Embedding theorems of function classes, IV

EMBEDDING THEOREMS OF FUNCTION CLASSES, IV SERGEY TIKHONOV Abstract. We study the interrelation between the strong class Sp(λ) and the Nikol’skii class WrHω β. 1. Introduction Let f(x) be a 2π-periodic continuous function and let a0 2+ ∞ X n=1 (ancos nx +bnsin nx) (1) be its Fourier series. The modulus of smoothness of order β(β > 0) of a function f∈Cis given by ωβ(f, t) = sup |h|≤t° ° ° ° ° ∞ X ν=0 (−1)νµβ ν¶f(x+ (β−ν)h)° ° ° ° °, where ¡β ν¢=β(β−1)···(β−ν+1) ν!for ν≥1, ¡β ν¢= 1 for ν= 0 and kf(·)k= max x∈[0,2π]|f(x)|. Denote by Sn(x) = Sn(f, x) the n-th partial sum of (1). Let En(f) be the best approximation of f(x) by trigonometric polynomials of order nand let f(r)be the derivative of the function fof order r > 0 (f(0) := f) in the sense of Weyl. We will write I1¿I2, if there exists a positive constant Csuch that I1≤C I2. If I1¿I2and I2¿I1hold simultaneously, then we will write I1³I2. 2000 Mathematics Subject Classification. 42A24, 42A25. Key words and phrases. Strong approximation, Fourier series, Embedding theorems. The research was funded by an EU Marie Curie fellowship (Contract MIF1-CT-2004- 509465). The paper was partially written while the author was staying at the Centre de Recerca Matem`atica (CRM) in Barcelona, Spain. He is grateful to the staff of CRM for their hospitality. 1 2 SERGEY TIKHONOV A sequence γ:= {γn}of positive terms will be called almost increasing (almost decreasing), if there exists a constant K:= K(γ)≥1 such that Kγn≥γm(γn≤Kγm) holds for any n≥m. Let Ωβbe the set of nondecreasing continuous functions on [0,2π] such that ω(0) = 0, ω(δ) is nondecreasing and δ−βω(δ) is nonincreasing. Define the following function classes: WrHω β=½f∈C:ωβ(f(r), δ) = O[ω(δ)]¾, Sp(λ) = (f∈C:° ° ° ° ° ∞ X ν=1 λν|f(x)−Sν(x)|p° ° ° ° °<∞), where ω(δ)∈Ωβ,λ={λn}∞ n=1 is a sequence of positive numbers, r∈ [0,∞), and β, p ∈(0,∞). We also define Hω β:= W0Hω β. We say that the sequence λ={λn}∞ n=1 satisfies the 42 2-condition if λn³λkfor n≤k≤2n. (2) We will need the following Definition. The sequence of positive numbers a={an}∞ n=1 is said to be general monotone, or a∈GM, if the relation 2n−1 X ν=n |aν−aν+1| ≤ Can holds for all integer n, where the constant Cis independent of n. It was proved in [11] that a∈GM if and only if asatisfies aν≤Canfor n≤ν≤2n(3) and N X k=n |4ak| ≤ CÃan+ N X k=n+1 ak k!for any n < N. (4) We remark that M QM ∪RBV S ORV QM ∪RBV S GM, where Mis a class of monotone sequences, QM is a class of quasi monotone sequences (see [7], [9]), ORV