A Note on Derivative of Sine Series with Square Root

Abstract

Chaundy and Jolliffe proved that if is a nonnegative, nonincreasing real sequence, then series converges uniformly if and only if . The purpose of this paper is to show that if is nonincreasing and , then the series can be differentiated term-by-term on for . However, may not exist.

1. Introduction

Chaundy and Jolliffe [1] proved the following.

Theorem 1. If is decreasing to zero, then converges uniformly in if and only if as .

Theorem 1 has had numerous generalizations.

Leindler [7] verified that in Theorem 1, the monotonicity assumption can be replaced by , i.e., if the conditions and hold for all with constant which depends only upon .

The next theorem was indicated in [11].

Theorem 2. If belongs to the class MVBVS, i.e., if there exist constants and , depending only on the sequence such that for all , then series converges uniformly in if and only if .

Móricz [8] proves the following theorem.

Theorem 3. Assume with property . If is nonincreasing on , then the integral , , converges uniformly in if and only if as .

A result due to Žak and Šneider [10] holds for double sine series.

Theorem 4. If is monotonically decreasing double sequences, i.e., a sequence of real numbers such that for , , , and , then is uniformly regularly convergent in if and only if as .

Theorem 4 was generalized by Kórus [6]. He has defined new classes of double sequences () to obtain those generalizations.

Duzinkiewicz and Szal [2] introduce a new class of double sequences called , which is a generalization of the class considered by Kórus, and they obtain sufficient and necessary conditions for uniform convergence of double sine series.

Dyachenko et al. [3] proved the Chaundy-Jolliffe theorem for sequences with majorant having the following form: , where is admissible.

In the recent paper [5], it was proved that converges uniformly on if and only if .

One of the results of paper [9] is that for any , converges uniformly if and only if .

2. Main Results

Lemma 5. Let where .
Let such that . Then, for all and for all ,

Proof. Let Note that for all , In view of (4), the following inequality is satisfied: Let In analogy with (5), we have If , then In view of (5) and (8)–(11), the following inequality is satisfied: Then, for all and , we have

In the recent paper [4], it was proved, among others, that if is a nonincreasing sequence and is convergent at any point , then . We show that if is nonincreasing and , then converges uniformly on , where .

Theorem 6. Let such that . If is nonincreasing and , then the series can be differentiated term-by-term on . The series converges uniformly on .

Proof. We can find such that , , and . If is a nonnegative monotone sequence and , then the series (14) converges uniformly [9]. We show that if, in addition, is nonincreasing, then (for ) the series converges uniformly on . Let , , , and . We denote by the minimal odd number for which there holds the following: and . Then, Let Note that for all such that and for all , the following conditions are fulfilled: and On the other hand, for all such that and for all , we have and We show that the sequence is bounded for and . Note that . Hence, Moreover, Furthermore, Note that In view of Lemma 5 and (22)–(26), the following inequality is satisfied: for all . Using (17)–(19), (21), and (27), we obtain In view of (16) and (28), we get We know that is nonincreasing and and (29) holds. Hence, the series converges uniformly. This follows from Dirichlet’s test.

Remark 7. The conditions ( is nonincreasing, and ) are not sufficient for the differentiability of (14) at the point .

Example 1. Let . It converges uniformly on [5]. We show that is not differentiable at . Let for and . Then, by Lagrange’s theorem, The function has a relative minimum at the point . Then, In view of (30)–(34), we obtain If and are the relative minimum and maximum (respectively) points of , such that for , then In view of (36), we obtain In view of (37)–(40), we get Hence, This follows from (35) and (41).

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they do not have any conflicts of interest.