Embedding theorems of funcitional classes, III

EMBEDDING THEOREMS OF FUNCTION CLASSES, III B. SIMONOV AND S. TIKHONOV Abstract. In this paper we obtain the necessary and sufficient conditions for embedding results of different function classes. The main result is a criterion for embedding theorems for the so-called generalized Weyl-Nikol’skii class and the generalized Lipschitz class. To define the Weyl-Nikol’skii class, we use the concept of a (λ, β)-derivative, which is a generalization of the derivative in the sense of Weyl. As corollaries, we give estimates of norms and moduli of smoothness of transformed Fourier series. 1. Introduction History of the question. One of the main problems of constructive approximation1is in finding a relationship between differential properties of a function and its structural or constructive characteristics. This topic started to develop more than a century ago and in many cases the research was conducted as follows; authors considered a given functional class and by investigating the properties of its elements obtained embedding theorems with other functional classes. We recommend the articles by A. Pinkus [Pi] and V. V. Zhuk and G. I. Natanson [Zh-Na] for a historical review. The following three classical results gave rise to development of new areas within the approximation theory: (A) f(r)∈Lipα⇐⇒ En(f) = Oµ1 nr+α¶(0 < α < 1, r ∈Z+), (B) f(r)∈Lipα⇐⇒ ωr+1³f, 1 n´=Oµ1 nr+α¶(0 < α < 1, r ∈Z+), (C) f∈Lipα=⇒e f∈Lipα(0 < α < 1). 2000 Mathematics Subject Classification. 46E35, 26A16, 26A33, 42A45. Key words and phrases. Embedding theorems, Lipschitz class, Weyl-Nikol’skii class, (λ, β)-derivative, Moduli of smoothness of fractional order. 1The concept became well known through Bersntein’s paper [Be,S,4, V. 2, p.295-300; p. 349-360]. 1 2 B. SIMONOV AND S. TIKHONOV Result (A) was proved2by D. Jackson (1911, [Ja]) in the necessity part, and by S. Bernstein (1912, [Be,S,1], [Be,S,2]) and Ch. de la Vall´ee-Poussin (1919, [Va]) in the sufficiency part. The theorems of this type are called direct and inverse theorems for trigonometric approximation. Direct theorems for Lp,1≤p≤ ∞ (see the review [Zh-Na]) are written as follows: En(f)p≤C(k)ωk³f, 1 n+ 1´p, k, n ∈N,(1) En(f)p≤C(k) nrωk³f(r),1 n+ 1´p, k, n, r ∈N.(2) Inverse theorems for Lp,1≤p≤ ∞ (see the review [Zh-Na]): ωk³f, 1 n+ 1´p≤C(k) (n+ 1)k n X ν=0 (ν+ 1)k−1Eν(f)p, k, n ∈N,(3) ωk³f(r),1 n+ 1´p≤(4) C(k)Ã1 (n+ 1)k n X ν=0 (ν+ 1)k+r−1Eν(f)p+ ∞ X ν=n+1 (ν+ 1)r−1Eν(f)p!, k, n ∈N. Here and further, the best trigonometric approximation En(f)pand the modulus of smoothness ωk(f, δ)pare defined by En(f)p= min ³kf−Tkp;T∈Tn´, Tn=span {cos mx, sin mx :|m| ≤ n} and ωk(f, δ)p= sup |h|≤δ°°4k hf(x)°°p,(5) 4k hf(x) = 4k−1 h(4hf(x)) and 4hf(x) = f(x+h)−f(x) respectively. Note that the theorems on existence of the r-th derivative of ffrom a given space have been initiated by Bernstein [Be,S,1]. He proved that the condition ∞ P ν=1 νr−1Eν(f)∞<∞implies f(r)∈C. Later, for Lp(1 ≤p≤ ∞), the following results were obtained (see the review [Zh-Na] and the 2See [Be,S,3] for a detailed review of the question before the 30-ths XX century. EMBEDDING THEOREMS OF FUNCTION CLASSES, III 3 paper by O.V. Besov [Be,O]). For convenience, we write these results in terms of the Besov space Br p,θ and the Sobolev space Wr p: Br p,1⊂Wr p⊂Br p,∞p= 1,∞, Br p,p ⊂Wr p⊂Br p,21< p ≤2, Br p,2⊂Wr p⊂Br p,p 2≤p < ∞. Result (B) was proved by A. Zygmund (1945, [Zy,1]). He was one of the first to use the modulus of smoothness concept of integer order introduced by S. Bernstein in 1912 ([Be,S,1]). At present, the moduli of smoothness properties are well-studied ([Jo], [Zh-Na]) and the result (B) follows from inequalities (see [De-Lo, Chapters 2 and 6], [Jo-Sc]): ωk+r³f, 1 n´p≤C(k, r) nrωk³f(r),1 n´p, k, r, n ∈N(6) ωk³f(r),1 n´p≤C(k, r) ∞ X ν=n+1 νr−1ωk+r³f, 1 ν´p, k, r, n ∈N.(7) Comparing the last two inequalities and inequalities (2) and (4) we see that from (6) and (7), using (1) and (3), it is easy to get (2) and (4). Result (C) was proved by I.I. Privalov (1919, [Pr]). The most complete version of the inequality, from which embedding (C) follows, was obtained by A. Zygmund ([Zy,1]) and N.K. Bary and S.B. Stechkin ([Ba-St]) and is the following one (p= 1,∞) ωk³e f(r),1 n´p≤C(k, r)Ãn−k n X ν=1 νk+r−1ωk³f, 1 ν´p+ ∞ X ν=n+1 νr−1ωk³f, 1 ν´p!, (8) k, n ∈N, r ∈Z+. Finally, we note the paper by G.H. Hardy and J.S. Littlewood [Ha-Li] in which seemingly, for the first time, some problems were formulated and solved in the same setting as in the present paper. For the historical aspects of this approach see also [Og]. Embedding theorems for functional classes. The results (A) - (C) as well as their generalizations mentioned above can be written as the embedding theorems of the following functional classes: Wr p=nf∈Lp:f(r)∈Lpo, f Wr p=nf∈Lp:e f(r)∈Lpo, Wr pHα[ϕ] = nf∈Wr p:ωα³f(r), δ´p=O[ϕ(δ)]o, 4 B. SIMONOV AND S. TIKHONOV