# Spectral Properties of Unbounded Jacobi Matrices with Almost Monotonic Weights

- 755 Downloads
- 3 Citations

## Abstract

We present a unified framework to identify spectra of Jacobi matrices. We give applications of the long-standing problem of Chihara (Mt J Math 21(1):121–137, 1991, J Comput Appl Math 153(1–2):535–536, 2003) concerning one-quarter class of orthogonal polynomials, to the conjecture posed by Roehner and Valent (SIAM J Appl Math 42(5):1020–1046, 1982) concerning continuous spectra of generators of birth and death processes, and to spectral properties of operators studied by Janas and Moszyńki (Integral Equ Oper Theory 43(4):397–416, 2002) and Pedersen (Proc Am Math Soc 130(8):2369–2376, 2002).

## Keywords

Jacobi matrix Continuous spectrum One-quarter class of orthogonal polynomials## Mathematics Subject Classification

Primary: 47B36 Secondary: 42C05 60J80## 1 Introduction

*A*is defined on the domain \(Dom(A) = \{ x \in \ell ^2 :A x \in \ell ^2\}\), where

*Jacobi matrix*.

The study of Jacobi matrices is motivated by connections with orthogonal polynomials and the classical moment problem (see, e.g., [23]). Also, every self-adjoint operator can be represented as a direct sum of Jacobi matrices. In particular, generators of birth and death processes may be seen as Jacobi matrices acting on weighted \(\ell ^2\) spaces.

There are several approaches to the problem of the identification of the spectrum of unbounded Jacobi matrices. A method often used is based on subordination theory (see, e.g., [6, 15, 19]). Another technique uses the analysis of a commutator between a Jacobi matrix and a suitable chosen matrix (see, e.g., [22]). The case of Jacobi matrices with monotonic weights was considered mainly by Dombrowski (see, e.g., [8]), where the author developed commutator techniques which enabled qualitative spectral analysis of examined operators.

The present article is motivated by commutator techniques of Dombrowski and some ideas of Clark [6]. In fact, commutators do not appear here directly but are hidden in some of our expressions.

Let *A* be a Jacobi matrix, and assume that the matrix *A* is self-adjoint. The spectrum of the operator *A* will be denoted by \(\sigma (A)\), the set of all its eigenvalues by \(\sigma _\mathrm {p}(A)\), and the set of all accumulation points of \(\sigma (A)\) by \(\sigma _\mathrm {ess}(A)\). For a real number *x*, we define \(x^- = \max (-x,0)\).

Our main result is the following theorem.

### **Theorem 1.1**

*A*be a Jacobi matrix. If there is a positive sequence \(\{ \alpha _n \}\) such that then the Jacobi matrix

*A*is self-adjoint and satisfies \(\sigma _\mathrm {p}(A) = \emptyset \), and \(\sigma (A) = \mathbb {R}\).

The importance of Theorem 1.1 lies in the fact that we have flexibility in the choice of the sequence \(\alpha _n\). Some choices of the sequence \(\alpha _n\) are given in Sect. 4.

Let us concentrate now on the simplest case of the theorem, i.e., when \(\alpha _n = a_n\). In [11, Lemma 2.6], it was proven that if the nonnegative sequence \(a_n^2 - a_{n-1}^2\) is bounded and \(b_n \equiv 0\), then the matrix *A* has no eigenvalues. Our theorem gives additional information that in this case \(\sigma (A) = \mathbb {R}\) holds. Moreover, our assumptions are weaker than the conditions of [11, Lemma 2.6].

In Sect. 6, we provide examples showing sharpness of the assumptions in the case \(\alpha _n = a_n\). In particular, condition (e) is necessary in the class of monotonic sequences \(\{ a_n \}\), and condition (b) could not be replaced by \([(a_{n+1}/a_n)^2 - 1]^- \rightarrow 0\). Corollary 1.3 shows that in general, condition (g) is necessary.

In Sect. 5, we apply the theorem for \(\alpha _n = a_n\) to resolve a conjecture (see [21]) about continuous spectra of generators of birth and death processes. We also present there applications to the following problem.

### **Problem 1.2**

**(Chihara**[4, 5]

**)**Assume that a Jacobi matrix

*A*is self-adjoint, \(b_n \rightarrow \infty \), the smallest point \(\rho \) of \(\sigma _\mathrm{ess}(A)\) is finite, and

A direct consequence of the theorem in the case \(\alpha _n = a_n\), providing easy to check additional assumptions to Problem 1.2, is the following result.

