I. CLASSES OF L1-COVERGENCES OF FOURIER SERIES
1.1. CLASSICAL AND NEOCLASSICAL RESULTS
Denote by L1(T ) Banach space of all complex, Lebesgue integrable functions on the unit circle T. To every function f L1(T ) corresponds the Fourier series of f ∈ int 1 int S(f) fˆ(n)e , where fˆ(n) = f(t)e− dt, n < ∼ 2π T∫ | | ∞ n < | X| ∞ are Fourier coefficients of f. A sequence of partial sums be denoted by
ikt Sn(f) = Sn(f, t) = fˆ(k)e , n = 0, 1, 2, 3, ... , k n |X|≤ while the (C, 1)-means (Fejer sums) of the sequances of partial sums will be written as: n 1 σ (f) = σ (f, t) = S (f, t), n = 0, 1, 2, ... . n n n + 1 k kX=0 The Dirichlet kernel is denoted by 1 1 n sin n + t D (t) = + cos kt = 2 n 2 t k=1 2 sin X 2 and the Fejer kernel is denoted by t 2 n 1 1 sin(n + 1) F = D (t) = 2 . n n + 1 k 2(n + 1) t k=0 sin X 2 Note that 4 D = log +O(1) , n , k nk1 π2 → ∞ F = 1, for every n, where denotes the L1(T ) norm. k nk1 k k1 − 1 Let t 1 n cos cos n + t D˜ (t) = sin kt = 2 − 2 n t k=1 2 sin X 2 1 1 t cos n + t D (t) = ctg + D˜ (t) = 2 , n n t −2 2 − 2 sin 2 n 1 1 sin(n + 1)t K˜n(t) = D˜ k(t) = sin t n + 1 2 t − n + 1 k=0 4 sin X 2 denote the conjugate Dirichlet’s kernel, modified Dirichlet’s kernel and conjugate Fejer’s kernel, respectively. The Banach space in the real case will be denoted by L1(0, π) and norm by . k · k Let
a ∞ 0 + a cos nx (C) 2 n n=1 X ∞ an sin nx (S) n=1 X be the cosine and sine trigonometric series.
The partial sums of a real cosine and sine series will be denoted by Sn(x) and
S˜n(x) respectively. The problem of L1-convergence, via Fourier coefficients, consists of finding the properties of Fourier coefficients such that the cosine series (C) is a Fourier series of its sum f and S (f) f = o(1), n if and only if fˆ(n) lg n = o(1), k n − k → ∞ | | n . → ∞ In general L1(T ) does not admit convergence in L1(T )-norm, (Zygmund [63], Vol.1. p.67) since the operator norm S L1(T ) is unbounded, i.e. k nk 1 4 S L (T ) = D = log n + O(1) , n . k nk k nk π2 → ∞ A classical result concerning the integrability and L1-convergence of a cosine series (C) is the following well-known theorem of Young, see [61].
2 Theorem 1.1 (Young). If a ∞ is a convex (∆ a = ∆(∆a ) = ∆a { n}n=0 n n n − ∆a = a 2a + a 0, n) null sequence, then the cosine series (C) is n+1 n − n+1 n+2 ≥ ∀ 2 the Fourier series of its sum f, and
S (f) f = o(1) n iff a log n = o(1), n . (1.1) k n − k → ∞ n → ∞
∞ The sequences a that satisfy the condition (n + 1) ∆2a < are called { n} | n| ∞ n=1 quasi-convex. X The next theorem of Kolmogorov extends Young’s result, since every convex null sequence is also quasi-convex null sequence.
Theorem 1.2 [21] (Kolmogorov). If a is a quasi-convex null sequence then { n} the cosine series (C) is the Fourier series of its sum f and (1.1) hold. We say that a sequence a is of bounded variation and we write a BV { k} { k} ∈ ∞ if ∆a < . Several authors (Sidon, Telyakovskii, Fomin, Stanojevi´cand | k| ∞ k=0 others)X have extended these classical rasults by answering one or both of the following two questions: (i) For which classes of coefficients of (C), subclasses of BV , does it follows that (C) is the Fourier series of its sum function and that the sum is an integrable function? (ii) If (C) is the Fourier series of some function f L1 under what conditions ∈ does (1.1) hold? In other words, for which classes of Fourier coefficients of even and integrable functions does (1.1) hold.
Theorem 1.3 [35] (Sidon). Let αn n∞=1 and Pn ∞n=1 be the sequences such that { } { } k ∞ ∞ p α 1, for every n and let p < , If a = k α , n = 1, 2, 3, ... | n| ≤ | n| ∞ n k l n=1 k=n l=n then the cosine series (C) is theX Fourier series of its sumX f.X It’s obvious that Sidon’s conditions [35] imply that a BV . { n} ∈ Telyakovskii [44] defined an extension of the class of the quasi-convex sequences. It’s denoted by S.
The class S is defined as follows: a null sequence a ∞ belongs to the class S if { n}n=0 ∞ there exists a monotonically decreasing sequence A ∞ such that A < { n}n=0 n ∞ n=0 and ∆a A , for all n. X | n| ≤ n Telyakovskii proved [44] that the Sidon’s class is equevalent to the class S. Thus, the class S is usually called as Sidon-Telyakovskii class.
3 Theorem 1.4 [44] (Telyakovskii). Let a ∞ S. Then the cosine series (C) { n}n=0 ∈ is the Fourier series of its sum f and (1.1) hold. On the other hand, Kano [18] have extended the classical result of Kolmogorov by answering of the first question (i).
∞ a Theorem 1.5 [8] (Kano). If a is a null sequence such that n2 ∆2 n < { n} n n=1 , then (C) is a Fourier series, or equivalently it represents an integrableX function. ∞ The following Lemma was proved by Telyakovskii in [46].
∞ a Lemma 1.1 [46]. Condition n2 ∆2 n < is equivalent to the simultane- n ∞ n=1 X ∞ a ∞ ous fulfillment of conditions | n| < and (n + 1) ∆2a < . n ∞ | n| ∞ n=1 n=1 X X Remark 1.1 Using this Lemma we obtain that the Theorem 1.5 is a corollary of the Theorem 1.2. Very later, S. Kumari and Baby Ram [22] have proved the following theorem: a Theorem 1.6 [22] (S. Kumari, B. Ram). Let (k + 1)2 ∆2 k 0. Then k ↓ n 1 a n a h(x) = lim (k + 1)2 ∆2 k + (v + 1)2 ∆2 v cos kx n "2 k v # →∞ k=1 v=k X X ∞ a exists for x (0, π] and h L(0, π] iff (k + 1)2 ∆2 k < . ∈ ∈ k ∞ k=1 In [27] C. N. Moore generalized quasi-convexityX of null sequences in the following way, ∞ nk ∆k+1a < , for k > 0 , (M) | n| ∞ n=1 X where the order od differences is fractional and proved the corresponding integra- bility result. It is well-known [7] that if a is a null sequence satisfyng the condition (M), { n} ∞ then nr ∆r+1a < , for 0 r < k. In particular it is of bounded variation. | n| ∞ ≤ n=1 N. SinghX and K. M. Sharma [33] proved the following generalized theorem of Kolmogorov.
