Zeros of Jacobi-Sobolev orthogonal polynomials following non-coherent pair of measures

Zeros of Jacobi-Sobolev orthogonal polynomials following non-coherent pair of measures

Eliana X.L. De Andrade^I; Cleonice F. Bracciali^I; Mirela V. De Mello^I; Teresa E. Pérez^II

^IDCCE, IBILCE, UNESP - Universidade Estadual Paulista 15054-000 São José do Rio Preto, SP, Brazil

IIDepartamento de Matemática Aplicada and Instituto Carlos I de Física Teórica y Computacional Universidad de Granada, 18071 Granada, Spain E-mails: [e-mail] / [e-mail] / [e-mail] / [e-mail]

ABSTRACT

Zeros of orthogonal polynomials associated with two different Sobolev inner products involving the Jacobi measure are studied. In particular, each of these Sobolev inner products involves a pair of closely related Jacobi measures. The measures of the inner products considered are beyond the concept of coherent pairs of measures. Existence, real character, location and interlacing properties for the zeros of these Jacobi-Sobolev orthogonal polynomials are deduced.

MATHEMATICAL SUBJECT CLASSIFICATION: 33C45, 33C47, 26C10.

Key words: Sobolev orthogonal polynomials, Jacobi orthogonal polynomials, Zeros of orthogonal polynomials.

1 Introduction

Consider the inner product

where d ψ_j, for j = 0,1,...,k, are positive measures supported on . This inner product is known as Sobolev inner product and the associated sequence of monic orthogonal polynomials, {S_n}, is known as a sequence of monic Sobolev orthogonal polynomials.

This kind of inner product is non-standard in the sense that the shift operator, i.e., the multiplication operator by x, is not self-adjoint

where f and g are polynomials with real coefficients. Therefore, some of the usual properties of standard orthogonal polynomials are not true. In fact, the usual three term recurrence relation and the properties about the zeros (real and simple characters, interlacing, etc.) are no longer valid.

In 1991, A. Iserles et al. [9] studied Sobolev inner products as (1) for k = 1 when the two measures dψ₀ and dψ₁ are related. If we denote by {} (i = 0,1) the sequence of monic orthogonal polynomials with respect to the standard inner product

then {dψ₀, dψ₁} is a coherent pair of measures if

where σ_n are non-zero constants. As a consequence (see [9]), the sequence of monic Sobolev orthogonal polynomials {S_n} associated with the Sobolev inner product

satisfies

H.G. Meijer [13] has shown that if {dψ₀, dψ₁} is a coherent pair of measures, then both measures are closely related and at least one of them must be classical.

Sobolev orthogonal polynomials associated with coherent pairs have been exhaustively studied. Algebraic and differential properties, as well as properties about zeros, have been investigated. In particular, in [6], [8], and [11], severalresults about existence, location and interlacing properties of the zeros of orthogonal polynomials with respect to Gegenbauer-Sobolev and Hermite-Sobolev inner products are shown. Moreover, in [14], the authors considered special Jacobi-Sobolev and Laguerre-Sobolev inner products, where the pair of measures forms a coherent pair and they proved interlacing properties of the zeros of Sobolev orthogonal polynomials.

In [2] and [3] it has been introduced an alternative approach to study Sobolev inner products such that the corresponding orthogonal polynomials still satisfy a relation of the form (3). This kind of Sobolev inner products generalizes Sobolev inner products defined from a coherent pair of measures and it allows to extend the results about Sobolev orthogonal polynomials beyond the concept of coherent pairs. In [7] the authors have considered the inverse problem: starting from the relation (3) to obtain a pair of quasi-definite moment functionals such that (3) holds. Their results show that the pair of measures involved being coherent is not necessary for (3) to hold.

In the present paper, properties for the zeros of Sobolev orthogonal polynomials associated with inner products of the form (2) have been investigated. The measures of the inner products involve Jacobi measures and they are beyond the concept of coherent pairs of measures.

For α, β > -1, let dψ^(α,β) denotes the classical Jacobi measure on [-1,1] given by

be the sequence of classical monic Jacobi orthogonal polynomials and

It is well known that the zeros of are all real, distinct and lie inside (-1,1). We denote the zeros of by , i = 1,2, ..., n, in increasing order. For more details about these polynomials see, for instance, [5] and [15].

