of Variations, Measure Theory, Real Analysis, Functional Analysis, Probability Theory, which place him in a central role in Perugia, one who is always willing to help.

At a few meetings with Mimmo, during my first years at Perugia from 1990 onward and my participations in the conference of the Research Group Real Analysis and Measure Theory, conducted by Calogero Vinti and colleagues in Capri, Ischia, Maiori, and Grado, together with my friends Carlo, Gianluca and Anna Rita, Mimmo was in the company of his spouse, Prof. Doretta Vivona. He seemed happy that she carried on the conversation (in English). Both were very friendly, Mimmo having an engaging and modest personality.

Preliminaries

Let \(\mathbb {H}:= \{(r,\theta ) \in \mathbb {R}^+ \times \mathbb {R}\}\) be the right half-plane and let \(\mathcal{D}\) be a domain in \(\mathbb {H},\) which means that \(\mathcal {D}\) is open, connected and non-empty.

Definition 1

We say that \(f:\mathcal{D}\rightarrow \mathbb {C}\) is polar-analytic on \(\mathcal{D}\) with polar derivative \(D_\mathrm{pol}f\) if for any \((r_0, \theta _0) \in \mathcal{D}\) the limit

$$\begin{aligned} \lim _{(r,\theta ) \rightarrow (r_0, \theta _0)}\frac{f(r, \theta ) - f(r_0, \theta _0)}{re^{i\theta } - r_0e^{i\theta _0}} =: (D_{\mathrm{pol}}f)(r_0, \theta _0) \end{aligned}$$

exists and is the same howsoever \((r, \theta )\) approaches \((r_0, \theta _0)\) within \(\mathcal{D}.\)

For the properties of the polar-analytic functions and their connections with the analytic functions in classical sense, see [3]. In particular, \(f = u + iv\) with real-valued functions u and v is polar-analytic on \(\mathcal{D}\) if and only if u and v have continuous partial derivatives on \(\mathcal{D}\) that satisfy the classical Cauchy–Riemann differential equations in polar form, and we have

$$\begin{aligned} (D_{\mathrm{pol}}f)(r, \theta )&= e^{-i\theta }\bigg (\frac{\partial }{\partial r}u(r, \theta ) + i \frac{\partial }{\partial r}v (r, \theta ) \bigg ) \nonumber \\&= \frac{e^{-i\theta }}{r}\bigg (\frac{\partial }{\partial \theta }v (r, \theta ) - i \frac{\partial }{\partial \theta }u (r, \theta ) \bigg ). \end{aligned}$$
(1)

As we anticipated in the introduction, for a fixed real number \(c \in \mathbb {R},\) we define the polar derivative of a polar-analytic function f in Mellin setting by the formula

$$\begin{aligned} \widetilde{\varTheta }_cf(r,\theta ):= re^{i\theta }(D_{\mathrm{pol}} f)(r, \theta ) + cf(r, \theta ). \end{aligned}$$
(2)

For \(\varphi (r):= f(r, 0),\) we have \(\widetilde{\varTheta }_cf(r,0) = (\varTheta _c\varphi )(r) := r \varphi '(r)+c\varphi (r),\) where \(\varTheta _c\) is the usual Mellin differential operator; see [6].

The higher order polar Mellin derivatives may be defined through the representation formula for (usual) Mellin derivatives in terms of Stirling numbers of the second type \(S_c(k,j)\) (see [6, Lemma 9]), namely

$$\begin{aligned} \widetilde{\varTheta }^{k}_cf(r,\theta ):= \sum _{j=0}^{k} S_c(k,j)r^je^{ij\theta } D_\mathrm{pol}^{j}f(r,\theta ) \qquad \bigl (k\in \mathbb {N}, \,(r,\theta ) \in \mathbb {H}). \end{aligned}$$
(3)

A general Cauchy integral formula

We begin with the following theorem whose proof is a slight extension of that of [3, Theorem 6.1]. For \((r_0, \theta _0) \in \mathbb {H}\) and \(\rho >0\), the polar disk centered at \((r_0, \theta _0)\) with radius \(\rho \) is defined by

$$\begin{aligned} E((r_0,\theta _0), \rho ):= \left\{ (r,\theta ) \in \mathbb {H}: \bigg (\log \frac{r}{r_0}\bigg )^2 + (\theta -\theta _0)^2 < \rho ^2\right\} . \end{aligned}$$

Theorem 1

Let \(f\,:\,\mathcal {D} \rightarrow \mathbb {C}\) be polar-analytic on a domain \(\mathcal {D}\subset \mathbb {H}\) and let \(c\in \mathbb {R}\). If \((r_0,\theta _0)\in \mathcal {D}\), then there holds the expansion

$$\begin{aligned} \bigl (re^{i\theta }\bigr )^c f(r,\theta )\,=\, \bigl (r_0e^{i\theta _0}\bigr )^c \sum _{k=0}^\infty \frac{\bigl (\widetilde{\varTheta }_c^kf\bigr )(r_0,\theta _0)}{k!} \left( \log \frac{r}{r_0} + i(\theta -\theta _0)\right) ^k, \end{aligned}$$
(4)

converging uniformly on every polar disk \(E\bigl ((r_0,\theta _0),\rho \bigr )\subset \mathcal {D}\).