Theorem 1.7 [33] (Singh, Sharma) Let k be a real number such that k > 0. If
4 (i) lim an = 0 n →∞ ∞ (ii) nk ∆k+1a < | n| ∞ n=1 thenX for the convergence of the series (C) in the metric L1 it is necessary and sufficient that a log n = o(1), n . n → ∞ Remark 1.2 The Theorem 1.7 is a corollary of Theorem 1.4. It suffices to show that the conditions (i) and (ii) of the Theorem 1.7 implies the Sidon-Telyakovskii
∞ k 1 k+1 type condition. Indeed, if we denote A = C m − ∆ a , where C is a n k | m| k m=n some positive constant depend only on k, whichX will be later on defined. We obtain A 0, n and n ↓ → ∞ N N N N k 1 k+1 ∞ k 1 k+1 A = C m − ∆ a + C m − ∆ a = n k | m| k | m| n=1 n=1 m=n n=1 X X X X m=XN+1 N m k 1 k+1 ∞ k 1 k+1 = C m − ∆ a + NC m − ∆ a k | m| k | m| ≤ m=1 n=1 X X m=XN+1 N ∞ ∞ C mk ∆k+1a + C mk ∆k+1a , i.e. A < . ≤ k | m| k | m| n ∞ m=1 n=1 X m=XN+1 X a+n Now, we need the following properties for binomial coefficients a (see [2], page 885): (a + 1)(a + 2) . . . (a + n) a) a > 1 a+n = > 0 − ⇒ a n! b) a+n C na, C > 0. a ≤ a a ∞ i n + k 1 For 0 < k < 1, we have: ∆a = − − ∆k+1a , i.e. n i n i i=n X −
∞ i n + k 1 k+1 ∞ k 1 k+1 ∆a − − ∆ a C i − ∆ a = A . | n| ≤ i n | i| ≤ k | i| n i=n i=n X − X The case k 1 is evidently. Finally a S. ≥ { n} ∈ In [37] Caslavˇ V. Stanojevi´cand Vera B. Stanojevi´cgeneralized the Telyakovskii theorem [44]. They defined a stronger class S , p > 1 as follows: a null sequence a of real p { n} numbers belongs to the class Sp if for some p > 1 and some monotone sequence
5 ∞ A such that A < the following condition holds { n} n ∞ n=1 X n p 1 ∆ak | p | = O(1) . n Ak kX=1 There exists a null-sequence a such that a S but a / S. { n} { n} ∈ p { n} ∈ 1 Example: Let us define a sequence a as follows: let ∆a = for n = m2 { n} n m2 and ∆a = 0 for n = m2. Firstly, we shall show that a / S. We have: n 6 { n} ∈ ∞ ∞ a 2 = ∆a = ∆a 2 + ∆a 2 + + ∆a 2 + = ∆a 2 = m i m m +1 ··· (m+1) ··· i 2 i=m iX=m X ∞ 1 = 0 , m , i2 → → ∞ i=m X i.e. an 0 as n . Let we denote An∗ =max ∆ai . Then An∗ 0 and → → ∞ i n | | ↓ ≥ ∞ 1 2 2 A∗ = . Really, A∗ = for (m 1) + 1 n m and n ∞ n m2 − ≤ ≤ n=1 X m2 m2 ∞ ∞ ∞ Ak∗ = A∗k = Am∗ 2 = k=1 m=1 k=(m 1)2+1 m=1 k=(m 1)2+1 X X X− X X− ∞ 1 ∞ 2m 1 = [m2 (m 1)2] = − = . m2 − − m2 ∞ m=1 m=1 X X Therefore for arbitraty positive sequence A such that A A∗ , we have { n} n ≥ n ∞ A = , i.e. a / S. n ∞ { n} ∈ n=1 X 1 ∞ Now, let A = , for all n. Then A 0, A < and for n = m2, we n n1+1/2p n ↓ n ∞ n=1 have: X p 2 1 m p m p m 1 ∆ai 1 ∆ak2 1 k2 2 | p| = 2 | | = 2 = m A m A 2 m 1 i=1 i k=1 k k=1 X X X k2+1/p 1 m p = k1/p = O(1) . m2 kX=1 6 Theorem 1.8 [37] (C.ˇ V. Stanojevi´cand V. B. Stanojevi´c). Let a S , { n} ∈ p 1 < p 2. Then the cosine series (C)is the Fourier series of its sum f and (1.1) ≤ hold. On the other hand, Fomin [12] have extended the Sidon-Telyakovskii’s class. He defined a class F , 1 < p 2 of Fourier coefficients as follows: a sequence a p ≤ { n} belongs to F , p > 1 if a 0 as k and p k → → ∞ 1/p ∞ 1 ∞ p ∆ai < . (1.2) k | | ! ∞ kX=1 Xi=k We note that Fomin has given an equivalent form of the condition (1.2). Namely, he proved the following lemma.
∞ Lemma 1.2 [12]. A sequence a F , p > 1 iff 2s∆(p) < , where { n} ∈ p s ∞ s=1 X s 1/p 1 2 ∆(p) = ∆a p . s s 1 k 2 − s 1 | | k=2X− +1 In [12], Fomin note that it is easy to see that the classFp is wider when p is closer to 1. But now we shall present the proof of this fact. Corollary 1.1 [67] For any 1 < r < p the following embedding relation holds F F . p ⊂ r 1 1 1 1 1 Proof. By inequality > , we have = + , where q > 0. This equality r p r p q 1 1 p q implies that + = 1, where p0 = and q0 = . p0 q0 r r Applying the Holder’s inequality, we have:
1/p0 1/q0 2s+1 2s+1 2s+1 2s+1 r r rp0 q0 ∆ak = ∆ak 1 ∆ak 1 = s | | s | | · ≤ s | | s k=2X+1 k=2X+1 k=2X+1 k=2X+1 1/p0 2s+1 s 1/q0 p = (2 ) ∆ak . s | | k=2X+1 Then
1/rp0 2s+1 ∞ ∞ s (r) s s/r s/q0r p 2 ∆ 2 2− 2 ∆a = s ≤ · · | k| s=1 s=1 s X X k=2X+1 7 1/p 2s+1 ∞ 1 1/r 1/q ∞ = 2s − ∆a p = 2s∆(p) . 2s | k| s s=1 s s=1 X k=2X+1 X Applying the Lemma 1.2, Fomin has proved that for the class F , 1 < p 2 we p ≤ have positive answers to both questions (i) and (ii). Theorem 1.9 [12] (Fomin). Let a F , 1 < p 2. Then the cosine series { n} ∈ p ≤ (C) is the Fourer series of its sum f and (1.1) hold.
Next we shall proved that Sp is a subclass of Fp, for all p > 1 Theorem 1.10. [67] For every p > 1 the following embedding relation holds S F . p ⊂ p Proof. Applyng the Abel’s transformation we have: s s 2 2 p p p ∆ak ∆ak = Ak | p | = s 1 | | s 1 Ak k=2X− +1 k=2X− +1 s s s 1 2 1 k p 2 p 2 − p − p ∆aj p ∆aj p ∆aj = ∆(A ) | p | + A2s | p | A s 1 | p | = k A A − 2 − +1 A s 1 j=1 j j=1 j j=1 j k=2X− +1 X X X s s 2 1 k p 2 p − p 1 ∆aj s p 1 ∆aj = k ∆(A ) | | + 2 A s | | k k Ap 2 2s Ap − s 1 j=1 j j=1 j k=2X− +1 X X s 1 2 − p s 1 p 1 ∆aj 2 − A s 1 | p | = − 2 − +1 2s 1 A − j=1 j X 2s 1 − p s p s 1 p = O(1) k ∆(A ) + 2 A s + 2 − A s 1 = k 2 2 − +1 s 1 k=2X− +1 2s p s 1 p s p s p s 1 p = O(1) A + 2 − A s 1 2 A s + 2 A s + 2 − A s 1 = k 2 − +1 2 2 2 − +1 s 1 − k=2X− +1 2s p s p s 1 p = O(1) A + 2 A s 1 = O(2 − A s 1 ) . k 2 − +1 2 − s 1 k=2X− +1 Firstly applying the Fomin’s lemma, then the Cauchy type theorem, we obtain
1/p ∞ ∞ s (p) ∞ s 1 s 1 p s 1 2 ∆s O(1) 2 2 − A s 1 = O 2 − A2s 1 < . ≤ 2s 1 2 − − ∞ s=1 s=1 − s=1 ! X X X 8 Very recenthly, L. Leindler [24] proved very important result. Namely, he proved that the Fomin’s class Fp is a subclass of the class Sp and also he given another proof of the Theorem 1.10. Precisely he proved the following Theorem.
Theorem 1.11 (Leindler) [24]. For all p > 1, the classes Fp and Sp are identical. A still larger class that answers both questions, but expressed in terms of a con- dition difficult to apply, is the class BV C, where C was defined by Garrett and ∩ Stanojevi´c[16] as follows: a null sequences of real numbers satisfy the condition C if for every ε > 0 there exists δ(ε) > 0 independent of n, such that
δ ∞ ∆akDk(x) dx < ε , for every n . ∫0 k=n X N. Singh and K. M. Sharma [33] proved that the Garrett-Stanojevi´cclass C is a stronger than that of Moore’s class (M). Theorem 1.12 [16] (Garrett and Stanojevi´c). Let a BV C. Then the { n} ∈ ∩ series (C) is the Fourier series of its sum f and (1.1) hold. In [15] Garrett, Rees and Stanojevi´cproved the following theorem. Theorem 1.13. For the classes S, BV and C, the following embedding relation holds S BV C. ⊂ ∩ Now, we shall prove the extension theorem of this theorem.
Theorem 1.14 [49]. For the classes Sp, BV and C, the following embedding relation holds S BV C. p ⊂ ∩ For the prove of this theorem we need the following lemma. Lemma 1.3 [61] (Hausdorff-Young). Let the sequence of complex numbers c { n} ∈ p l , 1 < p 2. Then cn is the sequence of Fourier coefficients of some ϕ 1 1 ≤ { } ∈ Lq + = 1 and p q 1/q 1/p 1 2π ∞ ϕ(x) qdx c p . 2π ∫ | | ≤ | n| 0 n= ! X−∞ Proof of Theorem 1.14.