f Wr pHα[φ] = ½f∈f Wr p:ωα³e f(r), δ´p=O[φ(δ)]¾, Wr pE[ξ] = nf∈Wr p:En³f(r)´p=O[ξ(1/n)]o. We will study more general classes for which Wr p,f Wr p,Wr pHα[ϕ], f Wr pHα[φ], WrEp[ξ] are particular cases. Transformed Fourier series. Let Lp=Lp[0,2π] (1 ≤p < ∞) be a space of 2π-periodic measurable functions for which |f|pis integrable, and L∞≡ C[0,2π] be the space of 2π-periodic continuous functions with kfk∞= max {|f(x)|,0≤x≤2π}. Let summable function f(x) have the Fourier series f(x)∼σ(f) := a0(f) 2+ ∞ X ν=1 (aν(f) cos νx +bν(f) sin νx)≡ ∞ X ν=0 Aν(f, x). (9) By the transformed Fourier series of (9) we mean the series σ(f, λ, β) := ∞ X ν=1 λν·aνcos µνx +πβ 2¶+bνsin µνx +πβ 2¶¸, where β∈Rand λ={λn}is a given sequence of positive numbers. Studies of the transformed Fourier series are naturally related to the problems of Fourier multipliers theory (see [Be-L¨o], [Be-Il-Ni], [Zy,2, Vol 1, Chapter III]), summability methods (see [Bu-Ne, Chapter 1.2], [Zy,2, Vol 1, Chapter III]) and3so-called the fractional Sobolev classes or the Weyl classes. We call the class Wλ,β p=nf∈Lp:∃g∈Lp, g(x)∼σ(f, λ, β)o the Weyl class. It is because λn=nr, r > 0 and β=rthe class Wλ,β pcoincides with the class Wr p, which is defined in the term of fractional derivatives f(r)in the Weyl sense ([Zy,2, Vol. 2, Chapter XII]). In the case λn=nr, r > 0 and β=r+ 1 the class Wλ,β pcoincides with the class f Wr p. We call the function g(x)∼σ(f, λ, β) the (λ, β)-derivative of the function f(x) and denote it by f(λ,β)(x). Generalized Weyl-Nikolskii class. In the definition of this functional class we use the modulus of smoothness concept ωα(f, δ)pof fractional4order of a 3See also references to [Ba,2, §13, Chapter II]. 4The term ”fractional” can be found in earlier papers ([Bu-Dy-G¨o-St] and [Ta]) which used this definition. As in the case of fractional derivatives, the positive number αthat defines the modulus order is not necessarily rational. EMBEDDING THEOREMS OF FUNCTION CLASSES, III 5 function f(x)∈Lp, i.e., ωα(f, δ)p= sup |h|≤δ k4α hf(x)kp, where 4α hf(x) = ∞ X ν=0 (−1)νµα ν¶f(x+ (α−ν)h), α > 0 is the α-th difference5of a function fwith step hat the point x. It is clear that for α∈Nthis definition is the same as (5). Let Φα(α > 0) be the class of functions ϕ(δ), defined and non-negative on (0, π] such that 1. ϕ(δ)→0 (δ→0), 2. ϕ(δ) is non-decreasing, 3. δ−αϕ(δ) is non-increasing. For such α > 0, ϕ ∈Φαand λ={λn}the generalized Weyl-Nikolskii class is defined similarly to the classes Wr pHα[ϕ] and f Wr pHα[φ]: Wλ,β pHα[ϕ] = nf∈Wλ,β p:ωα³f(λ,β), δ´p=O[ϕ(δ)] , δ →+0o. It is clear that if λn=nr, r > 0 and β=r, then Wλ,β pHα[ϕ]≡Wr pHα[ϕ] and if λn=nr, r > 0 and β=r+ 1, then Wλ,β pHα[ϕ]≡f Wr pHα[ϕ]. In the case λn≡1 and β= 0 the class Wλ,β pHα[ϕ] coincides with the generalized Lipschitz class Hϕ α, i.e., Hp α[ϕ] = nf∈Lp:ωα(f, δ)p=O[ϕ(δ)] δ→+0o. In particular, Lip (γ, Lp)≡Hp 1[δγ] = nf∈Lp:ω1(f, δ)p=O[δγ]δ→+0o. The problem setting and the structure of the paper. In this paper, we obtain embedding theorems for the Weyl class Wλ,β p, for the generalized Weyl- Nikolskii class Wλ,β pHα[ϕ] and for the generalized Lipschitz class Hp γ[ω]. We show a relation between the parameters αand γdepending on the behavior of the sequence {λn}and on the metric Lp. The remainder of the paper is organized as follows. In section 2 we present the main theorem. Sections 3 and 4 contain the proofs of the sufficiency and necessity parts of the main theorem respectively. In section 5 we provide several corollaries. In particular, we describe the difference in results for metrics Lp,1< p < ∞and Lp, p = 1,∞. The estimates 5As usual,  β ν  =β(β−1)···(β−ν+1) ν!for ν > 1,  β ν  =βfor ν= 1, and  β ν  = 1 for ν= 0. 6 B. SIMONOV AND S. TIKHONOV ωγ(f(r), δ)pand ωγ(e f(r), δ)pare written in terms of ωβ(f, δ)pfor different values of r, γ and β. The concluding remarks are given in section 6. 2. Embedding theorems for generalized Lipschitz and Weyl-Nikolskii classes For λ={λn}n∈Nwe define 4λn:= λn−λn+1;42λn:= 4(4λn). Theorem 1. Let θ= min(2, p),α∈R+, β ∈R, and λ={λn}be a nondecreasing sequence of positive numbers. Let ρbe a non-negative number such that the sequence {n−ρλn}is non-increasing. I. Let 1< p < ∞. Then Hp α+ρ[ω]⊂Wλ,β p⇐⇒ ∞ X n=1 ¡λθ n+1 −λθ n¢ωθµ1 n¶<∞,(10) Hp α+ρ[ω]⊂Wλ,β pHα[ϕ]⇐⇒ ½n−αθ n X ν=1 ν(ρ+α)θ¡ν−ρθλθ ν−(ν+ 1)−ρθλθ ν+1¢ωθµ1 ν¶ + ∞ X ν=n+2 ¡λθ ν+1 −λθ ν¢ωθµ1 ν¶+λθ n+1ωθµ1 n+ 1¶¾1 θ =O·ϕµ1 n+ 1¶¸,(11) Wλ,β p⊂Hp α+ρ[ω]⇐⇒ 1 λn =O·ωµ1 n¶¸,(12) Wλ,β pHα[ϕ]⊂Hp α+ρ[ω]⇐⇒ ϕ¡1 n¢ λn =O·ωµ1 n¶¸.(13) II. Let p= 1 or p=∞. (a) If 42λn≥0or 42λn≤0, then Hp α+ρ[ω]⊂Wλ,β p⇐⇒ | cos βπ 2| ∞ X n=1 (λn+1 −λn)ωµ1 n¶ +|sin βπ 2| ∞ X n=1 λn ω¡1 n¢ n<∞; (14) and if, additionally, for some τ > 0the following inequality holds, 42µλn nr¶≥0with r=ρ+τ sign ¯¯¯¯sin (β−ρ)π 2¯¯¯¯, then Hp α+r[ω]⊂Wλ,β pHα[ϕ]⇐⇒ EMBEDDING THEOREMS OF FUNCTION CLASSES, III 7 ⇐⇒ n−α n X ν=1 νr+α¡ν−rλν−(ν+ 1)−rλν+1¢ωµ1 ν¶ +|cos βπ 2| ∞ X ν=n+2 (λν+1 −λν)ωµ1 ν¶ +|sin βπ 2| ∞ X ν=n+2 λν ω¡1 ν¢ ν+λn+1ωµ1 n+ 1¶ =O·ϕµ1 n+ 1¶¸.