### **Corollary 1.3**

*A*satisfies \(\sigma _\mathrm {ess}(A) = [-M, \infty )\). Moreover, if \(a_{n+1}/a_n \rightarrow 1\), then

*J*be defined by

*K*on finite sequences by the formula

*A*, and \(\langle \cdot , \cdot \rangle \) is the scalar product on \(\ell ^2\), proved to be a useful tool to show that the matrix

*A*has continuous spectrum (see, e.g., [8, 11, 13]).

An important observation is that we can give closed form for \(S_n\) [see (3.1)]. To the author’s knowledge, this closed form has been known only for \(\alpha _n = a_n\) (see [7]). A related expression for \(\alpha _n \equiv 1\) was analyzed in [6]. Adaptation of techniques from [6] allow us to circumvent technical difficulties present in Dombrowski’s approach. Extending the definition of \(S_n\) to generalized eigenvectors [see (2.1)] enables us to show that \(\sigma (A) = \mathbb {R}\).

The article is organized as follows. In Sect. 2, we present definitions and well-known facts important for our argument. In Sect. 3, we prove Theorem 1.1, whereas in Sect. 4 we show its variants. In particular, we identify spectra of operators considered in [20] and [14]. In Sect. 5, we present applications of our theorem to some open problems. Finally, in the last section, we discuss the necessity of the assumptions in the case \(\alpha _n = a_n\). We also present examples showing that in some cases, our results are stronger than results known in the literature.

## 2 Tools

*A*, \(\lambda \in \mathbb {R}\) and real numbers \((a,b) \ne (0, 0)\), we introduce a generalized eigenvector

*u*by

*A*associated with an eigenvalue \(\lambda \).

Observe that \(\{ p_n(\cdot ) \}_{n=0}^\infty \) is a sequence of polynomials. Moreover, the sequence is orthonormal with respect to the measure \(\mu (\cdot ) = \langle E(\cdot ) \delta _0, \delta _0 \rangle \), where *E* is the spectral resolution of the matrix *A*, \(\langle \cdot , \cdot \rangle \) is the scalar product on \(\ell ^2\), and \(\delta _0 = (1, 0, 0, \ldots )\).

The following propositions are well known. We include them for the sake of completeness.

### **Proposition 2.1**

Let \(\lambda \in \mathbb {R}\). If every generalized eigenvector *u* does not belong to \(\ell ^2\), then the matrix *A* is self-adjoint, \(\lambda \notin \sigma _\mathrm {p}(A)\), and \(\lambda \in \sigma (A)\).

### *Proof*

[23, Theorem 3] asserts that *A* is self-adjoint provided that at least one generalized eigenvector \(\{u_n\} \notin \ell ^2\). Direct computation shows that \(\lambda \in \sigma _\mathrm {p}(A)\) if and only if \(\{ p_n(\lambda ) \} \in \ell ^2\). Therefore the matrix *A* is self-adjoint and \(\lambda \notin \sigma _\mathrm {p}(A)\).

*x*such that \((A - \lambda I) x = \delta _0\) satisfies the following recurrence relation:

*x*is a generalized eigenvector, thus \(x \notin \ell ^2\). Therefore, the operator \(A - \lambda I\) is not surjective, i.e., \(\lambda \in \sigma (A)\). \(\square \)

The following proposition is well known. For the proof we refer to, e.g., [12, Corollary 3.6].

### **Proposition 2.2**

*A*and \(\widehat{A}\) be Jacobi matrices defined by sequences \(\{a_n\}\), \(\{b_n\}\) and \(\{a_n\}\), \(\{-b_n\}\), respectively. Then

The following proposition has been used many times in the literature (see, e.g., [11, 12]).

### **Proposition 2.3**

*A*be a self-adjoint Jacobi matrix associated with the sequence \(b_n \equiv 0\). Let \(A_e\) and \(A_o\) be restrictions of \(A \cdot A\) to the subspaces \(span\{\delta _{2k} :k \in \mathbb {N}\}\) and \(span\{\delta _{2k+1} :k \in \mathbb {N}\}\), respectively. Then \(A_e\) and \(A_o\) are Jacobi matrices associated with

*X*, we define \(X^2 = \{ x^2 :x \in X \}\).