9 If suffices to show that
π ∞ Tn = ∆akDk(x) dx = o(1) , n . ∫0 → ∞ k=n X For each n, let k be the least natural number such that n 2kn 1. n ≤ − Then Tn can be majorized by
2kn 1 2l+1 1 π − ∞ π − Tn ∆ajDj(x) dx + ∆ajDj(x) dx = I1 + I2 . ≤ ∫0 ∫0 j=n l=kn j=2l X X X The second term is written as follows: l+1 l+1 2 1 ∞ 1/2 π − I2 = + ∆ajDj(x) dx = Σ1 + Σ2 . ∫0 1/2∫l+1 l=kn ( ) j=2l X X 1 For the first term, the uniform estimate D (x) n + , is applied, i.e. | n | ≤ 2
2l+1 1 2l+1 1 ∞ 1 − 1 ∞ 1 − Σ ∆a j + ∆a 2l+1 = 1 2l+1 j 2 2l+1 j ≤ l | | ≤ l | | lX=kn jX=2 lX=kn jX=2 2l+1 1 ∞ − ∞ = ∆aj = ∆aj . l | | k | | lX=kn jX=2 j=2Xn By summation by parts, and by generalized arithmetic-square mean inequality, we have:
i 2kn 1 ∞ ∞ ∆ai ∞ ∆aj − ∆aj ∆ai = | | Ai = ∆Ai | | A2kn | | | | Ai Aj − Aj ≤ kn kn kn j=1 j=1 i=2X i=2X i=2X X X 1/p 1/p i 2kn ∞ p p 1 ∆aj kn 1 ∆aj i(∆Ai) | p | + 2 A2kn | p | = ≤ i A 2kn A kn j=1 j j=1 j i=2X X X ∞ kn = O(1) i(∆Ai) + 2 A2kn . " k # i=2Xn ∞ Since A < , and A 0 both terms on the right-hand side of the above n ∞ n ↓ n=1 inequalityX are o(1) as n . Thus Σ = o(1), n . → ∞ 1 → ∞ 10 Let 2l+1 1 π ∞ − ∆aj Σ2 = AjDj(x) dx . 1/2∫l+1 Aj l=kn j=2l X X Applying Abel’s transformation, we get:
2l+1 1 2l+1 2 j π π − ∆aj − ∆ar AjDj(x) ∆Aj Dr(x) dx+ l+1 A l+1 A 1/2∫ l j ≤ l 1/2∫ r j=2 j=2 r=1 X X X 2l 1 2l+1 1 π π − ∆ar − ∆ar + A2l Dr(x) dx + A2l+1 1 Dr(x) dx . ∫l+1 A − ∫l+1 A 1/2 r=1 r 1/2 r=1 r X X Applying the Holder type inequality, we get:
2l 1 2l 1 π π − ∆ar 1 − ∆ar 1 Vl = Dr(x) dx = x sin r + x dx ∫l+1 A ∫l+1 A 2 ≤ 1/2 r=1 r 1/2 2 sin r=1 r X 2 X 1/p q 1/q 2l 1 π π dx − ∆ar 1 p sin r + x dx , ≤ ∫l+1 x ∫l+1 A 2 1/2 2 sin 1/2 r=1 r 2 X 1 1 where + = 1. Since p q
π dx πp π dx M (2l+1)p 1, x p p p p − 1/2∫l+1 2 sin ≤ 2 1/2∫l+1 x ≤ 2 where Mp is an absolute constant depending on p, it follows that
q 1/q 2l π l+1 1/q 1/p ∆ar 1 Vl (2 ) (Mp) sin r + x dx . ≤ ∫0 A 2 r=1 r X Applying the Hausdorff-Young inequality to the last integral we get:
q 1/q 1/p 2l 2l 1 π p ∆ar 1 − ∆ar sin r + x dx Bp | p | . ∫0 A 2 ≤ Ar r=1 r r=1 X X 11 Thus l+1 1/p 2 p l+1 1 ∆ar Vl 2 Cp l+1 | p | ,Cp > 0 . ≤ 2 Ar r=1 X Then 2l+1 2 j π ∞ − ∆ar Σ2 ∆Aj Dr(x) dx+ l+1 A ≤ l 1/2∫ r l=kn j=2 r=1 X X X 2l 1 π ∞ − ∆ar + A2l Dr(x) dx+ 1/2∫l+1 Ar l=kn r=1 X X 2l+1 1 π ∞ − ∆ar + A2l+1 1 Dr(x) dx = − 1/2∫l+1 Ar l=kn r=1 X X 2l+1 2 ∞ − ∞ l = Op(1) j ∆Aj + 4 2 A2l . l lX=kn jX=2 lX=kn Now, applying the Cauchy condensation test, we get:
∞ l 2 A l = o(1) , n . 2 → ∞ lX=kn But
2l+1 2 2l+1 1 − − l+1 l l l j ∆Aj = Aj 2 A2l+1 1 + 2 A2l + A2l+1 1 2 A2l + 2 A2l + A2l . − − l l − ≤ jX=2 j=2X+1 Thus 2l+1 2 ∞ − ∞ l ∞ j∆Aj 2 2 A2l + A2l = o(1) , n , l ≤ → ∞ lX=kn jX=2 lX=kn lX=kn i.e. Σ = o(1), n . Finally, I = o(1), n . 2 → ∞ 2 → ∞ The same method applied to I1, yields the estimate:
2kn 1 2kn 2 − − kn 1 k 1 I1 O(1) ∆aj + Op(1) j ∆Aj + 4(2 − A2 n− ) . ≤ k 1 | | k 1 l=2Xn− j=2Xn− Letting n , the proof of the theorem follows. → ∞ 12 Remark 1.3. Theorem 1.8 is a corollary of Theorem 1.14 and Theorem 1,12. Thus by proving of the Theorem 1.14 we obtained a new proof of Theorem 1.8. On the other hand, Stanojevi´c[36] proved the following inclusion connecting by the classes Fp, C and BV . Theorem 1.15. For all 1 < p 2 the following embedding relation holds ≤
F BV C. p ⊂ ∩
In [15] Garrett, Rees and Stanojevi´cdefined an extension class of null sequences of bounded variation of order m 1. ≥ Namely, a null sequence a belongs to the class (BV )(m) if for some integer { k} ∞ m m m 1 m 1 m 1 m 1, ∆ a < , where ∆ a = ∆(∆ − a ) = ∆ − a ∆ − a . ≥ | k| ∞ k k k − k+1 k=1 For m =X 1 the class (BV )1 is the class BV .
Theorem 1.16 [15] (Garrett, Rees, Stanojevi´c). Let a (BV )(m) and { n} ∈ a log n = o(1), n . Then S f = o(1), n iff a C n → ∞ k n − k → ∞ { n} ∈ Both Fomin [11] and Stanojevi´c[36] considered the following natural extension of the class Fp. Let p 1. A sequence a belongs to C , if a 0 as k and ≥ { k} p k → → ∞
p 1 ∞ p n − ∆a = o(1) as n . (1.3) | k| → ∞ kX=n Answering the question (ii) Fomin and Stanojevi´cproved the following result:
Theorem 1.17 (Fomin [11], Stanojevi´c[36]). If (C) is a Fourier series of f L1 ∈ and a C BV for some 1 < p 2 then (1.1) holds. { n} ∈ p ∩ ≤ Later, see [13], Fomin extended the above result by considering a still larger class:
Theorem 1.18 (Fomin). If (C) is a Fourier series of f L1 and for each sequence m ∈ of natural numbers m such that n 0 as n there exists p, 1 < p 2, { n} n → → ∞ ≤ independent of m such that { n}
n+mn p 1 p m − ∆a = o(1) , n (1.4) n | k| → ∞ kX=n then (1.1) holds.
13 The same statement holds for the sine series (S), i.e. the Fourier series of odd functions. Remark 1.4. It is trivial to see that Theorem 1.17 is a corollary of Th. 1.18, that is that (1.3) implies (1.4) for each sequence of natural numbers mn such m { } that n 0, n . n → → ∞ The class Cp has an interesting subclass Cp∗. A null sequence a belongs to the class C∗ if for some 1 < p 2, { k} p ≤
∞ p 1 p k − ∆a < . | k| ∞ kX=1 The next theorem is a corollary to Theorem 1.17. Theorem 1.19 (Fomin [11], Stanojevi´c[36]). Let (C) be a Fourier series of some 1 f L (0, π) and let a C∗ BV , 1 < p 2. Then (1.1) hold. ∈ { n} ∈ p ∩ ≤ A natural extension of BV is the following class: a null-sequence a belongs to { k} the class P if 1 n k ∆a = o(1) , n . n | k| → ∞ kX=1 Combining the class P with the condition n ∆an = O(1), Stanojevi´cobtained a theorem for L1-convergence of Fourier-Stiltjes series. Theorem 1.20 [36] (Stanojevi´c). Let (C) be a Fourier-Stiltjes seris with a P { k} ∈ and let n ∆a = O(1) holds. Then it converges in L1 iff a log n = o(1), n . n n → ∞ On the other hand Bojani´cand Stanojevi´c[4] defined a subclass of P as follows: a null sequence a belongs to the class V if for some p > 1, { k} p 1 n kp ∆a p = o(1), n . n | k| → ∞ kX=1 They proved the following therems. Theorem 1.21 [4] (Bojani´c-Stanojevi´c). If (C) is a Fourier series of f L1 and ∈ a V for some 1 < p 2 then (1.1) holds. { k} ∈ p ≤ Theorem 1.22 [4] (Bojani´c,Stanojevi´c). If a V BV for some 1 < p 2, { k} ∈ p ∩ ≤ then (C) is a Fourier series iff a C. { k} ∈ But N. Tanovi´c-Miller[39] considered the problem of integrability of the series (C) in regard to the classes Cp, p > 1 and C1 = BV .
14 Theorem 1.23 [39] (N. Tanovi´c-Miller). (i) If a C : p 1 then (C) converges a.e. to the function { k} ∈ ∪{ p ≥ }
∞ f(x) = ∆akDk(x) kX=0 and (C) is a Fourier series iff for some δ > 0,
δ ∞ ∆akDk(x) dx < ∫0 ∞ k=0 X in which case (C) is the Fourier series of f. (ii) If a C : p > 1 then (C) is a Fourier series iff a C. { k} ∈ ∪{ p } { k} ∈ These results extend the Theorem 1.22 and show that the classical question on integrability of the series (C) need not be restricted to series with coefficients of bounded variation. J. W. Garrett and C.ˇ V. Stanojevi´chave obtained a theorem for L1-convergence of Fourier series with monotone coefficients. Theorem 1.24 [16](J. W. Garrett, C.ˇ V. Stanojevi´c) Let
a ∞ 0 + (a cos nx + b sin nx) (CS) 2 n n n=1 X be the Fourier series with monotone coefficients. Then (1.1) hold, where Sn is the partial sums of this series. On the other hand, S. A. Telyakovskii and G. A. Fomin [45] obtained a similar result for Fourier series with quasi-monotone coefficients. A null sequence of positive numbers is called quasi-monotone if for some α 0 a α ≥ n 0, n 0 or equivalently a a 1 + . nα ↓ → n+1 ≤ n n Theorem 1.25 [45] (Fomin, Telyakovskii). Let a be a quasi-monotone se- { n} quence. If (C) is the Fourier series of its sum f, then (1.1) hold. The proof of sufficiency in the theorem of Fomin-Telyakovskii is simplified by Garrett-Rees-Stanojevi´c[14] using more refined estimations of S σ . k n − nk Telyakovskii and Fomin [45] also proved a corresponding result for the sine series, namely if a is a quasi-monotone sequence and (S) is the Fourier series of its { k} sum g then the same conclusion holds for the sine series.
