Jacobi-Sobolev Orthogonal Polynomials: Asymptotics for N-Coherence of Measures

Let us introduce the Sobolev-type inner product , where and , , with and for all A Mehler-Heine-type formula and the inner strong asymptotics on as well as some estimates for the polynomials orthogonal with respect to the above Sobolev inner product are obtained. Necessary conditions for the norm convergence of Fourier expansions in terms of such Sobolev orthogonal polynomials are given.

For a nontrivial probability measure , supported on , we define the linear space of all measurable functions on such that , where

(11)

Let us now introduce the Sobolev-type spaces (see, e.g., [1, Chapter  3] in a more general framework)

(12)

where and , with , , , and , for all . We denote by the vector of dimension with components .

Let and in . We can introduce the Sobolev-type inner product

(13)

where and

(14)

(15)

where , , , , and , for all . In the sequel, we will assume that , and, therefore, for all , and .

Using the standard Gram-Schmidt method for the canonical basis in the linear space of polynomials, we obtain a unique sequence (up to a constant factor) of polynomials orthogonal with respect to the above inner product. In the sequel, they will called Jacobi-Sobolev orthogonal polynomials.

For and , the pair of measures is a 0-coherent pair, studied in [2–4] (see also [5] in a more general framework). In [6], the authors established the distribution of the zeros of the polynomials orthogonal with respect to the above Sobolev inner product (1.3) when and . Some results concerning interlacing and separation properties of their zeros with respect to the zeros of Jacobi polynomials are also obtained assuming we are working in a coherent case. More recently, for a noncoherent pair of measures, when , , , and , the distribution of zeros of the corresponding Sobolev orthogonal polynomials as well as some asymptotic results (more precisely, inner strong asympttics, outer relative asymptotics, and Mehler-Heine formulas) for these sequences of polynomials are deduced in [7–9]. In the Jacobi case, some analog problems have been considered in [10, 11].

The aim of this contribution is to study necessary conditions for -norm convergence of the Fourier expansion in terms of Jacobi-Sobolev orthogonal polynomials. In order to prove it, we need some estimates and strong asymptotics for the polynomials as well as for their derivatives . A Mehler-Heine-type formula, inner strong asymptotics, upper bounds in , and norms of Jacobi-Sobolev orthonormal polynomials are obtained. Thus, we extend the results of [10] for generalized -coherent pairs of measures.

The structure of the manuscript is as follows. In Section 2, we give some basic properties of Jacobi polynomials that we will use in the sequel. In Section 3, an algebraic relation between the sequences of polynomials and Jacobi orthonormal polynomials is stated. It involves (where ) consecutive terms of such sequences in such a way that we obtain a generalization of the relations satisfied in the coherent case. Upper bounds for the polynomials and their derivatives in are deduced. The inner strong asymptotics as well as a Mehler-Heine-type formula are obtained. Finally, the asymptotic behavior of these polynomials with respect to the norm is studied. In Section 4, necessary conditions for the convergence of the Fourier expansions in terms of the sequence of Jacobi-Sobolev orthogonal polynomials are presented.

Throughout this paper, positive constants are denoted by and they may vary at every occurrence. The notation means that the sequence converges to 1 and notation means for sufficiently large .

For , , we denote by the sequence of Jacobi polynomials which are orthonormal on with respect to the inner product

(21)

We will denote by the leading coefficient of any polynomial , and . Now, we list some properties of the Jacobi orthonormal polynomials which we will use in the sequel.

Proposition 2.1.

1. (a)

   The leading coefficient of is (see [12, formulas   (4.3.4) and (4.21.6)])

(22)

(b)The derivatives of Jacobi polynomials satisfy (see [12, formula   (4.21.7)])

(23)

(c)For , and

(24)

where if and if (see [12, Theorem  7.32.1]).

(d)For the polynomials , we get the following estimate (see [12, formula  (7.32.6)], [13, Theorem  1]):

(25)

where and .

(e)Mehler-Heine formula (see [12, Theorem  8.1.1])

(26)

where are real numbers and is the Bessel function of the first kind. This formula holds locally uniformly, that is, on every compact subset of the complex plane.

(f)Inner strong asymptotics. For , when and , we get (see [12, Theorem  8.21.8])

(27)

where , , , and .

(g)For ,, , and (see [12, p.391. Exercise  91], [14,  (2.2)], [15,   Theorem  2]),

(28)

Let be the sequence of orthonormal polynomials with respect to the inner product (1.5), and let

(29)

be the th polynomial orthonormal with respect to , where and , , are the Tchebychev polynomials of the first kind.

