Embedding theorems of function classes, II

EMBEDDING THEOREMS OF FUNCTION CLASSES, II SERGEY TIKHONOV Abstract. In this paper the embedding results in the questions of strong approximation on Fourier series are considered. We prove several theorems on the interrelation between class WrHω βand class H(λ, p, r, ω) which was defined by L. Leindler. Previous related results from Leindler’s book [2] and the paper [5] are particular cases of our results. 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 function f∈Cis given by ωβ(f, t) = sup |h|≤t     ∞ X ν=0 (−1)νβ 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 function fof order r > 0 (f(0) := f) in the sense of Weyl (see [12]). 2000 Mathematics Subject Classification. 42A24, 42A25. Key words and phrases. Strong approximation, Fourier series, Embedding theorems. The research was funded by a 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 We shall write I1≪I2, if there exists a positive constant Csuch that I1≤C I2. If I1≪I2and I2≪I1hold simultaneously, then we shall write I1≍I2. 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. This concept was introduced by S.N. Bernstein. For any sequence λ={λn}∞ n=1 of positive numbers we set Λn= n P ν=1 λν and for any positive pwe define the following strong mean: hn(f, λ, p) :=       1 Λn n X ν=1 λν|f(x)−Sν(x)|p!1 p      . Let Ω be the set of nondecreasing continuous functions on [0,2π] such that ω(0) = 0 and ω(δ)→0 as δ→0. Further, we define the following function classes: H(λ, p, r, ω) = f∈C:hn(f, λ, p) = On−rω1 n, WrHω β=f∈C:ωβ(f(r), δ) = O[ω(δ)], Eω r=f∈C:En(f) = On−rω1 n. One of the first results concerning the interrelation between classes H(λ, p, r, ω) with λn≡1 and WrHω βwas obtained by G. Alexits and D. Kr´alik [1]: Lip α≡W0Hδα 1⊂H(λ, 1,0, δα),if 0 < α < 1. In his book L. Leindler (see [2, Chapter III], see also [3]) proved embedding theorems for classes H(λ, p, r, ω), W rHω βand Eω r, where λn=nα−1, α > 0, r ∈N∪ {0}and β= 1. In present article we continue investigating this topic and prove the following theorems. Theorem 1.1. Let β, p > 0, r ≥0,ω∈Ωand λnbe a positive sequence such that Λ2n≪Λn≪nλn.(2) EMBEDDING THEOREMS OF FUNCTION CLASSES, II 3 (i). Then H(λ, p, r, ω)⊂Eω r. (ii). If, additionally, ωsatisfies (B) ∞ X k=n+1 1 kω1 k=Oω1 n, (Bβ) n X k=1 kβ−1ω1 k=Onβω1 n, then H(λ, p, r, ω)⊂Eω r≡WrHω β. Theorem 1.2. Let β, p > 0, r ≥0,ω∈Ωand λnbe a positive sequence which satisfies condition (2). If ωsatisfies the following condition: there exists ε > 0such that λnωp1 nn1−rp−εis almost increasing sequence. (3) Then WrHω β⊂H(λ, p, r, ω). It was observed by L. Leindler in [5] that for a certain subclass of continuous functions the condition (3) can be relaxed. We need the following definitions. Let c:= {cn}be the positive sequence. The sequence cis called quasimonotonic (c∈QM) if there exists ρ≥0 such that n−ρcn↓0. A sequence c:= {cn}of positive numbers tending to zero is called