Proof

Setting \(A\,:=\,\left\{ z=x+iy\in \mathbb {C}\,:\, (e^x,y)\in \mathcal {D}\right\} ,\) we have shown in [3, proof of Theorem 6.1] that \( g : z=x+iy \, \longmapsto \, f(e^x,y)\) is analytic on A and for \(z_0:=\log r_0+i\theta _0\), we have \( g^{(k)}(z_0)\,=\,\bigl (\widetilde{\varTheta }_0^kf\bigr )(r_0,\theta _0).\) Now define \(h(z):= e^{cz} g(z)\). Then h is also analytic on A and

$$\begin{aligned} h^{(k)}(z_0) = \bigl (r_0e^{i\theta _0}\bigr )^c \sum _{j=0}^k {k\atopwithdelims ()j} \bigl (\widetilde{\varTheta }_0^jf\bigr )(r_0,\theta _0) c^{k-j}. \end{aligned}$$

It can be shown by induction on k that the sum on the right-hand side is equal to \(\bigl (\widetilde{\varTheta }_c^kf\bigr )(r_0,\theta _0).\) Since by Taylor expansion

$$\begin{aligned} e^{cz} g(z)\,=\, \sum _{k=0}^\infty \frac{h^{(k)}(z_0)}{k!}\,(z-z_0)^k, \end{aligned}$$

we obtain (4) by substituting \(z=x+iy\) with \(x=\log r\) and \(y=\theta \). The statement on convergence is seen as in [3]. \(\square \)

As a consequence of the series expansion (4), we deduce the following identity theorem for polar-analytic functions.

Theorem 2

Let \(\mathcal {D}\) be a domain in \(\mathbb {H}\) and let \(f\,:\,\mathcal {D} \rightarrow \mathbb {C}\) be polar-analytic. Suppose that \((r_0, \theta _0)\in \mathcal {D}\) is an accumulation point of distinct zeros of f. Then f is identically zero.

Proof

As an auxiliary result, we first prove that \(\bigl (\widetilde{\varTheta }_0^kf\bigr )(r_0, \theta _0)=0,\) for all \(k\in \mathbb {N}_0.\) Indeed, assume that there exists a smallest non-negative integer \(k_0\) such that \(\bigl (\widetilde{\varTheta }_0^{k_0}f\bigr )(r_0, \theta _0)\ne 0\). Then, by Theorem 1 for \(c=0\), f has an expansion

$$\begin{aligned} f(r,\theta )&=\left( \log \frac{r}{r_0} +i(\theta -\theta _0)\right) ^{k_0}\\&\quad \times \left( \frac{\bigl (\widetilde{\varTheta }_0^{k_0}f\bigr ) (r_0,\theta _0)}{k_0!} + \sum _{k=k_0+1}^\infty \frac{\bigl (\widetilde{\varTheta }_0^kf\bigr ) (r_0,\theta _0)}{k!}\left( \log \frac{r}{r_0}+i(\theta -\theta _0)\right) ^{k-k_0}\right) \end{aligned}$$

converging in a polar disk \(E((r_0,\theta _0),\rho )\) with \(\rho >0\). On the right-hand side, the first factor vanishes for \((r,\theta )=(r_0,\theta _0)\) only if \(k_0\ge 1\); otherwise it is identically 1. The second factor is different from zero at \((r,\theta )=(r_0,\theta _0)\). By continuity, it is also different from zero in a sufficiently small neighborhood of \((r_0, \theta _0)\). Hence \((r_0, \theta _0)\) cannot be an accumulation point of distinct zeros of f. A contradiction!

Now, let \((r^*, \theta ^*)\) be an arbitrary point in \(\mathcal {D}\). It suffices to show that \(f(r^*, \theta ^*)=0.\) Let \(\gamma \,:\,[0,1] \rightarrow \mathcal {D}\) be a Jordan arc in \(\mathcal {D}\) such that \(\gamma (0)=(r_0, \theta _0)\) and \(\gamma (1)= (r^*, \theta ^*)\). Since the trace of \(\gamma \) has a positive distance from the boundary of \(\mathcal {D}\), there exists a number \(\rho _\mathrm{inf}>0\) such that for each point \((r, \theta )\) on \(\gamma \) there is a polar disk \(E((r, \theta ),\rho )\subset \mathcal {D}\) with \(\rho \ge \rho _\mathrm{inf}\). First consider a polar disk \(E_0:=E((r_0, \theta _0), \rho _0)\subset \mathcal {D}\) with \(\rho _0\ge \rho _\mathrm{inf}.\) By Theorem 1, we have a representation

$$\begin{aligned} f(r, \theta )\,=\, \sum _{k=0}^\infty \frac{\bigl (\widetilde{\varTheta }_0^kf\bigr )(r_0, \theta _0)}{k!}\left( \log \frac{r}{r_0} + i(\theta -\theta _0)\right) ^k \end{aligned}$$

holding for all \((r,\theta )\in E_0\). Now, by the auxiliary result, it follows that the restriction of f to \(E_0\) is identically zero. Thus, if \((r^*, \theta ^*)\in E_0\), we have reached our aim. Otherwise, let \((r_1, \theta _1):=\gamma (t_1)\) with \(0<t_1\le 1\) be a point where \(\gamma \) intersects the boundary of \(E_0\). By our previous observations, there exists a polar disk \(E_1:=E((r_1, \theta _1), \rho _1)\subset \mathcal {D}\) with \(\rho _1\ge \rho _\mathrm{inf}\) such that

