We present the proper definitions of the pointwise best approximation of which implies the correctness of the proof of Lemma 5 as well as the other results of the paper.
In the paper entitled “Pointwise Analog of the Stečkin Approximation Theorem,’’ there was given an incorrect proof of Lemma 5 based on improper definitions of the pointwise best approximation of . Here, we define the mentioned and in the following way: where is the polynomial of the best approximation of in the usual norm and . Under the above definitions, we correct Lemma 5 as follows.
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 (see, e.g., [1] p. 96). We can also observe that for such and any Hence and our proof is complete.
Now, the proof is correct as well as other lemmas and all other results of the paper. Mainly, we have the following.
Theorem 2. If , then, for any positive integer and all real , where , which immediately yields the known result of Stečkin [2], by the obvious relation , where when and .
Remark. By our previous definitions (see (16)), with instead of , the polynomial was any trigonometric polynomial of the degree at most and therefore the polynomial considered in the proof of Lemma 5 in the above mentioned paper (see (16)) was essentially dependent on variable , that is, . Thus, inequality (22) from the proof of Lemma 5 in the above mentioned paper (see (16)) was not true because change of order of norms was not possible in general.