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Trigonometric series with general monotone coefficients
J. Math. Anal. Appl. 326 (2007) 721–735 www.elsevier.com/locate/jmaa
Trigonometric series with general monotone coefficients ✩ S. Tikhonov Centre de Recerca Matemàtica (CRM), Bellaterra (Barcelona) E-08193, Spain Received 3 October 2005 Available online 18 April 2006 Submitted by D. Waterman
Abstract We study trigonometric series with general monotone coefficients. Convergence results in the different metrics are obtained. Also, we prove a Hardy-type result on the behavior of the series near the origin. © 2006 Elsevier Inc. All rights reserved. Keywords: Trigonometric series; Convergence; Hardy–Littlewood theorem
1. Introduction In this paper we consider the series ∞
an cos nx
(1)
an sin nx,
(2)
n=1
and ∞ n=1
✩ The paper was partially written while the author was staying at the Centre de Recerca Matemàtica in Barcelona, Spain. The research was funded by a EU Marie Curie fellowship (Contract MIF1-CT-2004-509465) and was partially supported by the Russian Foundation for Fundamental Research (Project 06-01-00268) and the Leading Scientific Schools (Grant NSH-4681.2006.1). E-mail address: [email protected].
0022-247X/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2006.02.053
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where {an }∞ n=1 is given a null sequence of complex numbers. We define by f (x) and g(x) the sums of the series (1) and (2) respectively at the points where the series converge. 1.1. Pointwise and uniform convergence We start with the following well-known theorem [34, V.1, p. 182]. Theorem 1.1. Let an an+1 · · · and an → 0. Then series (1) and (2) converge for all x, except possibly x = 2πk, k ∈ Z, in the case of (1), and converge uniformly on any interval [ε, 2π − ε], where 0 < ε < π . It is clear that for uniform convergence of series (1) on the whole period the condition ∞ n=1 an < ∞ is necessary and sufficient. For series (2) we rewrite the classical result of Chaundy and Joliffe [9], [34, V.1, p. 183]. Theorem 1.2. Let an an+1 · · · and an → 0. Then a necessary and sufficient condition for the uniform convergence of series (2) is nan → 0
as n → ∞.
(3)
Condition (3) is both necessary and sufficient for (2) be the Fourier series of a continuous function. 1.2. L1 and Lp -convergence We note that in the case when the coefficients of series (1) and (2) are monotone one has f, g ∞ ∈ aLn1 iff series (1) and (2) are the Fourier series of f and g, respectively. The condition n=1 n < ∞ guarantees f, g ∈ L1 . Also, it is clear that if series (1) is the Fourier series then the condition σn (f, x) − Sn (f, x) = o(1) or O(1) 1 is equivalent to convergence (or boundedness) of Sn (f, x) in L1 . Here Sn (f, x) = nν=1 aν cos νx, 1 n σn (f, x) = n+1 ν=1 Sν (f, x). The same result holds for series (2) as well. The following theorem provides a criterion of the convergence of Sn (f, x) in the terms of the coefficients of (1) and (2) (see [11,28]). Theorem 1.3. Let an an+1 · · · and an → 0. Let f, g ∈ L1 , then f (x) − Sn (f, x) = o(1) or O(1) iff an log n = o(1) 1
or O(1) .
The same result holds for series (2) as well. In the case Lp -convergence, 1 < p < ∞, we refer to the following well-known theorem by Hardy and Littlewood (see [13], [34, V.2, XII, §6]). Theorem 1.4. Let an an+1 · · · and an → 0. A necessary and sufficient condition that the function f (or g) should belong to Lp , 1 < p < ∞, is that ∞ n=1
p
an np−2 < ∞.
(4)
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1.3. Order of magnitude of functions represented by series with monotone coefficients Hardy [10], [34, V.1, V, §2] proved the following result. Theorem 1.5. Let an an+1 · · · and an → 0. Let 0 < α < 1. Then nα an → A > 0 as n → ∞ if and only if either f (x) =
∞
an cos nx ∼ A sin
πα Γ (1 − α)x α−1 2
as x → 0+,
an sin nx ∼ A cos
πα Γ (1 − α)x α−1 2
as x → 0+.
n=1
or g(x) =
∞ n=1
The main goals of this paper are the following. We introduce a natural general notion of monotonicity which encompasses the previously known criteria, in the context of L∞ , L1 , and Lp , 1 < p < ∞. In addition, we study Hardy-type results on the behavior of trigonometric series near the origin. This paper is organized as follows. Section 2 contains the definition and a description of properties of general monotone sequences. Sections 3–5 are devoted to: pointwise and uniform convergence, L1 and Lp -convergence, and order of magnitude of functions represented by series with general monotone coefficients. We give a brief history of the topics covered. Section 6 provides the concluding remarks. 2. Definition and properties of general monotone sequence We start with the description of two directions of generalization of sequences from the class M = {an ∈ R: an > 0, an ↓ 0, i.e., an an+1 · · · → 0}. The first way goes back to the papers [22] and [24] where the concept of quasi-monotone sequence was introduced: an (5) QM = an ∈ R: ∃τ 0 such that τ ↓ 0 . n an We note that a = {an }∞ n=1 ∈ QM if and only if an := an − an+1 −α n for some α > 0. The second step was the replacing of nτ in (5) by a sequence λn satisfying certain conditions [26]. Moreover, the analog of (5) was given for complex sequences [25]. We need here two definitions of regular variation (see [3, pp. 52, 65]).
Definition 2.1. A positive sequence r = {rn }∞ n=1 is said to be regularly varying if r[ξ n] → ψ(ξ ) ∈ (0, ∞) as n → ∞ for any ξ 1. rn Definition 2.2. A positive sequence r = {rn }∞ n=1 is said to be O-regularly varying if the inequalities r[ξ n] r[ξ n] lim sup <∞ (6) 0 < lim inf n→∞ rn n→∞ rn hold for any ξ 1.
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Now we recall the definitions of RVQM [25] and ORVQM [26,31] classes. As usual, we define an := an − an+1 . Definition 2.3. Let {an }∞ n=1 be a complex sequence. Suppose that there exists a positive nondecreasing sequence {rn }∞ n=1 such that for all integers n the following relation holds:
an π . ∈ K(θ0 ) := z ∈ C: | arg z| θ0 < rn 2 ∞ If {rn }∞ n=1 is regularly varying (or O-regularly varying), then we call {an }n=1 regularly quasimonotone (or O-regularly quasi-monotone). We set
RVQM = a = {an }∞ n=1 : an ∈ C, an → 0, and a is regularly quasi-monotone , ORVQM = a = {an }∞ n=1 : an ∈ C, an → 0, and a is O-regularly quasi-monotone . It is clear that M ⊂ QM ⊂ RVQM ⊂ ORVQM. The second way to generalize M is to consider the sequences of rest bounded variation (see [14]).
∞ ∞ RBVS = a = {an }n=1 : an ∈ C, an → 0 and |aν − aν+1 | C|an | ∀n ∈ N . (7) ν=n
We suppose in (7) that the constant C is independent of n. Now we introduce the main definition. Definition 2.4. The sequence of complex numbers a = {an }∞ n=1 is said to be general monotone, or a ∈ GM, if the relation 2n−1
|aν − aν+1 | C|an |
ν=n
holds for all integer n, where the constant C is independent of n. We remark that a definition of this type was considered in [2]. We now establish some properties of GM. 2.1. Properties of GM 1◦ .
If a ∈ GM, then |aν | C|an |
This follows from C|an |
2n−1 k=n
for n ν 2n. ν−1 | ak | ak = |aν − an | |aν | − |an |. k=n
(8)
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2◦ .