of rest bounded variation (c∈R+ 0BV S), if it possesses the property ∞ X n=m |cn−cn+1| ≤ K(c)cm for all natural numbers m, where K(c) is a constant depending only on c. Answering to S.A. Telyakovsky’s question, L.Leindler [6] proved that the class R+ 0BV S was not comparable to the class QM. We define the following two subclasses of C: C1=(f∈C:f(x) = ∞ X n=1 bnsin nx, {bn} ∈ QM), C2=(f∈C:f(x) = ∞ X n=1 bnsin nx, {bn} ∈ R+ 0BV S). Now we can formulate the theorem for classes C1and C2, which gives more general conditions of the embedding WrHω β⊂H(λ, p, r, ω) than Theorem 2. 4 SERGEY TIKHONOV Theorem 1.3. Let β, p > 0, r ≥0,ω∈Ωand λnbe a positive sequence, which satisfies condition (2). If ωsatisfies the following condition: λnωp1 nn1−rpis almost increasing sequence. (4) Then WrHω β∩Cj⊂H(λ, p, r, ω),where j= 1 or j= 2.(5) Remarks. 1. Theorem 1.3 in the case λn=nα−1, α > 0, r= 2k, k ∈N∪ {0},β= 1 and f∈C2was proved in [5]. 2. In the book [2, p. 161-162] (see also [3]) for his version of Theorem 1.1 (ii) Leindler used the following conditions: ω∈Band there exists a natural number µsuch that the inequalities 2µω1 2n+µ>2ω1 2n(6) hold for all n. And for his version of Theorem 1.2: there exists a natural number µsuch that the inequalities 2µ(α p−r)ω1 2n+µ>21 pω1 2n(7) hold for all n. It follows from [10] that (6) is equivalent to ω∈B1and (7) is equivalent to (3) with λn=nα−1, α > 0. 2. Auxiliary results Lemma 2.1. ([9]). Let β > 0and f(x)∈C. (a): En(f)≪ωβf, 1 n≪n−βn P k=1 kβ−1Ek(f); (b): ωα+β(f, δ)p≤C(α)ωβ(f, δ)pfor α≥0. Lemma 2.2. ([4], [7]). Let an≥0, λn>0. (a): If p≥1, then ∞ P n=1 λn∞ P ν=n aνp ≪ ∞ P n=1 λ1−p nap nn P ν=1 λνp , (b): If 0< p < 1and an↓, then ∞ P n=1 λn∞ P ν=n aνp ≪ ∞ P n=1 np−1ap nnλn+ n−1 P ν=1 λν. EMBEDDING THEOREMS OF FUNCTION CLASSES, II 5 Lemma 2.3. Let p > 0, r ≥0,ω∈Ωand λnbe a positive sequence, which satisfies condition (2). Let f(x) = ∞ P n=1 bnsin nx ∈Cj(j= 1 or j= 2) and bn≪n−r−1ω1 n.(8) If ωsatisfies the condition (4), then f(x)∈H(λ, p, r, ω). Proof of Lemma 2.3. Let x∈(0, π) and mbe an integer such that π m+1 < x ≤π m.Applying Abel’s transformation, we obtain for k≤m |f(x)−Sk(x)| ≤  m X ξ=k+1 bξsin ξx+ ∞ X ξ=m+1 bξsin ξx ≪ x m X ξ=k+1 ξbξ+ ∞ X ξ=m+1 |bξ−bξ+1|e Dξ(x) ≪1 m m X ξ=k+1 ξbξ+m ∞ X ξ=m+1 |bξ−bξ+1|,(9) where e Dk(x) are the conjugate Dirichlet kernels, i.e. e Dk(x) := k P n=1 sin nx, k∈N,and we used e Dk(x)=O1 x. Therefore, from (9), if k≤mand {bn} ∈ R+ 0BV S, then |f(x)−Sk(x)| ≪ 1 m m X ξ=k ξbξ+mbm,(10) and if k≤mand {bn} ∈ QM, then |f(x)−Sk(x)| ≪ 1 m m X ξ=k ξbξ+mbm+m ∞ X ξ=m bξ ξ.(11) If k≥m, then we have |f(x)−Sk(x)| ≪ m ∞ X ξ=k |bξ−bξ+1|, and if {bn} ∈ R+ 0BV S, we write |f(x)−Sk(x)| ≪ mbk,(12) 6 SERGEY TIKHONOV and if {bn} ∈ QM, we write |f(x)−Sk(x)| ≪ mbk+m ∞ X ξ=k bξ ξ.