We show the results, corresponding to theorem of Lal (2009), on the rate of pointwise
approximation of functions from the pointwise integral Lipschitz
classes by matrix summability means of their Fourier series as well as
the theorems on norm approximations.

1. Introduction

Let Lp(1≤p<∞)be the class of all 2π-periodic real-valued functions integrable in the Lebesgue sense with pth power over Q=[-π,π] with the norm‖f‖:=‖f(⋅)‖Lp=(∫Q|f(t)|pdt)1/p,
and consider the trigonometric Fourier seriesSf(x):=ao(f)2+∑ν=1∞(aν(f)cosνx+bν(f)sinνx),
and conjugate oneS̃f(x):=∑ν=1∞(bν(f)cosνx-aν(f)sinνx)
with the partial sums Skf and S̃kf, respectively. We know that if f∈L, thenf̃(x):=-1π∫0πψx(t)12cott2dt=limϵ→0f̃(x,ϵ),
wheref̃(x,ϵ):=-1π∫ϵπψx(t)12cott2dt
withψx(t):=f(x+t)-f(x-t)
exists for almost all x [1, Th. (3.1)IV].

Let A:=(an,k) be an infinite lower triangular matrix of real numbers such thatan,k≥0whenk=0,1,2,…,n,an,k=0whenk>n,∑k=0nan,k=1,wheren=0,1,2,…,
and let the A-transformationsof (Skf) and (S̃kf) be given byTn,Af(x)∶=∑k=0nan,kSkf(x)(n=0,1,2,…),T̃n,Af(x)∶=∑k=0nan,kS̃kf(x)(n=0,1,2,…),
respectively. Denote, for m=0,1,2,…,n,An,m=∑k=0man,k,A¯n,m=∑k=mnan,k.

We define two classes of sequences (see [2]).

A sequence c∶=(cn) of nonnegative numbers tending to zero is called the Rest Bounded Variation Sequence, or briefly c∈RBVS, if it has the property∑k=m∞|ck-ck+1|≤K(c)cm
for all natural numbers m, where K(c) is a constant depending only on c.

A sequence c∶=(cn) of nonnegative numbers will be called the Head Bounded Variation Sequence, or briefly c∈HBVS, if it has the property∑k=0m-1|ck-ck+1|≤K(c)cm
for all natural numbers m, or only for all m≤N if the sequence c has only finite nonzero terms and the last nonzero terms is cN.

Now, we define another class of sequences.

Followed by Leindler [3], a sequence c∶=(cn) of nonnegative numbers tending to zero is called the Mean Rest Bounded Variation Sequence, or briefly c∈MRBVS, if it has the property∑k=m∞|ck-ck+1|≤K(c)1m+1∑k≥m/2mck
for all natural numbers m, where K(c) is a constant depending only on c.

Analogously, a sequence c∶=(ck) of nonnegative numbers will be called the Mean Head Bounded Variation Sequence, or briefly c∈MHBVS, if it has the property∑k=0n-m-1|ck-ck+1|≤K(c)1m+1∑k=n-mnck,
for all positive integer m<n, where the sequence c has only finite nonzero terms and the last nonzero term is cn and where K(c) is a constant depending only on c.

It is clear that (see [4])RBVS⊊MRBVS,HBVS⊊MHBVS.

Consequently, we assume that the sequence (K(αn))n=0∞ is bounded, that is, that there exists a constant K such that0≤K(αn)≤K
holds for all n, where K(αn) denotes the sequence of constants appearing in the inequalities (1.12) or (1.13) for the sequences αn=(an,k)k=0n.

Now, we can give the conditions to be used later on. We assume that, for all n and 0≤m<n,∑k=mn-1|an,k-an,k+1|≤K1m+1∑k≥m/2man,k,∑k=0n-m-1|an,k-an,k+1|≤K1m+1∑k=n-mnan,k,
where K is the same as above, hold if αn=(an,k)k=0n belong to MRBVS or MHBVS, for n=0,1,2,…, respectively.