15 Theorem 1.26 [14] (Garrett-Rees-Stanojevi´c). Let (CS) be the Fourier series with quasi-monotone coefficients. Then S σ = o(1), n iff k n − nk → ∞ (a + b ) log n = o(1), n . n n → ∞ The class P extends not only BV , but the class of quasi-monotone sequences. The next theorem is a slightly weaker form of a theorem of Telyakovskii and Fomin.
Theorem 1.27 [36] (Stanojevi´c). Let (C) be a Fourier series with quasi-monotone coefficients and let n ∆an = O(1) holds. Then (1.1) hold. Later, Bray and Stanojevi´c[8] considered the question of L1-convergence for more general Fourier series of so called asymptotically even functions. Concerning the Fourier series of even functions one of the results in [8] can be stated as follows:
Theorem 1.28 (Bray, Stanojevi´c). If (C) is a Fourier series of f L1 and for ∈ some 1 < p 2 ≤ [λn] p 1 p lim lim k − ∆ak = 0 , λ 1+ n →∞ | | → nX=k then (1.1) holds.
Remark 1.5. Theorem 1.28 is corollary of Theorem 1.18.
1.2. GENERALIZATIONS OF THE SIDON-FOMIN’S LEMMA
Sidon [35] proved the inequality named after him in 1939 year. It is an upper estimate for the integral norm of a linear combination of trigonometric Dirichlet kernels expressed in terms of the coefficients. Since the estimate has many ap- plications for instance in L1-convergence problems and summation method with respects to trigonometric series, newer and newer improvements of the original in- equality has been proved by several authors. Fomin [9] applying the linear method for summing of Fourier series has given another proof if this inequality. Thus the inequality is known as Sidon-Fomin’s inequality. Also, S. A. Telyakovskii in [44] has given an elegant proof of Sidon-Fomin’s in- equality.
Lemma 1.4 (Sidon-Fomin). Let α n be a sequence of real numbers such that { k}k=0 16 α 1 for all k. Then there exists a positive constant C such that for any n 0, | k| ≤ ≥ n
αkDk(x) C(n + 1) . ≤ k=0 X For the proof of our new result we need the following lemma.
Lemma 1.5 [63]. If Tn(x) is trigonometric polynomial of order n, then
T (r) nr T . k n k ≤ k nk This is S. Bernstein’s inequality in the L1(0, π)-metric (see [63], vol. 2, p. 11). Lemma 1.6 [51]. Let α n be a sequence of real numbers such that α 1 { k}k=0 | k| ≤ for all k. Then there exists a constant C > 0, such that for any n 0, ≥ n (r) r+1 αkDk (x) C(n + 1) , ≤ k=0 X (r) where Dk (x), k = 0, 1, 2, ... , n is the r-th derivate of the Dirichlet’s kernel. Proof. Since
n n 1 n n α D (x) = α + α cos kx , k k 2 i i i=0 ! kX=0 X kX=1 Xi=k we have that n
αkDk(x) kX=0 is a cosine trigonometric polynomial of order n. Applyyng first Bernstein’s inequality, then Sidon-Fomin’s lemma, yields:
n n (r) r r+1 αkDk (x) (n + 1) αkDk(x) C(n + 1) , C > 0 . ≤ ≤ k=0 k=0 X X n Lemma 1.7 [10] (Fomin-Steˇckin). Let 1 < p 2 and αk k=0 be a sequence of n ≤ { } real numbers such that αp Ap(n + 1). Then there exists a positive constant k ≤ kX=0 Cp depends only on p such that the following inequality holds:
n
αiDi(x) CpA(n + 1) . ≤ i=0 X 17 Lemma 1.8 [4] (Bojani´c-Stanojevi´c). Let α n be a sequence of real numbers. { k}k=0 Then for any 1 < p 2 and n 0 the following inequality holds: ≤ ≥
n n 1/p 1 p αkDk(x) Cp(n + 1) αk , (1.5) ≤ n + 1 | | ! k=0 k=0 X X where the constant Cp depends only on p. Remark 1.6. We note that this estimate is essentially contained (case p = 2) in Fomin [9]. Remark 1.7. It’s casy to see that Bojani´c-Stanojevi´ctype inequality is not valid for p = 1. log n Indeed, if α = 1 and α = 0 (k = n, k N) then the left side is of order n k 6 ∈ n 1 while the right side is of order as n . n → ∞ Remark 1.8. Sidon-Fomin’s inequality is a special case of the Bojani´c-Stanojevi´c inequality, i.e. it can easily be deduced from Lemma 1.8. (r) Now, we will prove a counterpart of inequality (1.5) in the case where Dk is used instead of Dk(x). Lemma 1.9 [57]. Let α n be a sequence of real numbers. Then for any { k}k=0 1 < p 2, and r = 0, 1, 2, ..., n 0 the following inequality holds: ≤ ≥
n n 1/p (r) r+1 1 p αkDk (x) Cp(n + 1) αk ≤ n + 1 | | ! k=0 k=0 X X where the constant Cp depends only on p. Proof. Applying first Bernstein’s inequality, then Bojani´c-Stanojevi´cinequality, yields
n n n 1/p (r) r r+1 1 p αkDk (x) (n + 1) αkDk(x) Cp(n + 1) αk . ≤ ≤ n + 1 | | ! k=0 k=0 k=0 X X X
18 1.3. THE EXTENSIONS ON SOME CLASSES OF FOURIER COEFFICIENTS
In this parth we shall give the extensions of the Garrett-Stanojevi´cclass C, Sidon-
Telyakovskii class S and the class Sp, p > 1 defined by V. B. Stanojevi´c-Cˇ V. Stanojevi´crespectively. A null sequence a belongs to the class C , r = 0, 1, 2, . . . if for every ε > 0 there { k} r is a δ > 0 such that
π ∞ (r) ∆akDk (x) dx < ε , for all n , ∫0 k=n X (r) where Dk (x) is the r-th derivate of the Dirichlet’s kernel.
When r = 0, we denote Cr = C. A null sequence a belongs to the class r = 0, 1, 2, . . . if there exists a mono- { k} =r ∞ tonically decreasing sequence A such that krA < and ∆a A , for { k} k ∞ | k| ≤ k k=1 all k. When r = 0 it is clear that = S. X =r A null sequence a belongs to the class S , 1 < p 2, r = 0, 1, 2, . . . if there { k} pr ≤ ∞ exists a monotonically decreasing sequence A such that krA < and { k} k ∞ kX=1
n p 1 ∆ak | p | = O(1) . n Ak kX=1
When r = 0, we denote Sp = Spr. The following lemma was proved by Valee Poussin Ch.J.de la, (see [60]), but we shall present a direct proof of this lemma.