QM is a class of O-regularly varying quasi monotone sequences (see [8]), and RBV S is a class of sequences of rest bounded variation (see [6]). EMBEDDING THEOREMS OF FUNCTION CLASSES, IV 3 We define the following two subclasses of C: Ccos =(f∈C:f(x) = ∞ X n=1 ancos nx, {an} ∈ GM), Csin =(g∈C:g(x) = ∞ X n=1 ansin nx, {an} ∈ GM). In this paper we study the interrelation between WrHω βand Sp(λ).Our investigation continues the findings from the book [3] and the papers [2], [5], [6] of L. Leindler. 2. Results First we study the embedding Sp(λ)⊂WrHω β. Related results can be found in [2] and [3]. Theorem 2.1. Let β, p > 0, r ≥0,ω∈Ωβand let {λn}satisfy 42 2- condition. Suppose λnωpµ1 n¶n1−rp ≥C. (5) (i). If r > 0and ωsatisfies the conditions (B) ∞ X k=n+1 1 kω³1 k´=O·ω³1 n´¸, (Bβ) n X k=1 kβ−1ω³1 k´=O·nβω³1 n´¸, then Sp(λ)⊂WrHω β.(6) (ii). If r= 0 and ωsatisfies the condition (Bβ), then Sp(λ)⊂Hω β.(7) We note that for certain subclasses of continuous functions the conditions on ωcan be relaxed. Theorem 2.2. Let β, p > 0, r ≥0,ω∈Ωβ∩B. Suppose {λn}satisfy 42 2-condition and condition (5); then Sp(λ)∩Ccos ⊂WrHω βfor r+β= 2l−1,(8) Sp(λ)∩Csin ⊂WrHω βfor r+β= 2l. (9) 4 SERGEY TIKHONOV Remark 2.3. Theorems 2.1 and 2.2 provide, in particular, an answer for the following question (see [3]): When does the condition ° ° ° ° ° ∞ X ν=1 ν(r+α)p−1|f(x)−Sν(x)|p° ° ° ° °<∞(10) imply the condition ωβ(f(r), δ) = O[δα] ? (11) In particular, the answer is: 0< α < β (by Theorem 2.1), and α=βif f∈Ccos and r+α= 2l−1or if f∈Csin and r+α= 2l(by Theorem 2.2). We note that in general, if α=β, the answer is negative. Indeed, for p, α, β = 1, (10) implies only ωβ(f(r), δ) = Oµδlog 1 δ¶(12) and this result is the best possible (see [3]). Remark 2.4. Let β > 0, r ≥0,ω∈Ωβ∩B, and let ω∗(δ) := δrω(δ). We have WrHω β≡Hω∗ r+β⊂Eω∗:= ½f∈C:En(f) = Ohω∗³1 n´i¾. Moreover, Ccos ∩WrHω β≡Ccos ∩Hω∗ α≡Ccos ∩Eω∗,where α≥r+β= 2l−1, and Csin ∩WrHω β≡Csin ∩Hω∗ α≡Csin ∩Eω∗,where α≥r+β= 2l. We remark that for some strong classes one can write the embedding into WrHω βwithout the conditions (B) and (Bβ) on ω. Remark 2.5. Let β, p > 0, r ≥0,ω∈Ωβ. Suppose {λn}satisfy 42 2- condition and condition (5); then ¯ Sp(λ) :=     f∈C: ∞ X n=1° ° °  2n+1 X ν=2n+1 λν|f(x)−Sν(x)|p  1 p° ° °<∞    ⊂WrHω β. (13) Now we study the converse embedding WrHω β⊂Sp(λ).A useful overview and a history of the question can be found in [3], [5], [6]. There the next Theorem 2.6 was proved for r= 0 and Theorem 2.7 was proved for r= 0 and g∈©g∈Csin :{an} ∈ RBV Sª Csin. EMBEDDING THEOREMS OF FUNCTION CLASSES, IV 5 Theorem 2.6. Let β, p > 0, r ≥0,ω∈Ωβand let {λn}satisfy 42 2- condition. Suppose ∞ X n=1 λnωp¡1 n¢ nrp <∞; (14) then WrHω β⊂Sp(λ).