As a measure of approximation of functions by the above means, we use the generalized pointwise moduli of continuity of f in the space Lp defined for β≥0 by the formulasw̃x,βf(δ)Lp:={1δ1+βp∫0δ(|ψx(t)||sint2|β)pdt}1/p,wx,βf(δ)Lp:={1δ1+βp∫0δ(|φx(t)||sint2|β)pdt}1/p,
whereφx(t)∶=f(x+t)+f(x-t)-2f(x).
It is clear that, for β>α≥0,wx,βf(δ)Lp≤wx,αf(δ)Lp,w̃x,βf(δ)Lp≤w̃x,αf(δ)Lp.
It is easily seen that wx,0f(·)Lp=wxf(·)Lp and w̃x,0f(·)Lp=w̃xf(·)Lp are the classical pointwise moduli of continuity.

The deviation Tn,Af-f with special form of matrix A was estimated in the norm of Lp by Lal [5, Theorem 2, page 347] as follows.

Theorem A.

If
f∈Lβp(ω)={f∈Lp:ωf(δ)Lβp:=sup0≤|t|≤δ{∫0π|φx(t)|p|sinx2|βpdx}1/p≤ω(δ)},ω(t)tisadecreasingfunctionoft,{∫0π/(n+1)(t|φx(t)|ω(t))psinβptdt}1/p=O((n+1)-1),{∫π/(n+1)π(t-γ|φx(t)|ω(t))psinβptdt}1/p=O((n+1)γ)(0<γ<1p),
then
‖1n+1∑ν=0n1Pν∑k=0νpν-kSkf-f‖Lp=O((n+1)β+(1/p)ω(1n+1)),
where Pn=∑ν=0npν,(pν) is a nonnegative and nonincreasing sequence, and the function ω of modulus of continuity type will be defined in the next section.

In this note we show that the conditions (1.21), (1.22), and (1.23) are superfluous when we use the pointwise modulus of continuity.

In our theorems, we will consider the pointwise deviations Tn,Af-f,T̃n,Af-f̃, T̃n,Af-f̃(·,2π/(n+2)) with the matrix whose rows belong to the classes of sequences MRBVS and MHBVS. Consequently, we also give some results on norm approximation.

We will write I1≪I2 if there exists a positive constant K, sometimes depending on some parameters, such that I1≤KI2.

2. Statement of the Results

Let us define, for a fixed x, a function wx (or w̃x) of modulus of continuity type on the interval [0,2π], that is, a nondecreasing continuous function having the following properties:wx(0)=0,wx(δ1+δ2)≤wx(δ1)+wx(δ2),(orw̃x(0)=0,w̃x(δ1+δ2)≤w̃x(δ1)+w̃x(δ2)),
for any 0≤δ1≤δ2≤δ1+δ2≤2π. It is easy to conclude that the function δ-1wx(δ) nondecreases in δ. LetLp(wx)β={f∈Lp:wx,βf(δ)Lp≤wx(δ)},Lp(w̃x)β={f∈Lp:w̃x,βf(δ)Lp≤w̃x(δ)},Lp(w)β={f∈Lp:‖w⋅,βf(δ)Lp‖Lp≤w(δ)},Lp(w̃)β={f∈Lp:‖w̃⋅,βf(δ)Lp‖Lp≤w̃(δ)},
where wx,w̃xw̃, and w are also the functions of modulus of continuity type. It is clear that, for β>α≥0,Lp(wx)α⊂Lp(wx)β,Lp(w)α⊂Lp(w)β,Lp(w̃x)α⊂Lp(w̃x)β,Lp(w̃)α⊂Lp(w̃)β.

Now, we can formulate our main results on the degrees of pointwise summability.

Theorem 2.1.

Let f∈Lp(wx)β with β<1-(1/p). If (an,k)k=0n∈MRBVS is such that An,τ=O(τ/(n+1)), where τ=[π/t](2π/(n+2)≤t≤π), then
|Tn,Af(x)-f(x)|=O((n+1)β+(1/p)wx(πn+1)),
for considered x.

Theorem 2.2.

Let f∈Lp(wx)β with β<1-(1/p). If (an,k)k=0n∈MHBVS is such that A¯n,n-2τ=O(τ/(n+1)), where τ=[π/t](2π/(n+2)≤t≤π), then
|Tn,Af(x)-f(x)|=O((n+1)β+(1/p)wx(πn+1)),
for considered x.