In this paper we consider two Sobolev inner products given in [3], namely

Type I. For |κ₁| < 1, κ₂> |κ₃|,

Type II. For |κ₃| > 1, κ₂, κ₄> 0, κ₁> - |κ₃|κ₂/(1+|κ₃|),

When κ₁≠ 0, κ₂ > 0 and κ₃ = 0, the measures involved in type I form a coherent pair (see [13]). In [14], the authors have proved that the zeros of the corresponding Sobolev orthogonal polynomials are real and simple, and they have interlacing properties. When κ₃≠ 0, the pair of measures no longer forms a coherent pair and, in this case, we will show in Section 2 that, under some conditions, the interlacing properties established in [14] for κ₃ = 0 still hold even when κ₃≠ 0.

When κ₁ = 0 the pair of measures of type II is also a coherent pair of measures (see [13]). In this case, properties about zeros for the corresponding Sobolev orthogonal appear in [14]. In [10] the authors have studied the special case when κ₁ = κ₄ = 0, κ₂ > 0 and κ₃ = -1. When κ₁≠ 0, the pair of measures of type II no longer forms a coherent pair. Section 3 is devoted to show interlacing properties for the zeros of Jacobi-Sobolev orthogonal polynomials of type II when κ₁≠ 0. Indeed, our results generalize the interlacing properties established in [14] for the particular case κ₁ = 0.

In [1] it has been studied properties of the zeros of orthogonal polynomials with respect to Gegenbauer-Sobolev inner product where the associated pair of measures does not form a symmetrically coherent pair.

2 Jacobi-Sobolev inner product of type I

Let us consider the following modification of the Jacobi weight

defined on [-1,1], where |κ₁| < 1. We denote by the sequence of monic orthogonal polynomials associated with , and

The following result is known (see, for instance, [3])

where

Observe that sgn (d_n) = sgn (κ₁) where, as usual,

For n > 1, the zeros of are real, simple and all lie in the interval (-1, 1). If we denote their zeros byi = 1,2, ..., n, in increasing order, then they interlace with the zeros of the classical Jacobi polynomials and their position depends on the sign of κ₁ (see [4]):

for -1 < κ₁ < 0 and 1 < i < n-1,

and for 0 < κ₁< 1 and 1 < i < n-1,

Consider the following Sobolev inner product, introduced in [3],

with |κ₁| < 1 andκ₂> |κ₃|. Then ‹·, ·›JS₁ is positive definite and we will refer to it as Jacobi-Sobolev inner product of type I.

Let denote the sequence of monic orthogonal polynomials with respect to (7), we will refer to it as sequence of Jacobi-Sobolev orthogonal polynomials of type I. In addition, we denote .

Furthermore, (x)= 1 and for n > 0

where is given by the expression ([3])

Observe that, if κ₁ = 0, then sgn() = sgn(κ₃). Otherwise, if sgn(κ₁) = sgn(κ₃), then sgn() = sgn(κ₁) = sgn(κ₃).

2.1 Zeros of

and

For n, i > 0 we define

and

Since ‹ , [x + sgn(κ₃)]ⁱ›JS₁ = 0 for n > 1 and 0 < i < n-1, we obtain

Using integration by parts in (10) we have for n > 1 and i > 0,

where η = (β+1)[1+ sgn(κ₃)] -(α+1)[1- sgn(κ₃)]. Since

then sgn(η) = sgn(κ₃).

Substituting (12) in (11), the following three term recurrence relation holds

for n > 2 and 1 < i < n-1, where

Since κ₂> |κ₃| and sgn(η) = sgn(κ₃), we observe that sgn(B_i) = sgn(κ₃) for i > 1.

Moreover, in order to assure the positivity of A_i for i > 1, we need some additional conditions.

Lemma 2.1. Supposeα, β> 0, κ₂> 3|κ₃| and sgn(κ₁) = sgn(κ₃) if κ₁≠ 0. Then A_i > 0 for i > 1.

Proof. Observe that we can write

for κ₃ < 0, and

for κ₃ > 0. Then, the result holds.