$$\begin{aligned} f(r, \theta )\,=\, \sum _{k=0}^\infty \frac{\bigl (\widetilde{\varTheta }_0^kf\bigr )(r_1, \theta _1)}{k!}\left( \log \frac{r}{r_1} + i(\theta -\theta _1)\right) ^k \end{aligned}$$

for all \((r,\theta )\in E_1\). Since \((r_1, \theta _1)\), being a boundary point of \(E_0\), is an accumulation point of distinct zeros of f, we conclude with the help of the auxiliary result that the restriction of f to \(E_1\) vanishes identically. If \((r^*, \theta ^*)\not \in E_1\), we continue this procedure with a point \((r_2, \theta _2):=\gamma (t_2)\), where \(t_1< t_2 \le 1\), lying on the boundary of \(E_1\). Since \(\rho _\mathrm{inf}>0\), we arrive after a finite number of steps at a polar disk \(E_n\), say, such that the restriction of f to \(E_n\) vanishes identically and \((r^*, \theta ^*)\in E_n\). This completes the proof. \(\square \)

Now we state the announced Cauchy integral formula for polar-analytic functions in its general form. Here a curve will be called regular if it is piecewise continuously differentiable.

Theorem 3

Let \(\mathcal {D}\) be a convex domain in \(\mathbb {H}\), and let \(f\,:\, \mathcal {D} \rightarrow \mathbb {C}\) be polar-analytic on \(\mathcal {D}\). Let \(\gamma \) be a positively oriented, closed, regular curve that is the boundary of a convex domain \(\mathrm{int}(\gamma )\subset \mathcal {D}\). Then, for \((r_0, \theta _0) \in \mathrm{int}(\gamma )\), \(c\in \mathbb {R}\) and \(k\in \mathbb {N}_0\), we have

$$\begin{aligned} \frac{1}{2\pi i} \int _\gamma \frac{(re^{i\theta })^{c-1} f(r,\theta ) e^{i\theta }}{\bigl (\log (r/r_0)+i (\theta -\theta _0)\bigr )^{k+1}} (dr +ir d\theta )\,=\, \bigl (r_0e^{i\theta _0}\bigr )^c \frac{(\widetilde{\varTheta }^k_cf)(r_0, \theta _0)}{k!}\,. \end{aligned}$$
(5)

Proof

As in [3, Lemma 5.1], we draw a positively oriented circle \(\lambda _\varepsilon \) of sufficiently small radius \(\varepsilon >0\) around \((r_0, \theta _0)\) and decompose \(\text{ int }(\gamma )\) into four parts using line segments which lie on rays starting at \((r_0, \theta _0)\) and intersecting under an angle \(2\pi /3.\) Denote the positively oriented boundaries of the three parts other than the disk by \(\beta _1, \beta _2\) and \(\beta _3.\) These are closed regular curves. It is geometrically evident that each \(\beta _j,\) lies in a convex subset of \(\mathcal {D}\) on which the function

$$\begin{aligned} F\,:\, (r, \theta )\,\longmapsto \, \frac{(re^{i\theta })^{c-1} f(r,\theta )}{\bigl (\log (r/r_0)+i(\theta -\theta _0)\bigr )^{k+1}} \end{aligned}$$
(6)

is polar-analytic. By [3, Theorem 4.1], we have

$$\begin{aligned} \int _{\beta _j} F(r,\theta ) e^{i\theta } (dr+ ir d\theta )\,=\,0 \quad (j=1,2,3). \end{aligned}$$

This enables us to conclude that

$$\begin{aligned} \frac{1}{2\pi i} \int _\gamma F(r,\theta ) e^{i\theta } (dr + ir d\theta )\,=\, \frac{1}{2\pi i} \int _{\lambda _\varepsilon } F(r,\theta ) e^{i\theta }(dr + ir d\theta ). \end{aligned}$$
(7)

From Theorem 1 we know that

$$\begin{aligned} F(r,\theta )\,=\,\bigl (r_0e^{i\theta _0}\bigr )^{c-1} \sum _{\ell =0}^\infty \frac{\bigl (\widetilde{\varTheta }_{c-1}^\ell f\bigr )(r_0,\theta _0)}{\ell !} \left( \log \frac{r}{r_0} + i(\theta -\theta _0)\right) ^{\ell -k-1}. \end{aligned}$$

Note that on the right-hand side all terms with \(\ell \ge k+1\) are polar-analytic as functions of \((r, \theta )\). Thus

$$\begin{aligned} \frac{1}{2\pi i} \int _{\lambda _\varepsilon } F(r,\theta ) e^{i\theta } (dr+ ir d\theta ) \,=\, \bigl (r_0 e^{i\theta _0}\bigr )^{c-1} \sum _{\ell =0}^k \frac{\bigl (\widetilde{\varTheta }_{c-1}^\ell f\bigr )(r_0, \theta _0)}{\ell !} I_{k+1-\ell }, \end{aligned}$$
(8)

where

$$\begin{aligned} I_j\,:=\, \frac{1}{2\pi i} \int _{\lambda _\varepsilon }\frac{e^{i\theta } (dr+ ird\theta )}{\bigr (\log (r/r_0) + i(\theta -\theta _0)\bigr )^j} \qquad (j=1, \ldots , k+1). \end{aligned}$$