If a ∈ GM, then N N |ak | | ak | C |an | + for any n < N. k k=n
725
(9)
k=n+1
Indeed, let l ∈ N ∪ {0} such that 2l n N < 2l+1 n. Then, by definition, N
s+1
| ak |
l 2 n−1 s=0 k=2s n
k=n
By 1◦ , we write I C |an | +
l s=1
3◦ .
5◦ .
l
|a2s n | = I.
s=0
s
2 n
|a2s n |
k=2s−1 n+1
1 k
N |ak | C |an | + . k k=n+1
a ∈ GM if and only if a satisfies (8) and (9). Indeed, (8) and (9) imply 2n−1 2n−1 2n−1 |ak | 1 | ak | C |an | + C |an | + |an | C|an |. k k k=n
4◦ .
| ak | C
k=n
k=n
RBVS ⊂ GM. This follows immediately from the definitions. ORVQM ⊂ GM. Let a = {an }∞ n=1 ∈ ORVQM. Using the following two facts from [6],
(1) | ak | C ( ak + 2( rk+1 rk − 1)ak ), where C is independent of k; is nonnegative and nonincreasing, (2) { ( arkk )}∞ n=1 we obtain 2n−1 ν=n
2n−1
aν | aν | C (an − a2n ) + C (rν+1 − rν ) rν ν=n C an + C an
r2n − rn r2n C|an | C|an |. rn rn
6◦ .
RBVS \ ORVQM = {0}. −n −n−2 , n ∈ N. Let {r }∞ be a positive Define a = {an }∞ n n=1 n=1 such that a2n−1 a= e , a2n = e an n+1 nondecreasing sequence such that rn rn+1 . Then we have n r2n+1 r2n e, and, therefore, e[ 2 ] , n ∈ N. r2n rn Then condition (6) does not hold for ξ = 2 and therefore for any ξ > 1. Thus, a ∈ / ORVQM. On the other hand, ∞ ∞ ∞ n 2 2 | aν | 1 + e aν 2 1 + e e−ν Ce−[ 2 ] Can ,
ν=n
i.e., a ∈ RBVS.
ν=n
ν=[ n2 ]
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7◦ . ORVQM \ (QM ∪ RBVS) = {0}. m 1 Let b = {bn = + n−2 }∞ , 2m n < 2m+1 , m = 0, 1, 2, . . . . Define c = {cn }∞ n=1 2m m2 n=1 m log2 m as follows cn := bn if n = 2m + 1 and cn := 2bn if n = 2m + 1. We have b ∈ QM and c ∈ ORVQM \ QM. Also,
∞ ∞ ∞ 1 1 2 | cν | | c2ν | = + ν 2 C . 2 log m 2 ν m ν=m ν=m ν log ν ν=2
Hence there is no such constant C for which the inequality
C log m
c2m is true. Thus, c ∈ / RBVS.
8◦ . GM \ (ORVQM ∪ RBVS) = {0}. Here we use the sequence dn := an + icn , where a and c are the sequences defined in 6◦ and 7◦ , respectively. 3. Pointwise and uniform convergence First, we study pointwise convergence. The generalizations of Theorem 1.1 were proved by Shah [23] for QM and by Leindler [14] for RBVS. We prove the following theorem. Theorem 3.1. Let a = {an }∞ n=1 ∈ GM and ∞ |an |
n
n=1
< ∞.
(10)
Then series (1) and (2) converge for all x, except possibly x = 2πk, k ∈ Z, in the case of (1), and converge uniformly on any interval [ε, 2π − ε], where 0 < ε < π . For any x ∈ (0, π] we have
1 |aν | |an | + ∞ if x1 n, ν=n+1 x ν f (x) − Sn (f, x) C (11) [1/x]−1 |aν | 1 if x1 > n, |aν | + |a[1/x] | + ∞ ν=n ν=[1/x]+1 ν x x and g(x) − Sn (g, x) C
1
∞ |aν | ν=n+1 ν x |an | + |aν | 1 2 [1/x]−1 ν|aν | + |a[1/x] | + ∞ ν=n ν=[1/x]+1 ν x x
if if
1 x 1 x
n, > n. (12)
To prove Theorem 3.1, we need a lemma. Lemma 3.1. Let a = {an }∞ ∈ GM and suppose (10) holds. Let {ξν } satisfy |Ξn | := n=1 n | ν=0 ξν | C ∗ . Then the series ∞ ν=1 aν ξν converges and ∞ ∞ |aν | ∗ aν ξν CC |an | + . ν ν=n
ν=n+1
Proof. By Abel’s transformation, we have N ν=n
aν ξν =
N −1 ν=n
aν Ξν + aN ΞN − an Ξn−1 .
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Then by 1◦ and 2◦ , N N −1 ∞ |a | ν aν ξν C ∗ | aν | + |an | + |aN | CC ∗ |an | + , ν ν=n ν=n ν=n+1
2
which establishes the statement.
n (x), Now define ξν = cos νx, ξ0 = 12 or ξν = sin νx. We have Ξn (x) = Dn (x) or Ξn (x) = D where Dn (x) is the Dirichlet kernel and Dn (x) is the conjugate Dirichlet kernel. Then using n (x)| 1/(2| sin x/2|) ≡ C ∗ and Lemma 3.1, we obtain convergence, uniform con|Dn (x)|, |D vergence on interval [ε, 2π − ε], and (11)–(12). This completes the proof of Theorem 3.1. Now let us discuss the case of L∞ -convergence. After the Chaundy–Joliffe result on uniform convergence on whole period (Theorem 1.2) the topic was studied in [20] for QM, in [6,27,31] for ORVQM, in [14] for RBVS. We extend the result for general monotone coefficients. Theorem 3.2. Let a = {an }∞ n=1 be a positive sequence and a ∈ GM. Then a necessary and sufficient condition for the uniform convergence of series (2) (or for (2) to be a Fourier series of a continuous function) is condition (3). Moreover, g(x) − Sn (g, x) C max(νaν ), (13) ∞ νn
where g∞ = max{|g(x)|, 0 x 2π}. Furthermore, series (2) has uniformly bounded partial sums if and only if nan C. Proof. Sufficiency. Note that (3) implies εn := maxνn (νaν ) → 0 as n → ∞. Let x > 0. By (12), we have for 1 xn, ∞ 1 g(x) − Sn (g, x) C nan + nεn Cεn ν2 ν=n+1
and for 1 > xn, [1/x]−1 g(x) − Sn (g, x) C x εν + ε[1/x] + ε[1/x] ν=n
∞ ν=[1/x]+1
1 xν 2
Cεn .
