# On Pointwise Approximation of Conjugate Functions by Some Hump Matrix Means of Conjugate Fourier Series

Authors: W. Łenski; B. Szal
Journal: Journal of Function Spaces
 (2015)

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## Abstract

The results generalizing some theorems onN,pnE,γ summability are shown. The same degrees of pointwise approximation as in earlier papers by weaker assumptions on considered functions and examined summability methods are obtained. From presented pointwise results, the estimation on norm approximation is derived. Some special cases as corollaries are also formulated.


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## Body

## 1. Introduction

LetLp(1≤p<∞) (resp., L∞) be the class of all 2π-periodic real-valued functions integrable in the Lebesgue sense with pth power (essentially bounded) over Q=[-π,π] with the norm(1)f≔f·Lp=∫Qftpdt1/pwhen1≤p<∞,ess⁡supt∈Q⁡ftwhenp=∞and consider the conjugate trigonometric Fourier series(2)S~fx≔∑ν=1∞aνfsin⁡νx-bνfcos⁡νxwith the partial sums S~kf. We know that if f∈L1 then (3)f~x≔-1π∫0πψxt12cot⁡t2dt=limϵ→0+⁡f~x,ϵ,where (4)f~x,ϵ≔-1π∫ϵπψxt12cot⁡t2dt,with (5)ψxt≔fx+t-fx-t,exists for almost all x [1, Th.(3.1)IV].LetA≔an,k and B≔bn,k be infinite lower triangular matrices of real numbers such that(6)an,k≥0,bn,k≥0,whenk=0,1,2,…,n,an,k=0,bn,k=0,whenk>n,∑k=0nan,k=1,∑k=0nbn,k=1,wheren=0,1,2,…,and let, for m=0,1,2,…,n,(7)An,m=∑k=0man,k,A¯n,m=∑k=mnan,k.Let the AB-transformation of S~kf be given by(8)T~n,A,Bfx≔∑r=0n∑k=0ran,rbr,kS~kfxn=0,1,2,….We define two classes of sequences (see [2]).Sequencec≔ck of nonnegative numbers tending to zero is called theRest Bounded Variation Sequence, or briefly c∈RBVS, if it has the property (9)∑k=m∞ck-ck+1≤Kccm,for all positive integer m, where Kc is a constant depending only on c.Sequencec≔ck of nonnegative numbers will be called theHead Bounded Variation Sequence, or briefly c∈HBVS, if it has the property(10)∑k=0m-1ck-ck+1≤Kccm,for all positive integer m, or only for all m≤n if the sequence c has only finite nonzero terms and the last nonzero term is cn.Now, we define the other classes of sequences.Following Leindler [3], sequence c≔cr of nonnegative numbers tending to zero is called theMean Rest Bounded Variation Sequence, or briefly c∈MRBVS, if it has the property(11)∑r=m∞cr-cr+1≤Kc1m+1∑r≥m/2mcr,for all positive integer m.Analogously as in [4], sequence c≔cr of nonnegative numbers will be called theMean Head Bounded Variation Sequence, or briefly c∈MHBVS, if it has the property(12)∑r=0n-m-1cr-cr+1≤Kc1m+1∑r=n-mncr,for all positive integers m<n, where the sequence c has only finite nonzero terms