On the Uniform Convergence of Sine Series with Square Root

Abstract

Chaundy and Jolliffe proved that if is a nonincreasing real sequence with , then the series converges uniformly if and only if . The purpose of this paper is to show that is a necessary and sufficient condition for the uniform convergence of series in . However for it is not true in .

1. Introduction

Chaundy and Jolliffe [1] proved the following.

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

Theorem 1 has had numerous generalizations.

Leindler [2] 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 [3].

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 x if and only if .

Móricz [4] proves the following theorem.

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

A result due to ak and neider [5] holds for double sine series. is regularly convergent in case of a fixed if the rectangular sums converge to a finite number as and independently tend to infinity; moreover the row and column series , , and , , are convergent.

Theorem 4. If is a monotonically decreasing double sequence, i.e., 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 [7] introduce a new class of double sequence 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.

A serieswas motivation for the generalization of the Theorem 1. Such series were studied by Paley and Wiener who called them nonharmonic Fourier series. They proved the following [8].

Theorem 5. If for , then the sequence is closed in and possesses a unique biorthogonal set , such that the series
converges uniformly to zero over interval for any positive , and over any such interval the summability properties of
are uniformly the same as those of the Fourier series of .

We will consider a special case of the series (1) for and , , which does not meet the assumptions of the above theorem.

2. Main Results

Theorem 6. If is nonincreasing, then the series converges uniformly in if and only if .

Proof (necessary condition). Suppose that a series converges uniformly on . Let . We consider for and : .
Hence,After considering that converges uniformly we obtain
for sufficiently large . Thus, in view of inequality (2), we obtain for sufficiently large , soAfter considering that the sequence is nonincreasing we haveThusby (3). In view of (5), we obtain and .

Proof (sufficient condition). Let .
Case 1:After considering and we obtainThis follows from (6). Note that the following condition is fulfilled: In view of (8) the following inequality is satisfied for :Thus the sequence is increasing with respect to and . This follows from (9). Finally for and ,This follows from (7), (10).
To prove the case we first observe the following.
Lemma 2.2. Let and . Let denote the floor function, i.e., and . ThenThe proof of (12).
Note that for the following condition is fulfilled:which follows from the relationship: For ,This follows from (16).
The proof of (14).
Note that for the following condition is fulfilled:which follows from the relationship:for .
For ,This follows from (19). The proof of the rest of Lemma 2.2 is obvious.
Case 2:Case 2’:Therefore,Case 2":In view of (23), for all there areNote thatwhich follows from (23) and (27). The proof ofis similar. This follows from (23) and (28). Moreover for all there areNote thatLet us observe that . This follows from (23) and (26). Hence,Therefore,This follows from (27), (31), and (34). The proof ofis similar. This follows from (28), (32), and (34).
Furthermore . This follows from (27), (35). Thus,Note thatwhich follows from the relationship and (this follows from (23), (26), and (31)).
Therefore, for , This follows from (33), (37), and (38). In analogy with (33), (37), and (38) we havethis follows from (28) and (36);which follows from the relationship
and (this follows from (23), (26), and (32)). Therefore, for , This follows from (40), (41), and (42). Denote by the sum . Let us observe thatThen,This follows from (29) and (39). Note that for any odd number and we havewhich follows from the relationshipOn the other hand, for any odd number and , we havewhich follows from the relationshipThusThis follows from (46) and (48). Note that we haveWe see that , and ,This follows from (12). Thus , , and for any odd number we haveMoreover , and , we getThis follows from (13). Thus , , and for any odd number we haveIn view of (51), (53), and (55), for any odd number , the following inequality is satisfied:Note that We see that , and ,This follows from (14). Thus , , and for any odd number we haveWe see that , and ,This follows from (15). Thus , , and for any odd number we haveIn view of (57), (59), and (61), for any odd number , the following inequality is satisfied:Now we see that which follows from the relationshipNote that, for some numbers or , some components will cease to exist in formula (63). As an example, let . Then there are not ,, , in formula (63). However, an estimation of the number of components of (63) shall be sufficient for further consideration. Denote by the setWe calculateand thusNote that and This follows from (56), (62), (63), and (67).
Note thatThis follows from (27), (35), (50), and (68). After considering that
we obtain ThusThis follows from (23). In view of (45) and (71) the following inequality is satisfied:On the other hand We see thatThis follows from (30) and (43). Note that for any even number and which follows from the relationship On the other hand, and for any even number and, we havewhich follows from the relationshipThusThis follows from (75), (77). Note that for and for any even number This follows from (52) and (54). Hence if is an even number, thenThis follows from (51), (80), and (81). Note that for and for any even number andbelong to . This follows from (58) and (60). In view of (57), (83), and (84) and for any even number , the following inequality is satisfied: Hence, if is an even number, thenThis follows from (63), (67), (82), and (85). ThusThis follows from (79) and (86). In analogy with (69) and (71) we haveThis follows from (23), (28), and (36). In view of (87) and (88), the following inequality is satisfied:HenceThis follows from (74) and (89). ThusThis follows from (72) and (90). In view of (25) and (91) the following inequality is satisfied:Case 3:In view of (24) and (25) there is one case to consider, namely, ThenNote thatWe now give an estimate for . Note that and for . This follows from (94). HenceThis follows from (9) and (94).
We now give an estimate for . There are two cases to consider:
Case 3’: .
Then, by (94) and (96), we haveCase 3”: .
Note that the replacement of with in (23) and (26) gives usand thus in case 3 we proved that , where , and This follows from (96), (97), and (98). In view of cases 1, 2, and 3 we obtainThis follows from (11), (92), and (99). If , then
, soprovided that . This finishes the proof of the sufficient condition. This completes the proof of Theorem 6.

Remark 7. If is nonincreasing, then converges uniformly in if and only if is convergent.

Proof. Proof of the sufficient condition is obvious.
Proof of the necessary condition. Letbe a convergent series. Note that for any even number we haveThusThis follows from (103). After considering that is convergent, we obtain the convergent series . This follows from (104). After considering that is nonincreasing, we have that
is convergent and is convergent.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares no conflicts of interest.