Now consider the mapping \((r,\theta ) \mapsto z=re^{i\theta }\). If \((r, \theta )\) traverses \(\lambda _\varepsilon \) once, then, for sufficiently small \(\varepsilon >0\), the point z runs along a piecewise continuously differentiable, closed curve \(\gamma _\varepsilon \), say, in \(\mathbb {C}\), surrounding \(z_0=r_0e^{i\theta _0}\) exactly once in the mathematically positive sense. Therefore,

$$\begin{aligned} I_j = \frac{1}{2\pi i} \int _{\gamma _\varepsilon } \frac{dz}{(\log z - \log z_0)^j} = \frac{1}{2\pi i} \int _{\log \circ \gamma _\varepsilon }\frac{e^w dw}{(w-\log z_0)^j} =\frac{r_0e^{i\theta _0}}{(j-1)!}\,. \end{aligned}$$

With this result (8) becomes

$$\begin{aligned} \frac{1}{2\pi i} \int _{\lambda _\varepsilon } F(r,\theta ) e^{i\theta } (dr+ ir d\theta ) =\frac{\bigl (r_0 e^{i\theta _0}\bigr )^c}{k!} \bigl (\widetilde{\varTheta }_c^kf\bigr )(r_0, \theta _0). \end{aligned}$$

The last conclusion is seen by noting that \(\widetilde{\varTheta }_cf=\widetilde{\varTheta }_{c-1}f +f\) and using this relation repeatedly. This completes the proof. \(\square \)

A polar residue theorem

First we introduce some notions.

Definition 2

Let \((r_0,\theta _0)\in \mathbb {H}\) and let \(\mathcal {U}\subset \mathbb {H}\) be an open neighborhood of \((r_0,\theta _0)\).

  1. (i)

    If \(f\,:\,\mathcal {U}{\setminus }\bigl \{(r_0,\theta _0)\bigr \} \rightarrow \mathbb {C}\) is polar-analytic, then \((r_0,\theta _0)\) will be called an isolated singularity of f.

  2. (ii)

    An isolated singularity \((r_0,\theta _0)\) is said to be a logarithmic pole of order k if \(k\in \mathbb {N}\) and there exists a polar-analytic function \(g\,:\,\mathcal {U}\rightarrow \mathbb {C}\) with \(g(r_0,\theta _0)\ne 0\) such that

    $$\begin{aligned} f(r,\theta )\,=\, \frac{g(r,\theta )}{\bigl (\log (r/r_0)+i(\theta -\theta _0)\bigr )^k} \quad \hbox { for } (r,\theta )\in \mathcal {U}{\setminus }\bigl \{(r_0,\theta _0)\bigr \}. \end{aligned}$$
    (9)

    In this case

    $$\begin{aligned} \bigl (\mathrm{res}_cf\bigr )(r_0,\theta _0)\,:=\, \bigl (r_0e^{i\theta _0}\bigr )^c \, \frac{\bigl (\widetilde{\varTheta }_c^{k-1}g\bigr )(r_0, \theta _0)}{(k-1)!} \end{aligned}$$
    (10)

    will be called the c-residue of f at \((r_0,\theta _0)\).

We are now in a position to formulate a residue theorem for logarithmic poles which will be suitable for many applications in Mellin analysis.

Theorem 4

Let \(\mathcal {D}\) be a convex domain in \(\mathbb {H}\) and let f be polar-analytic on \(\mathcal {D}\) except for isolated singularities which are all logarithmic poles. Let \(\gamma \) be a positively oriented, closed, regular curve that is the boundary of a convex domain \(\mathrm{int}(\gamma )\subset \mathcal {D}\). Suppose that no isolated singularity lies on \(\gamma \) while \((r_j, \theta _j)\) for \(j=1, \ldots , m\) are the isolated singularities lying in \(\mathrm{int}(\gamma ).\) Then, for \(c\in \mathbb {R}\), there holds

$$\begin{aligned} \int _\gamma \bigl (re^{i\theta }\bigr )^{c-1} f(r,\theta )e^{i\theta }\,(dr+ird\theta )\,=\, 2\pi i \sum _{j=1}^m \bigl (\mathrm{res}_cf\bigr )(r_j,\theta _j). \end{aligned}$$

Proof

For \((x_0, y_0)\in \mathbb {R}^2\) and \(\varepsilon >0\), we consider the vertical grid

$$\begin{aligned} V(x_0, \varepsilon )\,:=\, \bigl \{(r,\theta )\in \mathbb {R}^2\,:\, \theta \in \mathbb {R},\, r=x_0+n\varepsilon ,\, n\in \mathbb {Z}\bigr \} \end{aligned}$$

and the horizontal grid

$$\begin{aligned} H(y_0, \varepsilon )\,:=\, \bigl \{(r,\theta )\in \mathbb {R}^2\,:\, r\in \mathbb {R}, \, \theta =y_0+n\varepsilon ,\, n\in \mathbb {Z}\bigr \}. \end{aligned}$$