Hence, (13) follows. This implies the uniform convergence of (2). We also obtain that g ∈ C and (2) is the Fourier series of g. Necessity. If series (2) converges uniformly and if xn = 1, then we have, by 1◦ , nan C1
n ν=[ n2 ]
ν ν C2 aν sin → 0 as n → ∞. n n n
aν
ν=1
If (2) is the Fourier series of a continuous function, then the (C, 1)-means of (2) converge uniformly, i.e.,
n k 1− ak sin kx → 0 as n → ∞ uniformly in x. n+1 k=1
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Then
2n
k kπ 1− ak sin →0 2n + 1 4n k=1 and 1◦ imply n−1 nk=1 kak → 0 and nan → 0 as n → ∞. The same reasoning establishes the statement about the uniformly boundedness of partial sums of (2). This completes the proof of Theorem 3.2. 2 The next theorem provides a quantitative version of the condition for uniform convergence of a general trigonometric series. Theorem 3.3. Let c = {cn }n∈Z be a null sequence of complex numbers such that {cn }∞ n=1 ∈ GM, ncn = o(1) as n → ∞, and ∞
|ck + c−k | < ∞.
k=1
Then Sn (x) =
cν eiνx
|ν|n
converges uniformly on [−π, π] to a continuous function h(x). Moreover, ∞ h(·) − Sn (·) C |ck + c−k | + max ν|cν | . ∞
k=n
νn
The proof follows immediately from 2◦ and Theorem 3.2 of [6]. 4. L1 and Lp -convergence We first discuss L1 -convergence. The topic has been considered by many authors, in particular [12] and [28] for the case QM, [26] and [33] for the cases RVQM, ORVQM. We prove the following more general theorem. Theorem 4.1. Let a = {an }n∈Z be a null sequence of complex numbers such that {an }∞ n=1 ∈ GM. Let (1) be the Fourier series of a function f (x) ∈ L1 . Then lim f (·) − S(f, ·)1 = 0 if and only if lim |an | ln n = 0, n→∞ n→∞ Sn (f, ·) C ∀n ∈ N if and only if |an | ln n C ∀n ∈ N. 1 inx . The same results hold for series (2) and ∞ n=−∞ an e The proof follows directly from 1◦ and Theorem 1 of [2]. For the case Lp , 1 < p < ∞, we prove the following generalization of the Hardy–Littlewood Theorem 1.4. Theorem 4.2. Let a = {an }∞ n=1 be a positive sequence and a ∈ GM. A necessary and sufficient condition that the function f (or g) should belong to Lp , 1 < p < ∞, is that inequality (4) holds.
S. Tikhonov / J. Math. Anal. Appl. 326 (2007) 721–735
729
Proof. Sufficiency. Suppose inequality (4) holds. Then for the cosine series, say, we write ∞ 2 −1 n+1
ak cos kx =:
n=0 k=2n
∞
In (x).
n=0
By the Littlewood–Paley theorem [34, V. II, p. 233], we have for 1 < p < 2, ∞ ∞ 1 1 2 2 p p In (x) In (x) . f p C C p Using
n=0
M
In (x)p C
(M
2n+1 −1
p
l=2n
C2
n=0
p
p k=m cos kxp
n(p−1)
− m)p−1
and 1◦ , we have
| al |Dl (x)p + |a2n |D2n −1 (x)p + |a2n+1 −1 |D2n+1 −1 (x)p
2n+1 −1
p | al | + a2n
n
p C2n(p−1) a2n
2
C
l=2n
p f p
p
p
al l p−2 .
l=2n−1 +1
∞
p p−2 . n=1 an n
Finally, we have C For p 2 the same inequality follows from Paley’s theorem on Fourier coefficients [34, V. 2, p. 121]. Necessity. Paley’s theorem implies the result for 1 < p < 2. By 1◦ , we have for p 2,
∞ In (x)p f p C p
1
p
∞ p C In (x)p
n=0
C
∞
1
p
1 n+2 ∞ 2 p p C In (x) dx
n=0
2n+1 −1
p a2n+1 −1
n=0
0
n=0 0
p
π 2n+2
νx
1
p
dx
ν=2n
This completes the proof of Theorem 4.2.
π
C
∞
1
p
p a2n+1 −1 2n(p−1)
.
n=0
2
We note that one can use the same method for general monotone coefficients according to either the Walsh system or some generalization of the Walsh system from [30]. For Walsh coefficients from the class M the result was proved in [18]. For trigonometric series the case QM was considered in [1] and the case of RBVS was considered in [15]. For trigonometric series we can also write more general result. Theorem 4.3. Let a = {an }∞ n=1 be a positive sequence and a ∈ GM. Then ∞ 1 p p p(1−α)−2 f α,p gα,p an n , n=1
where f (x)α,p := x α f (x)p , 1 p < ∞, and − p1 < α < 1 − p1 . Proof. The result follows from 2◦ and from [1] and [4, p. 37]. Indeed, by 1◦ , n p ∞ ∞ p p −αp−2 ak n C an np(1−α)−2 =: I. f α,p C n=1
k=[n/2]
n=1
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On the other hand, by 2◦ and the Hardy-type inequality, we have p ∞ ∞ p ∞ ∞ ak p | ak | np−αp−2 C I + np−αp−2 CI. f α,p C k n=1
k=n
n=1
2
k=n
5. Behavior near the origin In this section we present Hardy-type theorems for general monotone sequence. First we give the following definition. Definition 5.1. A positive sequence r = {rn }∞ n=1 is said to be almost increasing (almost decreasing) if there exists a positive constant C 1 such that the inequalities rn Crm
(Crn rm )
hold for all n < m. Theorem 5.1. Let A ∈ C and let c = {cn }∞ n=0 be a null sequence of complex numbers such that c ∈ GM. Let ω = {ωn }∞ n=1 be a positive monotone sequence and suppose that there exists ε > 0 such that (a) the sequence {nε ωn } is almost decreasing, (b) the sequence {n1−ε ωn } is almost increasing. If cn ∼ Aωn as n → ∞, then h(x) =
∞
cn einx ∼ A
n=0
∞
ωn einx
as x → 0.
(14)
n=1
The converse result does not hold. Remark. We note that conditions (a) and (b) are equivalent to the following conditions (see for instance [29]): ∞ ωk = O[ωn ] k
and
k=n+1
n
ωk = O[nωn ],
k=1
respectively. The typical example is {ωn = n−α l(n)}, where 0 < α < 1 and l(x) is a positive slowly varying function on [1, ∞), i.e., l(λx) l(x) → 1 as x → ∞ for any λ > 1. ∞ ck ωk Proof. First, the convergence of ∞ k=1 k implies the convergence of k=1 k . By Theorem 3.1, ∞ the series n=0 cn einx converges to a function h(x) on 0 < |x| π . Let ξ > 1, m ∈ N and ξ )). Let M = [ xξ ]. We write x ∈ (0, min(π, m ∞ ∞ M ∞ inx inx inx h(x) = A ωn e − A ω n e + c0 + (cn − Aωn )e cn einx + n=1
=: I1 + I2 + I3 + I4 .
n=M+1
n=1
n=M+1
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Let us estimate I4 . By (11) and (12), we have ∞ ∞ |cν | ων C C ωM . |I4 | |cM | + ωM + C x ν x ν x ν=M+1
ν=M+1
Further, condition (a) implies ωM Cξ −ε ω[1/x] . Thus |I4 | Cξ −ε x −1 ω[1/x] . Using (11) and (12), we also have |I2 | Cξ −ε x −1 ω[1/x] . Now we estimate I3 . By condition (b), we have m M cn cn |I3 | |c0 | + ω − Aωn + ω − Aωn n n n=1
n=m+1
cn cn m M |c0 | + sup − A ωn + sup − A ωn n∈N ωn nm ωn n=1 n=m+1
cn C |c0 | + mωm + sup − AMωM nm ωn
cn 1 − Aξ 1−ε ω[1/x] . C |c0 | + mωm + sup x nm ωn It is clear that we can always assume that 0 < ε < 1 in (a) and (b). Now we choose ξ and m so large that ξ −ε and ξ 1−ε supnm | ωcnn − A| are as small as possible. On the other hand, by condition (b), we have ω1 Cx ε−1 ω[1/x] = o(x −1 ω[1/x] ) as x → 0+. Thus, collecting the estimates for I1 , I2 and I3 , we get ∞ inx (15) ωn e = o x −1 ω[1/x] as x → 0+. h(x) − A n=1
Finally, (14) follows from the following inequalities ∞ −1 ∞ −1 −1 inx −1 |x| ω[1/|x|] ωn e |x| ω[1/|x|] ωn sin nx C n=1
as x → 0.