(15) (b) If for β= 2k, k ∈Z, the condition 42(1/λn)≥0holds, and for β6= 2k, k ∈Z,conditions 42(1/λn)≥0and ∞ P ν=n+1 1 νλν≤C λnare fulfilled, then Wλ,β p⊂Hp α+ρ[ω]⇐⇒ 1 λn =O·ωµ1 n¶¸; (16) and if, additionally, for some τ > 0the following inequality holds, 42µnr λn¶≥0or 42µnr λn¶≤0with r=ρ+τ sign ¯¯¯¯sin (β−ρ)π 2¯¯¯¯, then Wλ,β pHα[ϕ]⊂Hp α+r[ω]⇐⇒ ϕ¡1 n¢ λn =O·ωµ1 n¶¸.(17) 3. Proof of sufficiency in Theorem 1. We will use the following notations. Let a function f(x)∈Lhave Fourier series (9). Then Sn(f) denotes the n-th partial sum of (9), Vn(f) denotes the de la Vall´ee-Poussin mean and Kn(x) is the Fej´er kernel, i.e., Sn(f) = n P ν=0 Aν(x), Vn(f) = 1 n 2n−1 P ν=n Sν(f), Kn(x) = 1 n+1 n P ν=0 µ1 2+ ν P m=1 cos mx¶. The following lemmas play the central role in the proof of Theorem 1. Lemma 3.1. Let f(x)∈Lp,1≤p≤ ∞, and α > 0. Then C1(p, α)ωα³f, 1 n´p≤µn−α°°°V(α) n(f, x))°°°p+kf(x)−Vn(f, x)kp¶ ≤C2(p, α)ωα³f, 1 n´p.(18) 8 B. SIMONOV AND S. TIKHONOV If f(x)∈Lp,1< p < ∞, then C1(p, α)ωα³f, 1 n´p≤µn−α°°°S(α) n(f, x))°°°p+kf(x)−Sn(f, x)kp¶ ≤C2(p, α)ωα³f, 1 n´p.(19) Proof of Lemma 3.1. The estimate from above ωα³f, 1 n´pfollows from the inequality (see [Bu-Dy-G¨o-St]) ωα³Tn,1 n´p≤C(p, α)n−α°°°T(α) n°°°p, where Tnis a trigonometric polynomial of order n. We get ωα³f, 1 n´p≤C(p, α)µωα³Tn,1 n´p+kf−Tnkp¶ ≤C(p, α)µn−α°°°T(α) n°°°p+kf−Tnkp¶. Now we will estimate ωα³f, 1 n´pfrom below. We need the generalized Nikol’skii-Stechkin inequality (see [Ta]) n−α°°°T(α) n°°°p≤C(p, α)ωα³Tn,1 n´p and the generalized Jackson inequality (see [Bu-Dy-G¨o-St]) En(f)p≤C(α)ωαµf, 1 n+ 1¶p .(20) It is well known that the Vall´ee-Poussin mean is the nearly best approximant, i.e., kf−Vn(f)kp≤CEn(f)p.(21) Then, n−α°°°V(α) n(f, x)°°°p+kf(x)−Vn(f, x)kp ≤C(p, α)µωα³Vn,1 n´p+En(f)p¶ ≤C(p, α)µωα³f, 1 n´p+ωα³f−Vn,1 n´p¶≤C(p, α)ωα³f, 1 n´p, and (18) is proved. Using kf−Sn(f)kp≤C(p)En(f)p(22) for 1 <p<∞, we will have (19) similarly. This completes the proof of Lemma 3.1. We note that (18) and (19) are realization results (see paper [Di-Hr-Iv] by Z. Ditzian, V. H. Hristov, K. G. Ivanov). EMBEDDING THEOREMS OF FUNCTION CLASSES, III 9 Lemma 3.2. ([St,S]). Let f(x)∈Lp, p = 1,∞, and let ∞ P k=1 k−1Ek(f)p<∞ be true. Then ˜ f(x)∈Lpand En(˜ f)p≤CÃEn(f)p+ ∞ X k=n+1 k−1Ek(f)p!, n ∈N. Lemma 3.3. Let p= 1,∞and {λn}be monotonic concave (or convex) sequence. Let Tn(x) = n X ν=0 aνcos νx +bνsin νx, Tn(λ, x) = n X ν=0 λν(aνcos νx +bνsin νx). Then for M > N ≥0one has kTM(λ, x)−TN(λ, x)kp≤µ(M, N)kTM(x)−TN(x)kp, where µ(M, N) =      2M(λM−λM−1)+λN+1 −(N+ 1)(λN+2 −λN+1),if λn↑(n↑),42λn≥0; 2λM+(N+ 1)(λN+2 −λN+1)−λN+1,if λn↑(n↑),42λn≤0; (N+ 1)(λN+1 −λN+2) + λN+1,if λn↓(n↑),42λn≥0. Proof of Lemma 3.3. Applying two times Abel’s transformation we write kTM(λ, x)−TN(λ, x)kp=1 π°°° π Z −π (TM−TN) (x+u) M X ν=N+1 λνcos νu du°°°p= =1 π°°° π Z −π (TM−TN) (x+u)nM−2 X ν=N+1 (λν−2λν+1 +λν+2) (ν+ 1)Kν(u) + + (λN+2 −λN+1) (N+ 1)KN(u) + + (λM−1−λM)MKM−1(u)odu +λM(TM−TN)°°°p≤ ≤ kTM(x)−TN(x)kp· ·nM−2 X ν=N+1 |λν−2λν+1 +λν+2|(ν+ 1) + |λM−1−λM|M+λMo =: kTM(x)−TN(x)kpI(M, N). 16 B. SIMONOV AND S. TIKHONOV As in (34), we have °°°°° M X ν=N (V2ν+1 −V2ν) (λ, f)°°°°°p ≤CÃλ2Nωα+rµf, 1 2N¶p + + ∞ X ν=2N+1 (λν+1 −λν)ωα+rµf, 1 ν¶p ; °°°°° M X ν=N³g V2ν(λ, f)− ] V2ν+1 (λ, f)´°°°°°p ≤°°°°° M X ν=N λ2ν+2 ³g V2ν(f)− ] V2ν+1 (f)´°°°°°p + M X ν=N°°°g V2ν(f)− ] V2ν+1 (f)°°°p  2ν+2−1 X m=2ν |42λm|(m+ 1) + 2ν+2|4λ2ν+2 |  =: I21 +I22. By Lemma 3.2, I21 ≤λ2N+1 °°° ^ V2M+1 (f)−g V2N(f)°°°p + M X ν=N (λ2ν+2 −λ2ν+1 )°°° ] V2ν+1 (f)−g V2ν(f)°°°p ≤CÃλ2N+1 ∞ X ν=N E2ν(f)p+ ∞ X ν=N (λ2ν+2 −λ2ν+1 ) ∞ X s=ν E2s(f)p! ≤C ∞ X ν=2N λν νωα+rµf, 1 ν¶p , I22 ≤C M X ν=N (λ2ν+3 −λ2ν−1)E2ν(˜ f)p≤C ∞ X ν=2N λν νωα+rµf, 1 ν¶p , and (35) follows. Repeating the argument in (34), we estimate n−α°°°V(α) n(f(λ,β))°°°p. By Lemma 3.3 and by inequalities (20) and (21), we write °°°V(α) n(f(λ,β))°°°p≤C(β, α, r)Ãnαλnωα+rµf, 1 n¶p + (36) + n X ν=1 µλν νr−λν+1 (ν+ 1)r¶να+rωα+rµf, 1 ν¶p!. EMBEDDING THEOREMS OF FUNCTION CLASSES, III 17 By means of (35) and (36) we get f∈Wλ,β pHα[ϕ],because of the condition in (15). Let us prove (16). Let f∈Wλ,β p.To establish f∈Hp α+ρ(ω), we shall first estimate En(f)pfor the case sin πβ 26= 0. Repeating the reasoning in (34), by (21), we get ( 2m≤n < 2m+1 ) En(f)p≤C ∞ X ν=m 1 λ2ν°°°¡V[2ν−1]−V2ν¢(f(λ,β))°°°p ≤CE[2m−1](f(λ,β))p ∞ X ν=m 1 λ2ν ≤C(ρ, p)kf(λ,β)kp λn . If sin πβ 2= 0, it is easy to see that En(f)p≤C λn En(f(λ,β))p≤C λn kf(λ,β)kp. Substituting the bound for En(f)pinto (3) and using the fact that nρλ−1 n↑ (n↑), we can write ωα+ρµf, 1 n¶p ≤C(α, ρ)1 nα+ρ n X ν=0 να+ρ−1Eν−1(f)p≤C(α, ρ) λn =O·ωµ1 n¶¸, i.e., f∈Hp α+ρ(ω).This completes the proof of the sufficiency part in (16). Let the right part of (17) be true, and f∈Hp α+ρ[ω]. First let us prove that °°°V(α+r) n(f)°°°p≤C(α, r)nr λn°°°V(α) 2n(f(λ,β))°°°p.