 Research Article
 Open access
 Published:
A Cohen Type Inequality for Fourier Expansions of Orthogonal Polynomials with a Nondiscrete JacobiSobolev Inner Product
Journal of Inequalities and Applications volume 2010, Article number: 128746 (2010)
Abstract
Let denote the sequence of polynomials orthogonal with respect to the nondiscrete Sobolev inner product , where and with , . In this paper, we prove a Cohen type inequality for the Fourier expansion in terms of the orthogonal polynomials Necessary conditions for the norm convergence of such a Fourier expansion are given. Finally, the failure of almost everywhere convergence of the Fourier expansion of a function in terms of the orthogonal polynomials associated with the above Sobolev inner product is proved.
1. Introduction
Let with be the Jacobi measure supported on the interval We say that if is measurable on and where
Let us introduce the Sobolevtype spaces (see, for instance, [1, Chapter III], in a more general framework) as follows:
where as well as the linear space of all bounded linear operators with the usual operator norm
Let and be in Let us consider the following Sobolevtype inner product:
where Let denote the sequence of polynomials orthogonal with respect to (1.4), normalized by the condition that has the same leading coefficient as the following classical Jacobi polynomial:
We call them the JacobiSobolev orthogonal polynomials.
The measures and constitute a particular case of the socalled coherent pairs of measures studied in [2]. In [3] (see also [4]), the authors established the asymptotics of the zeros of such JacobiSobolev polynomials.
The aim of our contribution is to obtain a lower bound for the norm of the partial sums of the Fourier expansion in terms of JacobiSobolev polynomials, the wellknown Cohen type inequality in the framework of Approximation Theory. A Cohen type inequality has been established in other contexts, for example, on compact groups or for classical orthogonal expansions. See [5–10] and references therein.
Throughout the paper, positive constants are denoted by and they may vary at every occurrence. The notation means that the sequence converges to 1 and means for sufficiently large where and are positive real numbers.
The structure of the paper is as follows. In Section 2, we introduce the basic background about Jacobi polynomials to be used in the paper. In particular, we focus our attention in some estimates and the strong asymptotics on for such polynomials as well as the MehlerHeine formula. In Section 3, we analyze the polynomials orthogonal with respect to the inner product (1.4). Their representation in terms of Jacobi polynomials yields estimates, inner strong asymptotics, and a MehlerHeine type formula. Some estimates of the weighted Sobolev norm of these polynomials will be needed in the sequel and we show them in Proposition 3.12. In Section 4, a Cohentype inequality, associated with the Fourier expansions in terms of the JacobiSobolev orthogonal polynomials, is deduced. In Section 5, we focus our attention in the norm convergence of the above Fourier expansions. Finally, Section 6 is devoted to the analysis of the divergence almost everywhere of such expansions.
2. Jacobi Polynomials
For we denote by the sequence of Jacobi polynomials which are orthogonal on with respect to the measure They are normalized in such a way that We denote the th monic Jacobi polynomial by
where (see [11, formula ])
Now, we list some basic properties of Jacobi polynomials which will be used in the sequel. The following integral formula for Jacobi polynomials holds (see (2.1) and [11, formula ]):
They satisfy a connection formula (see [11, formula ], [3, formula ]) as follows:
where
as well as the following relation for the derivatives (see [12, formula (4.21.7)]):
The following estimate for holds (see [12, formula ], [13]):
where and
The formula of MehlerHeine for Jacobi orthogonal polynomials is (see [12, Theorem ]) as follows:
where are real numbers, and is the Bessel function. This formula holds locally uniformly, that is, on every compact subset of the complex plane.
The inner strong asymptotics of , for and are read as follows (see [12, Theorem ]):
where and
For and (see [12, page 391. Exercise ], as well as [10, ])
3. Asymptotics of JacobiSobolev Orthogonal Polynomials
Let us denote by the monic JacobiSobolev polynomial of degree that is, From (2.4) and [3, formula ] (see also [4, 14] in a more general framework), we have the following relation between the JacobiSobolev and Jacobi monic orthogonal polynomials.
Proposition 3.1.
For
where is given in (2.5) and
Proposition 3.2.
One gets:
In particular, for defined in (3.2) one obtains
Proof.