Proposition 2.2 ([16, Lemma  2.1]).

For , there exist constants such that

(210)

and , where

(211)

Next, we will consider the polynomials

(212)

where , . Notice that

(213)

Taking into account that the zeros of the polynomial orthogonal with respect to on the interval are real, simple, and located in , we have . Therefore, for large enough.

On the other hand, using (b) in Proposition 2.1, we have

(214)

From Proposition 2.1 and (2.12), we get the following.

Proposition 2.3.

1. (a)

   For ,, and ,

   

   (215)

where if and if .

1. (b)

   When and , , we get the following estimate for the polynomials :

   

   (216)

1. (c)

   Mehler-Heine type formula. We get

   

   (217)

where are real numbers, and is the Bessel function of the first kind. This formula holds locally uniformly, that is, on every compact subset of the complex plane.

1. (d)

   Inner strong asymptotics. When and , we get

   

   (218)

where , , , and .

1. (e)

   For ,, , and ,

   

   (219)

Let denote the sequence of polynomials orthogonal with respect to (1.3) normalized by the condition that they have the same leading coefficient as , that is, .

The following relation between and holds.

Proposition 3.1.

For ,

(31)

where, for ,

(32)

Moreover, and for .

Proof.

Expanding with respect to the basis of the linear space of polynomials with degree at most , we get

(33)

where, for ,

(34)

For ,

(35)

Therefore,

(36)

As a conclusion,

(37)

(38)

Using the extremal property for monic orthogonal polynomials with respect to the corresponding norm (see [12, Theorem  3.1.2]),

(39)

we get

(310)

Thus,

(311)

Finally, from (3.8), we find that

(312)

and from Schwarz inequality,

(313)

Thus,

(314)

Using (3.1) in a recursive way, we get the representation of the polynomial in terms of the elements of the sequence . More precisely we get the following.

Proposition 3.2.

For , , it holds that

(315)

where , , and , , . Moreover, for , and for , .

Proof.

Let denote by , , and , , . First, we prove that

(316)

where , and, by convention, , .

We will prove (3.16) by induction. When , it is a trivial result. On the other hand, applying (3.1) in a recursive way, we get

(317)

Taking into account (3.7), we have . Thus, (3.16) follows for . Now, we assume (3.16) holds for . Again, from (3.1),

(318)

Now, we prove that for . For , this follows from (3.7). Since and , for the statement follows by induction. Thus, (3.16) holds for . Now taking in (3.16), we get (3.15).

Finally, we prove that for , and for , . First, the following inequality holds:

(319)

. Indeed, for , (3.19) follows from Proposition 3.1 and (3.7). Now, we assume that the relation (3.19) holds for . Thus, for ,

(320)

for

(321)

for

(322)

and for

(323)

Therefore, from

(324)

the relation (3.19) holds for . As consequence, for and .

Now, we will prove by induction that for and .

The case follows from (3.19). We assume that for and . For ,

(325)

and for

(326)

Therefore, from

(327)

the statement holds for .

Next, we will give some properties of the Jacobi-Sobolev orthogonal polynomials.

Proposition 3.3.

1. (a)

   For the polynomials , we get

   

   (328)

where , and .

1. (b)

   For the polynomials , we get

   

   (329)

where , and , .

Proof.

1. (a)

   Using Proposition 3.2, we have

   

   (330)

Therefore, from Proposition 2.3, the statement follows immediately.

On the other hand, taking into account Proposition 2.1, Proposition 2.2, (2.14), and (3.15), the proof of can be done in a similar way.

Now, we show that, like for the classical Jacobi polynomials, the polynomial attains its maximum in at the end-points. More precisely,

Proposition 3.4.

1. (a)

   For , , and

   

   (331)

where if and if .

1. (b)

   For , and

   

   (332)

where if and if .

Proof.

Here, we will prove only the case when . The case when can be done in a similar way.

1. (a)

   From Proposition 2.3,

   

   (333)

for and . Therefore, according to (3.30),

(334)

for and . From Proposition 3.1, we get

(335)

Finally, from Proposition 2.3(a), the statement follows.

1. (b)

   Taking into account Proposition 2.1, Proposition 2.2, (2.14), (3.1), and (3.15), we can conclude the proof in the same way as we did in (a).

Corollary 3.5.

For , ,

(336)

and for , ,

(337)

where

(338)

Proof.

The inequality

(339)

holds for , as well as

(340)

for . Therefore, from Propositions 3.3 and 3.4, the statement follows immediately.

Next, we deduce a Mehler-Heine-type formula for and (see Theorem  4.1 in [10]).

Proposition 3.6.