(37) If β=ρ+ 2m, and therefore, r=ρ, then V(α+r) n(f) = V(α) n³{νr λν}, f(λ,β)´ and, by Lemma 3.3 °°°V(α+r) n(f)°°°p≤C(α, r)nr λn°°°V(α) n(f(λ,β))°°°p≤C(α, r)nr λn°°°V(α) 2n(f(λ,β))°°°p. If β6=ρ+ 2m, and therefore, r > ρ, then by Lemma 3.4 °° °V(α+r) 2n(f)°°°p≤ ≤ n X ν=0 °°°V(α+r) 2ν(f)−V(α+r) 2ν−1(f)°°°p+°°°V(α+r) 1(f)°°°p ≤C n X ν=0 2νr λ2ν°°°V(α) 2ν(f(λ,β))−V(α) 2ν−1(f(λ,β))°°°p+1 λ1°°°V(α) 1(f(λ,β))°°°p 18 B. SIMONOV AND S. TIKHONOV ≤C°°°V(α) 2n+1 (f(λ,β))°°°pÃn X ν=0 2νr λ2ν +1 λ1! ≤C(α, r)2nr λ2n°°°V(α) 2n+1 (f(λ,β))°°°p. Thus, by (37) and the estimate En(f)p≤C λnE[n 4](f(λ,β))p,we can write ωα+rµf, 1 n¶p ≤C(α, r)µn−(α+r)°°°V(α+r) n(f)°°°p+En(f)p¶ ≤C(α, r) λnµn−α°°°V(α) n(f(λ,β))°°°p+E[n 4](f(λ,β))p¶ ≤C(α, r) λn ωαµf(λ,β),1 n¶p =O"ϕ¡1 n¢ λn#=O·ωµ1 n¶¸. Thus, ϕ(1 n) λn=O£ω¡1 n¢¤is sufficient for Wλ,β pHα[ϕ]⊂Hp α+r[ω]. 4. Proof of necessity in Theorem 1. First we define the trigonometric polynomials τn+1(x): τn+1(x) = n+1 X j=1 αn jsin jx, where αn j=(j n+2 ,1≤j≤n+2 2 1−j n+2 ,n+2 2≤j≤n+ 1. We will use the following lemmas as well as Lemmas 3.1-3.2. Lemma 4.1. ([Te]). Let f(x)∈L1have the Fourier series (9). Then En(f)1≥C¯¯¯¯¯ ∞ X ν=n+1 bν ν¯¯¯¯¯. Lemma 4.2. ([Zy,2, V. 1, p. 215]) Let 1≤p < ∞. (a) If a function f(x)∈Lphas the Fourier series ∞ P ν=1 (aνcos 2νx+bνsin 2νx), then (∞ X ν=1 ¡a2 ν+b2 ν¢)1 2 ≤Ckfkp. (b) Let an, bn(n∈N)be real numbers such that ∞ P ν=1 ¡a2 ν+b2 ν¢<∞.Then ∞ P ν=1 (aνcos 2νx+bνsin 2νx)be the Fourier series of a function f(x)∈Lp, EMBEDDING THEOREMS OF FUNCTION CLASSES, III 19 and kfkp≤C(∞ X ν=1 ¡a2 ν+b2 ν¢)1 2 . Lemma 4.3. ([Ba,2, Vol.2, p. 269]). Let f(x)∈L∞have the Fourier series ∞ P ν=1 (aνcos 2νx+bνsin 2νx),where aν, bν≥0. Then C1 ∞ X ξ=n (aξ+bξ)≤E2n−1(f)∞≤C2 ∞ X ξ=n (aξ+bξ). We need the following definitions. Let ω(·)∈Φα. We call ψaQα, θ(ω)-sequence if 0< ψn≤nαωµ1 n¶, ψn↑(n↑) (38) C1ωµ1 n¶≤(∞ X ν=n ν−αθ−1ψθ ν)1 θ ≤C2ωµ1 n¶.(39) We call εaqα, θ(ω)-sequence if 0< εn≤ωµ1 n+ 1¶, εn↓(n↑) (40) C1ωµ1 n+ 1¶≤((n+ 1)−αθ n+1 X ν=1 ναθ−1εθ ν)1 θ ≤C2ωµ1 n+ 1¶.(41) Necessity in (10) - (17). We prove the necessity part by constructing corresponding examples. The proof is in eight steps. I. 1< p < ∞.Step 1. Let us show the necessity part in (10). Let ω(·)∈Φα+ρand θ= min(2, p). We will construct a sequence ψthat will be a Qα+ρ, θ(ω)-sequence. Let us assume that we have chosen integers 1 = n1< n2<· · · < ns. Then, we define ns+1 as minimum of integers N > nssuch that ωµ1 N¶<1 2ωµ1 ns¶≤ωµ1 N−1¶. We set ψn=(nρ+α sω³1 ns´,if ns≤n < ns+1, 0,if n= 0. It is easy to see that this sequence is required. 20 B. SIMONOV AND S. TIKHONOV Let Hp α+ρ[ω]⊂Wλ,β p, and let the series in (10) be divergent. By means of properties of sequence {ψn}, we have ∞= ∞ X ν=1 ¡λθ ν+1 −λθ ν¢ωθµ1 ν¶ ≤C(α, ρ, θ) ∞ X ν=1 ¡λθ ν+1 −λθ ν¢∞ X m=ν m−(α+ρ)θ−1ψθ m ≤C(α, ρ, θ) ∞ X ν=1 λθ νν−(α+ρ)θ−1ψθ ν. Step 1(a): 2≤p < ∞. We consider the series ∞ X ν=1 2−ν(α+ρ)¡ψ2 2ν−ψ2 2ν−1¢1 2cos 2νx. (42) Since ∞ X ν=1 2−2ν(α+ρ)¡ψ2 2ν−ψ2 2ν−1¢≤ ∞ X ν=1 ¡ψ2 2ν−ψ2 2ν−1¢∞ X ξ=ν 2−2ξ(α+ρ) ≤ ∞ X ν=1 2−2ν(α+ρ)ψ2 2ν≤Cω2(1),(43) then, by Zygmund’s Lemma 4.2, series (42) is the Fourier series of a function f1(x)∈Lp. By Lemmas 3.1 and 4.2, C(α, ρ)ωα+ρ³f1,1 2n´p≤ ≤2−n(α+ρ)Ãn X ν=1 a2 ν22(α+ρ)ν!1 2 +Ã∞ X ν=n+1 a2 ν!1 2 =: I1+I2, where aν= 2−(α+ρ)¡ψ2 2ν−ψ2 2ν−1¢1 2.Repeating the argument in (43), we get I1≤2−n(α+ρ)ψ2n≤ω³1 2n´, because of (38) and I2≤C(α, ρ)Ã∞ X ν=n+1 2−2ν(α+ρ)ψ2 2ν!1 2 ≤C(α, ρ)ω³1 2n´, because of (39). EMBEDDING THEOREMS OF FUNCTION CLASSES, III 21 Thus, f1(x)∈Hp α+ρ[ω].Then from our assumption, f1(x)∈Wλ,β p. On the other hand, °°°f(λ,β) 1°°°p≥C(α, ρ, θ)Ã∞ X ν=1 λ2 νν−2(α+ρ)−1ψ2 ν!1 2 =∞. This contradiction proves the convergence of series in (10). Step 1(b): 1< p ≤2. Consider series6 ψ1cos x+ ∞ X ν=1 2−ν(α+ρ)2ν(1 p−1) ¡ψp 2ν−ψp 2ν−1¢1 p 2ν X µ=2ν−1+1 cos µx. (44) Using Jensen inequality µ∞ P n=1 aα n¶1/α ≤µ∞ P n=1 aβ n¶1/β (an≥0 and 0 < β ≤ α < ∞), we write 2π Z0    ∞ X ν=1  2−ν(α+ρ)2ν(1 p−1) ¡ψp 2ν−ψp 2ν−1¢1 p 2ν X µ=2ν−1+1 cos µx  2   p 2 dx ≤ 2π Z0   ∞ X ν=1 2−νp(α+ρ)2ν(1−p)¡ψp 2ν−ψp 2ν−1¢¯¯¯¯¯¯ 2ν X µ=2ν−1+1 cos µx¯¯¯¯¯¯ p dx ≤C(p) ∞ X ν=1 ¡ψp 2ν−ψp 2ν−1¢2−νp(α+ρ)≤C(p)ωp(1), because of C1(p)2ν(p−1) ≤°°°°° 2ν P µ=2ν−1+1 cos µx°°°°° p p ≤C2(p)2ν(p−1). By the Littlewood-Paley theorem, there exists a function f2∈Lpwith Fourier series (44). One can see that f2∈Hp α+ρ[ω].Then f2∈Wλ,β p. On the other hand, Paley’s theorem on Fourier coefficients [Zy,2, V.2, p. 121] implies that for f2∈Lp °°°f(λ,β) 2°°° p p≥ ≥C(p) ∞ X ν=1 2−νp(α+ρ)2ν(1−p)¡ψp 2ν−ψp 2ν−1¢2ν X µ=2ν−1+1 λp µµp−2 ≥C(α, ρ, p) ∞ X ν=1 ¡ψp 2ν−ψp 2ν−1¢∞ X ξ=ν³2−ξ(α+ρ)pλp 2ξ−2−(ξ+1)(α+ρ)pλp 2ξ+1 ´ 6Series of this type was considered in [Ti,M,1]. 22 B. SIMONOV AND S. TIKHONOV ≥C1(α, ρ, p) ∞ X ν=1 ψp νλp νν−p(α+ρ)−1−C2(α, ρ, p)ψp 1λp 22−p(α+ρ)−1=∞. This contradiction shows that the series in the right part of (10) converges. This completes the proof of the necessity part of (10). Step 2. Let us prove the necessity in (11) for 2 ≤p < ∞. We notice that by Lemmas 3.1 and 4.2, we have for f(x)∼ ∞ P ν=1 (aνcos 2νx+ bνsin 2νx) ωαµf, 1 2m¶p ³Ã2−2mα m X ν=1 (a2 ν+b2 ν)22να!1 2 +Ã∞ X ν=m+1 (a2 ν+b2 ν)!1 2 .