Their union constitutes a net \( N\bigl ((x_0,y_0), \varepsilon \bigr )\,:=\, V(x_0, \varepsilon ) \cup H(y_0,\varepsilon ).\) The intersection \(\mathrm{int}(\gamma )\cap N\bigl ((x_0,y_0), \varepsilon \bigr )\) creates a tessellation of \(\mathrm{int}(\gamma )\) into squares with edges of length \(\varepsilon \) and further convex sets whose boundary contains a piece of \(\gamma \). The latter may be called boundary sets. The collection of all subsets of the tessellation shall be denoted by \(\mathcal {T}\). If \(\varepsilon >0\) is sufficiently small, then the boundary sets will be free of isolated singularities and different isolated singularities in \(\mathrm{int}(\gamma )\) will lie in different squares of the tessellation. Furthermore, if \((r_j, \theta _j)\) lies in the interior of a square, then it will remain there under all sufficiently small variations of \((x_0,y_0)\). But if \((r_j, \theta _j)\) lies on the boundary of a square, then there exist arbitrarily small variations such that \((r_j, \theta _j)\) goes inside. We may therefore assume that each \((r_j, \theta _j)\) lies inside a square \(Q_j \in \mathcal {T}\) for \(j=1, \ldots , m\). For each set \(P\in \mathcal {T}\), we denote by \(\partial P\) its positively oriented boundary. Note that each line segment in \(\mathrm{int}(\gamma )\) which comes from a mesh of the net belongs to the boundaries of exactly two subsets of the tessellation where it occurs with opposite orientations. Therefore

$$\begin{aligned} \int _\gamma \bigl (re^{i\theta }\bigr )^{c-1} f(r,\theta ) e^{i\theta } (dr +ird\theta ) = \sum _{j=1}^m \int _{\partial Q_j}\bigl (re^{i\theta }\bigr )^{c-1} f(r,\theta ) e^{i\theta }(dr + ir d\theta ) \end{aligned}$$

since by [3, Theorem 4.1] the integral along \(\partial P\) vanishes if P does not contain an isolated singularity. As all our isolated singularities are logarithmic poles, we conclude with the help of Theorem 3 and by using the notion of residue that

$$\begin{aligned} \int _{\partial Q_j}\bigl (re^{i\theta }\bigr )^{c-1} f(r,\theta ) e^{i\theta }(dr + ir d\theta )\,=\, 2\pi i \bigl (\mathrm{res}_cf\bigr )(r_j,\theta _j) \quad (j=1, \ldots , m). \end{aligned}$$

This completes the proof. \(\square \)

Remark 1

The proof of Theorem 4 via Theorem 3 reveals that if (9) holds with \(g(r_0,\theta _0)\) being not necessarily different from zero, which implies that the order of the logarithmic pole is at most k, then formula (10) still provides the correct value of the c-residue of f at \((r_0,\theta _0)\).

Boas’ differentiation formula and Bernstein’s inequality for polar Mellin derivatives

For \(p\in [1, \infty {[}\), denote by \(\Vert \cdot \Vert _p\) the norm of the Lebesgue space \(L^p(\mathbb {R}^+)\). In Mellin analysis, the analogue of \(L^p(\mathbb {R}^+)\) are the spaces \(X^p_c\), where \(c \in \mathbb {R},\) comprising all functions \(f: \mathbb {R}^+\rightarrow \mathbb {C}\) such that \(f(\cdot ) (\cdot )^{c-1/p}\in L^p(\mathbb {R}^+)\) with the norm \(\Vert f\Vert _{X^p_c} := \Vert f(\cdot ) (\cdot )^{c-1/p}\Vert _p.\) Furthermore, for \(p=\infty \), we define \(X^\infty _c\) as the space of all measurable functions \(f : \mathbb {R}^+\rightarrow \mathbb {C}\) such that \(\Vert f\Vert _{X^\infty _c}:= \sup _{x>0}x^{c}|f(x)| < \infty .\)

We recall (see [1]) that the Mellin–Bernstein space \(\mathscr {B}^p_{c,T}\) comprises all functions \(f\,:\, \mathbb {H} \rightarrow \mathbb {C}\) with the following properties:

  1. (i)

      f is polar-analytic on \(\mathbb {H}\);

  2. (ii)

      \(f(\cdot ,0) \in X_c^p\);

  3. (iii)

      there exists a constant \(C_f>0\) such that \(r^c \left| f(r, \theta )\right| \le C_f e^{T|\theta |}\) for all \((r, \theta )\in \mathbb {H}.\)

Next we state three useful assertions on transformations in Mellin–Bernstein spaces. They are verified by straightforward calculations. To show polar analyticity, we simply check that the Cauchy–Riemann equations in polar form are satisfied. Concerning statement (c) below, for verifying property (ii) in the definition of Mellin–Bernstein spaces, we will use Theorem 4.2 in [1].

Proposition 1

Let \(f\in \mathscr {B}^p_{c,T}\), where \(p \in [1, +\infty ]\), \(c\in \mathbb {R}\) and \(T>0\).