(16)
n=1
To prove (16), we use one result of [19]: π π , 2n ], If ωn ↓ 0, then for |x| ∈ ( 2(n+1) ∞ ∞ 1 ωn sin nx ωm 1 − cos(m + 1)x . x 2| sin 2 | n=1
(17)
m=1
By (a), there exist ε > 0 and A > 2 such that mε ωm Anε ωn for n m. Define s such that 2A = ( 3s )ε . From (17) we get that for small x, ∞ C (ε, A) 3n−1 C1 (ε, A) 1 ωm sin mx ωm = (ω[sn] − ω3n ) |x| |x| m=[sn] m=1
ω[1/x] C1 (ε, A) [sn] ε C1 (ε, A) ω[sn] 1 − A ω[sn] C2 (ε, A) . |x| 3n 2|x| |x|
732
S. Tikhonov / J. Math. Anal. Appl. 326 (2007) 721–735
The convergence result does not hold for a series with O-regularly varying quasi-monotone coefficients (see [7]). So, by 5◦ , it does not hold and for the series with general monotone coefficients. This completes the proof of Theorem 5.1. 2 In a similar way to the proof of Theorem 5.1, one can also obtain the following theorems. Theorem 5.2. Under the conditions of Theorem 5.1, if cn ∼ Aωn as n → 0, then g(x) =
∞
cn sin nx ∼ A
n=0
∞
ωn sin nx
as x → 0.
(18)
n=1
To prove Theorem 5.2, we use the same reasoning as in Theorem 5.1 and relation (16). Theorem 5.3. Let A ∈ C, 0 < α < 1 and l : [1, ∞) → (0, ∞) be quasi-monotone slowly varying. −α Let a complex sequence c = {cn }∞ n=0 be in GM. If cn ∼ An l(n) as n → 0, then πα as x → 0, (19) f (x) ∼ Ax α−1 l(1/x)Γ (1 − α) sin 2 πα g(x) ∼ Ax α−1 l(1/x)Γ (1 − α) cos as x → 0, (20)
2 πi(1 − α) as x → 0. (21) h(x) ∼ Ax α−1 l(1/x)Γ (1 − α) exp 2 Proof. We use {ωn = n−α l(n)} and we have, say, for cosine series (see (15)) ∞ ωn cos nx = o x −1 ω[1/x] as x → 0+. h(x) − A n=1
On the other hand (see [3, Theorem 4.3.2]), A
∞
ωn cos nx ∼ Ax α−1 l(1/x)Γ (1 − α) sin
n=1
πα 2
as x → 0,
which gives (19). We also remark that (20) follows directly from Theorem 5.2 and [3, Theorem 4.3.2]. This completes the proof of Theorem 5.3. 2 We note that the previous results generalize many results of this type starting with the paper of Hardy [10]; see, for example, the papers [21] for QM case [7,31,32] for ORVQM case. The next theorem extends the results of [8]. Theorem 5.4. Under the conditions of Theorem 5.1, cn = O(ωn )
as n → ∞,
(22)
implies f (x) = g(x) =
∞ n=1 ∞ n=1
cn cos nx = O x −1 ω[1/x] as x → 0,
(23)
cn sin nx = O x −1 ω[1/x] as x → 0.
(24)
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733
Moreover, if cn ∈ R, then (22) ⇔ (23) ⇔ (24). Proof. The implication (22) ⇒ (23), (24) follows from the similar reasoning as in the proof of Theorem 5.1 and the following inequalities: ∞ ∞ [1/x] + C ω cos kx ω sin kx ωk = O x −1 ω[1/x] . k k k=1
k=1
k=1
To prove (23) ⇒ (22), we integrate f (x): x x ∞ ck t −1 ω[1/t] dt = O(ω[1/x] ), sin kx = f (t) dt = O k k=1
0
0
where we have used the condition (a) on ω in the last inequality. Integrating one more time, we have, by monotonicity of ω, ∞ ck k=1
k2
(1 − cos kx) = O(xω[1/x] ).
(1 − cos kx C > 0 for [ n2 ] < k n), we use 1◦ :
n cn ck ωn C , = O 2 n n k n
Setting x =
2π 3n
k=[ 2 ]+1
which gives (22). To prove (24) ⇒ (22), we use a similar technique and ∞ ck k=1
k
x (1 − cos kx) =
g(t) dt = O(ω[1/x] ). 0
This completes the proof of Theorem 5.4.
2
6. Final remarks 1. We note that the comparison of QM and RBVS was studied in [16]. This was question γ δ < ∞, raised by Teljakovski˘ı. In [5], Boas proved that if cn is δ-quasi-monotonic and ∞ k k k=1 0 < γ < 1, then (1) and (2) converge for all x except possibly x = 2πk, k ∈ Z, in the case of (1). One can construct a positive sequence c ∈ GM such that c satisfies (10) but Boas’s condition does not hold (see 7◦ ). 2. We define a generalization of GM. Definition. Let β = {βn }∞ n=1 be a positive sequence. The sequence of complex numbers is said to be β-general monotone, or a ∈ GM(β), if the relation a = {an }∞ n=1 |an | +
2n−1
|aν − aν+1 | Cβn
ν=n
holds for all integer n, where the constant C is independent of n.
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We remark that this type of generalization of RBVS was considered in [17]. Many of the theorems of Sections 3 and 4 can be rewritten for β-general monotone sequences. The theorems about the necessary and sufficient conditions are rewritten only as sufficient conditions. For example, Theorem 6.1. Let a = {an }∞ n=1 ∈ GM(β) and ∞
β2n < ∞.
n=1
Then series (1) and (2) converge for all x except possibly x = 2πk, k ∈ Z, in the case of (1), and converge uniformly on any interval [ε, 2π − ε], where 0 < ε < π. Moreover, if nβn = o(1) as n → ∞, then series (2) converges uniform on [0, 2π] and g(x) − Sn (g, x) C max(νβν ), ∞ νn
3. The series with general monotone coefficients have many nice properties. We write only the following theorem which can be obtained, using Theorem 4.2: ∞ ∞ Theorem 6.2. Let a = {a n }n=1 , b = {bn }n=1 be positive sequences and a, b ∈ GM. Let f ∈ Lp , ∞ 1 < p < ∞, and f (x) ∼ n=1 an cos nx + bn sin nx. Then n 1 ∞ 1
p p 1 −k p kp+p−2 p p−2 n (am + bm ) m + (am + bm ) m , ωk f, n p m=1
where ωk (f,
1 n )p
m=n+1
is a modulus of smoothness of a function f of order k > 0 in Lp metric.
Further, Theorem 6.2 implies the following Marchaud-type result. Theorem 6.3. Under the conditions of Theorem 6.2, we have 1 1 1 [1/δ] θ θ θ θ ωk (f, δ)p δ k t −kθ−1 ωk+1 (f, t)p dt δk ν kθ−1 Eν−1 (f )p dt , δ
ν=1
where θ = p and En (f )p is the best approximation of a function f by trigonometrical polynomials of total degree n. Note that using of θ = p is an improvement compared with the classical parameter θ = min(2, p) in the Marchaud inequality. Acknowledgment The author is grateful to K.M. Dyakonov and S.A. Teljakovski˘ı for helpful discussions.