### *Proof*

By direct computation it may be proved that \(A_o\) and \(A_e\) satisfy (2.2).

Assume that \(0 \notin \sigma _\mathrm {p}(\widetilde{A})\). Observe that \(A_o = \widetilde{A}_e\). Therefore, the previous argument applied to \(\widetilde{A}\) implies also that \(A_o\) is self-adjoint.

The conclusion of spectra follows from, e.g., [12, Section 4]. \(\square \)

## 3 Proof of the Main Theorem

*u*and a positive sequence \(\{ \alpha _n \}\), we set

*bounded*ones (see, e.g., [7, 10]). In the case of unbounded operators, a sequence similar to \(S_n\) for \(\alpha _n \equiv 1\) was also used in [6].

The following proposition is an adaptation of [6, Lemma 3.1].

### **Proposition 3.1**

*u*be a generalized eigenvector associated with \(\lambda \in \mathbb {R}\) and

*n*,

### *Proof*

*n*, there is a positive upper and lower bound of the above expressions. This completes the proof. \(\square \)

### **Corollary 3.2**

### *Proof*

*n*sufficiently large we have

*u*cannot belong to \(\ell ^2\). \(\square \)

Now we are ready to prove Theorem 1.1.

### *Proof (of Theorem 1.1)*

By virtue of Corollary 3.2, it is sufficient to show that \(\liminf _n S_n > 0\) for every generalized eigenvector \(\{ u_n \}\).

*N*such that for every \(n \ge N, S_n > 0\) holds. Let us define \(F_n = (S_{n+1} - S_n) / S_n\). Then \(S_{n+1} / S_n = 1 + F_n\); thus

### *Remark 3.3*

*convergent*to a positive number. This shows that for every generalized eigenvector

*u*, there are constants \(c_1 > 0, c_2 > 0\) such that \(c_1/(a_n \alpha _n) \le u_n^2 + u_{n+1}^2 \le c_2 / (a_n \alpha _n)\). Hence for every generalized eigenvectors

*u*,

*v*associated with \(\lambda \in \mathbb {R}\), there is a constant \(c > 0\) such that

*A*is purely absolutely continuous.

## 4 Special Cases

In this section, we are going to show a few choices of the sequence \(\{ \alpha _n \}\) from Theorem 1.1. In this way, we show the flexibility of our approach.

The following theorem was proved in [14, Theorem 1.6] and is a generalization of [6, Theorem 1.10]. In the proof, the authors analyze transfer matrices. Therefore, our argument gives an alternative proof.

### **Theorem 4.1**

Then \(\sigma (A) = \mathbb {R}\), and the matrix *A* has purely absolutely continuous spectrum.

### *Proof*

Let \(\alpha _n \equiv 1\). By virtue of Remark 3.3, we need to check the assumptions (b’), (d), and (f) of Theorem 1.1.

Since the sequence \(\{ a_{n+1}/a_n \}\) is of bounded variation, it is convergent to a number *a*. From the condition (b), we have \(a \le 1\), whereas the condition (a) gives \(a \ge 1\). This proves the conditions (b’) and (f) of Theorem 1.1.

The sequence \(\{ b_{n+1}/a_n \}\) is of bounded variation because \(\frac{b_{n+1}}{a_n} = \frac{b_{n+1}}{a_{n+1}} \cdot \frac{a_{n+1}}{a_n}\). The proof is complete. \(\square \)

The next theorem imposes very simple conditions on Jacobi matrices. In Sect. 5, we show its applications; furthermore, in Sect. 6, we discuss sharpness of the assumptions.

### **Theorem 4.2**

Then the Jacobi matrix *A* is self-adjoint and satisfies \(\sigma _\mathrm {p}(A) = \emptyset \) and \(\sigma (A) = \mathbb {R}\).

### *Proof*

Apply Theorem 1.1 with \(\alpha _n = a_n\). \(\square \)

Special cases of the following theorem were examined in [20] and [14] using commutator methods.

### **Theorem 4.3**

*K*,

*N*and for a summable nonnegative sequence \(c_n\),

Then \(\sigma _\mathrm {p}(A) = \emptyset \) and \(\sigma (A) = \mathbb {R}\).