and the last nonzero term is cn. It is clear that (see [5]) (13)RBVSMRBVS,HBVSMHBVS.Consequently, we assume that the sequence Kαnn=0∞ is bounded, that is, that there exists a constant K such that(14)0≤Kαn≤Kholds for all n, where Kαn denote the constants for the sequences αn=an,rr=0n, n=0,1,2,… appearing in the inequalities (11) and (12) as Kc.Now we can give the conditions to be used later on. We assume that for alln and 0≤m<n(15)∑r=mn-1an,r-an,r+1≤K1m+1∑r≥m/2man,r,∑r=0n-m-1an,r-an,r+1≤K1m+1∑r=n-mnan,rhold if an,rr=0n belongs to MRBVS and MHBVS, for n=1,2,…, respectively.We also define two hump matrices in the following way: a lower triangular matrixC=cn,k  is called amaximal hump matrix if, for each n, there exists integer k0=k0(n), such that cn,kk=0k0-1  is nondecreasing for 0≤k<k0 and cn,kk=k0n  is nonincreasing for k0≤k≤n, but otherwise we will have aminimal hump matrix. The hump matrices were defined and considered in [6, 7].As a measure of approximation off~ by T~n,A,Bf, we use the pointwise modulus of continuity of f in the space Lp defined by the formula(16)w~xpfδβ=1δ∫0δψxusinβ⁡u2pdu1/pwhen1≤p<∞,ess⁡sup0<u≤δ⁡ψxusinβ⁡u2whenp=∞and the classical one (17)ω~βfδLp=sup0<t≤δ⁡sinβ⁡t2ψ·tLp.The deviationT~n,Af-f~=T~n,A,Bf-f~, with br,r=1 and 0 otherwise, was estimated at the point as well as in the norm of Lp by Qureshi [8] and Lal and Nigam [9]. These results were generalized by Qureshi [10]. The next generalization was obtained by Lal [11]. In the case(18)an,r=1n+1,whenr=0,1,2,…,n,an,r=0,whenr>n,br,k=rkγr1+γr,whenk=0,1,2,…,r,bn,r=0,whenk>rwithγ>0,the deviation T~n,A,Bf-f~ was estimated by Sonker and Singh [12] as follows.Theorem 1.
Letf(x) be a 2π-periodic, Lebesgue integrable function which belongs to the Lip⁡α,p-class with p≥1 and αp≥1. Then the degree of approximation of f~(x), the conjugate of f(x) by (C,1)(E,γ) means of series (2), is given by(19)1n+1∑r=0n11+γr∑k=0rrkγkS~kf·-f~·Lp=On+1-α+1/p,provided(20)∫0π/n+1ψxttαpdt1/p=On+1-1,∫π/n+1πt-δψxttαpdt1/p=On+1δ,where δ is an arbitrary positive number with (α+δ)q+1<0 and p-1+q-1=1, p>1.In this paper we will consider the deviationsT~n,A,Bf·-f~· and T~n,A,Bf·-f~·,π/n+1 in general form. In the theorems we formulate the general conditions for the functions and the modulus of continuity obtaining the same degrees of approximation as above and sometimes essentially better one. Finally, we also give some results on norm approximation with essentially better degrees of approximation. The obtained results generalize the results from [4, 9].We will writeI1≪I2 if there exists positive constant K, sometimes depending on some parameters, such that I1≤KI2.