References [1] R. Askey, S. Wainger, Integrability theorems for Fourier series, Duke Math. J. 33 (1966) 223–228. [2] A.S. Belov, Remarks on the convergence (boundedness) in the mean of partial sums of a trigonometric series, Mat. Zametki 71 (6) (2002) 807–817; translation in: Math. Notes 71 (5–6) (2002) 739–748.
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Trigonometric series with general monotone coefficients ✩ S. Tikhonov Centre de Recerca Matemàtica (CRM), Bellaterra (Barcelona) E-08193, Spain Received 3 October 2005 Available online 18 April 2006 Submitted by D. Waterman
Abstract We study trigonometric series with general monotone coefficients. Convergence results in the different metrics are obtained. Also, we prove a Hardy-type result on the behavior of the series near the origin. © 2006 Elsevier Inc. All rights reserved. Keywords: Trigonometric series; Convergence; Hardy–Littlewood theorem
1. Introduction In this paper we consider the series ∞
an cos nx
(1)
an sin nx,
(2)
n=1
and ∞ n=1
✩ The paper was partially written while the author was staying at the Centre de Recerca Matemàtica in Barcelona, Spain. The research was funded by a EU Marie Curie fellowship (Contract MIF1-CT-2004-509465) and was partially supported by the Russian Foundation for Fundamental Research (Project 06-01-00268) and the Leading Scientific Schools (Grant NSH-4681.2006.1). E-mail address: [email protected].
0022-247X/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2006.02.053
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where {an }∞ n=1 is given a null sequence of complex numbers. We define by f (x) and g(x) the sums of the series (1) and (2) respectively at the points where the series converge. 1.1. Pointwise and uniform convergence We start with the following well-known theorem [34, V.1, p. 182]. Theorem 1.1. Let an an+1 · · · and an → 0. Then series (1) and (2) converge for all x, except possibly x = 2πk, k ∈ Z, in the case of (1), and converge uniformly on any interval [ε, 2π − ε], where 0 < ε < π . It is clear that for uniform convergence of series (1) on the whole period the condition ∞ n=1 an < ∞ is necessary and sufficient. For series (2) we rewrite the classical result of Chaundy and Joliffe [9], [34, V.1, p. 183]. Theorem 1.2. Let an an+1 · · · and an → 0. Then a necessary and sufficient condition for the uniform convergence of series (2) is nan → 0
as n → ∞.
(3)
Condition (3) is both necessary and sufficient for (2) be the Fourier series of a continuous function. 1.2. L1 and Lp -convergence We note that in the case when the coefficients of series (1) and (2) are monotone one has f, g ∞ ∈ aLn1 iff series (1) and (2) are the Fourier series of f and g, respectively. The condition n=1 n < ∞ guarantees f, g ∈ L1 . Also, it is clear that if series (1) is the Fourier series then the condition σn (f, x) − Sn (f, x) = o(1) or O(1) 1 is equivalent to convergence (or boundedness) of Sn (f, x) in L1 . Here Sn (f, x) = nν=1 aν cos νx, 1 n σn (f, x) = n+1 ν=1 Sν (f, x). The same result holds for series (2) as well. The following theorem provides a criterion of the convergence of Sn (f, x) in the terms of the coefficients of (1) and (2) (see [11,28]). Theorem 1.3. Let an an+1 · · · and an → 0. Let f, g ∈ L1 , then f (x) − Sn (f, x) = o(1) or O(1) iff an log n = o(1) 1
or O(1) .
The same result holds for series (2) as well. In the case Lp -convergence, 1 < p < ∞, we refer to the following well-known theorem by Hardy and Littlewood (see [13], [34, V.2, XII, §6]). Theorem 1.4. Let an an+1 · · · and an → 0. A necessary and sufficient condition that the function f (or g) should belong to Lp , 1 < p < ∞, is that ∞ n=1
p
an np−2 < ∞.
(4)
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1.3. Order of magnitude of functions represented by series with monotone coefficients Hardy [10], [34, V.1, V, §2] proved the following result. Theorem 1.5. Let an an+1 · · · and an → 0. Let 0 < α < 1. Then nα an → A > 0 as n → ∞ if and only if either f (x) =
∞
an cos nx ∼ A sin
πα Γ (1 − α)x α−1 2
as x → 0+,
an sin nx ∼ A cos
πα Γ (1 − α)x α−1 2
as x → 0+.
n=1
or g(x) =
∞ n=1
The main goals of this paper are the following. We introduce a natural general notion of monotonicity which encompasses the previously known criteria, in the context of L∞ , L1 , and Lp , 1 < p < ∞. In addition, we study Hardy-type results on the behavior of trigonometric series near the origin. This paper is organized as follows. Section 2 contains the definition and a description of properties of general monotone sequences. Sections 3–5 are devoted to: pointwise and uniform convergence, L1 and Lp -convergence, and order of magnitude of functions represented by series with general monotone coefficients. We give a brief history of the topics covered. Section 6 provides the concluding remarks. 2. Definition and properties of general monotone sequence We start with the description of two directions of generalization of sequences from the class M = {an ∈ R: an > 0, an ↓ 0, i.e., an an+1 · · · → 0}. The first way goes back to the papers [22] and [24] where the concept of quasi-monotone sequence was introduced: an (5) QM = an ∈ R: ∃τ 0 such that τ ↓ 0 . n an We note that a = {an }∞ n=1 ∈ QM if and only if an := an − an+1 −α n for some α > 0. The second step was the replacing of nτ in (5) by a sequence λn satisfying certain conditions [26]. Moreover, the analog of (5) was given for complex sequences [25]. We need here two definitions of regular variation (see [3, pp. 52, 65]).
Definition 2.1. A positive sequence r = {rn }∞ n=1 is said to be regularly varying if r[ξ n] → ψ(ξ ) ∈ (0, ∞) as n → ∞ for any ξ 1. rn Definition 2.2. A positive sequence r = {rn }∞ n=1 is said to be O-regularly varying if the inequalities r[ξ n] r[ξ n] lim sup <∞ (6) 0 < lim inf n→∞ rn n→∞ rn hold for any ξ 1.
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Now we recall the definitions of RVQM [25] and ORVQM [26,31] classes. As usual, we define an := an − an+1 . Definition 2.3. Let {an }∞ n=1 be a complex sequence. Suppose that there exists a positive nondecreasing sequence {rn }∞ n=1 such that for all integers n the following relation holds:
an π . ∈ K(θ0 ) := z ∈ C: | arg z| θ0 < rn 2 ∞ If {rn }∞ n=1 is regularly varying (or O-regularly varying), then we call {an }n=1 regularly quasimonotone (or O-regularly quasi-monotone). We set
RVQM = a = {an }∞ n=1 : an ∈ C, an → 0, and a is regularly quasi-monotone , ORVQM = a = {an }∞ n=1 : an ∈ C, an → 0, and a is O-regularly quasi-monotone . It is clear that M ⊂ QM ⊂ RVQM ⊂ ORVQM. The second way to generalize M is to consider the sequences of rest bounded variation (see [14]).
∞ ∞ RBVS = a = {an }n=1 : an ∈ C, an → 0 and |aν − aν+1 | C|an | ∀n ∈ N . (7) ν=n
We suppose in (7) that the constant C is independent of n. Now we introduce the main definition. Definition 2.4. The sequence of complex numbers a = {an }∞ n=1 is said to be general monotone, or a ∈ GM, if the relation 2n−1
|aν − aν+1 | C|an |
ν=n
holds for all integer n, where the constant C is independent of n. We remark that a definition of this type was considered in [2]. We now establish some properties of GM. 2.1. Properties of GM 1◦ .