∞ Lemma 1.10. A 0 with nrA < , r 0 then nr+1A = o(1), n . n ↓ n ∞ ≥ n → ∞ n=1 X 19 Proof 1. Let 0 < m < n. Adding the inequalities
r+1 n ∆An 1 0 − ≥ r+1 (n 1) ∆An 2 0 − − ≥ r+1 (n 2) ∆An 3 0 − − ≥
············ (m + 1)r+1∆A 0 , m ≥ we obtain:
n 1 − A nr+1 + A [(k + 1)r+1 kr+1] + A (m + 1)r+1 0 . − n k − m ≥ k=Xm+1
∞ The summ on the left is o(1) becouse nrA < . n ∞ n=1 Hence, X A (m + 1)r+1 A nr+1 o(1) , m, n . m − n ≥ → ∞ Since mrA 0, this means that m → A mr+1 A nr+1 o(1) , m, n . (1.6) m − n ≥ → ∞
r+1 ∞ r We cannot have lim inf n An > 0, since otherwise n An could not converge. n →∞ n=1 Hence, in particular there is for each ε > 0 an infiniteX sequence of indicies m for which r+1 m Am < ε . (1.7)
r+1 Now suppose that lim sup n An > 0. Then there exists ε > 0, and there is an n infinite sequence of indicies→∞ n such that
r+1 n An > 2ε > 0 (1.8)
For each m satisfyng (1.7) take a larger n satisfyng (1.8), we get a contradiction r+1 r+1 of (1.6). Hence lim sup n An = 0, i.e. n An = o(1), n . n →∞ → ∞ Proof 2. By inequalities
∞ nr+1A nr(A + A + + A ) irA , 2n ≤ n+1 n+2 ··· 2n ≤ i i=n+1 X 20 we obtain: ∞ (2n)r+1A 2r+1 irA = o(1), n . 2n ≤ i → ∞ i=n+1 X Similarly, we can get:
1 r+1 ∞ (2n + 1)r+1A 2 + irA = o(1) , n . 2n+1 ≤ n i → ∞ i=n+1 X Finally nr+1A = o(1) , n . n → ∞
∞ ∞ Lemma 1.11. If A 0 with nrA < , r 0, then nr+1(∆A ) < . n ↓ n ∞ ≥ n ∞ n=1 n=1 X X Proof. By partial summation, n 1 n n − r+1 r+1 r+1 r+1 r r+1 k (∆Ak) = [k (k 1) ]Ak n An = O k Ak n An . − − − ! − kX=1 kX=1 kX=1 The series on the right converges; nr+1A = o(1), n , by Lemma 1.10; so the n → ∞ partial sums on the left are converges as n . → ∞ It is trivially to see that for all r = 1, 2, 3, .... Now, let a ∞ . =r+1 ⊂ =r { n}n=1 ∈ =1 For arbitrary real number a , we shall prove that sequence a ∞ belongs to 0 { n}n=0 S. We define A = max( ∆a ,A ). Then ∆a A , i.e. ∆a A , for all 0 | 0| 1 | 0| ≤ 0 | n| ≤ n n 0, 1, 2, ... and A ∞ is monotonically decreasing sequence. ∈ { } { n}n=0 On the other hand, ∞ ∞ A A + nA . n ≤ 0 n n=0 n=1 X X If A = ∆a , then 0 | 0|
∞ ∞ ∞ A = ∆a + A a + a + nA n | 0| n ≤ | 0| | 1| n ≤ n=0 n=1 n=1 X X X ∞ ∞ ∞ a + ∆a + nA a + 2 nA < . ≤ | 0| | n| n ≤ | 0| n ∞ n=1 n=1 n=1 X X X If A0 = A1, then
∞ ∞ ∞ A = A + A 2 nA < . n 1 n ≤ n ∞ n=0 n=1 n=1 X X X 21 Thus, a ∞ S, i.e. , for all r = 0, 1, 2, . . .. The next example { n}n=0 ∈ =r+1 ⊂ =r verifies that the implication
a a , r = 0, 1, 2, ... { n} ∈ =r+1 ⇒ { n} ∈ =r is not reversible. ∞ 1 Example [54]. For n = 1, 2, 3, . . . define a = . Then a 0 as n n k2 n → → ∞ k=n+1 1 X and for n = 1, 2, 3, . . ., ∆a = . Firstly we shall show that a / . n (n + 1)2 { n} ∈ =1 Let A ∞ is an arbitrary positive sequence such that A 0 and ∆a A . { n}n=1 ↓ n ≤ n ∞ ∞ n However, nA is divergent, i.e. a / . n ≥ (n + 1)2 { n} ∈ =1 n=1 n=1 X X 1 Now, for all n = 0, 1, 2, . . . let A = . Then A 0, ∆a A and n (n + 1)2 n ↓ | n| ≤ n ∞ ∞ 1 A = < , i.e. a S. n n2 ∞ { n} ∈ n=0 n=1 X X Our next example will show that there exists a sequence a ∞ such that { n}n=1 a ∞ but a ∞ / , for all r = 1, 2, 3, . . .. { n}n=1 ∈ =r { n}n=1 ∈ =r+1 ∞ 1 Namely, for all n = 1, 2, 3, . . . let a = . Then a 0 as n and for n kr+2 n → → ∞ k=n 1 X n = 1, 2, 3, ... , ∆a = . Let A ∞ is an arbitrary positive sequence such n nr+2 { n}n=1 that A 0 and ∆a A . However, n ↓ n ≤ n ∞ ∞ 1 ∞ 1 nr+1A nr+1 = n ≥ nr+2 n n=1 n=1 n=1 X X X is divergent, i.e. a / . On the other hand, for all n = 1, 2, . . . let A = { n} ∈ =r+1 n 1 ∞ ∞ 1 . Then A 0, ∆a A and A = < , i.e. a . nr+2 n ↓ | n| ≤ n n nr+2 ∞ { n} ∈ =r n=1 n=1 X X Theorem 1.29 [51] For all r = 0, 1, 2, ... the following embedding relation holds
BV C . =r ⊂ ∩ r
Proof. It is clear that a implies a BV . Now for x = 0 we consider { n} ∈ =r { n} ∈ 6 the identity:
k n 1 ∞ ∞ ∆a − ∆a ∆a D (x) = (∆A ) j D (x) A j D (x) . k k k A j − n A j j=0 j j=0 j kX=n kX=n X X 22 ∞ (r) In the proof of Theorem 3.8, we shall prove that series ∆akDk (x) is uniformly k=1 convergent on any compact subset of (0, π). This implyX that
δ ∞ (r) ∆akDk (x) dx ∫0 ≤ k=n X k n 1 π π ∞ ∆aj (r) − ∆aj (r) (∆Ak) Dj (x) dx + An Dj (x) dx . ≤ ∫0 Aj ∫0 Aj k=n j=0 j=0 X X X ∆a Since j 1, applying the Lemma 1.6, and Lemma 1.10, we can get: A ≤ j N 1 δ ∞ (r) − r+1 r+1 ∆akDk (x) dx O(1) lim (∆Ak)(k + 1) + Ann = ∫0 ≤ "N # k=n →∞ k=n X X N r+1 r+1 r+1 r+1 = O(1) lim [(k + 1) k ]Ak (N + 1) AN + O(n An) = N − − →∞ " # kX=n
∞ r = O k Ak + o(1) = o(1) , n . ! → ∞ kX=n Next we define a new class 2, r = 0, 1, 2, . . . as follows: a null sequence a =r { k} belongs to the class 2, if there exists a null monotonically decreasing sequence =r ∞ A of nonnegative numbers such that kr+1(∆A ) < and ∆a A for { k} k ∞ | k| ≤ k k=1 all k. X
Theorem 1.30. The class is equivalent to 2, for all r = 0, 1, 2, ... =r =r ∞ Proof. Let a . Applying the Lemma 1.11, we get nr+1(∆A ) < . { n} ∈ =r n ∞ n=1 Now, if a 2, we have: X { n} ∈ =r
∞ nr+1A = nr+1 ∆A n k ≤ kX=n ∞ kr+1(∆A ) = o(1), n , i.e. nr+1A = o(1), n . ≤ k → ∞ n → ∞ kX=n 23 Then n n 1 k n n 1 r − r r − r+1 r+1 k Ak = (∆Ak) j + An j = O k (∆Ak) + O(n An) . j=1 j=1 ! kX=1 kX=1 X X kX=1 r Letting n , we obtain ∞ k A < , i.e. a . → ∞ k=1 k ∞ { n} ∈ =r Lemma 1.12 [50]. Let α Pk be a sequence of real numbers. Then the following { j}j=1 relation holds for 1 < p 2, v = 0, 1, 2, ... , r and r = 0, 1, 2, ... ≤ 1 v 1 vπ k π j + sin j + x + V = α 2 2 2 dx = k j h r+1v i π/k∫ x − j=1 sin X 2 1/p k 1 = O kr+1 α p , p k | j| j=1 X where Op depends only on p. Proof. Applying first Holder inequality, yields:
k π 1 1 v 1 vπ V = α j + sin j + x + dx k ∫ x r+1 v j 2 2 2 ≤ π/k sin − j=1 2 X h i 1/p π dx ≤ ∫ x (r+1 v)p × π/k sin − 2 q 1/q k π 1 v 1 vπ αj j + sin j + x + dx . × ∫0 2 2 2 j=1 X h i Since π (r+1 v)p 1 dx πk − − π (r+1 v)p 1 k − − , ∫ x (r+1 v)p ≤ (r + 1 v)p 1 ≤ p 1 π/k sin − − − − 2 we have: 1/p π (r+1 v)p 1 1/p V (k − − ) k ≤ p 1 × − q 1/q k π 1 v 1 vπ αj j + sin j + x + dx . × ∫0 2 2 2 j=1 X h i 24 Then using the Hausdorff-Young inequality we get:
q 1/q 1/p k k π v 1 1 vπ p vp αj j + sin j + x + dx = Op αj j . ∫0 2 2 2 | | j=1 j=1 X h i X Finally, 1/p k (r+1 v)p 1 1/p p vp V = O (k − − ) α j = k p | j| j=1 X 1/p k (r+1)p 1 1/p p = O (k − ) α = p | j| j=1 X 1/p k 1 = O kr+1 α p , p k | j| j=1 X where Op depends only on p. Lemma 1.13 [30]. Let r be a nonnegative integer and x (0, π]. Then ∈ 1 k 1 kπ r 1 − n + sin n + x + D(r)(x) = 2 2 2 ϕ (x)+ n hx r+1 k i k k=0 sin − X 2 1 r 1 rπ n + sin n + x + + 2 2 2 , h x i 2 sin 2 where the same ϕk denotes various analytical function of x, independent of n. Lemma 1.14. Let the coefficients a k belong to the class S , 1 < p 2, { j}j=0 pr ≤ r = 0, 1, 2, ... . Then the following inequality holds
k π ∆aj (r) r+1 Dj (x) dx = Op(k ) . ∫0 A j=1 j X Proof. We have:
k π π/k π ∆aj (r) Dj (x) dx = + = Ik + Jk . ∫0 A ∫0 ∫ j=1 j π/k X 25 (r) r+1 Applying the inequality Dn (x) = O(n ), we have: k ∆a k ∆a I α jr | j| αkr | j| k ≤ A ≤ A ≤ j=1 j j=1 j X X 1/p 1 k ∆a p αkr+1 | j| = O(kr+1) ≤ k Ap j=1 j X where α is a positive constant. Applying the Lemma 1.13, let us estimate the second integral:
k π ∆a J = j D(r)(x) dx k ∫ A j ≤ π/k j=1 j X 1 v 1 vπ k r 1 π ∆a − j + sin j + x + j 2 2 2 ϕ (x) dx+ h r+1v i v ≤ π/k∫ Aj x − j=1 v=0 sin X X 2 1 r 1 rπ k π ∆a j + sin j + x + + j 2 2 2 dx = λ + µ . h x i k k π/k∫ Aj 2 sin j=1 X 2 Since ϕv are bounded, we have: 1 v 1 vπ k π ∆a j + sin j + x + j 2 2 2 ϕ dx h r+1v i v π/k∫ Aj x − ≤ j=1 sin X 2 1 v 1 vπ k π j + sin j + x + B α 2 2 2 dx , j h r+1v i ≤ π/k∫ x − j=1 sin X 2 ∆aj where B is a positive constant and αj = , j = 1, 2, ... , k. Aj Applying Lemma 1.12 to the last integral, we get: 1 v 1 vπ k π ∆a j + sin j + x + j 2 2 2 ϕ (x) dx = h r+1v i v π/k∫ Aj x − j=1 sin X 2 1/p k 1 ∆a p = O kr+1 | j| = O (kr+1) . p k Ap p j=1 j X 26 r+1 Since r is a finite value, we have: λk = Op(k ). Similarly, we can get: µk = r+1 Op(k ). Hence k π ∆aj (r) r+1 r+1 r+1 Dj (x) dx = O(k ) + Op(k ) = Op(k ) . ∫0 A j=1 j X Lemma 1.15. Let the coefficients a k belong to the class S , 1 < p 2, { j}j=0 pr ≤ r = 0, 1, 2, ... Then
k π ∆aj (r) An Dj (x) dx = o(1) , n . ∫0 A → ∞ j=1 j X Proof. Applying first the Lemma 1,14, then Lemma 1.10, we obtain
k π ∆aj (r) r+1 An Dj (x) dx = Op(n An) = o(1) , n . ∫0 A → ∞ j=1 j X Theorem 1.31[50]. For each 1< p 2 and r = 0, 1, 2, ... the following embed- ≤ ding relationholds S BV C . pr ⊂ ∩ r Proof. We have:
n n n 1 k n − ∆a ∆a ∆a kr ∆a = (∆A ) | j| jr + A | j| jr | k| ≤ | k| k A n A ≤ j=1 j j=1 j kX=1 kX=1 kX=1 X X n 1 k n − ∆a ∆a kr(∆A ) | j| + nrA | j| ≤ k A n A ≤ j=1 j j=1 j kX=1 X X 1/p 1/p n 1 k n − 1 ∆a p 1 ∆a p kr+1(∆A ) | j| + nr+1A | j| = ≤ k k Ap n n Ap j=1 j j=1 j kX=1 X X n 1 n − r+1 r+1 r = O(1) k (∆Ak) + n An = O k Ak . " # ! kX=1 kX=1 Letting n , we get a BV . → ∞ { n} ∈ 27 Then applying Abel’s transformation, Lemma 1.15, Lemma 1.14 and Lemma 1.11 we have:
k δ π ∞ (r) ∞ ∆aj (r) ∆akDk (x) dx (∆Ak) Dj (x) dx + o(1) ∫0 ≤ ∫0 Aj k=n k=n j=1 X X X ∞ r+1 = Op(1) k (∆Ak) = o(1) , n . " # → ∞ kX=n
28 II. CLASSIFICATION ON QUASI-MONOTONE SEQUENCES AND ITS APPLICATIONS FOR L1-CONVERGENCE OF TRIGONOMETRIC SERIES
2.1. REMARKS ON TRIGONOMETRIC SERIES WITH QUASI-MONOTONE COEFFICIENTS
Quasi-monotone sequences are known to share many of the numbers properties of decreasing sequences: for example Vallee Poussin’s [60] theorem: ∞ a < na 0 (see also in [38]), the Cauchy condensation test for n ∞ ⇒ n → n=1 convergenceX and a number of theorems about the trigonometric series. Some proofs of the convergence theorems about the trigonometric series are based on the use of modified cosine sums defined by Rees-Stanojevi´c[29] as follows:
1 n n n gn(x) = ∆ak + ∆ai cos kx = Sn(x) an+1Dn(x) . 2 " ! # − kX=0 kX=1 Xi=k Marzug [25] proved the following theorem for L1-convergence of trigonometric series with quasi-monotone coefficients. Theorem 2.1 [25] (Marzug). Let a be a nonnegative quasi-monotone sequence { k} ∞ ak ∞ tending to zero, < and (k +1)[ ∆ak ∆ak] < . Then lim gn(x) = n k ∞ | |− ∞ →∞ kX=1 kX=1 ∞ g(x) L1[ π, π] iff a < . ∈ − n ∞ n=1 N. Singh and K. M.X Sharma [33] defined a class of L1-convergence as follows.
Namely, a sequence a belongs to the class S0 if a 0 as k and { k} k → → ∞ ∞ there exists a sequence A such that A is quasi-monotone, A < , { k} { k} k ∞ k=1 and ∆a A , for all k. They proved the following theorem. X | k| ≤ k 1 Theorem 2.2 [33]. Let a S0, then g (x) convergens to g(x) in L -metric. { k} ∈ n 29 Let the null quasi-monotone sequence be denoted by A 0. n ⇓ For convience the following notations are used:
∞ M = A : A 0 and nαA < , for some α 0 α { n n ↓ n ∞ ≥ } n=1 X
∞ α M 0 = A : A 0 and n A < , for some α 0 α { n n ⇓ n ∞ ≥ } n=1 X S = a : a 0 , n , for some α 0, r 0, 1, 2, ...[α] , pαr { n n → → ∞ ≥ ∈ { } n p 1 ∆ak p(α r)+1 | p | = O(1) , p > 1 and An Mα n − Ak ∈ } kX=1 S0 = a : a 0 , n , for some α 0, r 0, 1, 2, ...[α] , pαr { n n → → ∞ ≥ ∈ { } n p 1 ∆ak p(α r)+1 | p | = O(1) , p > 1 and An Mα0 . n − Ak ∈ } kX=1
We note that the classes Mα0 and Spαr0 were defined by Sheng [30].
Theorem 2.3 [68], [70]. The classes Mα and Mα0 are identical.
Proof. It is obvious that M M 0 . For the proof of the inclusion M 0 M , we α ⊂ α α ⊂ α use an idea obtained by Telyakovskii [47], i.e. we construct the sequence:
∞ Am B = A + β , for some β 0 , where A M 0 . (2.1) k k m ≥ n ∈ α mX=k We have:
A B B = ∆B = ∆A + β k 0 , i.e.B 0 as k and k − k+1 k k k ≥ k ↓ → ∞
∞ ∞ α ∞ α ∞ α Am ∞ α ∞ ∞ α 1 k B = k A + βk k A + β m − A k k m ≤ k m kX=1 kX=1 kX=1 mX=k kX=1 kX=1 mX=k m ∞ α ∞ α 1 ∞ α ∞ α = k A + β m − A = k A + β m A < . k m k m ∞ m=1 n=1 m=1 kX=1 X X kX=1 X Thus M M 0 . α ≡ α
Theorem 2.4 [68], [70]. The classes Spαr and Spαr0 are identical.
30 Proof. It obvious that S S0 . Let a S0 . It suffices to show that pαr ⊂ pαr { n} ∈ pαr the sequence (2.1) satisfyes the condition
n p 1 ∆ak p(α r)+1 | p | = O(1) . n − Bk kX=1 Clearly,
n p n p 1 ∆ak 1 ∆ak p(α r)+1 | p | p(α r)+1 | p | = O(1) , i.e Spαr Spαr0 . n − Bk ≤ n − Ak ≡ kX=1 kX=1 Fomin [13], applying the following two theorems have given a new proof of the Theorem 1.25. Theorem 2.5 [13]. If (C) converges in mean, then for any sequence of natural numbers m such that m n, n = 1, 2, 3, ... the following limit holds: { n} n ≤ m n a lim n+k = 0 . n →∞ k kX=1
Theorem 2.6 [13]. Let mn be a sequence of natural numbers such that m { } lim n = 0. If the following limit holds: n →∞ n
m 1 n− lim ∆an+k log(k + 1) + an+m log(mn + 1) = 0 , n n →∞ " | | | | # kX=1 then the series (C) converges in mean.
2.2. TRIGONOMETRIC SERIES WITH δ-QUASI-MONOTONE COEFFICIENTS
As an extension of the quasi-monotonic sequence, R. P. Boas [3] defined δ-quasi- monotonic sequence as follows. A sequence a is called δ-quasi-monotonic if a 0, a > 0 ultimately and { n} n → n ∆an δn, where δn is sequence of positive numbers. A quasi-monotonic se- ≥ − a quence with a 0, is one that is δ-quasi-monotonic with δ = α n . n → n n Boas [3] proved the following lemmas about the δ-quasi-monotonic sequences.
31 ∞ Lemma 2.1. If a is δ-quasi-monotonic with nδ < then the convergence { n} n ∞ n=1 X ∞ of a implies that na = o(1), n . n n → ∞ n=1 X Remark 2.1. This lemma includes the corresponding result for classical quasi- monotone; indeed if a is quasi-monotonic we have { n} ∞ ∞ a ∞ nδ = nα n = α a < , which is assumed convergent in the hypoth- n n n ∞ n=1 n=1 n=1 esis.X X X
∞ ∞ Lemma 2.2. Let a be a δ-quasi-monotonic with nδ < . If a < , { n} n ∞ n ∞ n=1 n=1 X X ∞ then (n + 1) ∆a < . | n| ∞ n=1 X Ahmad and Zahid in [1] proved the following theorem.