## 2. Statement of the Results

Let(21)Lpw~xβ=f∈Lp:w~xpfδβ≤w~xδ,where w~x is a positive, with w~x0=0, and nondecreasing continuous function.We can now formulate our main results. At the beginning, we formulate the results on the degrees of pointwise summability of conjugate series.Theorem 2.
Letf∈L1. If matrix A is a maximal or minimal hump matrix with k0-1=On-1 such that an,kk=0n∈MHBVS∪MRBVS and(22)∑r=μν∑k=0rbr,kcos⁡2k+1t2≪τfor 0≤μ≤ν, then(23)T~n,A,Bfx-f~x,πn+1=On+1β1n+1∑k=0nw~x1fπk+1β,for almost all considered x, where β∈0,1.Theorem 3.
Letf∈Lpw~xβ with 1<p<∞, and let w~x satisfy(24)∫0π/n+1w~xttsinβ⁡t/2p/p-1dtp-1/p=On+1β+1/pw~xπn+1,(25)∫0π/n+1ψxtw~xtpsinβp⁡t2dt1/p=On+1-1/pwith some β≥0. If the entries of matrix B satisfy condition (22) for 0≤μ≤ν and if matrix A is a maximal or minimal hump matrix with k0-1=On-1 such that an,kk=0n∈MHBVS∪MRBVS, then (26)T~n,A,Bfx-f~x=On+1β1n+1∑k=0nw~xπk+1,for almost all considered x such that f~x exists.Next, we formulate the results on estimates ofLp norm of the deviation considered above. In case of the deviation T~n,A,Bf·-f~·, let (27)Lpω~β=f∈Lp:ω~βfδLp≤ω~δ,where ω~ is positive, with ω~0=0, and almost nondecreasing continuous function.Theorem 4.
Letf∈Lp  1≤p≤∞. If the entries of matrix B satisfy condition (22) for 0≤μ≤ν and if matrix A is a maximal or minimal hump matrix with k0-1=On-1 such that an,kk=0n∈MHBVS∪MRBVS, then(28)T~n,A,Bf·-f~·,πn+1Lp=On+1β1n+1∑k=0nω~βfπk+1Lp,where 0≤β≤1.Theorem 5.
Letf∈Lpω~β with 1<p<∞, where ω~ instead of w~x satisfies (24) with some β≥0. If the entries of matrix B satisfy condition (22) for 0≤μ≤ν and if matrix A is a maximal or minimal hump matrix with k0-1=On-1 such that an,kk=0n∈MHBVS∪MRBVS, then(29)T~n,A,Bf·-f~·Lp=On+1β1n+1∑k=0nω~πk+1.Finally, we give corollary and remarks as an application of our results.Takingan,r=pn-r/∑ν=0n‍pν when r=0,1,2,…,n and an,r=0 when r>n with pν>0, pν≤pν+1 and br,k=rkγk/1+γr when k=0,1,2,…,r and bn,r=0 when k>r with γ>0, Theorem 2 (Theorem 3 analogously) implies the following.Corollary 6.
Iff∈L1, then (30)1∑ν=0npν∑r=0npr1+γr∑k=0rrkγkS~kfx-f~x,πn+1=On+1β1n+1∑k=0nw~x1fπk+1β,1∑ν=0npν∑r=0npn-r1+γr∑k=0rrkγkS~kfx-f~x,πn+1=On+1β1n+1∑k=0nw~x1fπk+1β,for almost all considered x, where 0≤β≤1, pν>0, pν≤pν+1, and γ>0.Remark 7.
In special case, ifpr=pn-r=1, then(31)1n+1∑r=0n11+γr∑k=0rrkγkS~kfx-f~x,πn+1=On+1β1n+1∑k=0nw~x1fπk+1β,and if f∈Lip⁡α,p, then(32)1n+1∑r=0n11+γr∑k=0rrkγkS~kfx-f~x,πn+1=On+1β-αwhen0<α<1On+1β-1log⁡n+1whenα=1,for almost all considered x, where 0≤β≤1 and γ>0.Remark 8.
Takingβ=0, we have, by Theorem 3 with w~xδ=δα, for 0<α<1 and αp>1, the estimate like in [12] with the better order of approximation without any additional assumptions.Remark 9.
Analyzing the proofs of Theorems2–5, we can deduce that, taking the assumption an,kk=0n∈RBVS or an,kk=0n∈HBVS instead of an,kk=0n∈MRBVS or an,kk=0n∈MHBVS, respectively, we obtain the results like that from [13].


