A Cohen Type Inequality for Fourier Expansions of Orthogonal Polynomials with a Nondiscrete Jacobi-Sobolev Inner Product

Abstract

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.

1. Introduction

Let with be the Jacobi measure supported on the interval We say that if is measurable on and where

(1.1)

Let us introduce the Sobolev-type spaces (see, for instance, [1, Chapter III], in a more general framework) as follows:

(1.2)

where as well as the linear space of all bounded linear operators with the usual operator norm

(1.3)

Let and be in Let us consider the following Sobolev-type inner product:

(1.4)

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:

(1.5)

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 [2]. In [3] (see also [4]), 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

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

(2.1)

where (see [11, formula ])

(2.2)

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 ]):

(2.3)

They satisfy a connection formula (see [11, formula ], [3, formula ]) as follows:

(2.4)

where

(2.5)

as well as the following relation for the derivatives (see [12, formula (4.21.7)]):

(2.6)

The following estimate for holds (see [12, formula ], [13]):

(2.7)

where and

The formula of Mehler-Heine for Jacobi orthogonal polynomials is (see [12, Theorem ]) as follows:

(2.8)

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 ]):

(2.9)

where and

For and (see [12, page 391. Exercise ], as well as [10, ])

(2.10)

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.

Proposition 3.1.

For

(3.1)

where is given in (2.5) and

(3.2)

Proposition 3.2.

One gets:

(3.3)

In particular, for defined in (3.2) one obtains

(3.4)

Proof.

We apply the same argument as in the proof of Theorem in [15]. Using the extremal property

(3.5)

we get the following:

(3.6)

On the other hand, from the extremal property of (2.4), and (2.6), we have

(3.7)

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

(3.8)

where and

(3.9)

Corollary 3.4.

For and

(3.10)

and for

(3.11)

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

(3.12)

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

(3.13)

and for

(3.14)

Proposition 3.6.

For the polynomials one obtains

(3.15)

for and

For the polynomials one has the following estimate:

(3.16)

for and

Proof.

• (a) Using (3.12), we have the following:

  

  (3.17)

From (2.7), it is straightforward to prove that, for and

(3.18)

Thus, according to Proposition 3.5,

(3.19)

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

(3.20)

Proof.

Taking into account that the Jacobi polynomials satisfy the following (see [12, paragraph below Theorem ]):

(3.21)

for thus, for

(3.22)

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

(3.23)

and for and

(3.24)

where

(3.25)

Proof.

The inequality

(3.26)

holds for as well as

(3.27)

holds for Therefore, the statement follows from Propositions 3.6 and 3.7.

Next, we show that the Jacobi-Sobolev polynomial attains its maximum in at the end points. To be more precise, consider the following.

Proposition 3.9.

For and ,

(3.28)

where if and if  

For and ,

(3.29)

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 Mehler-Heine type formula for and

Proposition 3.10.

Let Uniformly on compact subsets of one gets 

(3.31)

(3.32)

Proof.

Multiplying in (3.8) by we obtain

(3.33)

where and according to (3.9).

Using the above relation in a recursive way, we obtain

(3.34)

where and Moreover, by using the same argument as in Proposition 3.5, we have for every and Thus,

(3.35)

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

(3.36)

Thus, the sequence is uniformly bounded on As a conclusion,

(3.37)

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

(3.38)

and for one has

(3.39)

where and

Proof.

From Proposition 3.6, the sequence is uniformly bounded on compact subsets of Multiplication by in (3.10) yields

(3.40)

Since

(3.41)

we have

(3.42)

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 Jacobi-Sobolev polynomials.

Proposition 3.12.

For and , one has

(3.43)

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

(3.44)

Using (3.8) in a recurrence way and then Minkowski's inequality, we obtain

(3.45)

On the other hand, for and (2.10) implies

(3.46)

Thus,

(3.47)

On the other hand, from Proposition 3.5,

(3.48)

Thus,

(3.49)

In the same way as above, we conclude that

(3.50)

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

(3.51)

Proof.

We will use a technique similar to [12, Theorem ]. According to (3.11),

(3.52)

On the other hand, Stempak's lemma (see [16, Lemma ]), for and implies

(3.53)

Thus, for and large enough, (3.51) follows.

Finally, from (3.39) we obtain the following:

(3.54)

For the proof of Proposition 3.12, from (3.51), for and we get

(3.55)

Thus, using (3.44) and (3.55), the statement follows.

4. A Cohen Type Inequality for Jacobi-Sobolev Expansions

For its Fourier expansion in terms of Jacobi-Sobolev polynomials is

(4.1)

where

(4.2)

The Cesàro means of order of the expansion (4.1) is defined by (see [17, pages 76-77]),

(4.3)

where

For a function and a fixed sequence of real numbers with we define the operators by

(4.4)

Let and let be the conjugate of Now, we can state our main result.

Theorem 4.1.

For and one has

(4.5)

Corollary 4.2.

Let and be as in Theorem 4.1. For and for outside the interval , one has

(4.6)

For Theorem 4.1 yields the following.

Corollary 4.3.

For and one has

(4.7)

and ,

(4.8)

We will use the following as test functions (see [10, formula ], and [11, formula ]):

(4.9)

where and

(4.10)

Applying the operator to for some we get

(4.11)

where

(4.12)

and using (2.3) and (3.3), we deduce

(4.13)

Taking into account (4.9), for

(4.14)

If then we get

(4.15)

If then

(4.16)

On the other hand, for

(4.17)

and for

(4.18)

Thus,

(4.19)

As a conclusion,

(4.20)

Now, we will estimate

(4.21)

From [10, formula ],

(4.22)

for

On the other hand, from (2.6), (4.9), and [12, formula ], one has

(4.23)

From (2.10), for ,

(4.24)

for and ,

(4.25)

and for and ,

(4.26)

Thus, for and ,

(4.27)

By using (4.22) and (4.27), we find from (4.21) that

(4.28)

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

(4.29)

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 Jacobi-Sobolev orthonormal polynomials, that is,

(5.1)

For the Fourier expansion in terms of Jacobi-Sobolev orthonormal polynomials is

(5.2)

where

(5.3)

Let be the th partial sum of the expansion (5.2) as follows:

(5.4)

Theorem 5.1.

Let and If there exists a constant such that

(5.5)

for every , then

Proof.

For the proof, we apply the same argument as in [19]. Assume that (5.5) holds, then

(5.6)

Therefore,

(5.7)

where is the conjugate of

On the other hand, from (3.43) we obtain the Sobolev norms of Jacobi-Sobolev orthonormal polynomials as follows:

(5.8)

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

(6.1)

Therefore,

(6.2)

almost everywhere on E. From Egorov's Theorem, it follows that there is a subset of positive measure such that

(6.3)

uniformly for On the other hand, from (3.39)

(6.4)

uniformly for . Using the Cantor-Lebesgue Theorem, as described in [24, Section ], (see also [17, page 316]), we obtain

(6.5)

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

(6.6)

on By using [1, Theorem ], we have

(6.7)

Thus, from (5.8),

(6.8)

As a consequence of the Banach-Steinhaus theorem, there exists such that

(6.9)

Since this result contradicts (6.5), then for this the Fourier series diverges almost everywhere on in the norm of