Lemma 2.2. Assume that the conditions of Lemma 2.1 hold. For n> 1, if κ₁ = 0, we get

and ifκ₁≠ 0

Proof. Suppose κ₁ = 0. From (11), we get = 0, n > 1. On the otherhand, using the well known property for the monic classical Jacobi polynomials (see [15])

in relation (8), we have

Therefore

Since sgn() = sgn(κ₃) and > 0, we deduce

By making i = 0 in (12), we get = (α+β+2) , and then

Now, suppose that κ₁≠ 0. Using (8), we obtain

and then . Since (x) > 0 for n > 0, we get

The substitution of i = 0 in (13) yields , then

Therefore, using mathematical induction on i in (13), we get the result.

Lemma 2.3. Under the hypotheses of Lemma 2.1, let π_r(x) be a monic polynomial of degree r, with 1 < r < n, such that all of its zeros are real and lie in [-1,1]. Let us define

Then sgn(I_r,n) = (-1)^n+r[sgn(κ₃)]^n+r.

Proof. Let -1 < t_r,1< t_r,2< ... < t_r,r < 1 be the zeros of π_r(x). Then

where c_r = 1 and, if C_i ≠ 0, then sgn(C_i) = (-1)^r-i[ sgn(κ₃)]^r-i, i = 0,1,...,r. Using (9),

and the result holds from Lemma 2.2.Now we will show that, under the hypotheses of Lemma 2.1, the n-th Jacobi-Sobolev orthogonal polynomial of type I, , has n real and simple zeros and they interlace with the zeros of the classical Jacobi polynomial .

Theorem 2.4. Suppose that the conditions of Lemma 2.1 hold. Then, for n > 2, has n real and simple zeros. If we denote

for 1 < i < n, the zeros of in increasing order, then they satisfy

forκ₃ < 0,

forκ₃ > 0,

Proof. Define

Then, deg = n-1 and, using Lemma 2.3, we get

where

Applying the Gaussian quadrature rule based on the n zeros of , weobtain

for j = 1,2, ..., n.

Suppose κ₃ < 0. In this case, for j = 1,2, ..., n, we get

Therefore, there is just one zero of in each interval for i = 2,3,...,n. Since and are monic, there is one zero of in (-∞,). Then, (14) holds.

For κ₃ > 0, we observe that

and a similar argument as above shows (15).Moreover, it is possible to show interlacing properties between the zeros of Jacobi-Sobolev orthogonal polynomials, , and the zeros of the classical Jacobi polynomials , and .

Theorem 2.5. Under the hypotheses of Lemma 2.1, for n > 2 and 1 < i < n-1, we get

i) the zeros of interlace with the zeros of and as follows

ii) the zeros of separate the zeros of in the following way

Proof. Defining

and

a similar argument used for Theorem 2.4 shows the result i).

To get the result ii), take

in Lemma 2.3. Then one obtains

for j = 1,2, ..., n-1.As a consequence of this theorem, the following result is established.

Corollary 2.6. For n> 1, Jacobi-Sobolev orthogonal polynomial

has n real and simple zeros inside (-1,1).

Finally, interlacing properties of the zeros of Jacobi-Sobolev orthogonalpolynomials of two consecutive degrees can be shown.

Theorem 2.7. Under conditions of Lemma 2.1, for n> 2, the n-1 zeros of interlace with the zeros of

as follows

Proof. From (8) we get

On the other hand, Theorem 2.4 provides

and then

Since sgn = sgn(κ₃), we deduce > 0.

Therefore, has a zero in each interval , for i = 2,3,...,n.

Remark 2.8. If we have sgn(κ₁) ≠ sgn(κ₃) instead of sgn(κ₁) = sgn(κ₃) in the hypotheses of Lemma 2.1, then sgn = sgn(κ₁) and there exists N ∈ such that

Numerical experiments allow us to conjecture that, also in this case, the zeros of interlace with the zeros of . Moreover, for 1 < i < n,

• if κ₁ < 0 and κ₃ > 0, then < for n < N and < for n > N,

• if κ₁ > 0 and κ₃ < 0, then < for n < N and < for n > N.

Table 1 describes an example of this fact. Notice, from (8), that if = 0 then (x).