If a ∈ GM, then |aν | C|an |
This follows from C|an |
2n−1 k=n
for n ν 2n. ν−1 | ak | ak = |aν − an | |aν | − |an |. k=n
(8)
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2◦ .
If a ∈ GM, then N N |ak | | ak | C |an | + for any n < N. k k=n
725
(9)
k=n+1
Indeed, let l ∈ N ∪ {0} such that 2l n N < 2l+1 n. Then, by definition, N
s+1
| ak |
l 2 n−1 s=0 k=2s n
k=n
By 1◦ , we write I C |an | +
l s=1
3◦ .
5◦ .
l
|a2s n | = I.
s=0
s
2 n
|a2s n |
k=2s−1 n+1
1 k
N |ak | C |an | + . k k=n+1
a ∈ GM if and only if a satisfies (8) and (9). Indeed, (8) and (9) imply 2n−1 2n−1 2n−1 |ak | 1 | ak | C |an | + C |an | + |an | C|an |. k k k=n
4◦ .
| ak | C
k=n
k=n
RBVS ⊂ GM. This follows immediately from the definitions. ORVQM ⊂ GM. Let a = {an }∞ n=1 ∈ ORVQM. Using the following two facts from [6],
(1) | ak | C ( ak + 2( rk+1 rk − 1)ak ), where C is independent of k; is nonnegative and nonincreasing, (2) { ( arkk )}∞ n=1 we obtain 2n−1 ν=n
2n−1
aν | aν | C (an − a2n ) + C (rν+1 − rν ) rν ν=n C an + C an
r2n − rn r2n C|an | C|an |. rn rn
6◦ .
RBVS \ ORVQM = {0}. −n −n−2 , n ∈ N. Let {r }∞ be a positive Define a = {an }∞ n n=1 n=1 such that a2n−1 a= e , a2n = e an n+1 nondecreasing sequence such that rn rn+1 . Then we have n r2n+1 r2n e, and, therefore, e[ 2 ] , n ∈ N. r2n rn Then condition (6) does not hold for ξ = 2 and therefore for any ξ > 1. Thus, a ∈ / ORVQM. On the other hand, ∞ ∞ ∞ n 2 2 | aν | 1 + e aν 2 1 + e e−ν Ce−[ 2 ] Can ,
ν=n
i.e., a ∈ RBVS.
ν=n
ν=[ n2 ]
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7◦ . ORVQM \ (QM ∪ RBVS) = {0}. m 1 Let b = {bn = + n−2 }∞ , 2m n < 2m+1 , m = 0, 1, 2, . . . . Define c = {cn }∞ n=1 2m m2 n=1 m log2 m as follows cn := bn if n = 2m + 1 and cn := 2bn if n = 2m + 1. We have b ∈ QM and c ∈ ORVQM \ QM. Also,
∞ ∞ ∞ 1 1 2 | cν | | c2ν | = + ν 2 C . 2 log m 2 ν m ν=m ν=m ν log ν ν=2
Hence there is no such constant C for which the inequality
C log m
c2m is true. Thus, c ∈ / RBVS.
8◦ . GM \ (ORVQM ∪ RBVS) = {0}. Here we use the sequence dn := an + icn , where a and c are the sequences defined in 6◦ and 7◦ , respectively. 3. Pointwise and uniform convergence First, we study pointwise convergence. The generalizations of Theorem 1.1 were proved by Shah [23] for QM and by Leindler [14] for RBVS. We prove the following theorem. Theorem 3.1. Let a = {an }∞ n=1 ∈ GM and ∞ |an |
n
n=1
< ∞.
(10)
Then series (1) and (2) converge for all x, except possibly x = 2πk, k ∈ Z, in the case of (1), and converge uniformly on any interval [ε, 2π − ε], where 0 < ε < π . For any x ∈ (0, π] we have
1 |aν | |an | + ∞ if x1 n, ν=n+1 x ν f (x) − Sn (f, x) C (11) [1/x]−1 |aν | 1 if x1 > n, |aν | + |a[1/x] | + ∞ ν=n ν=[1/x]+1 ν x x and g(x) − Sn (g, x) C
1
∞ |aν | ν=n+1 ν x |an | + |aν | 1 2 [1/x]−1 ν|aν | + |a[1/x] | + ∞ ν=n ν=[1/x]+1 ν x x
if if
1 x 1 x
n, > n. (12)
To prove Theorem 3.1, we need a lemma. Lemma 3.1. Let a = {an }∞ ∈ GM and suppose (10) holds. Let {ξν } satisfy |Ξn | := n=1 n | ν=0 ξν | C ∗ . Then the series ∞ ν=1 aν ξν converges and ∞ ∞ |aν | ∗ aν ξν CC |an | + . ν ν=n
ν=n+1
Proof. By Abel’s transformation, we have N ν=n
aν ξν =
N −1 ν=n
aν Ξν + aN ΞN − an Ξn−1 .
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Then by 1◦ and 2◦ , N N −1 ∞ |a | ν aν ξν C ∗ | aν | + |an | + |aN | CC ∗ |an | + , ν ν=n ν=n ν=n+1
2
which establishes the statement.
n (x), Now define ξν = cos νx, ξ0 = 12 or ξν = sin νx. We have Ξn (x) = Dn (x) or Ξn (x) = D where Dn (x) is the Dirichlet kernel and Dn (x) is the conjugate Dirichlet kernel. Then using n (x)| 1/(2| sin x/2|) ≡ C ∗ and Lemma 3.1, we obtain convergence, uniform con|Dn (x)|, |D vergence on interval [ε, 2π − ε], and (11)–(12). This completes the proof of Theorem 3.1. Now let us discuss the case of L∞ -convergence. After the Chaundy–Joliffe result on uniform convergence on whole period (Theorem 1.2) the topic was studied in [20] for QM, in [6,27,31] for ORVQM, in [14] for RBVS. We extend the result for general monotone coefficients. Theorem 3.2. Let a = {an }∞ n=1 be a positive sequence and a ∈ GM. Then a necessary and sufficient condition for the uniform convergence of series (2) (or for (2) to be a Fourier series of a continuous function) is condition (3). Moreover, g(x) − Sn (g, x) C max(νaν ), (13) ∞ νn
where g∞ = max{|g(x)|, 0 x 2π}. Furthermore, series (2) has uniformly bounded partial sums if and only if nan C. Proof. Sufficiency. Note that (3) implies εn := maxνn (νaν ) → 0 as n → ∞. Let x > 0. By (12), we have for 1 xn, ∞ 1 g(x) − Sn (g, x) C nan + nεn Cεn ν2 ν=n+1
and for 1 > xn, [1/x]−1 g(x) − Sn (g, x) C x εν + ε[1/x] + ε[1/x] ν=n
∞ ν=[1/x]+1
1 xν 2
Cεn .