Theorem 2.7 [1]. Let (CS) be a Fourier series with δ-quasi-monotone coefficients ∞ with nδ < . Then S σ = o(1), n iff (a + b ) log n = o(1), n ∞ k n − nk → ∞ n n n=1 n X. → ∞ Applying the Theorem 2.5, and 2.6, for the series (C) we shall present a new proof of this theorem, rewritten as follows:
Theorem 2.8. Let (C) be a Fourier series with δ-quasi-monotone coefficients with ∞ nδ < . Then the series (C) converges in mean iff a log n = o(1), n . n ∞ n → ∞ n=1 X Proof. Let S f = o(1), n . We have: k n − k → ∞
a2n 1 a2n δ2n 1 − − n ≥ n − n
a2n 2 a2n 1 δ2n 2 a2n δ2n 1 δ2n 2 − − − − − n 1 ≥ n 1 − n 1 ≥ n 1 − n 1 − n 1 − − − − − − a2n 3 a2n 2 δ2n 3 a2n δ2n 1 δ2n 2 δ2n 3 − − − − − − n 2 ≥ n 2 − n 2 ≥ n 2 − n 2 − n 2 − n 2 − − − − − − − ·················· an an+1 δn a2n δ2n 1 δ2n 2 δn − − . 1 ≥ 1 − 1 ≥ 1 − 1 − 1 − · · · − 1 32 Adding these inequalities, we obtain: n an+k 1 1 1 1 1 − 1 + + + a2n 1 + + + δ2n 1 k ≥ 2 ··· n − 2 ··· n − − kX=1 1 1 1 δn 1 + + + δ2n 2 1 + δn+1 > − 2 ··· n 1 − − · · · − 2 − 1 − 2n 1 1 1 1 1 − > 1 + + + a 1 + + + δ 2 ··· n 2n − 2 ··· n k kX=n 1 1 1 By inequalities, log n < 1 + + + + n, n N, we obtain 2 3 ··· n ≤ ∈ n 2n 1 a − a + n+k > a log n kδ , i.e. n k 2n − k kX=2 kX=n n a ∞ a log n < a + n+k + kδ 2n n k k kX=1 kX=n Letting n and applying Theorem 2.5 we get a log n = o(1), n . → ∞ n → ∞ ”Only if”: Let a log n = o(1), n . Applying Theorem 2.6, it suffices to show n → ∞ that m 1 n− A = ∆a log(k + 1) = o(1) , n . n | k+n| → ∞ kX=1 Indeed,
m 1 n− A (log m ) ∆a n ≤ n | k+n| ≤ kX=1 m 1 m 1 n− n− (log mn) ∆ak+n + 2 δk+n = ≤ ! kX=1 kX=1 m 1 n− = (log mn)(an+1 an+m 1) + 2(log mn) δk+n = − n− kX=1 ∞ = O(a log n) + O iδ = o(1) , n . n+1 i → ∞ i=n+1 ! X This generalizies a theorem of J. W. Garrett, C. S. Rees and C.ˇ V. Stanojevi´c of a [14]by replacing the condition of quasi-monotonicity with the condition of δ-quasi-monotonicity.
33 On the other hand, S. M. Mazhar [26] defined a class S(δ). A null sequence a belongs to the class S(δ) if there exists a sequence A { n} { n} ∞ ∞ such that A is δ-quasi-monotone, nδ < , A < and ∆a A , { n} n ∞ n ∞ | n| ≤ n n=1 n=1 for all n. X X Later, Husein Bor [6] showed that condition a S(δ) is a sufficient for the { n} ∈ integrability of the limit g(x) = lim gn(x). n →∞ Theorem 2.9 [6] (H. Bor). Let a be a sequence belongs to the class S(δ). Then { n} 1 ∞ 1 h(x) ∆a sin k + x = x k 2 x kX=1 h(x) converges for x (0, π] and L(0, π]. ∈ x ∈ But in [48] we defined a new class of positive sequences. Namely, we say that a null-sequence a belongs to the class S (δ), p > 1 if there exists a sequence of { k} p numbers A such that { k} ∞ a) A is δ-quasi-monotone and kδ < . { k} k ∞ kX=1 ∞ b) A < , k ∞ k=1 X n 1 ∆a p c) | k| = O(1). n + 1 Ap k=0 k Thus, inX view of the above definitions it is obvious that S(δ) S (δ). ⊂ p Applying the Holder-Hausdorff-Young technique (similarly as in the proof of The- orem 1.14) we can get the following Lemma. Lemma 2.3 [55]. Let the coefficients a k belong to the class S (δ), 1 < p 2. { j}j=0 p ≤ Then the following inequality holds
k π ∆aj Dj(x) dx = Op(k + 1) , ∫0 A j=0 j X where O depends only on p. p Lemma 2.4. [55]. Let the coefficients a n belong to the class S (δ), 1 < p 2. { j}j=0 p ≤ Then n π ∆aj An Dj(x) dx = o(1) , n , ∫0 A → ∞ j=0 j X 34 Proof. Applying first the Lemma 2.3, then Lemma 2.1, yelds
n π ∆aj An Dj(x) dx = Op (n + 1)An = o(1), n . ∫0 A → ∞ j=0 j X Theorem 2.10 [55]. Let a S (δ), 1 < p 2. Then (C) is a Fourier series { k} ∈ p ≤ of some f L1(0, π) and S f = o(1), n if and only if a log n = o(1), ∈ k n − k → ∞ n n . → ∞ Proof. By summation by parts, and by generalized arithmetic-square mean in- equality, we have:
n n ∆ak ∆ak = Ak | | | | Ak ≤ kX=1 kX=1 1/p 1/p n 1 k n − 1 ∆a p 1 ∆a p (k + 1) ∆A | j| + (n + 1)A | j| = ≤ | k| k+1 Ap n n+1 Ap j=0 j j=0 j kX=1 X X n 1 − = O(1) (k + 1) ∆Ak + (n + 1)An . " | | # kX=1
∞ Application of Lemma 2.1 and Lemma 2.2 yields, ∆a < , i.e. S (x) | n| ∞ n n=1 converges to f(x), for x = 0. Using Abel’s transformation,X we obtain: 6
∞ f(x) = ∆akDk(x) , kX=0 by the fact that lim anDn(x) = 0, if x = 0. Then, n →∞ 6 S f = g (x) f(x) + a D (x) , k n − k k n − n+1 n k where gn(x) is the Rees-Stanojevi´csums. Using Abel’s transformation, we have:
n g (x) = S (x) a D (x) = ∆a D (x) . n n − n+1 n k k kX=0 35 ∞ ∞ Since ∆an < , the series ∆akDk(x) converges. Hence lim gn(x) exists | | ∞ n n=1 k=0 →∞ for x =X 0. Then, X 6 ∞ 1 π ∞ f(x) gn(x) = ∆akDk(x) = ∆akDk(x) dx . k − k π ∫0 k=n+1 k=n+1 X X Application of Abel’s transformation, Lemma 2.4, Lemma 2.3 and Lemma 2.2 yields:
π ∞ ∆ak Ak Dk(x) dx ∫0 Ak ≤ k=n+1 X k π ∞ ∆aj ∆Ak Dj(x) dx + o(1) = o(1) , n . ≤ | | ∫0 Aj → ∞ k=n+1 j=0 X X Hence, f(x) gn(x) = o(1), n . k − k → ∞ ”If”: Let S f = o(1), n . Since D (x) = O(log n), by the estimate k n − k → ∞ k n k a D (x) = S g S f + f g = o(1) + o(1) , n , k n+1 n k k n − nk ≤ k n − k k − nk → ∞ we have, a log n = o(1), n . n → ∞ ”Only if”: Let a log n = o(1), n . Then, n → ∞ S f g f + a D (x) = o(1) + a O(log n) = o(1) , n . k n − k ≤ k n − k k n+1 n k n+1 → ∞ Corollary 2.1. Let the sequence a belongs to the class S (δ), 1 < p 2. Then { n} p ≤ 1 ∞ 1 h(x) ∆a sin k + x = x k 2 x kX=1 h(x) converges for x (0, π] and L(0, π], ∈ x ∈ Proof. Since x x ∞ x 2 sin f(x) = a sin + a 2 sin cos kx = 2 0 2 k 2 kX=1 x ∞ 1 1 = a sin + a sin k + x k x = 0 2 k 2 − − 2 kX=1 h i x 3x 5x = (a a ) sin + (a a ) sin + (a a ) sin + 0 − 1 2 1 − 2 2 2 − 3 2 ··· ∞ x = ∆a sin(2k + 1) = h(x) , k 2 kX=1 36 by Teorem 2.10, proof is obvious.
2.3. ON THE EQUIVALENCE OF CLASSES OF FOURIER COEFFICIENTS
In [5] Bor Husein considered the following class S2(δ). A sequence a belongs { k} to the class S2(δ) if a 0 as k , there exists a sequence of numbers A k → → ∞ { k} ∞ ∞ such that it is δ-quasi-monotone, kδ < , k ∆A < and ∆a A , k ∞ | k| ∞ | k| ≤ k k=1 k=1 for all k. Also, he proved the TheoremX 2.2 andX 2.9, considering the class S2(δ) instead of S0 and S(δ). Very recently, Telyakovskii [47] and Leindler [64] proved that these classes together with Sidon-Telyakovskii class S are equivalent. Now, we shall present the proof of S. A. Telyakovskii.