  1. (a)

    If \(t>0\) and \(g : (r, \theta ) \, \longmapsto \, t^c\,f(tr, \theta ),\) then \(g\in \mathscr {B}^p_{c,T};\) in particular, \(\Vert g(\cdot , 0)\Vert _{X_c^p} =\Vert f(\cdot , 0)\Vert _{X_c^p}\) and \(r^c \left| g(r,\theta )\right| \le C_f e^{T |\theta |}\) for all \((r,\theta )\in \mathbb {H}\). Furthermore, \(\bigl (\widetilde{\varTheta }_cg\bigr )(r,\theta )\,=\, t^c\bigl (\widetilde{\varTheta }_cf\bigr )(tr,\theta ).\)

  2. (b)

    If \(h : (r, \theta ) \, \longmapsto \, f(r^{1/T}, \theta /T),\) then \(h\in \mathscr {B}^p_{c/T,1};\) in particular, \(\Vert h(\cdot , 0)\Vert _{X_{c/T}^p} =\Vert f(\cdot , 0)\Vert _{X_c^p}\) and \(r^{c/T} \left| h(r,\theta )\right| \le C_f e^{|\theta |}\) for all \((r,\theta )\in \mathbb {H}\). Furthermore,

    $$\begin{aligned} \bigl (\widetilde{\varTheta }_{c/T}h\bigr )(r,\theta )\,=\, \frac{1}{T}\, \bigl (\widetilde{\varTheta }_cf\bigr )\left( r^{1/T}, \frac{\theta }{T}\right) . \end{aligned}$$
  3. (c)

    If \(\alpha \in \mathbb {R}\) and \( \phi : (r,\theta )\,\longmapsto \, f(r, \theta +\alpha ),\) then \(\phi \in \mathscr {B}^p_{c,T}\); in particular, \(\Vert \phi (\cdot ,0)\Vert _{X_c^p}\le e^{T |\alpha |}\,\Vert f(\cdot ,0)\Vert _{X_c^p}\) and \(r^c \left| \phi (r, \theta )\right| \le C_f e^{T(|\alpha |+|\theta |)}\) for all \((r, \theta )\in \mathbb {H}\). Furthermore, \( \bigl (\widetilde{\varTheta }_c\phi \bigr )(r,\theta )\,=\, \bigl (\widetilde{\varTheta }_cf\bigr )(r, \theta +\alpha ).\)

Now we establish a formula for the polar Mellin derivatives. It is analogous to a differentiation formula of Boas for bandlimited functions in Fourier analysis. An abstract version of this result was obtained in the setting of one-parameter groups of operators in [8, § 3.3].

Theorem 5

Let \(f\in \mathscr {B}^p_{c,T}\), where \(p \in [1, +\infty ]\), \(c\in \mathbb {R}\) and \(T>0\). Then

$$\begin{aligned} \bigl (\widetilde{\varTheta }_cf\bigr )(r, \theta )\,=\, \frac{4T}{\pi ^2} \sum _{k\in \mathbb {Z}} \frac{(-1)^k}{(2k+1)^2}\,\, e^{(k+1/2)\pi c/T} f\bigl (r e^{(k+1/2)\pi /T}, \theta \bigr ) \end{aligned}$$
(11)

for \((r,\theta )\in \mathbb {H}\). Multiplied by \(r^c\), the series converges absolutely and uniformly on strips of bounded width parallel to the r-axis in \(\mathbb {H}\).

Proof

For simplicity, we first suppose that \(T=1\). Consider the function

$$\begin{aligned} F\,:\, (r,\theta ) \, \longmapsto \, \frac{f(r,\theta )}{(\log r+i\theta )^2 \cos (\log r+i\theta )}\,. \end{aligned}$$

It is polar-analytic on \(\mathbb {H}\) except for isolated singularities at the points where the denominator vanishes. Writing \(r_k:= e^{(k+1/2)\pi }\) for short, we obtain the exceptional points as (1, 0) and \((r_k, 0)\) for \(k\in \mathbb {Z}\). The first one is a logarithmic pole of order at most two; all the others are logarithmic poles of order at most one.

For \(n\in \mathbb {N}\), let \(\mathcal {R}_n\) be the rectangle with vertices at \((e^{\pm n\pi }, \pm n\pi )\) and denote by \(\partial \mathcal {R}_n\) its positively oriented boundary. Employing Theorem 4, we find that

$$\begin{aligned} \frac{1}{2\pi i} \int _{\partial \mathcal {R}_n} \bigl (re^{i\theta }\bigr )^{c-1}F(r,\theta )e^{i\theta } (dr +ird\theta ) = \bigl (\mathrm{res}_cF\bigr )(1,0) + \sum _{k=-n}^{n-1} \bigl (\mathrm{res}_cF\bigr )(r_k,0). \end{aligned}$$
(12)

Introducing \( \phi (r,\theta )\,:=\, f(r, \theta )\,(\cos (\log r +i\theta ))^{-1}\,,\) we have in view of Remark 1 that \( \bigl (\mathrm{res}_cF\bigr )(1,0)\,=\,\bigl (\widetilde{\varTheta }_c\phi \bigr )(1,0)\,=\, \bigl (\widetilde{\varTheta }_cf\bigr )(1,0).\) For calculating the other c-residues, we factor the cosine as

$$\begin{aligned} \cos (\log r + i\theta )\,=\, \bigl (\log r+i\theta - (k+\frac{1}{2})\pi \bigr ) \,\psi _k(r,\theta ), \end{aligned}$$

where \(\psi _k\) is polar-analytic in a neighborhood of \((r_k, 0)\) and

$$\begin{aligned} \psi _k(r_k,0)\,=\, \lim _{(r,\theta )\rightarrow (r_k,0)} \frac{\cos (\log r+i\theta )}{\log r+i\theta -(k+\frac{1}{2}) \pi } \,=\, (-1)^{k+1}. \end{aligned}$$

