# On the boundedness of poles of generalized Padé approximants

**Part of the following topical collections:**

## Abstract

Given a function *F* holomorphic on a neighborhood of some compact subset of the complex plane, we prove that if the zeros of the denominators of generalized Padé approximants (orthogonal Padé approximants and Padé–Faber approximants) for some row sequence remain uniformly bounded, then either *F* is a polynomial or *F* has a singularity in the complex plane. This result extends the known one for classical Padé approximants. Its proof relies, on the one hand, on difference equations where their coefficients relate to the coefficients of denominators of these generalized Padé approximants and, on the other hand, on a curious property of complex numbers.

## Keywords

Padé approximation Orthogonal polynomials Faber polynomials Difference equations Inverse results## MSC

30E10 41A21 41A27## 1 Introduction

Currently, Padé approximation theory emphasizes inverse-type problems where we want to describe the analytic properties of the approximated function from the knowledge of the asymptotic behavior of the poles of the approximating functions. Moreover, the theory of higher order recurrence relations (difference equations) plays very important roles in solving inverse-type problems (see, e.g., [3, 6, 7, 8, 11]). The object of the present paper is to investigate the relation between the boundedness of poles of row sequences of orthogonal Padé approximants and Padé–Faber approximants and the analyticity of the approximated function. The results are of inverse type.

In order to state a known result related to our study, we need to remind the reader of the definition of classical Padé approximants. In what follows, \(\mathbb{N}:=\{1,2,3,\ldots \}\), \(\mathbb{N}_{0}:=\mathbb{N} \cup \{0\}\), and \(\mathbb{P}_{n}\) is the set of all polynomials of degree at most *n*.

### Definition 1.1

*classical Padé approximant of*

*F*.

*n*sufficiently large. In order to find such \(Q_{n,m}(z)=1+\sum_{j=1}^{m} q_{n,j} z ^{j}\), one has to solve for all \(k=n+1,\ldots ,n+m\),

Given a formal power series \(F(z)=\sum_{k=0}^{\infty } f_{k} z^{k}\), we denote by \(R_{0}(F)\) the radius of the largest disk centered at 0 to which *F* can be extended holomorphically. That is, \(R_{0}(F)\) is the radius of convergence of *F*. In this paper, we are interested in proving analogs of the following theorem (see [11, Theorem 1.1] or [9, Corollary 2.4]) for orthogonal Padé approximants and Padé–Faber approximants.

### Theorem A

*Let*\(m\in \mathbb{N}\)

*be fixed and let*\(\mathcal{P}_{n}\)

*be the set of all zeros of*\(Q_{n,m}\).

*Suppose that the cardinality of*\(\mathcal{P} _{n}\)

*is at least*1

*for all*

*n*

*sufficiently large*,

*and*

*Then*,

*F*

*is a polynomial or*\(0< R_{0}(F)<\infty \).

In other words, if *F* is not a polynomial and the poles of \(R_{n,m}\) stay far from the origin and bounded for all *n* sufficiently large, then \(0< R_{0}(F) <\infty \). Up to my knowledge, this result is the first one of this sort. Moreover, the sequence \((R_{n,m})_{n \geq n_{0}}\), where *m* remains fixed, is called the *mth row sequence*.

*E*be a bounded continuum with connected complement containing infinitely many points. From now on, the set

*E*will satisfy the above condition. Let

*μ*be a finite positive Borel measure with infinite support \(\operatorname{supp}(\mu )\) contained in

*E*. We write \(\mu \in \mathcal{M}(E)\) and the corresponding inner product is defined by

*E*. The first two definitions are generalized Padé approximants constructed from the sequence of orthogonal polynomials \((p_{n})_{n \geq 0}\).

### Definition 1.2

*standard orthogonal Padé approximant of*

*F*

*with respect to*

*μ*.

### Definition 1.3

*modified orthogonal Padé approximant of*

*F*

*with respect to*

*μ*.

*Φ*the unique Riemann mapping function from \(\overline{ \mathbb{C}}\setminus {E}\) to the exterior of the closed unit disk verifying \(\varPhi (\infty )=\infty \), \(\varPhi '(\infty )>0\). For each \(\rho >1\), the

*level curve of index*

*ρ*and the

*canonical domain of index*

*ρ*are defined by

*ρ*such that

*F*extends as a holomorphic function to \(D_{\rho }\).

*Faber polynomial*of

*E*of degree

*n*is

*E*. The

*n*th

*Faber coefficient*of \(F\in \mathcal{H}(E)\) with respect to \(\varPhi _{n}\) is defined by the formula

The next two definitions are the ones of generalized Padé approximants constructed from the sequence of Faber polynomials \((\varPhi _{n})_{n\geq 0}\).

