- Research Article
- Open access
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On Weighted
Integrability of Functions Defined by Trigonometric Series
Journal of Inequalities and Applications volume 2010, Article number: 485705 (2010)
Abstract
We introduce a new class of sequences called and give a sufficient and necessary condition for weighted
integrability of trigonometric series with coefficients to belong to the above class. This is a generalization of the result proved by M. Dyachenko and S. Tikhonov (2009). Then we discuss the relations among the weighted best approximation and the coefficients of trigonometric series. Moreover, we extend the results of B. Wei and D. Yu (2009) to the class
.
1. Introduction
Let ,
, be the space of all
-power integrable functions
of period
equipped with the norm
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ1_HTML.gif)
Write
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ2_HTML.gif)
for those where the series converge. Denote by
either
or
, and let
be its associated coefficients, that is,
is either
or
.
For and a sequence
let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ3_HTML.gif)
In Subsection 2.1 we generalize the following result.
Theorem 1.1.
Let a nonnegative sequence ,
and
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ4_HTML.gif)
In the case when denotes the class
of all decreasing sequences, this theorem was proved in [1–4]; for
, the class of quasimonotone sequences, in [5]; for
in [6, 7]; for
in [8]; and for
in [9], where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ5_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ6_HTML.gif)
In [15] Dyachenko and Tikhonov extended Theorem 1.1 to the class , where
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ7_HTML.gif)
We have (see [15])
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ8_HTML.gif)
Let be a nonnegative function defined on the interval
. Denote by
the best approximation of
by trigonometric polynomials of degree at most
in the weighted
-norm, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ9_HTML.gif)
where denotes the set of all trigonometric polynomials of degree at most
.
A sequence of nonnegative terms is called almost increasing (decreasing) if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ10_HTML.gif)
We say that a weight function if
is defined by the sequence
as follows:
,
, and there exist positive constants
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ11_HTML.gif)
for all , and the sequences
,
are almost decreasing and almost increasing, respectively.
In Subsection 2.2 we generalize and extend the following results [16].
Theorem 1.2.
Assume that . If
for some
and
, then for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ12_HTML.gif)
Theorem 1.3.
Assume that . If
for some
and
, then for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ13_HTML.gif)
If and
or
the above theorem has been obtained by Konyushkov [17] and Leindler [18] for
, respectively.
In order to formulate our new results we define the next class of sequences.
Definition 1.4.
Let and
. One says that a sequence
belongs to
, if the relation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ14_HTML.gif)
holds for all.
Note that for and
(see Theorem 2.1(i))
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ15_HTML.gif)
Throughout this paper, we use to denote a positive constant independent of the integer
;
may depend on the parameters such as
and
, and it may have different values in different occurrences.
2. Statement of the Results
We formulate our results as follows.
Theorem 2.1.
Suppose that . The following properties are true.
(i)For any , and
there exists a sequence
, which does not belong to the class
(ii)Let ,
and
. If
, then
(iii)Let and
. If
and
, then the classes
and
are not comparable.
2.1. Weighted
-Integrability
Let and
. We define on the interval
an even function
, which is given on the interval
by the formula
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ16_HTML.gif)
where if
is an odd number, and
if
is an even number;
for
, and
for
.
Theorem 2.2.
Let a nonnegative sequence , where
,
and
. If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ17_HTML.gif)
then if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ18_HTML.gif)
Theorem 2.3.
Let a nonnegative sequence ,
, and
. If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ19_HTML.gif)
then if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ20_HTML.gif)
Remark 2.4.
If we take (and
in Theorem 2.2), then the result of Dyachenko and Tikhonov [15, Theorems
and
] follows from Theorems 2.2 and 2.3. By the embedding relations (1.8) and (1.15) we can also derive from Theorem 2.2 the result of You, Zhou, and Zhou [9].
2.2. Relations between The Best Approximation and Fourier Coefficients
Theorem 2.5.
Let a nonnegative sequence , where
,
, and
. If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ21_HTML.gif)
and (2.3) holds, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ22_HTML.gif)
where .
Theorem 2.6.
Let a nonnegative sequence ,
and
. If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ23_HTML.gif)
and (2.5) holds, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ24_HTML.gif)
where .
Remark 2.7.
If we restrict our attention to the class then by (1.8) and (1.15) Wei and Yu's result [16] follows from Theorems 2.5 and 2.6.