Theorem 2.3.

Let f∈Lp(w̃x)β with β<2-(1/p). If (an,k)k=0n∈MRBVS is such that An,τ=O(τ/(n+1)), where τ=[π/t](2π/(n+2)≤t≤π), then
|T̃n,Af(x)-f̃(x,2πn+2)|=O((n+1)β+(1/p)w̃x(πn+1)),
for considered x.

Theorem 2.4.

Let f∈Lp(w̃x)β with β<2-(1/p). If (an,k)k=0n∈MHBVS is such that A¯n,n-2τ=O(τ/(n+1)), where τ=[π/t](2π/(n+2)≤t≤π), then
|T̃n,Af(x)-f̃(x,2πn+2)|=O((n+1)β+(1/p)w̃x(πn+1)),
for considered x.

Theorem 2.5.

Let f∈Lp(w̃x)β, and
{∫02π/(n+2)1t(w̃x(t)sinβ(t/2))qdt}1/q=O((n+1)βw̃x(πn+1))
holds with q=p(p-1)-1 when β>0 or with q=1 when β=0. If (an,k)k=0n∈MRBVS is such that An,τ=O(τ/(n+1)), where τ=[π/t](2π/(n+2)≤t≤π), then
|T̃n,Af(x)-f̃(x)|=O((n+1)β+(1/p)w̃x(πn+1)),
for considered x such that f̃(x) exists.

Theorem 2.6.

Let f∈Lp(w̃x)β, and (2.8) holds with q=p(p-1)-1 when β>0 or with q=1 when β=0. If (an,k)k=0n∈MHBVS and A¯n,n-2τ=O(τ/(n+1)), where τ=[π/t](2π/(n+2)≤t≤π), then
|T̃n,Af(x)-f̃(x)|=O((n+1)β+(1/p)w̃x(πn+1)),
for considered x such that f̃(x) exists.

Consequently, we formulate the results on norm approximation.

Theorem 2.7.

Let f∈Lp(w)β with β<1-(1/p),(an,k)k=0n∈MRBVS is such that An,τ=O(τ/(n+1)), where τ=[π/t](2π/(n+2)≤t≤π), then
‖Tn,Af(⋅)-f(⋅)‖Lp=O((n+1)β+(1/p)w(πn+1)).

Theorem 2.8.

Let f∈Lp(w)β with β<1-(1/p),(an,k)k=0n∈MHBVS is such that A¯n,n-2τ=O(τ/(n+1)), where τ=[π/t](2π/(n+2)≤t≤π), then
‖Tn,Af(⋅)-f(⋅)‖Lp=O((n+1)β+(1/p)w(πn+1)).

Theorem 2.9.

Let f∈Lp(w̃)β with β<1-(1/p),(an,k)k=0n∈MRBVS is such that An,τ=O(τ/(n+1)), where τ=[π/t](2π/(n+2)≤t≤π), then
‖T̃n,Af(⋅)-f̃(⋅,2πn+2)‖Lp=O((n+1)β+(1/p)w̃(πn+1)).

Theorem 2.10.

Let f∈Lp(w̃)β with β<1-(1/p),(an,k)k=0n∈MHBVS is such that A¯n,n-2τ=O(τ/(n+1)), where τ=[π/t](2π/(n+2)≤t≤π), then
‖T̃n,Af(⋅)-f̃(⋅,2πn+2)‖Lp=O((n+1)β+(1/p)w̃(πn+1)).

Theorem 2.11.

Let f∈Lp(w̃)β, and
{∫02π/(n+2)1t(w̃(t)sinβ(t/2))qdt}1/q=O((n+1)βw̃(πn+1))
holds with q=p(p-1)-1. If (an,k)k=0n∈MRBVS and An,τ=O(τ/(n+1)), where τ=[π/t](2π/(n+2)≤t≤π), then
‖T̃n,Af(⋅)-f̃(⋅)‖Lp=O((n+1)β+(1/p)w̃(πn+1)).

Theorem 2.12.