(45) Let ω(·)∈Φα+ρ. One can construct7a sequence εsuch that εis a qα+ρ, θ(ω)- sequence. We consider for this case ε0+¡ε2 1−ε2 2¢1 2cos x+ ∞ X ν=1 ¡ε2 2ν−ε2 2ν+1 ¢1 2cos 2νx. (46) Repeating the argument for series (42) we obtain that series (46) is the Fourier series of a function f3∈Lp. Since E2n−1(f3)p≤C(p)ε2n, then by (3), (41) implies f3∈Hp α+ρ[ω]. We define f13 := f1+f3.We have f13 ∈Hp α+ρ[ω]⊂Wλ,β pHα[ϕ]. It is easy to see from (45) that C(α, β)ωα³f(λ,β) 13 ,1 n+ 1´p≥ωα³f(λ,β) 1,1 n+ 1´p+ωα³f(λ,β) 3,1 n+ 1´p. (47) Let us estimate ωα³f(λ,β) 1,1 n+1 ´p.By (45), using the properties of the sequence {ψν}, we have (2m≤n+ 1 <2m+1) ω2 αµf(λ,β) 1,1 n+ 1¶p ≥ ≥C(α, β) ∞ X ν=m ψ2 2ν2−2ν(r+α)Ãν X k=m (λ2 2k−λ2 2k−1) + λ2 2m−1! ≥C(α, β)Ãλ2 2mω2µ1 2m¶+ ∞ X k=m (λ2 2k−λ2 2k−1)ω2µ1 2k¶! ≥C(α, β)Ãλ2 n+1ω2µ1 n+ 1¶+ ∞ X ν=n+2 ¡λ2 ν+1 −λ2 ν¢ω2µ1 ν¶!.(48) 7See, for example, [Ge]. EMBEDDING THEOREMS OF FUNCTION CLASSES, III 23 Now we estimate ωα³f(λ,β) 3,1 n+1 ´p.By (45), we have (2m≤n+ 1 < 2m+1): ω2 αµf(λ,β) 3,1 n+ 1¶p ≥ ≥C(α, β)2−2mα m X ν=0 22ναλ2 2ν¡ε2 2ν−ε2 2ν+1 ¢ ≥C1(α, β)2−2mα m X ν=0 22ναλ2 2νε2 2ν−C2(α, p)λ2 2m+1 ε2 2m+1 .(49) The Jackson inequality implies ω2 αµf(λ,β) 3,1 n+ 1¶p ≥C(α)E2 2m−1³f(λ,β) 3´p ≥C(α, β) ∞ X ν=m λ2 2ν¡ε2 2ν−ε2 2ν+1 ¢ ≥C(α, β)λ2 2mε2 2m.(50) Both estimates (49) and (50) imply ωαµf(λ,β) 3,1 n+ 1¶p ≥C(α, β)Ã(n+ 1)−2α n+1 X ν=1 λ2 νν2α−1ε2 ν!1 2 .(51) Using (41) and ν−ρλν↓, we get λ2 n+1ω2µ1 n+ 1¶+ + (n+ 1)−2α n+1 X ν=1 ν2(ρ+α)ω2µ1 ν¶¡λ2 νν−2ρ−λ2 ν+1(ν+ 1)−2ρ¢ ≤C(α, ρ)(n+ 1)−2α n+1 X ν=1 ν2α−1λ2 νε2 ν.(52) Combining estimates (47), (48), (51), (52), and ωα³f(λ,β) 13 ,1 n+1 ´p= O£ϕ¡1 n¢¤, we arrive at the condition in the right part of (11). Step 3. Let us prove the necessity in (11) for 1 <p<2.The proof for 1 < p ≤2 is similar to 2 ≤p < ∞. The only difference is that we use Paley’s theorem on Fourier coefficients instead of Zygmund’s theorem. In 24 B. SIMONOV AND S. TIKHONOV this case we consider the sum of f2(x) and the following function ε0+ (εp 1−εp 2)1 pcos x+ ∞ X ν=0 2ν(1 p−1)¡εp 2ν+1 −εp 2ν+2 ¢1 p 2ν+1 X µ=2ν+1 cos µx. (53) Step 4. To prove the necessity in (12) and (16), we consider the general case 1 ≤p≤ ∞.Let Φ be the class of all decreasing null-sequences. It is clear that 1 λn =O·ωµ1 n¶¸ ⇐⇒ ∀ γ={γn} ∈ Φγn λn =O·ωµ1 n¶¸. Let us assume that γn λn=O£ω¡1 n¢¤does not hold for all γ∈Φ and Wλ,β p⊂ Hp α+ρ[ω]. Then there exist γ={γn} ∈ Φ and {Cn↑ ∞} such that γmn λmn≥ Cnω³1 mn´.Further, we choose a subsequence {mnk}such that mnk+1 mnk ≥2 and γmnk≤2−k.Consider the series ∞ X k=0 γmnk λmnk cos(mnk+ 1)x. (54) Since ∞ P k=0 γmnk λmnk ≤1 λmn0 ∞ P k=0 1 2k<∞,there exists a function f4∈Lpwith Fourier series (54). Because ∞ P k=0 γmnk≤ ∞ P k=0 1 2k<∞,we have f(λ,β) 4∈Lp, i.e., f4∈Wλ,β p. On the other hand, by (20) and En−1(f)p≥C(|an|+|bn|), ωα+ρµf4,1 mnk¶p ≥C(α, ρ)Emnk(f4)p≥C(α, ρ)γmnk λmnk ≥C(α, ρ)Cnkωµ1 mnk¶, i.e., f4/∈Hp α+ρ[ω].This contradiction implies that the condition 1 λn= O£ω¡1 n¢¤is necessary for Wλ,β p⊂Hp α+ρ[ω]. Step 5. To prove the necessity in (13) and (17), we verify that for any ρ > 0 and for any 1 ≤p≤ ∞, Wλ,β pHα[ϕ]⊂Hp α+ρ[ω] =⇒ϕ¡1 n¢ λn =O·ωµ1 n¶¸.(55) First we remark that ϕ¡1 n¢ λn =O·ωµ1 n¶¸⇐⇒ ∀ γ={γn} ∈ Φγnϕ¡1 n¢ λn =O·ωµ1 n¶¸. EMBEDDING THEOREMS OF FUNCTION CLASSES, III 25 Assume that the relation in the right hand side of (55) does not hold. Then there exist γ={γn} ∈ Φ and {Cn↑ ∞} such that γmnϕ(1 mn) λmn≥Cnω³1 mn´. We choose a subsequence {mnk}such that mnk+1 mnk ≥2 and γmnk≤2−k. Because ∞ P k=0 γmnkϕ³1 mn´≤ϕ³1 m0´∞ P k=0 1 2k<∞,there exists a function f5∈Lpwith Fourier series ∞ X k=0 γmnkϕµ1 mnk¶cos(mnk+ 1)x. (56) For mnk≤n < mnk+1 , by Lemmas 3.1 and 4.3, we have ωαµf5,1 n¶p ≤Cωαµf5,1 n¶∞ ≤CÃn−α k X s=0 γmnsϕµ1 mns¶mα ns+ ∞ X s=k+1 γmnsϕµ1 mns¶! ≤CÃϕµ1 n¶k X s=0 γmns+ϕµ1 n¶∞ X s=k+1 γmns!≤Cϕ µ1 n¶. Therefore, f5∈Hp α[ϕ], i.e., setting 1 λ:= n1 λ1,1 λ2,1 λ3,· · · o, we have f(1 λ,−β) 5∈Wλ,β pHα[ϕ]. However, we have ωα+ρµf(1 λ,−β) 5,1 mnk¶p ≥CEmnk(f(1 λ,−β) 5)p≥C γmnkϕ³1 mnk´ λmnk ≥CCnkωµ1 mnk¶, i.e., f(1 λ,−β) 5/∈Hp α+ρ[ω].This contradicts our assumption. The proof of the necessity part in (12)-(13) and (16)-(17) is now complete. II. p= 1 or p=∞.Step 6. Let us prove the necessity in (14). Let Hp α+ρ[ω]⊂Wλ,β pand the series in (14) be divergent. Step 6(a): sin βπ 26= 0. Then the divergence of the series in (14) is equivalent to the divergence of the series ∞ P n=1 λn nω¡1 n¢. Let p= 1. We take a sequence εwhich is a qα+ρ, 1(ω)-sequence and consider the series ∞ X ν=1 (εν−εν+1)Kν(x).(57) 32 B. SIMONOV AND S. TIKHONOV ωα³f(λ,β) 10 ,1 n+1 ´p+ωα³f(λ,β) 11 ,1 n+1 ´p³ωα³(f10 +f11)(λ,β),1 n+1 ´p. Step 8(c): p= 1 and cos πα 26= 0. Since f(λ,β) 8(x)∼ ± ∞ P ν=1 λνbνsin νx, by Lemmas 3.1 and 4.1, we write (see Step 7(d)) ωα³f(λ,β) 8,1 n´p≥C(α)n−α°°°V(α) n(f(λ,β) 8−)(·)°°°p≥C(α, β)J4.