We apply the same argument as in the proof of Theorem in [15]. Using the extremal property
we get the following:
On the other hand, from the extremal property of (2.4), and (2.6), we have
Since by (2.3) and (2.5) we have and then (3.6) and (3.7) yield (3.3).
As a straightforward consequence of Propositions 3.1 and 3.2, using (2.1) we deduce the following.
Corollary 3.3.
For
where and
Corollary 3.4.
For and
and for
Proof.
The first statement follows from Proposition 3.1 and (2.4). The second one follows by taking derivatives in (3.10) and using (2.6).
Using (3.10) in a recursive way, the representation of the polynomials in terms of the elements of the sequence becomes
where and
Proposition 3.5.
There exists a constant such that the coefficients in (3.11) satisfy for all and
Proof.
From (3.9), we have Thus, there exist and a constant such that for all and for Therefore, for
and for
Proposition 3.6.
For the polynomials one obtains
for and
For the polynomials one has the following estimate:
for and
Proof.

(a)
Using (3.12), we have the following:
(3.17)
From (2.7), it is straightforward to prove that, for and
Thus, according to Proposition 3.5,
On the other hand, from (3.11), the proof of the case (b) can be done in a similar way.
Proposition 3.7.
Let then
Proof.
Taking into account that the Jacobi polynomials satisfy the following (see [12, paragraph below Theorem ]):
for thus, for
As a consequence, the statement follows from the latter estimates and arguments similar to those we used in the proof of Proposition 3.6.
Corollary 3.8.
For and
and for and
where
Proof.
The inequality
holds for as well as
holds for Therefore, the statement follows from Propositions 3.6 and 3.7.
Next, we show that the JacobiSobolev polynomial attains its maximum in at the end points. To be more precise, consider the following.
Proposition 3.9.
For and ,
where if and if
For and ,
where if and if
Proof.

(a)
We will prove only the case If the the proof can be done in a similar way. From (3.9), (3.10), and Proposition 3.7,
(3.30)
Now, from [12, Theorem ] and Proposition 3.7, the result follows.
Taking into account (2.6), the case (b) can be proved in a similar way.
Next, we deduce a MehlerHeine type formula for and
Proposition 3.10.
Let Uniformly on compact subsets of one gets
Proof.
Multiplying in (3.8) by we obtain
where and according to (3.9).
Using the above relation in a recursive way, we obtain
where and Moreover, by using the same argument as in Proposition 3.5, we have for every and Thus,
On the other hand, from (2.8), we have that 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 using (2.8), we obtain the result.
Since we have uniform convergence in (3.31), taking derivatives and using some properties of Bessel functions, we obtain (3.32).
Now, we give the inner strong asymptotics of on
Proposition 3.11.
Let and For one has
and for one has
where and
Proof.
From Proposition 3.6, the sequence is uniformly bounded on compact subsets of Multiplication by in (3.10) yields
Since
we have
Now, (3.38) follows from (2.9).
Concerning (3.39), it can be obtained in a similar way by using (3.11) and Proposition 3.6
Next, we obtain an estimate for the Sobolev norms of the JacobiSobolev polynomials.
Proposition 3.12.
For and , one has
Notice that if then we have Proposition 3.9 Thus, in the proof we will assume
Proof.
In order to establish the upper bound in (3.38), it is enough to prove that
Using (3.8) in a recurrence way and then Minkowski's inequality, we obtain
On the other hand, for and (2.10) implies
Thus,
On the other hand, from Proposition 3.5,
Thus,
In the same way as above, we conclude that
Thus, (3.44) follows from (3.49) and (3.50).
In order to prove the lower bound in relation (3.43), we will need the following.
Proposition 3.13.
For and one has
Proof.
We will use a technique similar to [12, Theorem ]. According to (3.11),
On the other hand, Stempak's lemma (see [16, Lemma ]), for and implies
Thus, for and large enough, (3.51) follows.
Finally, from (3.39) we obtain the following:
For the proof of Proposition 3.12, from (3.51), for and we get
Thus, using (3.44) and (3.55), the statement follows.