## 3. Auxiliary Results

We begin this section by some notations following A. Zygmund [1, Section 5 of Chapter II].It is clear that(33)S~kfx=-1π∫-ππfx+tD~ktdt,T~n,A,Bfx=-1π∫-ππfx+t∑r=0n∑k=0ran,rbr,kDk~tdt,where(34)Dk~t=∑ν=0ksin⁡νt=cos⁡t/2-cos⁡2k+1t/22sin⁡t/2.Hence(35)T~n,A,Bfx-f~x,πn+1=-1π∫0π/n+1ψxt∑r=0n∑k=0ran,rbr,kDk~tdt+1π∫π/n+1πψxt∑r=0n∑k=0ran,rbr,kDk∘~tdt,T~n,A,Bfx-f~x=1π∫0πψxt∑r=0n∑k=0ran,rbr,kDk∘~tdt,where (36)Dk∘~t=12cot⁡t2-∑ν=0ksin⁡νt=cos⁡2k+1t/22sin⁡t/2.Now, we formulate some estimates for the conjugate Dirichlet kernels.Lemma 10 (see [1]).
If0<t≤π/2, then(37)Dk∘~t≤π2tand for any real t one has (38)Dk~t≤12kk+1t,Dk~t≤k+1.Lemma 11.
Letbr,kk=0r be such that (22) for 0≤μ≤ν holds. If an,kk=0n∈MRBVS, then (39)∑r=0n∑k=0ran,rbr,kDk∘~t≪τAn,τ,and if an,kk=0n∈MHBVS, then(40)∑r=0n∑k=0ran,rbr,kDk∘~t≪τA¯n,n-τwith τ=π/t and t∈π/n+1,π, for n=0,1,2,….Proof.
Let(41)Knt≔∑r=0n∑k=0ran,rbr,kcos⁡2k+1t2.
The relationan,kk=0n∈MRBVS implies(42)an,s-an,m≤an,m-an,s≤∑k=ms-1an,k-an,k+1≪∑k=rn-1an,k-an,k+1≪1r+1∑k≥r/2ran,k,0≤r≤m<s≤nwhence(43)an,s≪an,m+1r+1∑k≥r/2ran,k,0≤r≤m<s≤n,and thus, by our assumption,(44)∑l=τ+1r∑k=0lbl,kcos⁡2k+1t2≪τ,we obtain(45)Knt=∑r=0τ+∑r=τ+1nan,r∑k=0rbr,kcos⁡2k+1t2≤∑r=0τan,r∑k=0rbr,k+∑r=τ+1n-1an,r-an,r+1·∑l=τ+1r∑k=0lbl,kcos⁡2k+1t2+an,n∑l=τ+1n∑k=0lbl,kcos⁡2k+1t2≪An,τ+1τ+1∑k≥τ/2τan,k+1τ+1∑k≥τ/2τan,nτ≤An,τ+1τ+1∑k≥τ/2τan,k+1τ+1∑k≥τ/2τan,k+1k+1∑l≥k/2kan,lτ≤An,τ+1τ+1∑k≥τ/2τan,k+1τ+1∑k≥τ/2τan,k+1τ/2+1∑l≥τ/4τan,lτ≤An,τ+1τ+1∑k≥τ/2τan,k+1τ+1∑k≥τ/2τan,k+1τ/2+1∑l≥τ/4τan,lτ≤An,τ+1τ+1∑k≥τ/2τan,k+1τ+1∑k≥τ/2τan,k+1τ+1∑l=0τan,l·τ≤An,τ+3∑l=0τan,l=4An,τ.The relation an,kk=0n∈MHBVS implies (46)an,m-an,s≤an,m-an,s≤∑k=ms-1an,k-an,k+1≪∑k=0r-1an,k-an,k+1≪1n-r+1∑k=rnan,k,0≤m<s≤r≤nwhence(47)an,m≪an,s+1n-r+1∑k=rnan,k,0≤m<s≤r≤n,and thus, by our assumption,(48)∑l=0r∑k=0lbl,kcos⁡2k+1t2≪τ,we get(49)Knt=∑r=n-τn+∑r=0n-τan,r∑k=0rbr,kcos⁡2k+1t2=A¯n,n-τ+∑r=0n-τ-1an,r-an,r+1·∑l=0r∑k=0lbl,kcos⁡2k+1t2+an,n-τ-1∑l=0n-τ∑k=0lbl,kcos⁡2k+1t2≪A¯n,n-τ+1τ+1∑k=n-τnan,k+an,n-τ-1τ≪A¯n,n-τ+1τ+1∑k=n-τnan,k+1τ+1∑k=n-τn-τ/2an,n-τ-1τ≪A¯n,n-τ+1τ+1∑k=n-τnan,k+1τ+1∑k=n-τn-τ/2an,k+1τ+1∑ν=n-τ/2nan,ντ≤A¯n,n-τ+∑k=n-τnan,k+∑k=n-τn1τ+1∑ν=n-τnan,ν≪A¯n,n-τ.
Now, our proof is complete.Lemma 12.
If matrixA is a maximal or minimal hump matrix with k0-1=On-1, then (50)n+1m+1∑r=0man,r=O1or (51)n+1m+1∑r=0man,n-r=O1for m=0,1,2,…, respectively.Proof.
Since the above formulas are similar, we prove the first one only. If0<m≤k0, then (52)1m∑r=0m-1an,r-1m+1∑r=0man,r=1mm+1∑r=0man,r-an,mm=1mm+1∑r=0man,r-an,m≤0,whence 1/m+1∑r=0m‍an,r nondecreases with respect to m and therefore (53)1m+1∑r=0man,r≤1k0+1∑r=0k0an,r≤1k0+1∑r=0nan,r=1k0+1=On-1.But if m≥k0, then(54)1m+1∑r=0man,r≤1k0+1∑r=0man,r≤1k0+1∑r=0nan,r=1k0+1=On-1and our proof is complete.Lemma 13.
Iff∈Lp1≤p≤∞ and β≥0, then(55)w~xpfπn+1β≤2n+1∑r=0nw~xpfπr+1βp1/pholds for every natural n and all real x.Proof.
The proof follows by the easy account(56)w~xpfπn+1βp=n+1π∫0π/n+1ψxupdu∑r=0n2r+1n+1n+2≤2n+1∑r=0nr+1n+1πn+2∫0π/n+1ψxupdu≤2n+1∑r=0nr+1π∫0π/r+1ψxupdu=21n+1∑r=0nw~xpfπr+1βp.Now, our proof is complete.