2.2 Zeros of and

In this section we relate the zeros of Jacobi-Sobolev orthogonal polynomial of type I with the zeros of the polynomial , κ₁≠ 0, orthogonal with respect to the first measure in (7).

For n, i > 0, we define

and

Observe that, since ‹ ,(1 + κ₁x)ⁱ›_JS1 = 0, for n > 1,

On the other hand, using integration by parts we get

for n > 1 and i > 0. Then, the following recurrence relation holds

for n > 1 and i = 1,2,...,n-1, where

In order to obtain A_i, B_i and C_i as non-negative coefficients, we need some additional conditions. Observe that the conditions given in the next lemma are sufficient.

Lemma 2.9. Suppose α, β> 0, κ₂> 3κ₃/κ₁> 0 and sgn(κ₁) = sgn(κ₃). Then A_i, B_i and C_i are non-negative for i> 1.

A similar argument used in Lemma 2.2 allows us to obtain the sign of and, using the hypotheses of Lemma 2.9.

Lemma 2.10. Assume that the conditions of Lemma 2.9 hold. For n > 1, we have

• i) = 0, sgn() = (-1)ⁿ⁺ⁱ[ sgn(κ₁)]ⁿ, i = 2,3,...,n,

• ii) sgn() = (-1)ⁿ⁺ⁱ⁺¹[ sgn(κ₁)]ⁿ⁺¹, i = 0,1,...,n-1.

The next lemma is analogous to Lemma 2.3 and it can be proved using the same technique. Again, we assume that the hypotheses of Lemma 2.9 are valid.

Lemma 2.11. Under conditions of Lemma 2.9, for n > 2 it follows

i) Let π_r be a monic polynomial of degree r, 1 < r < n-1, such that all of its zeros are real and lie in (-1,1). Define

then sgn(I_r,n) = (-1)^n+r+1[ sgn(κ₁)]^n+r+1.

ii) Let π_r be a monic polynomial of degree r, 1 < r < n, with all real zeros in (-1,1). If we define

then sgn(J_r,n) = (-1)^n+r+1[sgn(κ₁)]^n+r.

Proof. To prove i), let -1 < t_r,1< t_r,2< ... < t_r,r < 1 be the zeros of π_r. Then

with c_r = 1. When-1 < κ₁ < 0 we have t_r,j+ < 0 and when 0 < κ₁< 1 we have t_r,j+ > 0. Therefore

Hence,

and using Lemma 2.10 the result holds. A similar argument shows ii).Now, we have the necessary tools to get the announced interlacing property between the zeros of and the zeros of .

Theorem 2.12. Under the conditions of Lemma 2.9, for n > 2 and 1 < i < n-1, the zeros of satisfy

• i)

  If -1

  <

  κ

  ₁ < 0,

and, if 0 < κ₁< 1,

The zeros of separate the zeros of . That is,

Collecting all the interlacing properties given in (5), (6) and Theorems 2.4 and 2.12, we have for -1 < κ₁ < 0,

and, for 0 < κ₁< 1,

To finish this section, the extremal points of can be analyzed. Denote the extremal points of by , i = 1,2,...,n-1, in increasing order.

Theorem 2.13. Under the hypotheses of Lemma 2.9, for n > 3 the polynomial

has n-1 extremal points in the interval (-1,1) and they satisfy

3 Jacobi-Sobolev inner product of type II

Let dψ(x) be the measure defined on [-1,1] by means of

where |κ₃| > 1 and κ₄> 0, and let be the corresponding sequence of monic orthogonal polynomials. In Maroni [12] (see also [3]), the author has obtained the relation

where

Note that sgn(d_n-1) = - sgn(κ₃).

In this section we consider Jacobi-Sobolev inner product of type II, introduced in [3], given by the expression

where

We denote by the sequence of monic orthogonal polynomials associated with , and we call it sequence of monic Jacobi-Sobolev orthogonal polynomials of type II. These polynomials satisfy and

where

and

Since , we can also write

Observe that sgn(b_n) = sgn(d_n-1) = - sgn(κ₃) and, if κ₁ > 0, then sgn() = sgn(b_n) = - sgn(κ₃).