(71) Also, by Lemma 4.1 and the Jackson inequality (20), we have ωαµf(λ,β) 8,1 n¶1 ≥C(α, β) ∞ X ν=n+1 λν bν ν ≥C(α, β) ∞ X ν=n+1 λν ∞ X j=2ν−1 εj−εj+1 j+ 1 ≥C(α, β, r) ∞ X j=4n−1 (εj−εj+1)λν. Using the properties of modulus of smoothness, we get (15). Step 8(d): p= 1 and cos πα 2= 0. We use that f12 := e f8∈L1and En(f12)p≤ Cεn+1 (see [Ge]). Then f8+f12 ∈Hp α+r[ω] and ωα³f(λ,β) 8,1 n´1≥C(α, β)J3, ωα³f(λ,β) 12 ,1 n´1≥C(α, β)J4. This completes the proof of Theorem 1. 5. Corollaries. Estimates of transformed Fourier series. Theorem 1 actually provides estimates of the norms and moduli of smoothness of the transformed Fourier series, i.e., the estimates of kϕkpand ωα(ϕ, δ)p, where ϕ∼σ(f, λ), by ωγ(f, δ)p. Analyzing Theorem 1, one can see that the following two conditions play a crucial role. The first is the behavior of the transforming sequence {λn}and the second is the alternation between Lp,1< p < ∞and Lp, p = 1,∞. We will investigate in detail some important examples for Lp,1< p < ∞ and for Lp, p = 1,∞, separately. 1. The case 1 <p<∞. Theorem 2. Let 1< p < ∞,θ= min(2, p)τ= max(2, p),α∈R+,and λ={λn}be a non-decreasing sequence of positive numbers. Let ρbe a non-negative number such that the sequence {n−ρλn}is non-increasing. EMBEDDING THEOREMS OF FUNCTION CLASSES, III 33 I. If for f∈L0 pthe series ∞ X n=1 ¡λθ n+1 −λθ n¢ωθ α+ρµf, 1 n¶p converges, then there exists a function ϕ∈L0 pwith Fourier series σ(f, λ), and kϕkp≤ ≤C(p, λ, α, ρ)(λθ 1kfkθ p+ ∞ X n=1 ¡λθ n+1 −λθ n¢ωθ α+ρµf, 1 n¶p)1 θ ,(72) ωαµϕ, 1 n+ 1¶p ≤ ≤C(p, λ, α, ρ)(n−αθ n X ν=1 ν(ρ+α)θ¡ν−ρθλθ ν−(ν+ 1)−ρθλθ ν+1¢ωθ α+ρµf, 1 ν¶p + ∞ X ν=n+2 ¡λθ n+1 −λθ n¢ωθ α+ρµf, 1 ν¶p +λθ n+1ωθ α+ρµf, 1 n¶p)1 θ .(73) II. If for f∈L0 pthere exists a function ϕ∈Lpwith Fourier series σ(f, λ), then (λτ 1kfkτ p+ ∞ X n=1 ¡λτ n+1 −λτ n¢ωτ α+ρµf, 1 n¶p)1 τ ≤C(p, λ, α, ρ)kϕkp,(74) (n−ατ n X ν=1 ν(ρ+α)τ¡ν−ρτ λτ ν−(ν+ 1)−ρτ λτ ν+1¢ωτ α+ρµf, 1 ν¶p + (75) ∞ X ν=n+2 ¡λτ n+1 −λτ n¢ωτ α+ρµf, 1 ν¶p +λτ n+1ωτ α+ρµf, 1 n¶p)1 τ ≤C(p, λ, α, ρ)ωαµϕ, 1 n+ 1¶p , ωα+ρµf, 1 n¶p ≤C(p, λ, α, ρ)kϕkp λn ,(76) ωα+ρµf, 1 n¶p ≤C(p, λ, α, ρ)ωα¡ϕ, 1 n¢p λn .(77) 34 B. SIMONOV AND S. TIKHONOV Inequalities (72)-(73) and (76)-(77) were proved in Theorem 1 (see the sufficiency in part I). To prove (74)-(75), we use similar reasoning, applied to the theorems by Littlewood-Paley, Marcinkiewicz, and the Minkowski’s inequality. As an important corollary of Theorem 2, Corollary 1. Let 1<p<∞,θ= min(2, p)τ= max(2, p). Then, for any k, r > 0, C1(∞ X ν=n+1 νrτ−1ωτ k+rµf, 1 ν¶p)1 τ ≤ωkµf(r),1 n¶p (78) ≤C2(∞ X ν=n+1 νrθ−1ωθ k+rµf, 1 ν¶p)1 θ , where C1=C1(p, k, r), C2=C2(p, k, r), n ∈N. The last two inequalities are an improvement compared to (6) and (7). Indeed, by properties of the modulus of smoothness and by the Jensen inequality, nrωk+rµf, 1 n¶p ≤C(k, r)(∞ X ν=n+1 νrτ−1ωτ k+rµf, 1 ν¶p)1 τ ; (∞ X ν=n+1 νrθ−1ωθ k+rµf, 1 ν¶p)1 θ ≤C(k, r) ∞ X ν=n+1 νr−1ωk+rµf, 1 ν¶p . Example. Let ψ(t) = trln−A(1/t) and 2 ≤p < ∞,1 2<A<1.For ωk+r(f, t)p³ψ(t), (6) and (7) give only Cln−A(1/t)≤ωk¡f(r), t¢p, and (78) gives C1ln−A+1/p(1/t)≤ωk¡f(r), t¢p≤C2ln−A+1/2(1/t), which is superior. Proof of Corollary 1 is obvious from (73) and (75) with r=ρ, because if f∈Lp,1< p < ∞, then f(r)∼σ(f, λ) for {λn=nr}. From (6), (7) and (78), one can see that it is natural to estimate ωα¡f(γ), δ¢pby ωα+r(f, t)p. Further analysis allows the distinction between three different types of such estimates. It will be convenient for us to write inequalities in integral form: EMBEDDING THEOREMS OF FUNCTION CLASSES, III 35 1. γ=r(see Corollary 1)8    δ Z0 t−rτ−1ωτ r+α(f, t)pdt   1 τ ¿ωα(f(r), δ)p¿   δ Z0 t−rθ−1ωθ r+α(f, t)pdt   1 θ ; (79) 2. γ=r−ε, 0< ε < r (see Theorem 2 for ρ=rand λn=nr−ε):    δ Z0 t−(r−ε)τ−1ωτ r+α(f, t)pdt+δατ 1 Zδ t−(r−ε+α)τ−1ωτ r+α(f, t)pdt   1 τ ¿ ¿ωα(f(r−ε), δ)p,(80) ωα(f(r−ε), δ)p¿ ¿   δ Z0 t−(r−ε)θ−1ωθ r+α(f, t)pdt+δαθ 1 Zδ t−(r−ε+α)θ−1ωθ r+α(f, t)pdt   1 θ ;(81) 3. γ=r+ε, 0< ε < α, (see [Po-Si-Ti,1]):    δ Z0 t−(r+ε)τ−1ωτ r+α(f, t)pdt   1 τ ¿δα−ε   1 Zδ t−(α−ε)θ−1ωθ α(f(r+ε), t)pdt   1 θ , (82) δα−ε   1 Zδ t−(α−ε)τ−1ωτ α(f(r+ε), t)pdt   1 τ ¿   δ Z0 t−(r+ε)θ−1ωθ r+α(f, t)pdt   1 θ . (83) One can consider a more general sequence than {λn=nγ},γ > 0. Let us define {Λn(s) := Λ(s, 1 n)}, where Λ(s, t) = Λ(s, r, t) =   1 Zt ξ(u)du +t−rs t Z0 ursξ(u)du  1 s ,(84) and non-negative function ξ(u) on [0,1] is such that ursξ(u) is summable. The sequence {Λn= Λn(s)}is often considered a transforming sequence. Example. 1. {Λn=nγ}, 0 < γ < r; 2. {Λn=nrln−An},A > 0; 3. {Λn= lnAn},A > 0. 8Here and further τ= max(2, p), θ = min(2, p). We will write A1¿A2, if A1≤CA2, C≥1. Also, if A1¿A2and A2¿A1, then A1³A2. 