Pointwise Approximation of Functions from by Linear Operators of Their Fourier Series
WΕodzimierz Εenski1and Bogdan Szal1
Academic Editor: Henryk Hudzik
Received01 Sept 2010
Accepted24 Nov 2010
Published03 Jan 2012
Abstract
We show the results, corresponding to theorem of Lal (2009), on the rate of pointwise
approximation of functions from the pointwise integral Lipschitz
classes by matrix summability means of their Fourier series as well as
the theorems on norm approximations.
1. Introduction
Let be the class of all -periodic real-valued functions integrable in the Lebesgue sense with th power over with the norm
and consider the trigonometric Fourier series
and conjugate one
with the partial sums and , respectively. We know that if , then
where
with
exists for almost all [1, Th. (3.1)IV].
Let be an infinite lower triangular matrix of real numbers such that
and let the transformationsof and be given by
respectively. Denote, for ,
A sequence of nonnegative numbers tending to zero is called the Rest Bounded Variation Sequence, or briefly , if it has the property
for all natural numbers , where is a constant depending only on .
A sequence of nonnegative numbers will be called the Head Bounded Variation Sequence, or briefly , if it has the property
for all natural numbers , or only for all if the sequence has only finite nonzero terms and the last nonzero terms is .
Now, we define another class of sequences.
Followed by Leindler [3], a sequence of nonnegative numbers tending to zero is called the Mean Rest Bounded Variation Sequence, or briefly , if it has the property
for all natural numbers , where is a constant depending only on .
Analogously, a sequence of nonnegative numbers will be called the Mean Head Bounded Variation Sequence, or briefly , if it has the property
for all positive integer , where the sequence has only finite nonzero terms and the last nonzero term is and where is a constant depending only on .
Consequently, we assume that the sequence is bounded, that is, that there exists a constant such that
holds for all , where denotes the sequence of constants appearing in the inequalities (1.12) or (1.13) for the sequences .
Now, we can give the conditions to be used later on. We assume that, for all and ,
where is the same as above, hold if belong to or , for , respectively.
As a measure of approximation of functions by the above means, we use the generalized pointwise moduli of continuity of in the space defined for by the formulas
where
It is clear that, for ,
It is easily seen that and are the classical pointwise moduli of continuity.
The deviation with special form of matrix was estimated in the norm of by Lal [5, Theoremββ2, page 347] as follows.
Theorem A. If
then
where is a nonnegative and nonincreasing sequence, and the function of modulus of continuity type will be defined in the next section.
In this note we show that the conditions (1.21), (1.22), and (1.23) are superfluous when we use the pointwise modulus of continuity.
In our theorems, we will consider the pointwise deviations , with the matrix whose rows belong to the classes of sequences and . Consequently, we also give some results on norm approximation.
We will write if there exists a positive constant , sometimes depending on some parameters, such that .
2. Statement of the Results
Let us define, for a fixed , a function (or ) of modulus of continuity type on the interval , that is, a nondecreasing continuous function having the following properties:
for any . It is easy to conclude that the function nondecreases in . Let
where , and are also the functions of modulus of continuity type. It is clear that, for ,
Now, we can formulate our main results on the degrees of pointwise summability.
Theorem 2.1. Let with . If is such that , where , then
for considered .
Theorem 2.2. Let with . If is such that , where , then
for considered .
Theorem 2.3. Let with . If is such that , where , then
for considered .
Theorem 2.4. Let with . If is such that , where , then
for considered .
Theorem 2.5. Let , and
holds with when or with when . If is such that , where , then
for considered such that exists.
Theorem 2.6. Let , and (2.8) holds with when or with when . If and , where , then
for considered such that exists.
Consequently, we formulate the results on norm approximation.
Theorem 2.7. Let with is such that , where , then
Theorem 2.8. Let with is such that , where , then
Theorem 2.9. Let with is such that , where , then
Theorem 2.10. Let with is such that , where , then
Theorem 2.11. Let , and
holds with . If and , where , then
Theorem 2.12. Let , and (2.15) holds with . If is such that , where , then
Remark 2.13. In the case , we can suppose that the expression nondecreases in instead of the assumption .
Remark 2.14. Under additional assumptions , and , the degree of approximation in Theorem 2.1 is following . The same degree of approximation we obtain in Theorem 2.2 under the assumption . In case of the remaining theorems, we can use the same remarks.
Remark 2.15. If we consider the classical modulus of continuity of the function instead of the modulus , then the condition holds for every function and thus . The same remark for conjugate functions holds too.
Remark 2.16. Our theorems will be also true if we consider function from with the following norm:
Remark 2.17. We can observe that, taking , we obtain the mean considered by Lal [5], and if is monotonic with respect to , then . Therefore, if is a nonincreasing sequence such that , then, from Theorem 2.1, we obtain the corrected form of the result of Lal [5] (i.e., with condition instead of (1.22)).
3. Auxiliary Results
We begin this section by some notations following Zygmund [1].
It is clear that
where
Hence,
where
Now, we formulate some estimates for the conjugate Dirichlet kernel.
Lemma 3.1 (see [1]). If , then
and, for any real , we have
Lemma 3.2 (cf. [2, 6]). If , then
and if , then
for , where .
Lemma 3.3 (see [7]). If , then
and if , then
for , where .
The proof is the same as the proof of Theorem 2.1, the only difference is that, in place of , we have to write the quantity for which we suppose the same order.
The proof is the same as the proof of Theorem 2.3, the only difference is that, in place of , we have to write the quantity for which we suppose the same order.
We start with the obvious relations
Lemma 3.1 gives
If and , then, by the HΓΆlder inequality ,
or if , then
Therefore, using (2.8), we have
The same estimation as in the proof of Theorem 2.3 gives
Collecting these estimates, we obtain the desired result.
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