- Research Article
- Open access
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A Cohen Type Inequality for Fourier Expansions of Orthogonal Polynomials with a Nondiscrete Jacobi-Sobolev 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 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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ1_HTML.gif)
Let us introduce the Sobolev-type spaces (see, for instance, [1, Chapter III], in a more general framework) as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ2_HTML.gif)
where as well as the linear space
of all bounded linear operators
with the usual operator norm
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ3_HTML.gif)
Let and
be in
Let us consider the following Sobolev-type inner product:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ4_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ5_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ6_HTML.gif)
where (see [11, formula ])
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ7_HTML.gif)
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 ]):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ8_HTML.gif)
They satisfy a connection formula (see [11, formula ], [3, formula
]) as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ9_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ10_HTML.gif)
as well as the following relation for the derivatives (see [12, formula (4.21.7)]):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ11_HTML.gif)
The following estimate for holds (see [12, formula
], [13]):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ12_HTML.gif)
where and
The formula of Mehler-Heine for Jacobi orthogonal polynomials is (see [12, Theorem ]) as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ13_HTML.gif)
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
]):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ14_HTML.gif)
where and
For and
(see [12, page 391. Exercise
], as well as [10,
])
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ15_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ16_HTML.gif)
where is given in (2.5) and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ17_HTML.gif)
Proposition 3.2.
One gets:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ18_HTML.gif)
In particular, for defined in (3.2) one obtains
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ19_HTML.gif)
Proof.
We apply the same argument as in the proof of Theorem in [15]. Using the extremal property
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ20_HTML.gif)
we get the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ21_HTML.gif)
On the other hand, from the extremal property of (2.4), and (2.6), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ22_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ23_HTML.gif)
where and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ24_HTML.gif)
Corollary 3.4.
For and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ25_HTML.gif)
and for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ26_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ27_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ28_HTML.gif)
and for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ29_HTML.gif)
Proposition 3.6.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_IEq97_HTML.gif)
For the polynomials one obtains
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ30_HTML.gif)
for and
For the polynomials one has the following estimate:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ31_HTML.gif)
for and
Proof.
-
(a)
Using (3.12), we have the following:
(3.17)
From (2.7), it is straightforward to prove that, for and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ33_HTML.gif)
Thus, according to Proposition 3.5,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ34_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ35_HTML.gif)
Proof.
Taking into account that the Jacobi polynomials satisfy the following (see [12, paragraph below Theorem ]):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ36_HTML.gif)
for thus, for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ37_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ38_HTML.gif)
and for and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ39_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ40_HTML.gif)
Proof.
The inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ41_HTML.gif)
holds for as well as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ42_HTML.gif)
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.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_IEq121_HTML.gif)
For and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ43_HTML.gif)
where if
and
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_IEq129_HTML.gif)
For and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ44_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_IEq144_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ46_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_IEq145_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ47_HTML.gif)
Proof.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_IEq146_HTML.gif)
Multiplying in (3.8) by we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ48_HTML.gif)
where and
according to (3.9).
Using the above relation in a recursive way, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ49_HTML.gif)
where and
Moreover, by using the same argument as in Proposition 3.5, we have
for every
and
Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ50_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ51_HTML.gif)
Thus, the sequence is uniformly bounded on
As a conclusion,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ52_HTML.gif)
and using (2.8), we obtain the result.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_IEq165_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ53_HTML.gif)
and for one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ54_HTML.gif)
where and
Proof.
From Proposition 3.6, the sequence
is uniformly bounded on compact subsets of
Multiplication by
in (3.10) yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ55_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ56_HTML.gif)
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ57_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ58_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ59_HTML.gif)
Using (3.8) in a recurrence way and then Minkowski's inequality, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ60_HTML.gif)
On the other hand, for and
(2.10) implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ61_HTML.gif)
Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ62_HTML.gif)
On the other hand, from Proposition 3.5,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ63_HTML.gif)
Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ64_HTML.gif)
In the same way as above, we conclude that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ65_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ66_HTML.gif)
Proof.