4. A Cohen Type Inequality for JacobiSobolev Expansions
For its Fourier expansion in terms of JacobiSobolev polynomials is
where
The Cesàro means of order of the expansion (4.1) is defined by (see [17, pages 7677]),
where
For a function and a fixed sequence of real numbers with we define the operators by
Let and let be the conjugate of Now, we can state our main result.
Theorem 4.1.
For and one has
Corollary 4.2.
Let and be as in Theorem 4.1. For and for outside the interval , one has
For Theorem 4.1 yields the following.
Corollary 4.3.
For and one has
and ,
We will use the following as test functions (see [10, formula ], and [11, formula ]):
where and
Applying the operator to for some we get
where
and using (2.3) and (3.3), we deduce
Taking into account (4.9), for
If then we get
If then
On the other hand, for
and for
Thus,
As a conclusion,
Now, we will estimate
From [10, formula ],
for
On the other hand, from (2.6), (4.9), and [12, formula ], one has
From (2.10), for ,
for and ,
and for and ,
Thus, for and ,
By using (4.22) and (4.27), we find from (4.21) that
for and
Now, we can prove our main result.
Proof of Theorem 4.1.
By duality, it is enough to assume that From (4.11), (4.20), and (4.28), one has
Now from Proposition 3.12, the statement of the theorem follows.
5. Necessary Conditions for the Norm Convergence
The problem of the convergence in the norm of partial sums of the Fourier expansions in terms of Jacobi polynomials has been discussed by many authors. See, for instance, [18–20] and the references therein.
Let be the JacobiSobolev orthonormal polynomials, that is,
For the Fourier expansion in terms of JacobiSobolev orthonormal polynomials is
where
Let be the th partial sum of the expansion (5.2) as follows:
Theorem 5.1.
Let and If there exists a constant such that
for every , then
Proof.
For the proof, we apply the same argument as in [19]. Assume that (5.5) holds, then
Therefore,
where is the conjugate of
On the other hand, from (3.43) we obtain the Sobolev norms of JacobiSobolev orthonormal polynomials as follows:
for and Now, from (5.8) it follows that the inequality (5.7) holds if and only if
The proof of Theorem 5.1 is complete.
6. Divergence Almost Everywhere
For and Pollard [21] showed that for each there exists a function such that its Fourier expansion (4.27) diverges almost everywhere on . Later on, Meaney [22] extended the result to Furthermore, he proved that this is a special case of a divergence result for the Fourier expansion in terms of Jacobi polynomials. The failure of almost everywhere convergence of the Fourier expansions associated with systems of orthogonal polynomials on and Bessel systems has been discussed in [16, 23].
If the sequence is uniformly bounded on a set, say of positive measure in then
Therefore,
almost everywhere on E. From Egorov's Theorem, it follows that there is a subset of positive measure such that
uniformly for On the other hand, from (3.39)
uniformly for . Using the CantorLebesgue Theorem, as described in [24, Section ], (see also [17, page 316]), we obtain
Theorem 6.1.
Let and There is an whose Fourier expansion (5.2) diverges almost everywhere on in the norm of
Proof.
Consider the linear functionals
on By using [1, Theorem ], we have
Thus, from (5.8),
As a consequence of the BanachSteinhaus theorem, there exists such that
Since this result contradicts (6.5), then for this the Fourier series diverges almost everywhere on in the norm of
References
Adams RA: Sobolev Spaces, Pure and Applied Mathematics. Volume 6. Academic Press, New York, NY, USA; 1975:xviii+268.
Meijer HG: Determination of all coherent pairs. Journal of Approximation Theory 1997, 89(3):321–343. 10.1006/jath.1996.3062
Kim DH, Kwon KH, Marcellán F, Yoon GJ: Zeros of JacobiSobolev orthogonal polynomials. International Mathematical Journal 2003, 4(5):413–422.
Meijer HG, de Bruin MG: Zeros of Sobolev orthogonal polynomials following from coherent pairs. Journal of Computational and Applied Mathematics 2002, 139(2):253–274. 10.1016/S03770427(01)004216
Cohen PJ: On a conjecture of Littlewood and idempotent measures. American Journal of Mathematics 1960, 82: 191–212. 10.2307/2372731
Dreseler B, Soardi PM: A Cohen type inequality for ultraspherical series. Archiv der Mathematik 1982, 38(3):243–247.