(15) We remark that condition (14) implies λnωpµ1 n¶n1−rp ≤C. (16) As in the case of Theorem 2.2, one can assume only this weaker condition if we consider Csin. Theorem 2.7. Let β, p > 0, r ≥0,ω∈Ωβand let {λn}satisfy 42 2- condition and condition (16). Suppose there exists ε∈(0,1) such that {n1−ελn}is almost increasing; (17) then WrHω β∩Csin ⊂Sp(λ).(18) Remark 2.8. In general, condition (16) is not sufficient for embedding (15). Indeed, suppose ω(δ) = δαand α=β= 1; then there exists a function f such that f∈WrHω βbut f /∈Sp(λ∗), where λ∗ n=nrp−1ω−p¡1 n¢ satisfies (17) (see [3]). From Theorems 2.2 and 2.7 and Remark 2.4 we have Corollary 2.9. Let β, p > 0, r ≥0,ω∈Ωβ∩B, and let ω∗(δ) := δrω(δ). Suppose α≥r+β= 2l; then Ssin p(λ∗)≡Csin ∩WrHω β≡Csin ∩Hω∗ α≡Csin ∩Eω∗, where Ssin p(λ∗) := Ssin p³nνrp−1 ωp¡1 ν¢o´= =(f∈Csin :° ° ° ° ° ∞ X ν=1 νrp−1 ωp¡1 ν¢|f(x)−Sν(x)|p° ° ° ° °<∞). 6 SERGEY TIKHONOV 3. Proofs We start with the following lemmas. Lemma 1. ([3, Theorem 8.1]). Let p > 0and let {γn}be a positive sequence such that γ2n≤Cγ2n+i(C≥1, n ∈N,1≤i≤2n).Then Vn(f, p) := ° ° °³1 n n X ν=[n 2]+1 |f(x)−Sν(x)|p´1 p° ° °¿γn(19) implies En(f)¿γn.(20) Lemma 2. ([1]). If f(x)∈C, then ωβ+r³f, δ´¿δrωβ³f(r), δ´for r, β > 0. Lemma 3. ([13]). If f(x)∈Chas a Fourier series ∞ X k=1 aksin kx, ak≥0,(21) then n−β n X k=1 kβak¿ωβ³f, 1 n´,for β6= 2l, l = 1,2,· · · Lemma 4. ([4]). Let an≥0, λn>0. (a): If p≥1, then ∞ X n=1 λnÃ∞ X ν=n aν!p ¿ ∞ X n=1 λ1−p nap nÃn X ν=1 λν!p , (b): If 0< p < 1and aν+j≤Kaνfor 1≤j≤ν, then ∞ X n=1 λnÃ∞ X ν=n aν!p ¿ ∞ X n=1 np−1ap nÃnλn+ n−1 X ν=1 λν!. Proof of Theorem 2.1. Since {λn}satisfies 42 2-condition it is clear that f∈Sp(λ) implies Vn(f, p)¿(nλn)−1 p.By Lemma 1 and condition (5), we have En(f)¿(nλn)−1 p¿ωµ1 n¶n−r. Further we use the conditions (B) and (Bβ) and the following inequalities: in the case of r > 0 ωβµf(r),1 n¶¿1 nβ n X k=1 kr+β−1Ek(f) + ∞ X k=n kr−1Ek(f) (22) EMBEDDING THEOREMS OF FUNCTION CLASSES, IV 7 and in the case of r= 0 ωβµf, 1 n¶¿1 nβ n X k=1 kβ−1Ek(f).(23) Finally, we have ωβ¡f(r),1 n¢¿ω¡1 n¢,which finishes the proof. Proof of Theorem 2.2. As in the proof of Theorem 2.1 we have En(f)¿ ω¡1 n¢n−r. Let f∈Ccos. Then because of ∞ P k=2n ak¿En(f), we get ∞ X k=n+1 ak¿ωµ1 n¶n−r.