### Definition 1.4

*standard Padé–Faber approximant of*

*F*

*with respect to*

*E*.

### Definition 1.5

*modified Padé–Faber approximant of*

*F*

*with respect to*

*E*.

In order to find \(Q_{n,m}^{\mu }\), \(\tilde{Q}_{n,m}^{\mu }\), \(Q_{n,m}^{E}\), or \(\tilde{Q}_{n,m}^{E}\) in Definitions 1.2–1.5, one has to solve a system of *m* homogeneous linear equations on \(m + 1\) unknowns. Therefore, for any integers \(n\geq 0\) and \(m\geq 1\), polynomials \(Q_{n,m}^{\mu }\), \(\tilde{Q}_{n,m}^{\mu }\), \(Q_{n,m}^{E}\), and \(\tilde{Q}_{n,m}^{E}\) always exist but they may not be unique. Since \(Q_{n,m}^{\mu }\), \(\tilde{Q} _{n,m}^{\mu }\), \(Q_{n,m}^{E}\), and \(\tilde{Q}_{n,m}^{E}\) are not the zero function, we normalize them to be “monic” polynomials. Unlike the classical Padé approximants, for any integers \(n\geq 0\) and \(m\geq 1\), \(R_{n,m}^{\mu }\), \(\tilde{R}_{n,m}^{\mu }\), \(R_{n,m}^{E}\), and \(\tilde{R}_{n,m}^{E}\) may not be unique. The rational functions \(R^{\mu }_{n,m}\) and \(R^{E}_{n,m}\) are natural extensions of \(R_{n,m}\) and were introduced by Maehly [12] in 1960. Suetin [15, 16] was the first to give necessary and sufficient conditions for the convergence with geometric rate of the denominators of standard orthogonal Padé and standard Padé–Faber approximants on row sequences. The rational functions \(\tilde{R}^{\mu }_{n,m}\) and \(\tilde{R}^{E}_{n,m}\) were recently introduced (in the vector forms) in order to solve some inverse-type problems about detecting the poles of a vector of functions nearest the set *E* (see [5, 6] for more details). Note that in general, the approximations in Definitions 1.2 and 1.4 are not the same as the ones in Definitions 1.3 and 1.5, respectively.

An outline of this paper is as follows. In Sect. 2, we state analogs of Theorem A which are in our main result. We relegate all lemmas in Sect. 3. The proof of the main result is in Sect. 4.

## 2 Main result

*E*is convex. The measure \(\mu \in \mathbf{Reg}_{1}^{*}(E)\) when \(\mu \in \mathbf{Reg}_{1}(E)\) and

The main result of this paper is the following.

### Theorem 2.1

*Let*\(F\in \mathcal{H}(E)\)

*and*\(\mu \in \mathbf{Reg}_{1}^{*}(E)\).

*Fix*\(m\in \mathbb{N}\)

*and denote by*\(\mathcal{P}_{n}^{\mu }\), \(\mathcal{P}_{n}^{E}\), \(\tilde{\mathcal{P}}_{n}^{\mu }\),

*and*\(\tilde{\mathcal{P}}_{n}^{E}\)

*the sets of all zeros of*\(Q_{n,m}^{\mu }\), \(Q_{n,m}^{E}\), \(\tilde{Q}_{n,m} ^{\mu }\),

*and*\(\tilde{Q}_{n,m}^{E}\),

*respectively*.

*Assume that one of the following conditions holds*.

- (a)
*The cardinality of*\(\mathcal{P}_{n}^{\mu }\)*is at least*1*for all**n**sufficiently large and*$$ \inf_{N \geq m} \sup_{n\geq N} \bigl\{ \vert \zeta \vert :\zeta \in \mathcal{P}_{n} ^{\mu }\bigr\} < \infty . $$ - (b)
*The cardinality of*\(\mathcal{P}_{n}^{E}\)*is at least*1*for all**n**sufficiently large and*$$ \inf_{N \geq m} \sup_{n\geq N} \bigl\{ \vert \zeta \vert :\zeta \in \mathcal{P}_{n} ^{E}\bigr\} < \infty . $$ - (c)
*The cardinality of*\(\tilde{\mathcal{P}}_{n}^{\mu }\)*is at least*1*for all**n**sufficiently large and*$$ \inf_{N \geq m} \sup_{n\geq N} \bigl\{ \vert \zeta \vert :\zeta \in \tilde{\mathcal{P}}_{n}^{\mu } \bigr\} < \infty . $$ - (d)
*The cardinality of*\(\tilde{\mathcal{P}}_{n}^{E}\)*is at least*1*for all**n**sufficiently large and*$$ \inf_{N \geq m} \sup_{n\geq N} \bigl\{ \vert \zeta \vert :\zeta \in \tilde{\mathcal{P}}_{n}^{E} \bigr\} < \infty . $$

*Then either*

*F*

*is a polynomial or*\(\rho _{0}(F)<\infty \).