Dreseler B, Soardi PM: A Cohentype inequality for Jacobi expansions and divergence of Fourier series on compact symmetric spaces. Journal of Approximation Theory 1982, 35(3):214–221. 10.1016/00219045(82)90003X
Giulini S, Soardi PM, Travaglini G: A Cohen type inequality for compact Lie groups. Proceedings of the American Mathematical Society 1979, 77(3):359–364. 10.1090/S00029939197905455963
Hardy GH, Littlewood JE: A new proof of a theorem on rearrangements. Journal of the London Mathematical Society 1948, 23: 163–168. 10.1112/jlms/s123.3.163
Markett C: Cohen type inequalities for Jacobi, Laguerre and Hermite expansions. SIAM Journal on Mathematical Analysis 1983, 14(4):819–833. 10.1137/0514063
Abramowitz M, Stegun IA: Handbook of Mathematical Functions. Dover, New York, NY, USA; 1964.
Szegő G: Orthogonal Polynomials, American Mathematical Society, Colloquium Publications. Volume 23. 4th edition. American Mathematical Society, Providence, RI, USA; 1975:xiii+432.
Nevai P, Erdélyi T, Magnus AP: Generalized Jacobi weights, Christoffel functions, and Jacobi polynomials. SIAM Journal on Mathematical Analysis 1994, 25(2):602–614. 10.1137/S0036141092236863
Iserles A, Koch PE, Nørsett SP, SanzSerna JM: On polynomials orthogonal with respect to certain Sobolev inner products. Journal of Approximation Theory 1991, 65(2):151–175. 10.1016/00219045(91)90100O
MartínezFinkelshtein A, MorenoBalcázar JJ, PijeiraCabrera H: Strong asymptotics for GegenbauerSobolev orthogonal polynomials. Journal of Computational and Applied Mathematics 1997, 81(2):211–216. 10.1016/S03770427(97)000599
Stempak K: On convergence and divergence of FourierBessel series. Electronic Transactions on Numerical Analysis 2002, 14: 223–235.
Zygmund A: Trigonometric Series: Vols. I, II. 2nd edition. Cambridge University Press, London, UK; 1968:Vol. I. xiv+383 pp.; Vol. II: vii+364 pp..
Muckenhoupt B: Mean convergence of Jacobi series. Proceedings of the American Mathematical Society 1969, 23: 306–310. 10.1090/S00029939196902473605
Newman J, Rudin W: Mean convergence of orthogonal series. Proceedings of the American Mathematical Society 1952, 3: 219–222. 10.1090/S00029939195200478112
Pollard H: The mean convergence of orthogonal series. III. Duke Mathematical Journal 1949, 16: 189–191. 10.1215/S0012709449016191
Pollard H: The convergence almost everywhere of Legendre series. Proceedings of the American Mathematical Society 1972, 35: 442–444. 10.1090/S00029939197203029737
Meaney C: Divergent Jacobi polynomial series. Proceedings of the American Mathematical Society 1983, 87(3):459–462. 10.1090/S00029939198306846394
Guadalupe JJ, Pérez M, Ruiz FJ, Varona JL: Two notes on convergence and divergence a.e. of Fourier series with respect to some orthogonal systems. Proceedings of the American Mathematical Society 1992, 116(2):457–464.
Meaney Ch: Divergent Cesàro and Riesz means of Jacobi and Laguerre expansions. Proceedings of the American Mathematical Society 2003, 131(10):3123–3218. 10.1090/S0002993902068533
Acknowledgments
The authors thank the careful revision of the paper by the referees. Their remarks and suggestions have contributed to improve the presentation. The work of the second author (F. Marcellán) has been supported by Direcci ón General de Investigación, Ministerio de Ciencia e Innovación of Spain, Grant no. MTM200912740C0301.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
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.
About this article
Cite this article
Fejzullahu, B., Marcellán, F. A Cohen Type Inequality for Fourier Expansions of Orthogonal Polynomials with a Nondiscrete JacobiSobolev Inner Product. J Inequal Appl 2010, 128746 (2010). https://doi.org/10.1155/2010/128746
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2010/128746