We will use a technique similar to [12, Theorem ]. According to (3.11),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ67_HTML.gif)
On the other hand, Stempak's lemma (see [16, Lemma ]), for
and
implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ68_HTML.gif)
Thus, for and
large enough, (3.51) follows.
Finally, from (3.39) we obtain the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ69_HTML.gif)
For the proof of Proposition 3.12, from (3.51), for and
we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ70_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ71_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ72_HTML.gif)
The Cesàro means of order of the expansion (4.1) is defined by (see [17, pages 76-77]),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ73_HTML.gif)
where
For a function and a fixed sequence
of real numbers with
we define the operators
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ74_HTML.gif)
Let and let
be the conjugate of
Now, we can state our main result.
Theorem 4.1.
For and
one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ75_HTML.gif)
Corollary 4.2.
Let and
be as in Theorem 4.1. For
and for
outside the interval
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ76_HTML.gif)
For Theorem 4.1 yields the following.
Corollary 4.3.
For and
one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ77_HTML.gif)
and ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ78_HTML.gif)
We will use the following as test functions (see [10, formula ], and [11, formula
]):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ79_HTML.gif)
where and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ80_HTML.gif)
Applying the operator to
for some
we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ81_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ82_HTML.gif)
and using (2.3) and (3.3), we deduce
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ83_HTML.gif)
Taking into account (4.9), for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ84_HTML.gif)
If then we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ85_HTML.gif)
If then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ86_HTML.gif)
On the other hand, for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ87_HTML.gif)
and for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ88_HTML.gif)
Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ89_HTML.gif)
As a conclusion,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ90_HTML.gif)
Now, we will estimate
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ91_HTML.gif)
From [10, formula ],
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ92_HTML.gif)
for
On the other hand, from (2.6), (4.9), and [12, formula ], one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ93_HTML.gif)
From (2.10), for ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ94_HTML.gif)
for and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ95_HTML.gif)
and for and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ96_HTML.gif)
Thus, for and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ97_HTML.gif)
By using (4.22) and (4.27), we find from (4.21) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ98_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ99_HTML.gif)
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,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ100_HTML.gif)
For the Fourier expansion in terms of Jacobi-Sobolev orthonormal polynomials is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ101_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ102_HTML.gif)
Let be the
th partial sum of the expansion (5.2) as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ103_HTML.gif)
Theorem 5.1.
Let and
If there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ104_HTML.gif)
for every , then
Proof.
For the proof, we apply the same argument as in [19]. Assume that (5.5) holds, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ105_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ106_HTML.gif)
where is the conjugate of
On the other hand, from (3.43) we obtain the Sobolev norms of Jacobi-Sobolev orthonormal polynomials as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ107_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ108_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ109_HTML.gif)
almost everywhere on E. From Egorov's Theorem, it follows that there is a subset of positive measure such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ110_HTML.gif)
uniformly for On the other hand, from (3.39)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ111_HTML.gif)
uniformly for . Using the Cantor-Lebesgue Theorem, as described in [24, Section
], (see also [17, page 316]), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ112_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ113_HTML.gif)
on By using [1, Theorem
], we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ114_HTML.gif)
Thus, from (5.8),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ115_HTML.gif)
As a consequence of the Banach-Steinhaus theorem, there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F128746/MediaObjects/13660_2010_Article_2057_Equ116_HTML.gif)
Since this result contradicts (6.5), then for this the Fourier series diverges almost everywhere on
in the norm of
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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. MTM2009-12740-C03-01.
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Fejzullahu, B., Marcellán, F. A Cohen Type Inequality for Fourier Expansions of Orthogonal Polynomials with a Nondiscrete Jacobi-Sobolev Inner Product. J Inequal Appl 2010, 128746 (2010). https://doi.org/10.1155/2010/128746
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DOI: https://doi.org/10.1155/2010/128746