Note that we are not interested in proving \(\rho _{0}(F)>1\) because this is a direct consequence of \(F\in \mathcal{H}(E)\).

## 3 Auxiliary lemmas

*nth Fourier coefficient*of \(F\in \mathcal{H}(E)\) corresponding to \(p_{n}\) is defined as follows:

### Lemma 3.1

*Let*\(F\in \mathcal{H}(E)\)

*and*\(\mu \in \mathbf{Reg}_{1}(E)\).

*Then*

*Moreover*,

*the series*\(\sum_{n=0}^{\infty } \langle F \rangle _{n} p _{n}\)

*and*\(\sum_{n=0}^{\infty } [F]_{n} \varPhi _{n}\)

*converge uniformly to*

*F*

*on each compact subset of*\({D}_{\rho _{0}(F)}\).

An estimate of \(\Vert \varPhi _{n} \Vert _{\varGamma _{\rho }}\) is given in the succeeding lemma (see [10, p. 583] or [14, p. 43]).

### Lemma 3.2

*Fix*\(\rho >1\).

*Then there exists*\(c>0\)

*such that*,

*for all*\(n\in \mathbb{N}_{0}\),

A curious relation of complex numbers (see [4, Lemma 3] for its proof) which serves as the cornerstone for the proof of our main result is the following.

### Lemma 3.3

*If a sequence of complex numbers*\((A_{N})_{N \in \mathbb{N}}\)

*has the following properties*:

- (i)
\(\lim_{N \rightarrow \infty } \vert A_{N} \vert ^{1/N}=0\);

- (ii)
*there exist*\(N_{0}\in \mathbb{N}\)*and*\(C>0\)*such that*\(\vert A_{N} \vert \leq C \sum_{k=N+1}^{\infty } \vert A_{k} \vert \),*for all*\(N \geq N_{0}\),

*then there exists*\(N_{1}\in \mathbb{N}\)

*such that*\(A_{N}=0\)

*for all*\(N \geq N_{1}\).

## 4 Proof of main result

### Proof of Theorem 2.1

*F*is an entire function, then

*F*is a polynomial. Let

*n*sufficiently large and \(q_{n,m _{n}}=1\).

*k*and \(m_{n}\). Combining (5), (6), and (7), it is easy to check that (4) implies that, for all \(k=1,\ldots ,m\), and for all \(n\geq n_{0}\),

*n*,

*k*and \(m_{n}\). For each \(n \geq n_{0}\), we choose \(k=m_{n}\) in the previous inequality and we obtain, for all \(n \geq n_{0}\),

*n*and

*F*is a polynomial as desired.

*F*is an entire function, then

*F*is a polynomial. Let

*n*,

*k*and \(m_{n}\). For each \(n \geq n_{0}\), we choose \(k=m_{n}\) in the previous inequality and we obtain, for all \(n \geq n_{0}\),

*F*is a polynomial.

*F*is an entire function, then

*F*is a polynomial. Let

*F*is a polynomial.

*F*is an entire function, we want to show that

*F*is a polynomial. Let

## 5 Conclusion

We prove that if the zeros of the denominators of four generalized Padé approximations based on orthogonal and Faber polynomials stay uniformly bounded, then either the approximated function is a polynomial or it has a singularity in the complex plane. This result extends the well-known one for classical Padé approximants.

## Notes

### Acknowledgements

The author wishes to express gratitude toward to the anonymous referees for careful reading, helpful comments, and suggestions leading to improvements of this work. The author would like to thank Assoc. Prof. Chontita Rattanakul and Prof. Dr. Guillermo López Lagomasino for their invaluable guidance.

### Availability of data and materials

Not applicable.

### Authors’ contributions

The author was the only one to contribute to the writing of this paper. The author conceived of the study, participated in its design and coordination, and read and approved the final manuscript.

### Funding

The research of N. Bosuwan was supported by the Strengthen Research Grant for New Lecturer from the Thailand Research Fund and the Office of the Higher Education Commission (MRG6080133) and Faculty of Science, Mahidol University.

### Competing interests

The author declares to have no competing interests.

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