A Cohen Type Inequality for Fourier Expansions of Orthogonal Polynomials with a Nondiscrete Jacobi-Sobolev Inner Product
© Bujar Xh. Fejzullahu and Francisco Marcellán. 2010
Received: 5 May 2010
Accepted: 24 August 2010
Published: 30 August 2010
Let denote the sequence of polynomials orthogonal with respect to the non-discrete 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.
We call them the Jacobi-Sobolev orthogonal polynomials.
The measures and constitute a particular case of the so-called coherent pairs of measures studied in . In  (see also ), the authors established the asymptotics of the zeros of such Jacobi-Sobolev 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 Jacobi-Sobolev polynomials, the well-known 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 Mehler-Heine 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 Mehler-Heine 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 Cohen-type inequality, associated with the Fourier expansions in terms of the Jacobi-Sobolev 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
3. Asymptotics of Jacobi-Sobolev Orthogonal Polynomials
Let us denote by the monic Jacobi-Sobolev 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 Jacobi-Sobolev and Jacobi monic orthogonal polynomials.
As a straightforward consequence of Propositions 3.1 and 3.2, using (2.1) we deduce the following.
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).
On the other hand, from (3.11), the proof of the case (b) can be done in a similar way.
As a consequence, the statement follows from the latter estimates and arguments similar to those we used in the proof of Proposition 3.6.
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.
and using (2.8), we obtain the result.
Now, (3.38) follows from (2.9).
Next, we obtain an estimate for the Sobolev norms of the Jacobi-Sobolev polynomials.
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.
Thus, using (3.44) and (3.55), the statement follows.
4. A Cohen Type Inequality for Jacobi-Sobolev Expansions
Now, we can prove our main result.
Proof of Theorem 4.1.
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.
The proof of Theorem 5.1 is complete.
6. Divergence Almost Everywhere
For and Pollard  showed that for each there exists a function such that its Fourier expansion (4.27) diverges almost everywhere on . Later on, Meaney  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].
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. MTM2009-12740-C03-01.
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