Hence, (13) follows. This implies the uniform convergence of (2). We also obtain that g ∈ C and (2) is the Fourier series of g. Necessity. If series (2) converges uniformly and if xn = 1, then we have, by 1◦ , nan C1
n ν=[ n2 ]
ν ν C2 aν sin → 0 as n → ∞. n n n
aν
ν=1
If (2) is the Fourier series of a continuous function, then the (C, 1)-means of (2) converge uniformly, i.e.,
n k 1− ak sin kx → 0 as n → ∞ uniformly in x. n+1 k=1
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Then
2n
k kπ 1− ak sin →0 2n + 1 4n k=1 and 1◦ imply n−1 nk=1 kak → 0 and nan → 0 as n → ∞. The same reasoning establishes the statement about the uniformly boundedness of partial sums of (2). This completes the proof of Theorem 3.2. 2 The next theorem provides a quantitative version of the condition for uniform convergence of a general trigonometric series. Theorem 3.3. Let c = {cn }n∈Z be a null sequence of complex numbers such that {cn }∞ n=1 ∈ GM, ncn = o(1) as n → ∞, and ∞
|ck + c−k | < ∞.
k=1
Then Sn (x) =
cν eiνx
|ν|n
converges uniformly on [−π, π] to a continuous function h(x). Moreover, ∞ h(·) − Sn (·) C |ck + c−k | + max ν|cν | . ∞
k=n
νn
The proof follows immediately from 2◦ and Theorem 3.2 of [6]. 4. L1 and Lp -convergence We first discuss L1 -convergence. The topic has been considered by many authors, in particular [12] and [28] for the case QM, [26] and [33] for the cases RVQM, ORVQM. We prove the following more general theorem. Theorem 4.1. Let a = {an }n∈Z be a null sequence of complex numbers such that {an }∞ n=1 ∈ GM. Let (1) be the Fourier series of a function f (x) ∈ L1 . Then lim f (·) − S(f, ·)1 = 0 if and only if lim |an | ln n = 0, n→∞ n→∞ Sn (f, ·) C ∀n ∈ N if and only if |an | ln n C ∀n ∈ N. 1 inx . The same results hold for series (2) and ∞ n=−∞ an e The proof follows directly from 1◦ and Theorem 1 of [2]. For the case Lp , 1 < p < ∞, we prove the following generalization of the Hardy–Littlewood Theorem 1.4. Theorem 4.2. Let a = {an }∞ n=1 be a positive sequence and a ∈ GM. A necessary and sufficient condition that the function f (or g) should belong to Lp , 1 < p < ∞, is that inequality (4) holds.
S. Tikhonov / J. Math. Anal. Appl. 326 (2007) 721–735
729
Proof. Sufficiency. Suppose inequality (4) holds. Then for the cosine series, say, we write ∞ 2 −1 n+1
ak cos kx =:
n=0 k=2n
∞
In (x).
n=0
By the Littlewood–Paley theorem [34, V. II, p. 233], we have for 1 < p < 2, ∞ ∞ 1 1 2 2 p p In (x) In (x) . f p C C p Using
n=0
M
In (x)p C
(M
2n+1 −1
p
l=2n
C2
n=0
p
p k=m cos kxp
n(p−1)
− m)p−1
and 1◦ , we have
| al |Dl (x)p + |a2n |D2n −1 (x)p + |a2n+1 −1 |D2n+1 −1 (x)p
2n+1 −1
p | al | + a2n
n
p C2n(p−1) a2n
2
C
l=2n
p f p
p
p
al l p−2 .
l=2n−1 +1
∞
p p−2 . n=1 an n
Finally, we have C For p 2 the same inequality follows from Paley’s theorem on Fourier coefficients [34, V. 2, p. 121]. Necessity. Paley’s theorem implies the result for 1 < p < 2. By 1◦ , we have for p 2,
∞ In (x)p f p C p
1
p
∞ p C In (x)p
n=0
C
∞
1
p
1 n+2 ∞ 2 p p C In (x) dx
n=0
2n+1 −1
p a2n+1 −1
n=0
0
n=0 0
p
π 2n+2
νx
1
p
dx
ν=2n
This completes the proof of Theorem 4.2.
π
C
∞
1
p
p a2n+1 −1 2n(p−1)
.
n=0
2
We note that one can use the same method for general monotone coefficients according to either the Walsh system or some generalization of the Walsh system from [30]. For Walsh coefficients from the class M the result was proved in [18]. For trigonometric series the case QM was considered in [1] and the case of RBVS was considered in [15]. For trigonometric series we can also write more general result. Theorem 4.3. Let a = {an }∞ n=1 be a positive sequence and a ∈ GM. Then ∞ 1 p p p(1−α)−2 f α,p gα,p an n , n=1
where f (x)α,p := x α f (x)p , 1 p < ∞, and − p1 < α < 1 − p1 . Proof. The result follows from 2◦ and from [1] and [4, p. 37]. Indeed, by 1◦ , n p ∞ ∞ p p −αp−2 ak n C an np(1−α)−2 =: I. f α,p C n=1
k=[n/2]
n=1
730
S. Tikhonov / J. Math. Anal. Appl. 326 (2007) 721–735
On the other hand, by 2◦ and the Hardy-type inequality, we have p ∞ ∞ p ∞ ∞ ak p | ak | np−αp−2 C I + np−αp−2 CI. f α,p C k n=1
k=n
n=1
2
k=n
5. Behavior near the origin In this section we present Hardy-type theorems for general monotone sequence. First we give the following definition. Definition 5.1. A positive sequence r = {rn }∞ n=1 is said to be almost increasing (almost decreasing) if there exists a positive constant C 1 such that the inequalities rn Crm
(Crn rm )
hold for all n < m. Theorem 5.1. Let A ∈ C and let c = {cn }∞ n=0 be a null sequence of complex numbers such that c ∈ GM. Let ω = {ωn }∞ n=1 be a positive monotone sequence and suppose that there exists ε > 0 such that (a) the sequence {nε ωn } is almost decreasing, (b) the sequence {n1−ε ωn } is almost increasing. If cn ∼ Aωn as n → ∞, then h(x) =
∞
cn einx ∼ A
n=0
∞
ωn einx
as x → 0.
(14)
n=1
The converse result does not hold. Remark. We note that conditions (a) and (b) are equivalent to the following conditions (see for instance [29]): ∞ ωk = O[ωn ] k
and
k=n+1
n
ωk = O[nωn ],
k=1
respectively. The typical example is {ωn = n−α l(n)}, where 0 < α < 1 and l(x) is a positive slowly varying function on [1, ∞), i.e., l(λx) l(x) → 1 as x → ∞ for any λ > 1. ∞ ck ωk Proof. First, the convergence of ∞ k=1 k implies the convergence of k=1 k . By Theorem 3.1, ∞ the series n=0 cn einx converges to a function h(x) on 0 < |x| π . Let ξ > 1, m ∈ N and ξ )). Let M = [ xξ ]. We write x ∈ (0, min(π, m ∞ ∞ M ∞ inx inx inx h(x) = A ωn e − A ω n e + c0 + (cn − Aωn )e cn einx + n=1
=: I1 + I2 + I3 + I4 .
n=M+1
n=1
n=M+1
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Let us estimate I4 . By (11) and (12), we have ∞ ∞ |cν | ων C C ωM . |I4 | |cM | + ωM + C x ν x ν x ν=M+1
ν=M+1
Further, condition (a) implies ωM Cξ −ε ω[1/x] . Thus |I4 | Cξ −ε x −1 ω[1/x] . Using (11) and (12), we also have |I2 | Cξ −ε x −1 ω[1/x] . Now we estimate I3 . By condition (b), we have m M cn cn |I3 | |c0 | + ω − Aωn + ω − Aωn n n n=1
n=m+1
cn cn m M |c0 | + sup − A ωn + sup − A ωn n∈N ωn nm ωn n=1 n=m+1
cn C |c0 | + mωm + sup − AMωM nm ωn
cn 1 − Aξ 1−ε ω[1/x] . C |c0 | + mωm + sup x nm ωn It is clear that we can always assume that 0 < ε < 1 in (a) and (b). Now we choose ξ and m so large that ξ −ε and ξ 1−ε supnm | ωcnn − A| are as small as possible. On the other hand, by condition (b), we have ω1 Cx ε−1 ω[1/x] = o(x −1 ω[1/x] ) as x → 0+. Thus, collecting the estimates for I1 , I2 and I3 , we get ∞ inx (15) ωn e = o x −1 ω[1/x] as x → 0+. h(x) − A n=1
Finally, (14) follows from the following inequalities ∞ −1 ∞ −1 −1 inx −1 |x| ω[1/|x|] ωn e |x| ω[1/|x|] ωn sin nx C n=1
as x → 0.
