Jacobi-Sobolev Orthogonal Polynomials: Asymptotics for N-Coherence of Measures
© B. Xh. Fejzullahu and F. Marcellán. 2011
Received: 24 November 2010
Accepted: 7 March 2011
Published: 14 March 2011
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.
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  in a more general framework). In , 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  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.
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]).
Proposition 2.2 ([16, Lemma 2.1]).
From Proposition 2.1 and (2.12), we get the following.
3. Asymptotics of Jacobi-Sobolev Orthogonal Polynomials
Next, we will give some properties of the Jacobi-Sobolev orthogonal polynomials.
Next, we deduce a Mehler-Heine-type formula for and (see Theorem 4.1 in ).
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, 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.
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.
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 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.
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