36 B. SIMONOV AND S. TIKHONOV Properties of Λ(s, t): Λ(s, t) is non-increasing on t, Λ(s, r, t)tris nondecreasing on t. By these properties and Theorem 2, one can obtain the following estimates, which are more general than (80)–(81) (see also [Po-Si,1]). Theorem 3. Let 1<p<∞,θ= min(2, p)τ= max(2, p),α, r > 0, and Λ = {Λn(s) := Λ(s, 1 n)}, where Λ(s, t)and ξ(t)were denoted above. I. If for f∈Lpthe integral I1:=    1 Z0 ξ(t)ωs α+r(f, t)pdt   1 s is finite for s=θ, then there exists a function ϕ∈Lpwith Fourier series σ(f, Λ),kϕkp≤C(p, ξ, α, r, s)I1, and ωα(ϕ, δ)p≤C(p, ξ, α, r, s)   δ Z0 t−rs−1 t Z0 ursξ(u)du ωs r+α(f, t)dt +δαs 1 Zδ tαs−1 1 Zt ξ(u)du ωs r+α(f, t)pdt   1 s =: C I2.(85) II. If for f∈Lpthere exists a function ϕ∈Lpwith Fourier series σ(f, Λ), then for s=τkϕkp≥C(p, ξ, α, r, s)I1, and ωα(ϕ, δ)p≥C(p, ξ, α, r, s)I2. In a similar way we can generalize inequalities (82)–(83). Theorem 4. Let 1<p<∞,θ= min(2, p)τ= max(2, p),α, r > 0, and Λ = {Λn(s) := Λ(s, 1 n)}, where Λ(s, t)and ξ(t)were denoted above. I. If for f∈Lpthe integral I3:=    1 Z0 ξ(t)ωs α+r(f, t)pdt   1 s is finite for s=θ, then there exists a function ϕ∈Lpwith Fourier series σ(f, Λ),kϕkp≤C(p, ξ, α, r, s)I3, and for any α1> α I4(τ, s) =: =:   δ(r+α)τΛτ(s, r, δ) 1 Zδ t−(r+α)τ−1 Λτ(s, r, t)ωτ r+α1(ϕ, t)pdt   1 τ ≤C(p, ξ, α, α1, r, s) EMBEDDING THEOREMS OF FUNCTION CLASSES, III 37 ×  Λ(s, r, δ)ωr+α(f, δ)p+   δ Z0 t−rθ−1 t Z0 urτ ξ(u)duωθ r+α(f, t)pdt   1 θ   =: CI5(θ) (86) II. If for f∈Lpthere exists a function ϕ∈Lpwith Fourier series σ(f, Λ), then for s=τand for any α1> α kϕkp≥C(p, λ, α, r, s)I3and I4(s, θ)≥C(p, ξ, α, α1, r, s)I5(s). 2. The case p =1,∞. Estimates ωα(ϕ, t)pby ωr+α(f, t)pfor this case follow from Theorem 1 (see part II). We will write only the commonly used estimates of ωα(f(r), t)p and ωα(˜ f(r), t)pby ωr+α(f, t)p. Corollary 2. If p= 1,∞, then (6), (7) are true for any k, r > 0. As a corollary of Theorem 1 for {λn=nρ},ρ≥0 and β=ρ+ 1, Corollary 3. Let p= 1,∞. One has Hp α+ρ[ω]⊂f Wρ p⇐⇒ ∞ X n=1 nρ−1ωµ1 n¶<∞. For p= 1, ρ = 0 and α= 1 Corollary 3 gives the answer for the question by F. Moricz [Mo]. Also, we mention the papers [Be,S,1], [Ha-Sh], [St,S], where the embedding results were obtained in the necessity part. Corollary 4. Let p= 1,∞, and r, α, ε > 0. I. If for f∈Lpthe series ∞ P ν=1 νr−1ωr+α+ε¡f, 1 ν¢pconverges, then there exists ˜ f(r)∈Lp, and ωα³e f(r),1 n´p≤ ≤C(r, α, ε)Ãn−α n X ν=1 νr+α−1ωr+α+ε³f, 1 ν´p + ∞ X ν=n+1 νr−1 ωr+α+ε³f, 1 ν´p!, n∈N. II. If for f∈Lpthere exists ˜ f(r)∈Lp, then ωr+α+εµf, 1 n¶p ≤C(r, α, ε) nrωαµ˜ f(r),1 n¶p , n ∈N.(87) Using direct and inverse approximation theorems, we can rewrite inequality in Iin the following equivalent form (compare to [Ba-St]): 38 B. SIMONOV AND S. TIKHONOV Corollary 5. Let p= 1,∞, and r, α > 0. If for f∈Lpthe series ∞ P ν=1 νr−1Eν(f)pconverges, then there exists ˜ f(r)∈Lp, and ωα³e f(r),1 n´p≤C(r, α, ε)Ãn−α n X ν=1 νr+α−1Eν(f)p+ ∞ X ν=n+1 νr−1Eν(f)p!, n∈N. 6. Final remarks 1. The Weyl class Wλ,β pcoincides with the class of functions from L(0,2π) such that their Fourier series can be presented in the following form a0(f) 2+ ∞ X ν=1 1 πλν 2π Z0 ψ(x−t) cos³νt −πβ 2´dt, ψ(x)∈L0. Further, consider the case when ∞ X ν=1 1 λν cos³νt −πβ 2´ is the Fourier series of a summable function Dλ,β(t). For example, it is so if {λν↑ ∞} (n↑), and ∞ P ν=1 1 νλν<∞. Then the elements of Wλ,β pcan differ only by the mean value from functions f, which have the following representation by convolution, f(x) = 1 π 2π Z0 ψ(x−t)Dλ,β(t)dt, ψ(x)∈L0. Here, ψcoincides almost everywhere with f(λ,β).See, for example, [Bu-Ne, Ch. 11]. A representation by convolution was first considered in [Sz]. 2. The generalization of Wr pE[ξ] is the class Wλ,β pE[ξ] = nf∈Wλ,β p:En³f(λ,β)´p=O[ξ(1/n)]o. We do not consider in this paper the embedding results between Wλ,β pE[ξ], Wλ,β pHα[ϕ] and E[ϕ]≡W{1},0 pE[ϕ]. We only notice that some results of such types follow from direct and inverse theorems (1)-(4). Also, we mention the papers [Ha-Li], [Og], [Si-Ti], and [St,A]. 3. The results of sections 2 and 5 can be generalized to a multidimensional case. We only write here the following estimates for the mixed modulus of smoothness ωα1,α2(f, δ1, δ2)pof a function f(in the Lpmetric) of orders α1 EMBEDDING THEOREMS OF FUNCTION CLASSES, III 39 and α2(α1, α2>0) with respect to the variables x1and x2, respectively (see, for example, [Po-Si-Ti,1]). Theorem 5. Let f(x1, x2)∈L0 p,1< p < ∞,θ= min(2, p), τ = max(2, p), and let α1, α2,r1, r2>0, I. If J1(θ) :=   1 Z0 1 Z0 t−r1θ−1 1t−r2θ−1 2ωθ r1+α1,r2+α2(f, t1, t2)pdt1dt2  1 θ <∞, then fhas a mixed derivative in the sense of Weyl f(r1,r2)∈L0 p, kf(r1,r2)kp≤C(p, r1, r2)J1(θ), and ωα1,α2(f(r1,r2), δ1, δ2)p≤ ≤C(p, α1, α2, r1, r2)  δ1 Z0 δ2 Z0 t−r1θ−1 1t−r2θ−1 2ωθ r1+α1,r2+α2(f, t1, t2)pdt1dt2  1 θ =: C J2(θ). II. If fhas a mixed derivative in the sense of Weyl f(r1,r2)∈L0 p, then J1(τ)≤C(p, r1, r2)kf(r1,r2)kp, and J2(τ)≤C(p, α1, α2, r1, r2)ωα1,α2(f(r1,r2), δ1, δ2)p. Theorem 6. Let f(x1, x2)∈L0 p,p= 1,∞, and let α1, α2, r1, r2>0. I. If J3:= 1 Z0 1 Z0 t−r1−1 1t−r2−1 2ωr1+α1,r2+α2(f, t1, t2)pdt1dt2<∞, then fhas a mixed derivative in the sense of Weyl f(r1,r2)∈L0 p, kf(r1,r2)kp≤C(r1, r2)J3, and ωα1,α2(f(r1,r2), δ1, δ2)p≤ ≤C(α1, α2, r1, r2) δ1 Z0 δ2 Z0 t−r1−1 1t−r2−1 2ωr1+α1,r2+α2(f, t1, t2)pdt1dt2. 