Thus,

$$\begin{aligned} \bigl (\mathrm{res}_cF\bigr )(r_k,0)\,=\, \frac{r_k^c f(r_k,0)}{\bigl ((k+\frac{1}{2})\pi \bigr )^2 \psi _k (r_k,0)} \,=\, \frac{4}{\pi ^2}\,\frac{(-1)^{k+1}}{(2k+1)^2} \,r_k^c f(r_k,0). \end{aligned}$$

With these values of the c-residues, we may rewrite (12) as

$$\begin{aligned} \bigl (\widetilde{\varTheta }_c f\bigr )(1,0)&= \frac{4}{\pi ^2} \sum _{k=-n}^{n-1} \frac{(-1)^k}{(2k+1)^2} r_k^c f(r_k,0)\\&\quad + \frac{1}{2\pi i} \int _{\partial \mathcal {R}_n} \bigl (re^{i\theta }\bigr )^{c-1}F(r,\theta ) e^{i\theta } (dr + ir d\theta ). \end{aligned}$$

Noting that \(\left| \cos (\log r +i\theta )\right| > e^{|\theta |}/3\)  on \(\partial \mathcal {R}_n\) and employing property (iii) of functions belonging to \(\mathscr {B}^p_{c,1}\), we see that

$$\begin{aligned} r^{c-1}\left| F(r,\theta )\right| \,\le \, \frac{3C_f}{r(\log ^2r +\theta ^2)} \quad \hbox { for } (r,\theta )\in \partial \mathcal {R}_n. \end{aligned}$$

Considering the integrals along the vertical and the horizontal line segments of \(\partial \mathcal {R}_n\) separately, we find that

$$\begin{aligned} \left| \frac{1}{2\pi i} \int _{\partial \mathcal {R}_n} \bigl (re^{i\theta }\bigr )^{c-1} F(r,\theta ) e^{i\theta } (dr +ir d\theta )\right| \le \frac{6C_f}{n\pi ^2} \int _{-1}^1 \frac{dx}{1+x^2} \,\longrightarrow \, 0 \hbox { as } n\rightarrow \infty . \end{aligned}$$

Thus

$$\begin{aligned} \bigl (\widetilde{\varTheta }_c f\bigr )(1,0) \,=\, \frac{4}{\pi ^2} \sum _{k=-\infty }^\infty \frac{(-1)^k}{(2k+1)^2}\, r_k^c f(r_k,0)\quad \hbox { for } f\in \mathscr {B}^p_{c,1}. \end{aligned}$$

With the Mellin translation of Proposition 1(a), for \(f\in \mathscr {B}^p_{c,1},\) we deduce

$$\begin{aligned} \bigl (\widetilde{\varTheta }_c f\bigr )(t,0) \,=\, \frac{4}{\pi ^2} \sum _{k=-\infty }^\infty \frac{(-1)^k}{(2k+1)^2}\, r_k^c f(tr_k,0)\quad (t>0). \end{aligned}$$

Next we want to extend this formula to \(f \in \mathscr {B}^p_{c,T}\) with an arbitrary \(T>0\). For this, we note that the formula is valid for the function h defined in Proposition 1(b) provided that we replace c with c/T. Expressing h in terms of f, we obtain

$$\begin{aligned} \bigl (\widetilde{\varTheta }_c f\bigr )(t^{1/T},0) \,=\, \frac{4T}{\pi ^2} \sum _{k=-\infty }^\infty \frac{(-1)^k}{(2k+1)^2}\,\, r_k^{c/T} f(t^{1/T}r_k^{1/T},0)\quad (t>0), \end{aligned}$$

now valid for \(f \in \mathscr {B}^p_{c,T}.\) In this formula, we may replace the variable t with \(t^T\) on both sides. Furthermore, we may employ Proposition 1(c) for extending the argument of the polar Mellin derivative from (t, 0) to an arbitrary point \((t, \alpha )\in \mathbb {H}\). This leads us to

$$\begin{aligned} \bigl (\widetilde{\varTheta }_c f\bigr )(t,\alpha ) \,=\, \frac{4T}{\pi ^2} \sum _{k=-\infty }^\infty \frac{(-1)^k}{(2k+1)^2}\, \,r_k^{c/T} f(tr_k^{1/T},\alpha ) \end{aligned}$$

for \(f \in \mathscr {B}^p_{c,T}\) and all \((t,\alpha )\in \mathbb {H}\), which is identical with (11).

It remains to justify the statement on convergence. Since

$$\begin{aligned} \frac{4}{\pi ^2} \sum _{k=-\infty }^\infty \frac{1}{(2k+1)^2}\,=\,1, \end{aligned}$$
(13)

and by property (iii) in the definition of \(\mathscr {B}^p_{c,T}\), there holds

$$\begin{aligned} t^cr_k^{c/T} \left| f\bigl (tr_k^{1/T}, \alpha \bigr )\right| \,\le \, C_f\, e^{T|\alpha |}\quad (k\in \mathbb {Z}), \end{aligned}$$

we readily see absolute and uniform convergence as asserted in the theorem. \(\square \)

As a consequence of Theorem 5, we can establish a Bernstein inequality for polar Mellin derivatives.