Let f∈Lp(w̃x)β, and (2.15) holds with q=p(p-1)-1. If (an,k)k=0n∈MHBVS is such that A¯n,n-2τ=O(τ/(n+1)), where τ=[π/t](2π/(n+2)≤t≤π), then
‖T̃n,Af(⋅)-f̃(⋅)‖Lp=O((n+1)β+(1/p)w̃(πn+1)).

Remark 2.13.

In the case p≥1(speciallyifp=1), we can suppose that the expression t-βwx(t) nondecreases in t instead of the assumption β<1-(1/p).

Remark 2.14.

Under additional assumptions ann=O(1/n),β=0, and wx(t)=O(tα)(0<α≤1), the degree of approximation in Theorem 2.1 is following O(n(1/p)-α). The same degree of approximation we obtain in Theorem 2.2 under the assumption an0=O(1/n). In case of the remaining theorems, we can use the same remarks.

Remark 2.15.

If we consider the classical modulus of continuity ω0f(δ)Lp of the function f instead of the modulus w, then the condition ∥w·,βf(δ)Lp∥Lp≤ω0f(δ)Lp holds for every function f∈Lp and thus Lp(w)β=Lp. The same remark for conjugate functions holds too.

Remark 2.16.

Our theorems will be also true if we consider function f from Lβp(ω) with the following norm:
‖f‖Lβp:=‖f(⋅)‖Lβp=(∫Q|f(t)|p|sint2|βpdt)1/p.

Remark 2.17.

We can observe that, taking ank=1/(n+1)∑ν=knpν-k/Pν, we obtain the mean considered by Lal [5], and if (pν) is monotonic with respect to ν, then (an,k)k=0n∈MRBVS. Therefore, if (pν) is a nonincreasing sequence such that Pτ∑ν=τnPν-1=O(τ), then, from Theorem 2.1, we obtain the corrected form of the result of Lal [5] (i.e., with condition {∫0π/(n+1)(|φx(t)|/ω(t))psinβptdt}1/p=Ox((n+1)-1/p) instead of (1.22)).

3. Auxiliary Results

We begin this section by some notations following Zygmund [1].

It is clear thatS̃kf(x)=-1π∫-ππf(x+t)D̃k(t)dt,Skf(x)=1π∫-ππf(x+t)Dk(t)dt,T̃n,Af(x)=-1π∫-ππf(x+t)∑k=0nan,kDk̃(t)dt,Tn,Af(x)=1π∫-ππf(x+t)∑k=0nan,kDk(t)dt,
whereDk̃(t)=∑ν=0ksinνt=cos(t/2)-cos((2k+1)t/2)2sin(t/2),Dk(t)=12+∑ν=1kcosνt=sin((2k+1)t/2)2sin(t/2).
Hence,T̃n,Af(x)-f̃(x,2πn+2)=-1π∫02π/(n+2)ψx(t)∑k=0nan,kDk̃(t)dt+1π∫2π/(n+2)πψx(t)∑k=0nan,kDk∘̃(t)dt,T̃n,Af(x)-f̃(x)=1π∫0πψx(t)∑k=0nan,kDk∘̃(t)dt,
whereDk∘̃(t)=cos((2k+1)t/2)2sin(t/2),Tn,Af(x)-f(x)=1π∫0πφx(t)∑k=0nan,kDk(t)dt.
Now, we formulate some estimates for the conjugate Dirichlet kernel.

Lemma 3.1 (see [<xref ref-type="bibr" rid="B8">1</xref>]).

If 0<|t|≤π/2, then
|Dk∘̃(t)|≤π2|t|,|Dk̃(t)|≤π|t|,
and, for any real t, we have
|Dk̃(t)|≤12k(k+1)|t|,|Dk̃(t)|≤k+1.

If (an,k)k=0n∈MHBVS, then
|∑k=0nan,kDk(t)|=O(t-1A¯n,n-2τ),
and if (an,k)k=0n∈MRBVS, then
|∑k=0nan,kDk(t)|=O(t-1An,τ),
for 2π/n≤t≤π(n=2,3,…), where τ=[π/t].

Lemma 3.3 (see [<xref ref-type="bibr" rid="B6">7</xref>]).