3.1 Zeros of

and

For n, i > 0, we define

and

Because of the orthogonality property, we have = 0, for n > 1, and then

Integration by parts in (22) for n > 1 and i > 1 provides

and then, if we define = 0, for n > 1 and i = 2,3,...,n-1, the following three term recurrence relation can be deduced

where

The next lemma establishes sufficient conditions to determine the sign of the above coefficients.

Lemma 3.1. Forκ₂> 2κ₁> 0, α+β > 2 and

then sgn(A_i) = - sgn(κ₃) and B_i > 0, for i > 1. Moreover, if |κ₃| ≠ 1 then sgn(C_i) = - sgn(κ₃) and if |κ₃| = 1 then C_i = 0.

We remark that Lemma 3.1 establishes sufficient conditions in order to obtain the sign of A_i, B_i and C_i. Under conditions of Lemma 3.1, analogous techniques to those used in Lemmas 2.2 and 2.3 allow us to prove the next two lemmas.

Lemma 3.2. For n > 3, we have = 0 and

Lemma 3.3. For n > 3, let us consider π_r a monic polynomial of degree r, 1 < r < n-1, such that all its zeros are real, simple and lie in the interval (-1,1). Define

Then sgn(I_r,n) = .

Under the same restrictions given in Lemma 3.1 for the parameters and using the above two lemmas, we can show that the n-th Jacobi-Sobolev orthogonal polynomial of type II, , has n different real zeros and at least n-1 zeros lie inside (-1,1). We denote the real zeros of , in increasing order, by , i = 1,2,...,n.

Theorem 3.4. Under the conditions of Lemma 3.1, for n > 3,has n real zeros and at least n-1 of them lie inside the interval (-1,1). Moreover, denoting the zeros of , by , i = 1,2,...,n, in increasing order, then

• i) if κ₃< -1,

if κ₃> 1,

• ii) For 1 < i < n-1, the following interlacing property holds

We must point out that one zero of can be outside the interval (-1,1). Figure 1 shows the graphs of Jacobi-Sobolev orthogonal polynomial of type II and the classical Jacobi orthogonal polynomial . According to Theorem 3.4, in Figure 1(a), with κ₃ < -1, we can see that the smallest zero is outside the interval (-1,1). In Figure 1(b), with κ₃ > 1, we can see that the largest zero is outside (-1,1).

(a) Figure (b) Figure

3.2 Some conditions for all zeros ofto lie inside (-1,1)

In this section we obtain some conditions for the parameters in the inner product (18) to assure that all zeros of lie inside the interval (-1,1).

We now denote the polynomials by (x) and the coefficients by .

Let κ₂ tend to ∞ in . Then we find that the monic polynomials, (x), must satisfy

Since and, for a fixed κ₁, = -sgn(κ₃), from (20) we verify that = 0. Then we conclude from (19) that

with b_-1 = b₀ = 0.

It is well known that the sequence of monic Jacobi polynomials, , satisfies

with

Now, we can prove the following result.

Theorem 3.5. If the conditions of Lemma 3.1 are satisfied, n > 3 and κ₂ large enough, then the n zeros of lie inside the interval (-1,1) provided that

• forκ₃< -1, α and β are such that

• forκ₃> 1, α and β are such that

Proof. i) For x = -1 it is known that

and sgn((-1)) = (-1)ⁿ. Then, from (26),

Choosing α and β such that

we get sgn((-1)) = (-1)ⁿ.

From Theorem 3.4, for κ₃< -1, at most s_n,1 lies outside (-1,1). Since is monic and sgn((-1)) = (-1)ⁿ then s_n,1 > -1 and all zeros of lie inside (-1,1).

ii) For κ₃> 1, the proof is analogous using x = 1.

Acknowledgements. This research was supported by grants from CAPES, CNPq and FAPESP of Brazil and by grants from Ministerio de Ciencia e Innovación (Micinn) of Spain and European Regional Development Fund (MTM2008-06689-C02-02) and by Junta de Andalucía (G.I. FQM 0229). The authors would like to thank the referees for their valuable remarks, suggestions and references.

Received:04/VII/09.

Accepted:14/XII/09.

#CAM-117/09.

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Publication Dates

• Publication in this collection
  22 Nov 2010
• Date of issue
  2010

History

• Received
  04 July 2009
• Accepted
  14 Dec 2009