## 4. Proofs of the Results

### 4.1. Proof of Theorem2

We start with the obvious relations(57)T~n,A,Bfx-f~x,πn+1=-1π∫0π/n+1ψxt∑r=0n∑k=0ran,rbr,kDk~tdt+1π∫π/n+1πψxt∑r=0n∑k=0ran,rbr,kDk∘~t=I1~+I2∘~,T~n,Afx-f~x,πn+1≤I1~+I2∘~.By Lemmas10 and 13, we have(58)I1~≪n+12∫0π/n+1tψxtdt≤n+12∫0π/n+1tψxtsinβ⁡t2sin-β⁡t2dt≪n+12∫0π/n+1ψxtsinβ⁡t2dtπn+11-β≪n+1βw~x1fπn+1β≪n+1β-1∑k=0nw~x1fπk+1β,for β∈0,1.Using Lemmas11 and 12 we obtain(59)I2∘~≪∫π/n+1πψxtt∑k=0τan,kdt≪∫π/n+1πψxttτn+1dtor (60)I2∘~≪∫π/n+1πψxtt∑k=n-τnan,kdt=∫π/n+1πψxtt∑k=0τan,n-kdt≪∫π/n+1πψxttτn+1dt,whence (61)I2∘~≪1n+1∫π/n+1πt-2ψxtsinβ⁡t21sinβ⁡t/2dt≪n+1β1n+1∫π/n+1πt-2ψxtsinβ⁡t2dt=n+1β1n+1·∫π/n+1πt-2ddt∫0tψxusinβ⁡u2dudt=n+1β-1t-2∫0tψxusinβ⁡u2dut=π/n+1π+2∫π/n+1πt-3∫0tψxusinβ⁡u2dudt=n+1β-1·π-1w~x1fπβ-πn+1-1w~x1fπn+1β+2∫π/n+1πt-2w~x1ftβdt≪n+1β-1·∑k=0nw~x1fπk+1β+∫1n+1w~x1fπtβdt≪n+1β-1·∑k=0nw~x1fπk+1β.Collecting these estimates, we obtain the desired result.