(24) It was proved in [12] that if ω∈Band r+β= 2l−1, then condition (24) is equivalent to f∈WrHω β. If g∈Csin, then by inequality (23), En(f)¿ω¡1 n¢n−rgives for β < β1 ωβ1+rµf, 1 n¶¿n−r−β1 n X k=1 kβ1−1ωµ1 k¶ ¿ωµ1 n¶n−r+β−β1 n X k=1 kβ1−β−1¿n−rωµ1 n¶. Therefore, by this, Lemma 3, and inequality (3), we write nan¿ωµ1 n¶n−r.(25) From [12], if ω∈Band r+β= 2l, then condition (25) is equivalent to g∈WrHω β.This completes the proof. Proof of Remark 2.4 follows from [12] and Lemma 2. Proof of Remark 2.5. Let f∈¯ Sp(λ). Then Lemma 1 of [2] implies ∞ X n=1 nr−1 ω¡1 n¢En(f)<∞. Therefore because of ω∈Ωβ, we write 1 nβω¡1 n¢ n X k=1 kr+β−1Ek(f) + 1 ω¡1 n¢ ∞ X k=n kr−1Ek(f)<∞, and, by (22) and (23), we have f∈WrHω β. The proof is now complete. Proof of Theorem 2.6. Let f∈WrHω β. By the Jackson inequality and Lemma 2, we have En(f)¿ωβ+r³f, 1 n´¿n−rω³1 n´. 8 SERGEY TIKHONOV Further, we use the following important result of Leindler [3, Theorem 8.2, p. 32 and (2.75), p.65] 1 n 2n X ν=n+1 |Sn−f|p¿Ep n(f), p > 0, n ∈N. Then this inequality implies ∞ X ν=1 λν|f(x)−Sν(x)|p¿ ∞ X n=1 2nλ2nEp 2n(f) ¿ ∞ X n=1 2n(1−rp)λ2nωpµ1 2n¶¿ ∞ X n=1 λnωp¡1 n¢ nrp <∞. Thus, (14) implies (15). The proof is complete. Proof of Theorem 2.7. Let f∈WrHω β∩Csin. By Lemmas 2 and 3 and inequality (3), we get nan¿ωβ+r³f, 1 n´¿n−rω³1 n´. This and (16) give ap n¿1 n1+pλn .(26) Let us prove that (26) and {an} ∈ GM imply f∈Sp(λ). First we note that for x > 0 one has ¯¯¯¯¯ ∞ X k=n aksin kx¯¯¯¯¯¿1 xÃan+ ∞ X k=n+1 ak k!.(27) Using Abel’s transformation, (27) follows immediately from ¯¯¯e Dk(x)¯¯¯≡ ≡¯¯¯¯ k P n=1 sin nx¯¯¯¯=O¡1 x¢and inequality (4). Let now x > 0 and N∈Nsuch that π N+1 < x ≤π N.Then ∞ X ν=1 λν|f(x)−Sν(x)|p=ÃN X ν=1 + ∞ X ν=N+1!λν|f(x)−Sν(x)|p=: I1+I2. Using (27), we write I2¿Np ∞ X ν=N λνap ν+Np ∞ X ν=N λνÃ∞ X k=ν ak k!p =: I21 +I22. EMBEDDING THEOREMS OF FUNCTION CLASSES, IV 9 By (26), we have I21 ¿Np ∞ X ν=N λν 1 ν1+pλν ¿C. To estimate I22, we note that if {λn}satisfies 42 2-condition, then condition (17) is equivalent to the following condition n X k=1 λk¿nλn.(28) If p≥1,then by Lemma 4(a) and condition (28) I22 ¿Np ∞ X k=N³ak k´p λ1−p kÃk X ν=N λν!p ¿I21 ¿C. If 0 < p < 1, then we use Lemma 4(b): I22 ¿Np ∞ X k=N kp−1³ak k´pÃkλk+ k X ν=N λν!¿I21 ¿C. Now let us estimate I1. I1≤ N X ν=1 λν¯¯¯¯¯ N−1 X k=ν+1 aksin kx¯¯¯¯¯ p + N X ν=1 λν¯¯¯¯¯ ∞ X k=N aksin kx¯¯¯¯¯ p =: I11 +I12. By (27), we have I12 ¿Np N X ν=1 λν"ap N+Ã∞ X k=N ak k!p#. Because of {n1−ελn}is almost increasing, one can write Ã∞ X k=N ak k!p ¿Ã∞ X k=N 1 k2(kλk)1 p!p ¿1 N1−ελNÃ∞ X k=N 1 k2+ ε p!p ¿1 N1+pλN . Hence, we get I12 ¿N1+pλN·ap N+1 N1+pλN¸¿C. To estimate I11, we write I11 ¿xp N X ν=1 λνÃN−1 X k=ν kak!p .