Degrees of the approximations by some special matrix means of conjugate Fourier series

https://doi.org/10.1515/dema-2019-0014 Received October 3, 2018; accepted January 29, 2019 Abstract: In this paper we will present the pointwise and normwise estimations of the deviations considered by W. Łenski, B. Szal, [Acta Comment. Univ. Tartu. Math., 2009, 13, 11-24] and S. Saini, U. Singh, [Boll. UnioneMat. Ital., 2016, 9, 495-504] under general assumptions on the class considered sequences defining the method of the summability. We show that the obtained estimations are the best possible for some subclasses of Lp by constructing the suitable type of functions.

Recently, Saini and Singh [15] have proved the following theorem: Theorem C. Let f be 2 -periodic function belonging to Lip (︀̃︀ (t), p )︀ -class with p ≥ 1 and let A = (a n,k ) be a lower triangular regular matrix with nonnegative and nondecreasing (with respect to 0 ≤ k ≤ n) entries and A n,0 = 1. Then the degree of approximation of̃︀ f , conjugate of f , by matrix means of its conjugate Fourier series is given by We shall write J 1 ≪ J 2 , if there exists a positive constant C, depending on some parameters, such that J 1 ≤ CJ 2 .

Statement of the results
In this paper, we will present the estimations of the deviations̃︀ T n, )︀ under general assumptions and we will show that the obtained degrees of approximations are the best for some subclasses of L p .
︀ . Let f ∈ L p and let̃︀ be such that and hold with 0 < < + 1 p . Then holds for all x.
︀ . Let f ∈ L p and let̃︀ be such that (7) and hold with 0 < < + 1 p . Then holds for all x such that̃︀ f (x) exists.

If the function t −̃︀ (t) is nondecreasing and concave then
is a nondecreasing and concave function of t.

Corollaries
Finally, we give some corollaries as an application of our results.

Auxiliary results
We begin this section with some notation following Zygmund [1, Section 5 of Chapter II].
It is clear that̃︀ Hencẽ︀ Now, we formulate some estimates of the considered kernel.

Lemma 3. If t −̃︀ (t) is a concave and nondecreasing function of t, then the function
belongs to L p (̃︀) .
and 0 ≤ ka k ≤ k̃︀ From the relations (see equation [17]) and using the concavity of the function t −̃︀ (t) , we obtain i.e. 2k + 1 k + 1 k̃︀ Thus, we get Hence, we finally obtain estimation (16) . We know (see inequality [17]) that from the concavity of the function t −̃︀ (t) , we have which implies immediately the left side of inequality (17). Using the monotonicity of the function t −̃︀ (t) we get which gives the right side of inequality (17). Let us denote 0 Using the mean value theorem and the left side of inequality (17) , we get for x < z < x + t Thus by summation, we obtain the estimate For the terms |s 2 | and |s 3 |, with x ≠ 0, we get using the left side of inequality (17) and following Totik (see estimation [17]) Thus by inequalities (16) and (17) , we obtain and analogously |s 3 | ≤ 4|t| −̃︀ (|t|) .
Thus we have proved that f 0 belongs to L p (̃︀) .

Lemma 4. If t −̃︀ (t) is a nondecreasing function of t, then the function
belongs to L p (̃︀) .
Proof. We havẽ︀ Hence, we get̃︀ Thus, we have proved that f 1 belongs to L p (̃︀) .

Proof of Theorem 3
Let us fix a point x and let us consider the class L p (̃︀) , with > 0, of all functions f ∈ L p such that︀ f ( ) L p ≤̃︀ ( ) , (0 ≤ ≤ 2 ).
Then Theorem 1 implies the following estimate for < 1− 1 p . On the other hand, the function by Lemma 4 belongs to class L p (̃︀) , if t −̃︀ (t) is a nondecreasing function of t, and f 1 satisfies the conditions (6) and (7) of Theorem 1. Indeed, we have Moreover, there exists such that 1 p < < + 1 p and Let n < t < n−1 . We havẽ︀ We get Thus in a special case, for x = 0, we get Hence, we finally obtain equation (12) . When x = x 0 , we can consider the function fx 0 (·) = f 1 (· − x 0 ) instead of f 1 (·). Thus our proof is complete.

Proof of Theorem 4
Let us fix a point x and let us consider the class L p (̃︀) , with < 1 − 1 p , of all functions f ∈ L p such that︀ f ( ) L p ≤̃︀ ( ) , (0 ≤ ≤ 2 ).
The Theorem 2 implies the estimate On the other hand, the function by Lemma 3 belongs to class L p (̃︀) , if t −̃︀ (t) is a concave and nondecreasing function of t.
We can see that the function f 0 satisfies the condition (7) with 1 p < < + 1 p and the condition (9). Indeed, using the estimation (18) obtained in the proof of Lemma 3, we have |Ψ 0 Thus, we easily get Hence, by Theorem 2 the estimation (11) holds for the function f 0 . On the other hand using the fact︀ Thus in a special case, for x = 0, we get Using the inequality (18) , we have n ∑︁ k=2 a n,k (k + 1)̃︀ Hence, we finally obtain (13) . When x = x 0 , we can consider the function fx 0 (·) = f 0 (· − x 0 ) instead of f 0 (·). Thus our proof is complete.

Proof of Theorem 5
Note that if f ∈ L p (̃︀) , then Theorem 1 implies Thus, we easily get The conditions (6) and (7) from Theorem 1 are satisfied in the following form Hence our proof is complete.

Proof of Theorem 6
Similarly to the previous proof, note that if f ∈ L p (̃︀) , then Theorem 2 implies Thus, we easily get We know by the previous proof that the condition (7) is satisfied and the condition (9) , from Theorem 2, is satisfied in the following form Hence, our proof is complete.