### 4.2. Proof of Theorem3

We start with the obvious relations(62)T~n,Afx-f~x=1π∫0π/n+1ψxt∑r=0n∑k=0ran,rbr,kDk∘~tdt+1π∫π/n+1πψxt∑r=0n∑k=0ran,rbr,kDk∘~tdt=I1∘~+I2∘~,T~n,Afx-f~x≤I1∘~+I2∘~.By the Hölder inequality1/p+1/q=1, Lemma 10, (25), and (24),(63)I1∘~≪∫0π/n+1ψxttdt≤∫0π/n+1ψxtw~xtsinβ⁡t2pdt1/p·∫0π/n+1w~xttsinβ⁡t/2qdt1/q≪n+1β·w~xπn+1=n+1β-1w~xπn+1∑k=0n1≪n+1β-1∑k=0nw~xπk+1.We can estimate the termI2∘~ by the same way like in the proof of Theorem 2: (64)I2∘~≪∫π/n+1πψxtt∑k=n-τnan,kdt=∫π/n+1πψxtt∑k=0τan,n-kdt≪∫π/n+1πψxttτn+1dt≪n+1β-1∑k=0nw~x1fπk+1β≪n+1β-1∑k=0nw~xπk+1.Collecting these estimates, we obtain the desired result.


### 4.3. Proofs of Theorems4-5

The proofs are similar to these above and follow from the evident inequality(65)w~·pfδβLp≪ω~βfδLpand additionally in case of Theorem 5 from the estimate(66)∫0π/n+1ψ·tω~tpsinβp⁡t2dt1/pLp≤∫0π/n+1ψ·sinβ⁡t2Lpω~tdt1/p≤∫0π/n+1ω~βftLpω~tpdt1/p≤∫0π/n+1dt1/p=Oxn+1-1/pfor f∈Lpω~β.


### 4.4. Proof of Corollary6

First of all we note that our matrixA is the hump matrix with k0=n.Next, we have to verify the assumptions of Lemma11. For the first one, we note that any nondecreasing sequence belongs to the class MRBVS and any nonincreasing sequence belongs to the class MHBVS. The second one follows from the following calculations: (67)∑r=μν1+γ-r∑k=0rrkγkcos⁡2k+1t2=Re⁡∑r=μν1+γ-r∑k=0rrkγkexp⁡2k+1it2=Re⁡∑r=μν1+γ-rexp⁡it2∑k=0rrkγkexp⁡ikt=Re⁡∑r=μν1+γ-rexp⁡it21+γexp⁡itr=Re⁡exp⁡it2∑r=μν1+γexp⁡it1+γr=Re⁡exp⁡it2·1+γexp⁡it1+γμ·1-1+γexp⁡it/1+γν-μ+11-1+γexp⁡it/1+γ≤1-1+γexp⁡it/1+γν-μ+11-1+γexp⁡it/1+γ≤21-1+γexp⁡it/1+γ=21+γ1+γ-1-γexp⁡it=21+γγ1-exp⁡it=21+γγ1-cos⁡t2+sin2⁡t=21+γγ2-2cos⁡t=21+γγ22sin2⁡t/2=1+γγsin⁡t/2≤1+γγπt≪τ.Thus proof is complete.


## 4.1. Proof of Theorem2

We start with the obvious relations(57)T~n,A,Bfx-f~x,πn+1=-1π∫0π/n+1ψxt∑r=0n∑k=0ran,rbr,kDk~tdt+1π∫π/n+1πψxt∑r=0n∑k=0ran,rbr,kDk∘~t=I1~+I2∘~,T~n,Afx-f~x,πn+1≤I1~+I2∘~.By Lemmas10 and 13, we have(58)I1~≪n+12∫0π/n+1tψxtdt≤n+12∫0π/n+1tψxtsinβ⁡t2sin-β⁡t2dt≪n+12∫0π/n+1ψxtsinβ⁡t2dtπn+11-β≪n+1βw~x1fπn+1β≪n+1β-1∑k=0nw~x1fπk+1β,for β∈0,1.Using Lemmas11 and 12 we obtain(59)I2∘~≪∫π/n+1πψxtt∑k=0τan,kdt≪∫π/n+1πψxttτn+1dtor (60)I2∘~≪∫π/n+1πψxtt∑k=n-τnan,kdt=∫π/n+1πψxtt∑k=0τan,n-kdt≪∫π/n+1πψxttτn+1dt,whence (61)I2∘~≪1n+1∫π/n+1πt-2ψxtsinβ⁡t21sinβ⁡t/2dt≪n+1β1n+1∫π/n+1πt-2ψxtsinβ⁡t2dt=n+1β1n+1·∫π/n+1πt-2ddt∫0tψxusinβ⁡u2dudt=n+1β-1t-2∫0tψxusinβ⁡u2dut=π/n+1π+2∫π/n+1πt-3∫0tψxusinβ⁡u2dudt=n+1β-1·π-1w~x1fπβ-πn+1-1w~x1fπn+1β+2∫π/n+1πt-2w~x1ftβdt≪n+1β-1·∑k=0nw~x1fπk+1β+∫1n+1w~x1fπtβdt≪n+1β-1·∑k=0nw~x1fπk+1β.Collecting these estimates, we obtain the desired result.