Uniformly on compact subsets of ,

1. (a)
   

   (341)

1. (b)
   

   (342)

where are real numbers, and is the Bessel function of the first kind.

Proof.

To prove the proposition, we use the same technique as in [17].

1. (a)

   Multiplying in (3.1) by , we obtain

   

   (343)

where , and , . Moreover, and for .

Using the above relation in a recursive way as well as the same argument of Proposition 3.2, we have

(344)

where , for , and for ,. Thus,

(345)

On the other hand, from Proposition 2.3(c), is uniformly bounded on compact subsets of . Thus, for a fixed compact set , there exists a constant , depending only on , such that when ,

(346)

Thus, the sequence is uniformly bounded on . As a conclusion,

(347)

and from Proposition 2.3(c), we obtain the result.

1. (b)

   Since we have uniform convergence in (3.41), taking derivatives and using a well known property of Bessel functions of the first kind (see [12, formula  1.71.5]), we obtain (3.42).

Now, we give the inner strong asymptotics of on .

Proposition 3.7.

For and ,

(348)

(349)

where , ,  , , and .

Proof.

From Proposition 3.3(a), the sequence is uniformly bounded on compact subsets of ; thus, from Proposition 3.1,

(350)

Now, using Proposition 2.3(d), the relation (3.48) follows.

Concerning (3.49), it can be obtained in a similar way by using Propositions 2.1(f) and 2.2, (2.14), Propositions 3.1 and 3.3(b).

Now, we can give the sharp estimate for the Sobolev norms of the Jacobi-Sobolev polynomials.

Proposition 3.8.

For and ,

(351)

Proof.

Clearly, if , then we get Proposition 3.4(b). Thus, in the proof, we will assume . Since by Proposition 3.2 and(2.14)

(352)

where , , and are bounded because of the orthonormality condition, we obtain

(353)

where , and .

On the other hand, using (3.30), Minkowski's inequality, and Proposition 2.3(e), we deduce

(354)

In the same way as above, we get

(355)

Thus, from (3.53), (3.54), and (3.55), we have

(356)

Notice that the upper estimate in (3.54) and (3.55) can also be proved using the bounds for Jacobi-Sobolev polynomials given in Corollary 3.5.

In order to prove the lower bound in (3.51) we will need the following.

Proposition 3.9.

For and ,

(357)

Proof.

We will use a technique similar to [12, Theorem  7.34]. According to (3.42),

(358)

On the other hand, from (see [18, Lemma  2.1]), if and , we have

(359)

Thus, for and large enough, (3.57) follows.

Finally, from (3.49), we obtain

(360)

The proof of Proposition 3.9 is complete.

From (3.57), for and ,

(361)

Thus, using (3.56) and (3.61), the statement follows.

The analysis of the norm convergence of partial sums of the Fourier expansions in terms of Jacobi polynomials has been done by many authors. See, for instance, [19–21], and the references therein.

Let be the Jacobi-Sobolev orthonormal polynomials, that is,

(41)

For , its Fourier expansion in terms of Jacobi-Sobolev orthonormal polynomials is

(42)

where

(43)

Let be the th partial sum of the expansion (4.2)

(44)

Theorem 4.1.

Let , and . If there exists a constant such that

(45)

for every , then with

(46)

Proof.

For the proof, we apply the same argument as in [20]. Assume that (4.5) holds. Then,

(47)

Consider the linear functionals

(48)

on . Hence, for every in holds. From the Banach-Steinhaus theorem, this yields . On the other hand, by duality (see, for instance, [1, Theorem  3.8]), we have

(49)

where is the conjugate of . Therefore,

(410)

On the other hand, from (3.51), we obtain the Sobolev norms of Jacobi-Sobolev orthonormal polynomials

(411)

for and . Now, from (4.11), it follows that the inequality (4.10) holds if and only if .

The proof of Theorem 4.1 is complete.

The authors thank the referees for the careful revision of the manuscript. Their comments and suggestions have contributed to improve substantially its presentation. The work of F. Marcellán has been supported by Dirección General de Investigación, Ministerio de Ciencia e Innovación of Spain, Grant no. MTM2009-12740-C03-01.

Affiliations

1. Faculty of Mathematics and Sciences, University of Prishtina, Mother Teresa 5, 10000, Prishtina, Kosovo

   • Bujar Xh Fejzullahu

2. Departamento de Matemáticas, Escuela Politécnica Superior, Universidad Carlos III de Madrid, Avenida de la Universidad, 30, 28911, Leganes, Spain

   • Francisco Marcellán

Corresponding author

Correspondence to Francisco Marcellán.

Reprints and Permissions