2 Theorem 2.11. The following classes S, S0, S(δ) and S (δ) are equivalent. Proof. First we shall prove that classes S(δ) and S2(δ) are equivalent. ∞ Let a S(δ). It suffices to show that n ∆A < . The last condition is { n} ∈ | n| ∞ n=1 satisfied by Lemma 2.2. X If a S2(δ), then { n} ∈
∞ ∞ nA = n ∆A k ∆A = o(1) , n , i.e. nA = o(1) , n . n k ≤ | k| → ∞ n → ∞ kX=n kX=n But n n 1 n 1 − − A = k ∆A + nA k ∆A + nA , k k n ≤ | k| n kX=1 kX=1 kX=1 ∞ and this implies that A < i.e. a S(δ). n ∞ { n} ∈ n=1 Next we shall prove thatX the classes S and S(δ) are equivalent. It is obvious that S S(δ). If a S(δ), we construct the sequence ⊂ { n} ∈
∞ Bk = Ak + δm . mX=k Now we have: B B = ∆A + δ 0, i.e. B 0 as k . k − k+1 k k ≥ k ↓ → ∞ 37 On the other hand
m ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ B = A + δ = A + δ = A + mδ < , k k m k m k m ∞ m=1 m=1 kX=1 kX=1 kX=1 mX=k kX=1 X kX=1 kX=1 X and ∆a A < B , for all n, i.e. a S. Now we have: | n| ≤ n n { n} ∈
S S0 S(δ) S. ⊂ ⊂ ⊂ Consequently, 2 S S0 S(δ) S (δ) . ≡ ≡ ≡ Applying this result, inequality
n p n p 1 ∆ak 1 ∆ak | p | | p | = O(1) , n Bk ≤ n Ak kX=1 kX=1 and also Theorem 2.4, we can get the following corollary.
Corollary 2.2. For all p > 1, the classes Sp, Sp0 (case α = r = 0) and Sp(δ) are equivalent.
Remark 2.2. If cn an is real and even sequence (cn = c n = an, n = 0, 1, 2, . . .) ≡ − then the Theorem 1.8 of C.ˇ V. Stanojevi´cand V. B. Stanojevi´c,the Sheng’s theorem (see III, 3.3, Theorem 3.17) and Theorem 2.10 are equivalent.
2.4. TRIGONOMETRIC SERIES WITH REGULARLY QUASI-MONOTONE COEFFICIENTS
A positive measurable funkction L(u) is said to be slowly varying in the sence of L(λu) Karamata [19] if for every λ > 0, lim = 1. u →∞ L(u) A basic property of slowly varying functions is the asymptotic relation [19]:
α α u max s− L(s) L(u) , u , for any α > 0 . u s< ∼ → ∞ ≤ ∞
Slowly varying sequences are defined analogously: a positive sequence ln is said l { } to be slowly varying if, for every λ > 0, lim [λn] = 1. n →∞ ln The class of slowly varying sequences is denoted by SV (N).
38 A non-decreasing sequence r of positive numbers is regularly varying, i.e. { n} r (RV )(N) in the sence of J. Karamata [20], if for some α 0 { n} ∈ ≥ r lim [λn] = λα , λ > 1 . n →∞ rn Regularly varying sequences are characterized [20] in form as follows: r (RV )(N) if and only if r = nαl , for some α > 0 and some l { n} ∈ n n { n} ∈ (SV )(N). On the other hand, a sequence an is called a regularly quasi-monotone, { } an or briefly, written as an RQM, if for some rn (RV )(N), 0. It is { } ∈ { } ∈ rn ↓ obvious that the class of quasi-monotone sequences is a subclass of the class RQM. The next theorem is generalization of the Valle Poissin’s theorem (see II. 2.1, p.28).
∞ Theorem 2.12. If a RQM, a < then na = o(1), n . { n} ∈ n ∞ n → ∞ n=1 X Proof. We have.
1 α − l2n 1 a2n 1 1 + a2n − , − ≥ 2n 1 l2n − 1 2 α − l2n 2 α − l2n 2 a2n 2 1 + a2n 1 − 1 + a2n − , − ≥ 2n 2 − l2n 1 ≥ 2n 2 l2n − − − ····················· n α − ln an 1 + a2n . ≥ n l2n Adding these inequalities, we obtain:
2n 1 2n 1 − a − α (2n v) a 2n l 1 + − − . v ≥ l v v v=n 2n v=n X X But α 2n v α 2n v α n 1 + − 1 + − 1 + , v ≤ n ≤ n implies that 2n 1 2n 1 − a − α n a 2n l 1 + − . v ≥ l v n v=n 2n v=n X X Thus 2n 1 − lv n 2n 1 2n 1 v=n α − α − a2n X 1 + av e av . l ≤ n ≤ 2n v=n v=n X X 39 β β The asymptotic relation (2.2) lk k sup n− ln , k gives for large n, ∼ n k → ∞ ≥ 2n 1 2n 1 2n 1 − − β β β − β l v sup m− l sup m− l v v ≈ m ≥ m ≥ ≥ v=n v=n m v m 2n 1 v=n X X ≥ ≥ − X 2n 1 β β − n sup m− l 1 = ≥ m m 2n 1 v=n ≥ − X β n β β = (2n 1) sup m− lm n 2n 1 − m 2n 1 ∼ − ≥ − 1 nl2n 1 , ∼ 2β − for some β > 0. Consequently,
l2n 1 α β ∞ na − e 2 a . 2n l ≤ v 2n v=n X Letting n , we obtain na = o(1), n . → ∞ n → ∞ Sheng Shuyun [31] proved the following results for L1-approximation of trigono- metric series with regulary quasi-monotone coefficients. Theorem 2.13. Let the sequence of coefficients a , b RQM in (CS). { n} { n} ∈ Then there exists two positive constans C1 and C2, such that
2n 1 − a + b C v v S (x) τ (x) , 1 v n ≤ k n − n k v=n+1 X − 2n 1 − l S (x) τ (x) C (a + b ) v+1 + ε + k n − n k ≤ 2 v v (v n)l v v=n+1 v n X − 2n 1 1 − + (a + b ) log(v n + 1) n v v − v=n+1 X o 2n 1 1 − lv+1 where, τn(x) = Sk(x) and εv = 1. n lv − k=n The next theoremX generalizies Theorem 1.25. Theorem 2.14. Let (C) be a Fourier series with a RQM. Then (C) { n} ∈ converges in mean iff a log n = o(1), n . n → ∞ 40 n a Proof. For the necessity we applying the Theorem 2.5, i.e. n+k = o(1), k k=1 n . From the fact a RQM we obtain the inequalities:X → ∞ { n} ∈
n n α an+k an+k (n + k) ln+k = α k (n + k) ln+k k ≥ kX=1 kX=1 n α a2n (n + k) ln+k α ≥ (2n) l2n k ≥ kX=1 n + 1 α a n l 2n n+k . ≥ 2n l2n k kX=1 Applying the asymptotic relation (2.2) for large n, we have:
β β n 2n 2n k sup m− lm ln+k lk m k = ≥ k k n ≈ k n ≥ kX=1 k=Xn+1 − k=Xn+1 − 2n β β k sup m− lm ≥ m 2n k n ≥ ≥ k=Xn+1 − n β β 1 (n + 1) sup m− lm = ≥ m 2n k ≥ kX=1 β n n + 1 β β 1 1 = (2n) sup m− lm β l2n log n . 2n m 2n k ≈ 2 ≥ kX=1 Letting n in inequality → ∞ 2n α n a a log n < 2β n+k , 2n n + 1 k kX=1 the proof of necessity is complete. For sufficiency, we shall applying the Theorem 2.6. From the monotonicity of the a sequence n , we get: rn
m 1 n+m 1 n− n− A = ∆a log(k + 1) = ∆a log(i n + 1) = n | k+n| | i| − i=n+1 kX=1 X 41 n+m 1 n− a a r ∆ i + i+1 (r r ) log(i n + 1) i r r i − i+1 − ≤ i=n+1 i i+1 X n+mn 1 n+mn 1 − ai − ai+1 rn+m 1 log mn ∆ + (ri+1 ri) log(i + 1) = ≤ n− r r − i=n+1 i i=n+1 i+1 X X
an+1 an+mn 1 = rn+mn 1 log mn − + − rn+1 − rn+mn 1 − n+mn 1 − ri + max ai+1 log(i + 1) 1 . n+1 i n+mn 1 − r ≤ ≤ − i=n+1 i+1 X Since
n+m 1 n+m 1 n− r n− r r r 1 i log i+1 = log n+mn n+mn , − r ≤ r r ≤ r i=n+1 i+1 i=n+1 i n+1 n X Y we obtain
rn+mn 1 log mn rn+mn An − an+1 log(n+1) + max ai+1 log(i+1) . ≤ r log n r n+1 i n+mn 1 n n ≤ ≤ − The hypothesis a log n = o(1), n and r (RV )(N) implies that the first n → ∞ { n} ∈ and second term on the right side are o(1), n . → ∞ Finaly, A = o(1), n , i.e. the series (C) converges in mean. n → ∞ Remark 2.3. The proof of necessity of this theorem, we can simplified using the an monotonicity of the sequence rn and fact that . { } rn ↓ We have n o
n n n an+k an+k rn+k a2n 1 rn = rn (a2n log n) . k rn+k k ≥ r2n k ≈ r2n kX=1 kX=1 kX=1 n r2n an+k Taking n in the inequality a2n log n , the proof of the necessity → ∞ ≤ rn k k=1 is obvious. X
42