### *Proof*

*x*sufficiently large and a constant \(c > 0\). Therefore, the right-hand side of (4.1) is summable.

### *Remark 4.4*

When we compare Theorem 4.2 with Theorem 4.3, we see that Theorem 4.3 is interesting only in the case when \(\sum _{n=0}^\infty 1/a_n^2 < \infty \). In this case, the condition Theorem 4.3(d) is satisfied.

A sequence similar to \(\alpha _n = n a_n^{-1}\) was used in the proof of [20, Theorem 4.1] and [14, Theorem 2.1]. There it was shown that under the stronger assumptions (which in particular imply \(c_n \equiv 0\), \(b_n \equiv 0\) and \(K=0\)), the measure \(\mu \) is absolutely continuous. Whether \(\sigma (A) = \mathbb {R}\) was not investigated.

### *Example 4.5*

Let \(K > 0\). Fix *M* such that \(\log ^{(K)}(M) > 0\). Then for the sequences \(a_n = (n+M) g_K(n+M)\) and \(b_n \equiv 0\), the assumptions of Theorem 4.3 are satisfied.

## 5 Applications of Theorem 4.2

### 5.1 Birth and Death Processes

*Q*is well defined on the domain \(Dom(Q) = \{ x \in \ell ^2(\pi ) :Q x \in \ell ^2(\pi )\}\). Notice that any sequence with finite support belongs to \(Dom(Q)\). If the operator

*Q*is self-adjoint, it is of a probabilistic interest to examine the spectrum \(\sigma (Q)\) of the operator

*Q*(see, e.g., [17]).

### **Theorem 5.1**

Then the matrix *Q* is self-adjoint and satisfies \(\sigma _\mathrm {p}(Q) = \emptyset \) and \(\sigma (Q) = (-\infty , 0]\).

### *Proof*

Let *P* be a diagonal matrix with entries \(\sqrt{\pi _n}\) on the main diagonal. Then we have \(\bar{A} = P Q P^{-1}\), where \(\bar{A}\) is the Jacobi matrix associated with sequences \(\bar{a}_n = \sqrt{\lambda _n \mu _{n+1}}\) and \(\bar{b}_n = -(\lambda _n + \mu _n)\) (see [18, Section 2]). Since the matrix \(P :\ell ^2(\pi ) \rightarrow \ell ^2\) is an isometry (hence *P* and \(P^{-1}\) are bounded), it is enough to consider only the spectrum of \(\bar{A}\). By virtue of Proposition 2.2, it is sufficient to consider the spectrum of the matrix \(\widehat{A}\), corresponding with the sequences \(\{a_n\}\) and \(\{-b_n\}\).

In [16] it was shown that in the case when \(\lambda _n = \mu _{n+1} = n+a,\ a > 0\), the matrix *Q* has purely absolutely continuous spectrum and \(\sigma (Q) = (-\infty , 0]\). Theorem 5.1 is applicable in more general situations.

In [21], the following conjecture about spectral properties of operators of the form (5.1) was stated.

### **Conjecture 5.2**

In [3] it was shown that without additional assumptions, the conjecture is false. In Theorem 5.1, we provide sufficient conditions when Conjecture 5.2 holds.

*opposite*conclusion to results from [18].

*Q*is self-adjoint and \(\lambda _k \rightarrow \infty \), then \(\sigma _\mathrm{ess}(Q) = \emptyset \).

### 5.2 Chihara’s Problem

In [1] (see also [2, IV-Theorem 4.2]), the following result was proved.

### **Theorem 5.3**

**(Chihara**[1]

**)**Assume that a Jacobi matrix

*A*is self-adjoint, \(b_n \rightarrow \infty \), the smallest point \(\rho \) of \(\sigma _{ess}(A)\) is finite, and

This suggests the following problem stated in [4] and [5].

### **Problem 5.4**

**(Chihara** [4, 5]**)** Let the assumptions of Theorem 5.3 be satisfied. Find additional assumptions (if needed) to assure that \(\sigma _\mathrm{ess}(A) = [\rho , \infty )\).

The following theorem gives sufficient (and easy to verify) additional conditions to Problem 5.4. In fact *every* Jacobi matrix with \(b_n \equiv 0\) and \(a_{n+1}/a_n \rightarrow 1\) from this article provides an example (via Proposition 2.3) satisfying the conclusion of Problem 5.4.