(13) Further, we shall conduct the proof for f∈C1and use inequalities (11) and (13). Let n > m. Then n X k=1 λk|f(x)−Sk(x)|p= m X k=1 λk|f(x)−Sk(x)|p+ n X k=m+1 λk|f(x)−Sk(x)|p =: I1+I2. Using (11), we have I1≪1 mp m X k=1 λk  m X ξ=k ξbξ  p +mpbp m m X k=1 λk+ m X k=1 λk m ∞ X ξ=m bξ ξ  p =: I11 +I12 +I13. Using (13), we have I2≪ n X k=m λk(mbk)p+ n X k=m λk m ∞ X ξ=k bξ ξ  p =: I21 +I22. Now we shall estimate I11. Let p≥1.Then by Lemma 2.2(a) and conditions (2), (4) and (8), we have I11 ≪1 mp m X ξ=1 ξpbp ξλ1−p ξ ξ X k=1 λk!p ≪1 mp m X ξ=1 ξp−1λξξ1−rpωp1 ξ ≪m1−rp−pλmωp1 mm X ξ=1 ξp−1 ≪m1−rpλmωp1 m.(14) EMBEDDING THEOREMS OF FUNCTION CLASSES, II 7 If 0 < p < 1, then we shall use inequality (8), Lemma 2.2(b), and inequalities (2), (4): I11 ≪1 mp m X k=1 λk  m X ξ=k ξ−rω1 ξ  p ≪1 mp m X ξ=1 ξp−1−rpωp1 ξ ξλξ+ ξ X k=1 λk! ≪m1−rp−pλmωp1 mξ X k=1 kp−1 ≪m1−rpλmωp1 m.(15) The estimate of I12 evidently follows from inequalities (2) and (4): I12 ≪m1−rpλmωp1 m.(16) Using (2) and monotonicity of ω, we write I13 ≪ m X k=1 λk m1−rω1 m∞ X ξ=m 1 ξ2  p ≪m1−rpλmωp1 m.(17) From (8) and (4) we estimate I2. I21 ≪mp n X k=m k−p−rpλkωp1 k ≪mpn1−rpλnωp1 nn X k=m k−p−1 ≪n1−rpλnωp1 n.(18) Using (8) and the monotonicity of ω, we write I22 ≪mp n X k=m k−p−rpλkωp1 k. 8 SERGEY TIKHONOV The last expression we have already estimated in (18). Thus, collecting estimates (14) – (17) and the estimates for I21 and I22, we have I1≪m1−rpλmωp1 m, I2≪n1−rpλnωp1 n. and, by (4), I1+I2≪n1−rpλnωp1 n. If n≤m, then for the estimates of n P k=1 λk|f(x)−Sk(x)|pwe shall use the same reasoning as for estimates of I1with only one difference that instead of mthere should be n. Thus, we have for any n n X k=1 λk|f(x)−Sk(x)|p≪n1−rpλnωp1 n.(19) It is easy to see that the following conditions are equivalent: (a) Λ2n≪Λn≪nλn, (b) Λn≪nλnand λn≍λkfor n≤k≤2n, (c) Λ2n≍Λn≍nλn. Only one nontrivial part is (a)⇒(b).Let us prove it. From (4) and the monotonicity of ωone has for n≤k≤2n λk≫n1−rpλnωp1 nkrp−1ω−p1 k ≫n1−rpλnkrp−1 ≫λn.(20) Let (a) be true. Using (20), we can write λn≫1 nΛn ≫1 nΛ4n ≫1 n 2k X ν=k λν ≫λk. Hence, (b) holds. EMBEDDING THEOREMS OF FUNCTION CLASSES, II 9 From the assumption of our Lemma, condition (a) holds and, therefore, (c) holds. Then, by (19), we have f(x)∈H(λ, p, r, ω). For f∈C2we shall use estimates (10) and (12) and the comparison between couples of estimates (10), (12) and (11), (13) implies f(x)∈ H(λ, p, r, ω).The proof of Lemma 2.3 is complete. Lemma 2.4. ([11]). If g(x)∈Cand g(x)has a Fourier series ∞ P n=1 bnsin nx, bn≥0,then n−β n X k=1 kβbk≤Cωβg, 1 n,for β6= 2l, l = 1,2,··· Lemma 2.5. ([11]). If f(x)∈Cand f(x)has a Fourier series ∞ P n=1 ancos nx, an≥0, then n−β n X k=1 kβak≤Cωβf, 1 n,for β6= 2l−1, l = 1,2,··· 3. Proofs of Theorems We shall follow the method of proof of L. Leindler. Proof of Theorem 1.1. Part (i) immediately follows from the inequality (see [2, Theorem 8.2, p. 147]) En(f) = O(hn(f, λ, p)) . Let us prove part (ii). We need the following inequality (see [8]) ωβf(r), δ≪ δ Z0 t−r−1ωr+β(f, t)dt, r ≥0.(21) Then from (21), Lemma 2.1 (a), part (i) and conditions (B) and (Bβ) on ω we have