Journal of Inequalities and Applications volume 2011, Article number: 294134 (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.
For a nontrivial probability measure 
, supported on 
, we define the linear space 
of all measurable functions 
on 
such that 
, where
Let us now introduce the Sobolev-type spaces (see, e.g., [1, Chapter 3] in a more general framework)
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
where 
and
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
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.
The leading coefficient of 
is (see [12, formulas (4.3.4) and (4.21.6)])
(b)The derivatives of Jacobi polynomials satisfy (see [12, formula (4.21.7)])
(c)For 
, and 
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]):
where 
and 
.
(e)Mehler-Heine formula (see [12, Theorem 8.1.1])
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.
(f)Inner strong asymptotics. For 
, when 
and 
, we get (see [12, Theorem 8.21.8])
where 
, 
, 
, and 
.
(g)For 
,
, 
, and 
(see [12, p.391. Exercise 91], [14, (2.2)], [15, Theorem 2]),
Let 
be the sequence of orthonormal polynomials with respect to the inner product (1.5), and let
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
and 
, where
Next, we will consider the polynomials
where 
, 
. Notice that
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
From Proposition 2.1 and (2.12), we get the following.
Proposition 2.3.
For 
,
, and 
,
where 
if 
and 
if 
.
When 
and 
, 
, we get the following estimate for the polynomials 
:
Mehler-Heine type formula. We get
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.
Inner strong asymptotics. When 
and 
, we get
where 
, 
, 
, and 
.
For 
,
, 
, and 
,
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 
,
where, for 
,
Moreover, 
and 
for 
.
Proof.
Expanding 
with respect to the basis 
of the linear space of polynomials with degree at most 
, we get
where, for 
,
For 
,
Therefore,
As a conclusion,
Using the extremal property for monic orthogonal polynomials with respect to the corresponding norm (see [12, Theorem 3.1.2]),
we get
Thus,
Finally, from (3.8), we find that
and from Schwarz inequality,
Thus,
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
where 
, 
, and 
, 
, 
. Moreover, 
for 
, and 
for 
, 
.
Proof.
Let denote by 
, 
, and 
, 
, 
. First, we prove that
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
Taking into account (3.7), we have 
. Thus, (3.16) follows for 
. Now, we assume (3.16) holds for 
. Again, from (3.1),
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:
. Indeed, for 
, (3.19) follows from Proposition 3.1 and (3.7). Now, we assume that the relation (3.19) holds for 
. Thus, for 
,
for 
for 
and for 
Therefore, from
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 
,
and for 
Therefore, from
the statement holds for 
.
Next, we will give some properties of the Jacobi-Sobolev orthogonal polynomials.
Proposition 3.3.
For the polynomials 
, we get
where 
, and 
.
For the polynomials 
, we get
where 
, and 
, 
.
Proof.
Using Proposition 3.2, we have
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.
For 
, 
, and 
where 
if 
and 
if 
.
For 
, 
and 
where 
if 
and 
if 
.
Proof.
Here, we will prove only the case when 
. The case when 
can be done in a similar way.
From Proposition 2.3
,
for 
and 
. Therefore, according to (3.30),
for 
and 
. From Proposition 3.1, we get
Finally, from Proposition 2.3(a), the statement follows.
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 
, 
,
and for 
, 
,
where
Proof.
The inequality
holds for 
, as well as
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 
,
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].
Multiplying in (3.1) by 
, we obtain
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
where 
, 
for 
, and 
for 
,
. Thus,
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 
,
Thus, the sequence 
is uniformly bounded on 
. As a conclusion,
and from Proposition 2.3(c), we obtain the result.
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 
,
where 
, 
, 
, 
, and 
.
Proof.
From Proposition 3.3(a), the sequence 
is uniformly bounded on compact subsets of 
; thus, from Proposition 3.1,
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 
,
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)
where 
, 
, and 
are bounded because of the orthonormality condition, we obtain
where 
, and 
.
On the other hand, using (3.30), Minkowski's inequality, and Proposition 2.3(e), we deduce
In the same way as above, we get
Thus, from (3.53), (3.54), and (3.55), we have
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 
,
Proof.
We will use a technique similar to [12, Theorem 7.34]. According to (3.42),
On the other hand, from (see [18, Lemma 2.1]), if 
and 
, we have
Thus, for 
and 
large enough, (3.57) follows.
Finally, from (3.49), we obtain
The proof of Proposition 3.9 is complete.
From (3.57), for 
and 
,
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,
For 
, its Fourier expansion in terms of Jacobi-Sobolev orthonormal polynomials is
where
Let 
be the 
th partial sum of the expansion (4.2)
Theorem 4.1.
Let 
, and 
. If there exists a constant 
such that
for every 
, then 
with
Proof.
For the proof, we apply the same argument as in [20]. Assume that (4.5) holds. Then,
Consider the linear functionals
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
where 
is the conjugate of 
. Therefore,
On the other hand, from (3.51), we obtain the Sobolev norms of Jacobi-Sobolev orthonormal polynomials
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.
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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.
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Fejzullahu, B.X., Marcellán, F. Jacobi-Sobolev Orthogonal Polynomials: Asymptotics for N-Coherence of Measures. J Inequal Appl 2011, 294134 (2011). https://doi.org/10.1155/2011/294134
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DOI: https://doi.org/10.1155/2011/294134