### **Theorem 5.5**

*A*satisfies \(\sigma _\mathrm {ess}(A) = [-M, \infty )\). Moreover, if \(a_{n+1}/a_n \rightarrow 1\), then

### *Proof*

We show (5.2) by a direct computation. Without loss of generality, we may assume that \(M = 0\). Let \(-r_n = a_{n-1} - b_n + a_n\). Then \(a_{n-1} - (b_n-r_n) + a_n = 0\). Let \(\widetilde{A}\) be the Jacobi matrix for sequences \(\widetilde{a}_n = a_n, \ \widetilde{b}_n = b_n - r_n\). The matrix \(R = A - \widetilde{A}\) defines a compact self-adjoint operator on \(\ell ^2\) (because \(r_n \rightarrow 0\)). Hence, by the Weyl perturbation theorem (see [25]), \(\sigma _\mathrm{ess}(A) = \sigma _\mathrm{ess}(\widetilde{A})\). Theorem 5.1 implies that \(\sigma _\mathrm{ess}(\widetilde{A}) = (-\infty , 0]\). Finally, Proposition 2.2 applied to the matrix \(\widetilde{A}\) finishes the proof. \(\square \)

## 6 Examples

### *Example 6.1*

*A*is always self-adjoint. Moreover, 0 is its eigenvalue if and only if

In [19] it was shown that for \(\widetilde{a}_k = k^\alpha , \ (\alpha \in (0,1))\) the spectrum \(\sigma (C) = \mathbb {R}\). In the case \(\alpha \le 1/2\), the measure \(\mu (\cdot ) = \langle E(\cdot ) \delta _0, \delta _0 \rangle \) is absolutely continuous, whereas for \(\alpha > 1/2\), the measure \(\mu \) is absolutely continuous on the set \(\mathbb {R} \backslash \{ 0 \}\).

### *Example 6.2*

Let \(b_n \equiv 0\) and \(a_n = n^\alpha + c_n \ (0 < \alpha \le 2/3)\), where \(c_{2n} = 1\) and \(c_{2n+1} = 0\). Then (see [9]) \(\sigma (A) = \mathbb {R} \backslash (-1,1)\), and the measure \(\mu \) is absolutely continuous on \(\mathbb {R} \backslash [-1,1]\). It shows that the condition Theorem 4.2(c) could not be replaced by \([(a_{n+1}/a_n)^2 - 1]^- \rightarrow 0\).

### *Example 6.3*

## Notes

### Acknowledgments

The author would like to thank Ryszard Szwarc and Bartosz Trojan for their helpful suggestions concerning the presentation of this article.

