Pointwise Analog of the Stečkin Approximation Theorem
We show the pointwise version of the Stečkin theorem on approximation by de la Vallée-Poussin means. The result on norm approximation is also derived.
Let be the class of all -periodic real-valued functions integrable in the Lebesgue sense with th power (continuous) over , and let when or when .
Let us define the norms of as where We note additionally that Consider the trigonometric Fourier series of with the partial sums .
As a measure of approximation by the previous quantities, we use the pointwise characteristics compared to , and and also where constructed on the base of definition of -points (Lebesgue points (-points) or points of continuity (-points)). We also use the modulus of continuity of in the space defined by the formula and its arithmetic mean
We can observe that, for and , whence whence
Let us introduce one more measure of pointwise approximation analogical to the best approximation of function by trigonometric polynomials of the degree at most Namely, We will also use its arithmetic mean
Using these characteristics, we will show the pointwise version of the Stečkin  generalization of the Fejér-Lebesgue theorem. As a corollary, we will obtain the mentioned original result of Stečkin on norm approximation as well as the result of Tanović-Miller .
By we will designate either an absolute constant or a constant depending on some parameters, not necessarily the same of each occurrence.
2. Statement of the Results
At the beginning, we formulate the partial solution of the considered problem.
Theorem 1 (see ). If , then, for any positive integer and all real ,
Now, we can present the main result on pointwise approximation.
Theorem 2. If , then, for any positive integer and all real ,
Remark 3. Theorem 2, in case , immediately yields the result of Stečkin . The result of Stečkin type also holds if instead of we consider the spaces with . Then, in the proof, we need the Hardy-Littlewood estimate of the maximal function.
At every point of and thus from Theorem 1, we obtain the corollary which states the result of the Tanović-Miller type .
Corollary 4. If , then, for any positive integer at every -point of ,
3. Auxiliary Results
In order to prove our theorems, we require some lemmas.
Lemma 5. If is the trigonometric polynomial of the degree at most of the best approximation of with respect to the norm , then it is also the trigonometric polynomial of the degree at most of the best approximation of with respect to the norm for any .
Proof. From the inequalities where and are the trigonometric polynomials of the degree at most of the best approximation of with respect to the norms and , respectively, we obtain relation whence for any by uniqueness of the trigonometric polynomial of the degree at most of the best approximation of with respect to the norm (e.g. see  p. 96). We can also observe that for such and any , Hence, and our proof is complete.
Lemma 6. If and , then is nonincreasing function of and nondecreasing function of . These imply that for the function is nonincreasing function of and simultaneously.
Proof. The first part of our statement follows from the property of the norm and supremum. The second part is a consequence of the calculation
Lemma 7. Let such that and . If , then
Proof. It is clear that
Hence, by orthogonality of the trigonometric system,
with trigonometric polynomial of the degree at most of the best approximation of .
Using the notations we get We next evaluate the sums and using the partial integrating and Lemma 5. Thus, which proves Lemma 7.
Before formulating the next lemmas, we define a new difference. Let and . Denote that
Lemma 8. Let such that . If , then
Proof. The proof follows by the method of Leindler . Namely, By Lemma 6, for , and our proof is complete.
Lemma 9. Let and . If , then
Proof. Our proof runs parallel with the proof of Theorem 1 in .
If , then and if , then by Lemmas 6 and 7.
Next, we construct the same decreasing sequence of integers that was given by Stečkin. Let where denotes the integral part of . It is clear that there exists smallest index such that and By the definition of the numbers , we have whence follow.
Under these notations, we get the following equality: whence, by , follows.
It is easy to see that the terms in the sum , by Lemma 8, with and do not exceed and by Lemma 7, we get Thus, whence, by the monotonicity of ,
4. Proofs of the Results
Proof of Theorem 2. The proof follows the lines of the proofs of Theorem 4 in  and Theorem in . Therefore, let and be fixed. Let us define an increasing sequence of indices introduced by Stečkin in the following way. Set . Assuming that the numbers are already defined and , we define as follows. Let denote the smallest natural number such that
According to the magnitude of , we define
If , we continue the procedure, and if once , then we stop the construction and define .
By the previous definition of , we have the following obvious properties: and relations whenever .
Let us start with Using Theorem 1 and that , we get
The estimate of the sum in the right hand side of (55) we derive from the following one We split the proof of this inequality in two parts. If , then by Lemma 9, If , then, by Lemma 7, and since , we have
Consequently, Since for all , changing the order of summation, we get
Using the inequality we obtain where denotes the smallest index having the property . Hence, and our proof follows.
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