4. Proofs of The Main Results
4.1. Proof of Theorem 2.1
(i)Let ,
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ28_HTML.gif)
First, we prove that . Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ29_HTML.gif)
Then for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ30_HTML.gif)
and . If
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ31_HTML.gif)
and since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ32_HTML.gif)
the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ33_HTML.gif)
does not hold, that is, does not belong to
.
Let ,
and
. If
, then exists a natural number
such that
Supposing that
, we have for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ34_HTML.gif)
whence . Thus
.
Let and
and let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ35_HTML.gif)
Supposing that and
, we can prove, similarly as in
, that
,
,
and
Therefore the classes
and
are not comparable.
4.2. Proof of Theorem 2.2
We prove the theorem for the case when . The case when
can be proved similarly.
Sufficiency. Suppose that (2.3) holds. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ36_HTML.gif)
It is clear that for an odd
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ37_HTML.gif)
(for the last sum should be omitted), and for an even
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ38_HTML.gif)
First, we estimate the following integral:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ39_HTML.gif)
By (3.3), for we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ40_HTML.gif)
Using (3.2) with and the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ41_HTML.gif)
we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ42_HTML.gif)
If then by (3.3), for
we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ43_HTML.gif)
Now, we estimate the following integral:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ44_HTML.gif)
By (3.3), for we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ45_HTML.gif)
Using (3.2) with and the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ46_HTML.gif)
we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ47_HTML.gif)
If then by (3.3), for
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ48_HTML.gif)
Thus, combining (4.9), (4.12)–(4.13), (4.16)–(4.18), (4.21), and (4.10) or (4.11), we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ49_HTML.gif)
Necessity.
We follow the method adopted by . Tikhonov [15]. Note that if
, then
. Integrating
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ50_HTML.gif)
and consequently
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ51_HTML.gif)
If , then using (4.24),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ52_HTML.gif)
Using this and (3.3), for , we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ53_HTML.gif)
Defining we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ54_HTML.gif)
and by (3.3), for , we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ55_HTML.gif)
Applying Hölder's inequality, for we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ56_HTML.gif)
Finally,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ57_HTML.gif)
which completes the proof.
4.3. Proof of Theorem 2.3
The proof of Theorem 2.3 goes analogously as the proof of Theorem 2.2. The only difference is that instead of (4.13) (for ) and (4.18) (for
) we use the below estimations.
Applying the inequalities for
,
for
and using (3.3), for
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ58_HTML.gif)
This ends our proof.
4.4. Proof of Theorem 2.5
We prove the theorem for the case when . The case when
can be proved similarly.
If then by (2.3) we obtain that (2.7) obviously holds. Let
. It is clear that if
is an odd number, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ59_HTML.gif)
(for the last sum should be omitted), and if
is an even number, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ60_HTML.gif)
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ61_HTML.gif)
where and
if
is an even number, and
if
is an odd number.
Then, for , by (3.2) and (4.14), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ62_HTML.gif)
We immediately have for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ63_HTML.gif)
If and
, then by Hölder's inequality we have for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ64_HTML.gif)
When and
, an elementary calculation gives for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ65_HTML.gif)
If , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ66_HTML.gif)
We have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ67_HTML.gif)
and taking and
for
,
for
in (3.3), we get for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ68_HTML.gif)
If , then using (3.2) and (4.14), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ69_HTML.gif)
Set ,
for
and
for
. Then by (3.3), we have for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ70_HTML.gif)
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ71_HTML.gif)
where and
. Then, for
, using (3.2) and (4.19), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ72_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ73_HTML.gif)
If , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ74_HTML.gif)
Similarly as in the estimation of the quantity using (3.3) for
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ75_HTML.gif)
If , then using (3.2) and (4.19), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ76_HTML.gif)
Further, by (3.3), we have for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ77_HTML.gif)
Combining (4.32) or (4.33), (4.35)–(4.43) and (4.45)–(4.50) we complete the proof of Theorem 2.5.
4.5. Proof of Theorem 2.6
The proof of Theorem 2.6 goes analogously as the proof of Theorem 2.5. The only difference is that instead of (4.41) (for ) and (4.48) (for
) we use the below estimations.
Applying the inequalities for
and
for
and using (3.3), for
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F485705/MediaObjects/13660_2010_Article_2166_Equ78_HTML.gif)
This completes the proof.
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Szal, B. On Weighted Integrability of Functions Defined by Trigonometric Series.
J Inequal Appl 2010, 485705 (2010). https://doi.org/10.1155/2010/485705
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DOI: https://doi.org/10.1155/2010/485705