## References

- 1.Chihara, T.S.: Orthogonal polynomials whose zeros are dense in intervals. J. Math. Anal. Appl.
**24**, 362–371 (1968)MathSciNetCrossRefzbMATHGoogle Scholar - 2.Chihara, T.S.: An introduction to orthogonal polynomials. In: Mathematics and its Applications, vol. 13, pp. 1–249. Gordon and Breach Science Publishers, New York-London-Paris (1978)Google Scholar
- 3.Chihara, T.S.: On the spectra of certain birth and death processes. SIAM J. Appl. Math.
**47**(3), 662–669 (1987)MathSciNetCrossRefzbMATHGoogle Scholar - 4.Chihara, T.S.: The one-quarter class of orthogonal polynomials. Rocky Mt. J. Math.
**21**(1), 121–137 (1991)MathSciNetCrossRefzbMATHGoogle Scholar - 5.Chihara, T.S.: An analog of the Blumenthal–Nevai theorem for unbounded intervals. J. Comput. Appl. Math.
**153**(1–2), 535–536 (2003)CrossRefGoogle Scholar - 6.Clark, S.L.: A spectral analysis for self-adjoint operators generated by a class of second order difference equations. J. Math. Anal. Appl.
**197**(1), 267–285 (1996)MathSciNetCrossRefzbMATHGoogle Scholar - 7.Dombrowski, J.: Tridiagonal matrix representations of cyclic selfadjoint operators. II. Pac. J. Math.
**120**(1), 47–53 (1985)MathSciNetCrossRefzbMATHGoogle Scholar - 8.Dombrowski, J.: Cyclic operators, commutators, and absolutely continuous measures. Proc. Am. Math. Soc.
**100**(3), 457–463 (1987)MathSciNetCrossRefzbMATHGoogle Scholar - 9.Dombrowski, J., Janas, J., Moszyński, M., Pedersen, S.: Spectral gaps resulting from periodic perturbations of a class of Jacobi operators. Constr. Approx.
**20**(4), 585–601 (2004)MathSciNetCrossRefzbMATHGoogle Scholar - 10.Dombrowski, J., Nevai, P.: Orthogonal polynomials, measures and recurrence relations. SIAM J. Math. Anal.
**17**(3), 752–759 (1986)MathSciNetCrossRefzbMATHGoogle Scholar - 11.Dombrowski, J., Pedersen, S.: Spectral measures and Jacobi matrices related to Laguerre-type systems of orthogonal polynomials. Constr. Approx.
**13**(3), 421–433 (1997)MathSciNetCrossRefzbMATHGoogle Scholar - 12.Dombrowski, J., Pedersen, S.: Absolute continuity for unbounded Jacobi matrices with constant row sums. J. Math. Anal. Appl.
**267**(2), 695–713 (2002)MathSciNetCrossRefzbMATHGoogle Scholar - 13.Dombrowski, J., Pedersen, S.: Spectral transition parameters for a class of Jacobi matrices. Stud. Math.
**152**(3), 217–229 (2002)MathSciNetCrossRefzbMATHGoogle Scholar - 14.Janas, J., Moszyński, M.: Alternative approaches to the absolute continuity of Jacobi matrices with monotonic weights. Integral Equ. Oper. Theory
**43**(4), 397–416 (2002)MathSciNetCrossRefzbMATHGoogle Scholar - 15.Janas, J., Naboko, S.: Multithreshold spectral phase transitions for a class of Jacobi matrices. Oper. Theory Adv. Appl.
**124**, 267–285 (2001)MathSciNetCrossRefzbMATHGoogle Scholar - 16.Janas, J., Naboko, S.: Spectral properties of selfadjoint Jacobi matrices coming from birth and death processes. Oper. Theory Adv. Appl.
**127**, 387–397 (2001)MathSciNetzbMATHGoogle Scholar - 17.Karlin, S., McGregor, J.L.: The differential equations of birth-and-death processes, and the Stieltjes moment problem. Trans. Am. Math. Soc.
**85**, 489–546 (1957)MathSciNetCrossRefzbMATHGoogle Scholar - 18.Kreer, M.: Analytic birth-death processes: a Hilbert-space approach. Stoch. Process. Appl.
**49**(1), 65–74 (1994)MathSciNetCrossRefzbMATHGoogle Scholar - 19.Moszyński, M.: Spectral properties of some Jacobi matrices with double weights. J. Math. Anal. Appl.
**280**(2), 400–412 (2003)MathSciNetCrossRefzbMATHGoogle Scholar - 20.Pedersen, S.: Absolutely continuous Jacobi operators. Proc. Am. Math. Soc.
**130**(8), 2369–2376 (2002). (electronic)MathSciNetCrossRefzbMATHGoogle Scholar - 21.Roehner, B., Valent, G.: Solving the birth and death processes with quadratic asymptotically symmetric transition rates. SIAM J. Appl. Math.
**42**(5), 1020–1046 (1982)MathSciNetCrossRefzbMATHGoogle Scholar - 22.Sahbani, J.: Spectral theory of certain unbounded Jacobi matrices. J. Math. Anal. Appl.
**342**(1), 663–681 (2008)MathSciNetCrossRefzbMATHGoogle Scholar - 23.Simon, B.: The classical moment problem as a self-adjoint finite difference operator. Adv. Math.
**137**(1), 82–203 (1998)MathSciNetCrossRefzbMATHGoogle Scholar - 24.Szwarc, R.: Absolute continuity of spectral measure for certain unbounded Jacobimatrices. In: Buhmann, M.D., Mache, D.H. (eds.) Advanced Problems in Constructive Approximation, ISNM International Series of Numerical Mathematics, vol. 142, pp. 255–262. Birkhäuser, Basel (2003). doi: 10.1007/978-3-0348-7600-1_18
- 25.Weyl, H.: Über beschränkte quadratische Formen, deren Differenz vollstetig ist. Rend. Circ. Mat. Palermo
**27**, 373–392 (1909)CrossRefzbMATHGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.