Corollary 1

Let \(f\in \mathscr {B}^p_{c,T}\), where \(p\in [1, \infty ]\), \(c\in \mathbb {R}\) and \(T>0\). Then for any \(\theta \in \mathbb {R},\) we have

$$\begin{aligned} \left\| \bigl (\widetilde{\varTheta }_cf\bigr )(\cdot , \theta )\right\| _{X_c^p}\,\le \, T\,\left\| f(\cdot , \theta )\right\| _{X_c^p}. \end{aligned}$$
(14)

Proof

Let us write \(r_k=e^{(k+1/2)\pi }\), where \(k\in \mathbb {Z}\), for short. Multiplying (11) by \(r^c\), taking moduli on both sides, applying the triangular inequality on the right-hand side and using (13), we obtain

$$\begin{aligned} r^c \left| \bigl (\widetilde{\varTheta }_cf\bigr )(r,\theta )\right| \le \frac{4T}{\pi ^2}\sum _{k\in \mathbb {Z}} \frac{1}{(2k+1)^2} \,r^cr_k^{c/T} \left| f(rr_k^{1/T}, \theta )\right| = T\Vert f(\cdot , \theta )\Vert _{X_c^\infty } \end{aligned}$$
(15)

for any \((r, \theta )\in \mathbb {H}\). This implies (14) for \(p=\infty \).

Now let \(1\le p< \infty \) and let \(R>0\). Since \(a^c \Vert f(a \,\cdot , \theta )\Vert _{X_c^p}\,=\, \Vert f( \cdot ,\theta )\Vert _{X_c^p},\) for \(a>0\), we conclude from (15) using the triangular inequality for norms and B. Levi’s theorem that

$$\begin{aligned}&\left( \int _{1/R}^R r^{cp} \left| \bigl (\widetilde{\varTheta }_cf\bigr )(r,\theta )\right| ^p \frac{dr}{r}\right) ^{1/p} \le \frac{4T}{\pi ^2} \left\| \sum _{k\in \mathbb {Z}} \frac{r_k^{c/T}}{(2k+1)^2}\, f(r_k^{1/T} \,\cdot , \theta )\right\| _{X_c^p} \\&\qquad \le \frac{4T}{\pi ^2} \sum _{k\in \mathbb {Z}} \frac{r_k^{c/T}}{(2k+1)^2}\, \left\| f(r_k^{1/T} \,\cdot , \theta )\right\| _{X_c^p} = T\Vert f(\cdot , \theta )\Vert _{X_c^p}. \end{aligned}$$

This guarantees that the \(X_c^p\) norm of \(\bigl (\widetilde{\varTheta }_cf\bigr )(\cdot , \theta )\) exists. Letting \(R\rightarrow \infty \), we obtain (14) for \(p\in [1, \infty {[}\). \(\square \)

Corollary 2

The Mellin–Bernstein space is invariant under polar Mellin differentiation, that is, if \(f\in \mathscr {B}^p_{c,T},\) where \(p\in [1,+\infty ]\), \(c\in \mathbb {R}\) and \(T>0\), then \(\widetilde{\varTheta }_cf \in \mathscr {B}^p_{c,T}\).

Proof

We have to show that \(\widetilde{\varTheta }_c f\) satisfies properties (i)–(iii) in the definition of Mellin–Bernstein spaces. Since for a polar-analytic function f there exist polar derivatives of arbitrary order (see [3, Theorem 5.3]), it follows immediately that \(\widetilde{\varTheta }_cf\) is polar-analytic on \(\mathbb {H}\), and so (i) is satisfied. Property (ii) is guaranteed by Corollary 1. Finally, knowing that (iii) holds for f, we deduce from (11) that

$$\begin{aligned} r^c\left| \bigl (\widetilde{\varTheta }_{c}f\bigr )(r,\theta )\right| \le \frac{4T}{\pi ^{2}} \sum _{k\in \mathbb {Z}} \frac{C_{f}\,e^{T|\theta |}}{(2k+1)^{2}} = TC_{f} e^{T |\theta |}. \end{aligned}$$

Hence (iii) holds with \(C_{\widetilde{\varTheta }_cf} = T C_f.\) \(\square \)

Combining Corollaries 1 and 2, we can apply (14) repeatedly and obtain

$$\begin{aligned} \left\| \bigl (\widetilde{\varTheta }_c^kf\bigr )( \cdot , \theta )\right\| _{X_c^p} \,\le \, T^k\,\Vert f( \cdot , \theta )\Vert _{X_c^p} \qquad (\theta \in \mathbb {R}, k\in \mathbb {N}). \end{aligned}$$

This inequality and its special case (14) relate to [8, Theorem 3.4]; also see [9, Theorem 2.6].

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Open Access funding provided by Projekt DEAL. We wish to express our deep thanks to the reviewer for his/her interesting remarks on our papers, in particular, for exposing the connections with the theory of groups of operators.

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Bardaro, C., Butzer, P.L., Mantellini, I. et al. Integration of polar-analytic functions and applications to Boas’ differentiation formula and Bernstein’s inequality in Mellin setting. Boll Unione Mat Ital 13, 503–514 (2020). https://doi.org/10.1007/s40574-020-00226-9

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Keywords

  • Polar-analytic functions
  • Cauchy’s integral formulae
  • Identity theorems
  • Logarithmic poles
  • Boas’ differentiation formula

Mathematics Subject Classification

  • 30E20
  • 30F30
  • 44A05
  • 65D25