40 B. SIMONOV AND S. TIKHONOV II. If fhas a mixed derivative in the sense of Weyl f(r1,r2)∈L0 p, then ωr1+α1,r2+α2(f, δ1, δ2)p≤C(α1, α2, r1, r2)δr1 1δr2 2kf(r1,r2)kp, and ωr1+α1,r2+α2(f, δ1, δ2)p≤C(α1, α2, r1, r2)δr1 1δr2 2ωα1,α2(f(r1,r2), δ1, δ2)p. See [Po-Si-Ti,1] for the estimates of transformed series in a multidimensional case. 4. In view of inequalities (79) and (80) - (83), the problem of finding the estimates of ωα(ϕ, t)pby ωα+r(f, t)parises such as in the case ϕ∼σ(f, λ) with λn=nrlnAn. If A < 0 (that is the analogue of the case λn=nr−ε), then estimates ωα(ϕ, t)pfollow from Theorem 3. For example, if p= 2 and ϕ∼σ(f, nrlnAn), A < 0, then ω2 β(ϕ, δ)2³ δ Z0 t−2r−1 ln2|A|¡2 t¢ω2 r+β(f, t)2dt +δ2β 1 Zδ t−2(r+β)−1 ln1+2|A|¡2 t¢ω2 r+β(f, t)2dt. (88) Note that the only difference between (88) and (80)-(81) is related to the replacement of n−εby lnAn. The case A > 0 (that is the analogue of the case λn=nr+ε) is interesting. For p= 2 and ϕ∼σ(f, nrlnAn), A > 0 we have δ2βln2Aµ2 δ¶1 Zδ t−2β−1 ln1+2A¡2 t¢ω2 β(ϕ, t)2dt +ω2 β(ϕ, δ)2³ ³ δ Z0 t−2r−1ln2Aµ2 t¶ω2 r+β(f, t)2dt. (89) Comparing this with the estimates (82)-(83) one can remark that the new item ω2 β(ϕ, δ)2has appeared in (89). Thus, this case has essential distinctions. See for detail [Po-Si-Ti,2], [Ti,S,1]. 5. Defining the class Wλ,β pHα[ϕ] we assumed that ϕ∈Φα. This restriction is natural for a majorant of the modulus of smoothness of order α(see [Ti,S,3]). 6. In Theorem 1 (item II) we used the inequality ∞ P ν=n+1 1 νλν≤C λn. We remark that it is equivalent to the following condition: there exists ε > 0 such that the sequence {n−ελn}is almost increasing, i.e., n−ελn≤Cm−ελm, C≥1, n≤m. This and other equivalence results can be found in [Ba-St] and [Ti,S,2]. EMBEDDING THEOREMS OF FUNCTION CLASSES, III 41 7. Acknowledgements The first named author acknowledges financial support from Russian Fund of Basic Research (project 03-01-00080). The second named author acknowledges financial support from the European Commission (Contract MIF1-CT-2004-509465). The paper was written while they were staying at the Centre de Recerca Matematica (CRM) in Barcelona, Spain. The authors are grateful to the staff of CRM and in particular to its director, Manuel Castellet, for their hospitality. References [Ba,1] N. K. Bari, On best approximation of two conjugate functions by trigonometric polynomials // Izv. Akad. Nauk SSSR. Ser. Mat., 19, 285–302, 1955. [Ba,2] N. K. Bary, A treatise on trigonometric series. Vol. I, II. Pergamon Press, 1964. [Ba-St] N. K. Bari; S. B. Steˇckin, Best approximations and differential properties of two conjugate functions // Trudy Moskov. Mat. Obˇsˇc., 5, 483–522, 1956. [Be-L¨o] J. Bergh, J. L¨ofstr¨om, Interpolation spaces. An introduction. Springer-Verlag, 1976. [Be,S,1] S. Bernstein, On the best approximation of continuous functions by polynomials of a given degree // Comm. Soc. Math. Kharkow, Ser. 2, V. 13, 49–194, 1912; reprinted in Collected works (see [Be,S,4]), Vol. I, pp. 11–104; for French transl. see [Be,S,2]. [Be,S,2] S. Bernstein, Sur l’orde de la meilleure approximation des fonctions continues par les polynomes de degr´e donn´e // M´em. Cl. Sci. Acad. Roy. Belg., 4, 1–103, 1912. Available online at http://www.math.technion.ac.il/hat/fpapers/ber2.pdf [Be,S,3] S. Bernstein, Sur les recherches recentes relatives `a la meilleure approximation des fonctions continues par des polynˆomes. [Proc. 5. Intern. Math. Congr.] 1, 256– 266, 1913; reprinted in Collected works (see [Be,S,4]), Vol. II, pp. 112–124. Available online at http://www.math.technion.ac.il/hat/fpapers/ber.pdf [Be,S,4] S. N. Bernstein, Collected works. Vol. I and II. Izdat. Akad. Nauk, Moscow, 1952, 1954. [Be,O] O. V. Besov, Investigation of a class of function spaces in connection with imbedding and extension theorems // Trudy. Mat. Inst. Steklov, 60, 42–81, 1961. [Be-Il-Ni] O. V. Besov, V. P. Il’in, S. M. Nikol’skii, Integral representations of functions and imbedding theorems. Vol. I, II, Washington, D.C., 1978, 1979. [Bu-Ne] P. L. Butzer, R.J. Nessel, Fourier analysis and approximation. Vol. 1: Onedimensional theory. Academic Press, N.Y.-L., 1971. [Bu-Dy-G¨o-St] P.L. Butzer, H. Dyckhoff, E. G¨oerlich, R.L. Stens, Best trigonometric approximation, fractional order derivatives and Lipschitz classes // Can. J. Math., 29, 781–793, 1977. [De-Lo] R. A. DeVore, G. G. Lorentz, Constructive approximation. Berlin: Springer- Verlag, 1993. [Di-Hr-Iv] Z. Ditzian, V. H. Hristov, K. G. Ivanov, Moduli of smoothness and K- functionals in Lp,0<p<1 // Constr. Approx., 11, no. 1, 67–83, 1995. [Ge] V. ` E. Ge˘ıt, The accuracy of certain inequalities in approximation theory // Math. Notes, 10, 768-776, 1971-72; Transl. from Mat. Zametki, 10, 571–582, 1971. [Ha-Li] G.H. Hardy, J.S. Littlwood, Some properties of fractional integrals // Math. Z., 27, 565–606, 1928.