- Research Article
- Open Access
Jacobi-Sobolev Orthogonal Polynomials: Asymptotics for N-Coherence of Measures
- Bujar Xh Fejzullahu^{1} and
- Francisco Marcellán^{2}Email author
https://doi.org/10.1155/2011/294134
© B. Xh. Fejzullahu and F. Marcellán. 2011
- Received: 24 November 2010
- Accepted: 7 March 2011
- Published: 14 March 2011
Abstract
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.
Keywords
- Compact Subset
- Orthogonal Polynomial
- Fourier Expansion
- Jacobi Polynomial
- Sobolev Norm
1. Introduction
where and , with , , , and , for all . We denote by the vector of dimension with components .
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 .
2. Preliminaries
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.
- (a)
The leading coefficient of is (see [12, formulas (4.3.4) and (4.21.6)])
where if and if (see [12, Theorem 7.32.1]).
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.
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]).
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.
From Proposition 2.1 and (2.12), we get the following.
3. Asymptotics of Jacobi-Sobolev Orthogonal Polynomials
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.
Proof.
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.
where , , and , , . Moreover, for , and for , .
Proof.
where , and, by convention, , .
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).
the relation (3.19) holds for . As consequence, for and .
Now, we will prove by induction that for and .
Next, we will give some properties of the Jacobi-Sobolev orthogonal polynomials.
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,
Proof.
Corollary 3.5.
Proof.
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.
where are real numbers, and is the Bessel function of the first kind.
Proof.
- (a)
where , and , . Moreover, and for .
- (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.
Proof.
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.
Proof.
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.
Proof.
Thus, for and large enough, (3.57) follows.
The proof of Proposition 3.9 is complete.
Thus, using (3.56) and (3.61), the statement follows.
4. Necessary Conditions for the Norm Convergence
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.
Theorem 4.1.
Proof.
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.
Declarations
Acknowledgments
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.
Authors’ Affiliations
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