(16)
n=1
To prove (16), we use one result of [19]: π π , 2n ], If ωn ↓ 0, then for |x| ∈ ( 2(n+1) ∞ ∞ 1 ωn sin nx ωm 1 − cos(m + 1)x . x 2| sin 2 | n=1
(17)
m=1
By (a), there exist ε > 0 and A > 2 such that mε ωm Anε ωn for n m. Define s such that 2A = ( 3s )ε . From (17) we get that for small x, ∞ C (ε, A) 3n−1 C1 (ε, A) 1 ωm sin mx ωm = (ω[sn] − ω3n ) |x| |x| m=[sn] m=1
ω[1/x] C1 (ε, A) [sn] ε C1 (ε, A) ω[sn] 1 − A ω[sn] C2 (ε, A) . |x| 3n 2|x| |x|
732
S. Tikhonov / J. Math. Anal. Appl. 326 (2007) 721–735
The convergence result does not hold for a series with O-regularly varying quasi-monotone coefficients (see [7]). So, by 5◦ , it does not hold and for the series with general monotone coefficients. This completes the proof of Theorem 5.1. 2 In a similar way to the proof of Theorem 5.1, one can also obtain the following theorems. Theorem 5.2. Under the conditions of Theorem 5.1, if cn ∼ Aωn as n → 0, then g(x) =
∞
cn sin nx ∼ A
n=0
∞
ωn sin nx
as x → 0.
(18)
n=1
To prove Theorem 5.2, we use the same reasoning as in Theorem 5.1 and relation (16). Theorem 5.3. Let A ∈ C, 0 < α < 1 and l : [1, ∞) → (0, ∞) be quasi-monotone slowly varying. −α Let a complex sequence c = {cn }∞ n=0 be in GM. If cn ∼ An l(n) as n → 0, then πα as x → 0, (19) f (x) ∼ Ax α−1 l(1/x)Γ (1 − α) sin 2 πα g(x) ∼ Ax α−1 l(1/x)Γ (1 − α) cos as x → 0, (20)
2 πi(1 − α) as x → 0. (21) h(x) ∼ Ax α−1 l(1/x)Γ (1 − α) exp 2 Proof. We use {ωn = n−α l(n)} and we have, say, for cosine series (see (15)) ∞ ωn cos nx = o x −1 ω[1/x] as x → 0+. h(x) − A n=1
On the other hand (see [3, Theorem 4.3.2]), A
∞
ωn cos nx ∼ Ax α−1 l(1/x)Γ (1 − α) sin
n=1
πα 2
as x → 0,
which gives (19). We also remark that (20) follows directly from Theorem 5.2 and [3, Theorem 4.3.2]. This completes the proof of Theorem 5.3. 2 We note that the previous results generalize many results of this type starting with the paper of Hardy [10]; see, for example, the papers [21] for QM case [7,31,32] for ORVQM case. The next theorem extends the results of [8]. Theorem 5.4. Under the conditions of Theorem 5.1, cn = O(ωn )
as n → ∞,
(22)
implies f (x) = g(x) =
∞ n=1 ∞ n=1
cn cos nx = O x −1 ω[1/x] as x → 0,
(23)
cn sin nx = O x −1 ω[1/x] as x → 0.
(24)
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733
Moreover, if cn ∈ R, then (22) ⇔ (23) ⇔ (24). Proof. The implication (22) ⇒ (23), (24) follows from the similar reasoning as in the proof of Theorem 5.1 and the following inequalities: ∞ ∞ [1/x] + C ω cos kx ω sin kx ωk = O x −1 ω[1/x] . k k k=1
k=1
k=1
To prove (23) ⇒ (22), we integrate f (x): x x ∞ ck t −1 ω[1/t] dt = O(ω[1/x] ), sin kx = f (t) dt = O k k=1
0
0
where we have used the condition (a) on ω in the last inequality. Integrating one more time, we have, by monotonicity of ω, ∞ ck k=1
k2
(1 − cos kx) = O(xω[1/x] ).
(1 − cos kx C > 0 for [ n2 ] < k n), we use 1◦ :
n cn ck ωn C , = O 2 n n k n
Setting x =
2π 3n
k=[ 2 ]+1
which gives (22). To prove (24) ⇒ (22), we use a similar technique and ∞ ck k=1
k
x (1 − cos kx) =
g(t) dt = O(ω[1/x] ). 0
This completes the proof of Theorem 5.4.
2
6. Final remarks 1. We note that the comparison of QM and RBVS was studied in [16]. This was question γ δ < ∞, raised by Teljakovski˘ı. In [5], Boas proved that if cn is δ-quasi-monotonic and ∞ k k k=1 0 < γ < 1, then (1) and (2) converge for all x except possibly x = 2πk, k ∈ Z, in the case of (1). One can construct a positive sequence c ∈ GM such that c satisfies (10) but Boas’s condition does not hold (see 7◦ ). 2. We define a generalization of GM. Definition. Let β = {βn }∞ n=1 be a positive sequence. The sequence of complex numbers is said to be β-general monotone, or a ∈ GM(β), if the relation a = {an }∞ n=1 |an | +
2n−1
|aν − aν+1 | Cβn
ν=n
holds for all integer n, where the constant C is independent of n.
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We remark that this type of generalization of RBVS was considered in [17]. Many of the theorems of Sections 3 and 4 can be rewritten for β-general monotone sequences. The theorems about the necessary and sufficient conditions are rewritten only as sufficient conditions. For example, Theorem 6.1. Let a = {an }∞ n=1 ∈ GM(β) and ∞
β2n < ∞.
n=1
Then series (1) and (2) converge for all x except possibly x = 2πk, k ∈ Z, in the case of (1), and converge uniformly on any interval [ε, 2π − ε], where 0 < ε < π. Moreover, if nβn = o(1) as n → ∞, then series (2) converges uniform on [0, 2π] and g(x) − Sn (g, x) C max(νβν ), ∞ νn
3. The series with general monotone coefficients have many nice properties. We write only the following theorem which can be obtained, using Theorem 4.2: ∞ ∞ Theorem 6.2. Let a = {a n }n=1 , b = {bn }n=1 be positive sequences and a, b ∈ GM. Let f ∈ Lp , ∞ 1 < p < ∞, and f (x) ∼ n=1 an cos nx + bn sin nx. Then n 1 ∞ 1
p p 1 −k p kp+p−2 p p−2 n (am + bm ) m + (am + bm ) m , ωk f, n p m=1
where ωk (f,
1 n )p
m=n+1
is a modulus of smoothness of a function f of order k > 0 in Lp metric.
Further, Theorem 6.2 implies the following Marchaud-type result. Theorem 6.3. Under the conditions of Theorem 6.2, we have 1 1 1 [1/δ] θ θ θ θ ωk (f, δ)p δ k t −kθ−1 ωk+1 (f, t)p dt δk ν kθ−1 Eν−1 (f )p dt , δ
ν=1
where θ = p and En (f )p is the best approximation of a function f by trigonometrical polynomials of total degree n. Note that using of θ = p is an improvement compared with the classical parameter θ = min(2, p) in the Marchaud inequality. Acknowledgment The author is grateful to K.M. Dyakonov and S.A. Teljakovski˘ı for helpful discussions.
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