## 4.2. Proof of Theorem3

We start with the obvious relations(62)T~n,Afx-f~x=1π∫0π/n+1ψxt∑r=0n∑k=0ran,rbr,kDk∘~tdt+1π∫π/n+1πψxt∑r=0n∑k=0ran,rbr,kDk∘~tdt=I1∘~+I2∘~,T~n,Afx-f~x≤I1∘~+I2∘~.By the Hölder inequality1/p+1/q=1, Lemma 10, (25), and (24),(63)I1∘~≪∫0π/n+1ψxttdt≤∫0π/n+1ψxtw~xtsinβ⁡t2pdt1/p·∫0π/n+1w~xttsinβ⁡t/2qdt1/q≪n+1β·w~xπn+1=n+1β-1w~xπn+1∑k=0n1≪n+1β-1∑k=0nw~xπk+1.We can estimate the termI2∘~ by the same way like in the proof of Theorem 2: (64)I2∘~≪∫π/n+1πψxtt∑k=n-τnan,kdt=∫π/n+1πψxtt∑k=0τan,n-kdt≪∫π/n+1πψxttτn+1dt≪n+1β-1∑k=0nw~x1fπk+1β≪n+1β-1∑k=0nw~xπk+1.Collecting these estimates, we obtain the desired result.


## 4.3. Proofs of Theorems4-5

The proofs are similar to these above and follow from the evident inequality(65)w~·pfδβLp≪ω~βfδLpand additionally in case of Theorem 5 from the estimate(66)∫0π/n+1ψ·tω~tpsinβp⁡t2dt1/pLp≤∫0π/n+1ψ·sinβ⁡t2Lpω~tdt1/p≤∫0π/n+1ω~βftLpω~tpdt1/p≤∫0π/n+1dt1/p=Oxn+1-1/pfor f∈Lpω~β.


## 4.4. Proof of Corollary6

First of all we note that our matrixA is the hump matrix with k0=n.Next, we have to verify the assumptions of Lemma11. For the first one, we note that any nondecreasing sequence belongs to the class MRBVS and any nonincreasing sequence belongs to the class MHBVS. The second one follows from the following calculations: (67)∑r=μν1+γ-r∑k=0rrkγkcos⁡2k+1t2=Re⁡∑r=μν1+γ-r∑k=0rrkγkexp⁡2k+1it2=Re⁡∑r=μν1+γ-rexp⁡it2∑k=0rrkγkexp⁡ikt=Re⁡∑r=μν1+γ-rexp⁡it21+γexp⁡itr=Re⁡exp⁡it2∑r=μν1+γexp⁡it1+γr=Re⁡exp⁡it2·1+γexp⁡it1+γμ·1-1+γexp⁡it/1+γν-μ+11-1+γexp⁡it/1+γ≤1-1+γexp⁡it/1+γν-μ+11-1+γexp⁡it/1+γ≤21-1+γexp⁡it/1+γ=21+γ1+γ-1-γexp⁡it=21+γγ1-exp⁡it=21+γγ1-cos⁡t2+sin2⁡t=21+γγ2-2cos⁡t=21+γγ22sin2⁡t/2=1+γγsin⁡t/2≤1+γγπt≪τ.Thus proof is complete.



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*Source: 470205-2015-05-26.xml*