If (an,k)k=0n∈MHBVS, then
|∑k=0nan,kDk∘̃(t)|=O(t-1A¯n,n-2τ),
and if (an,k)k=0n∈MRBVS, then
|∑k=0∞an,kDk∘̃(t)|=O(t-1An,τ),
for 2π/n≤t≤π(n=2,3,…), where τ=[π/t].

4. Proofs of the Results4.1. Proof of Theorem <xref ref-type="statement" rid="thm1">2.1</xref>

As usual,Tn,Af(x)-f(x)=1π∫02π/(n+2)φx(t)∑k=0∞an,kDk(t)dt+1π∫2π/(n+2)πφx(t)∑k=0∞an,kDk(t)dt=I1+I2,|Tn,Af(x)-f(x)|≤|I1|+|I2|.
By the Hölder inequality ((1/p)+(1/q)=1) and Lemma 3.1, for β<1-(1/p),|I1|≤(n+1)π∫02π/(n+2)|φx(t)|dt≤(n+1)π{∫02π/(n+2)[|φx(t)|sinβt2]pdt}1/p{∫02π/(n+2)[1sinβ(t/2)]qdt}1/q≪(n+1)1-β-(1/p)wx(2πn+2){∫02π/(n+2)[1tβ]qdt}1/q≪wx(2πn+2)≤2wx(πn+1).
Using Lemma 3.2 and the Hölder inequality ((1/p)+(1/q)=1),|I2|≪1π∫2π/(n+2)π|φx(t)|tAn,τdt≪1π(n+1)∫2π/(n+2)π|φx(t)|t2dt≤1π(n+1)∫π/(n+2)π|φx(t)|sinβ(t/2)t2sinβ(t/2)dt≤πβ-1(n+1)∫π/(n+1)π|φx(t)|sinβ(t/2)t2+βdt=πβ-1n+1{[1t2+β∫0t|φx(u)|sinβu2du]t=π/(n+1)π+(2+β)∫π/(n+1)π∫0t|φx(u)|sinβ(u/2)dut3+βdt}≤1π3(n+1)∫0π|φx(u)|sinβu2du+πβ-1(2+β)n+1{∫π/(n+1)π[t-1∫0t|φx(u)|sinβ(u/2)duwx(t)]pdt}1/p⋅{∫π/(n+1)π[wx(t)t2+β]qdt}1/q≤1π2(n+1){1π∫0π(|φx(u)|sinβu2)pdu}1/p+πβ-1(2+β)n+1{∫π/(n+1)π[{t-1∫0t(|φx(u)|sinβ(u/2))pdu}1/pwx(t)]pdt}1/p⋅{∫π/(n+1)π[wx(t)t2+β]qdt}1/q≪1n+1wx(π)+1n+1{∫π/(n+1)πtβpdt}1/pwx(π/(n+1))π/(n+1){∫π/(n+1)π[1t1+β]qdt}1/q.
Since β+(1/p)>0, we have|I2|≪1n+1wx(π)+{[t(-1-β)q+1(-1-β)q+1]t=π/(n+1)π}1/qwx(πn+1)≪1n+1wx(π)+(n+1)β+(1/p)wx(πn+1)≪(n+1)β+(1/p)wx(πn+1).

Collecting these estimates, we obtain the desired result.

4.2. Proof of Theorem <xref ref-type="statement" rid="thm2">2.2</xref>

The proof is the same as the proof of Theorem 2.1, the only difference is that, in place of An,τ, we have to write the quantity A¯n,n-2τ for which we suppose the same order.

4.3. Proof of Theorem <xref ref-type="statement" rid="thm3">2.3</xref>

We start with the obvious relationsT̃n,Af(x)-f̃(x,2πn+1)=-1π∫02π/(n+2)ψx(t)∑k=0nankDk̃(t)dt+1π∫2π/(n+2)πψx(t)∑k=0nankDk∘̃(t)=I1̃+I2∘̃,|T̃n,Af(x)-f̃(x,2πn+2)|≤|I1̃|+|I2∘̃|.
By the Hölder inequality ((1/p)+(1/q)=1) and Lemma 3.1, we have|I1̃|≤(n+1)2∫02π/(n+2)t|ψx(t)|dt≤(n+1)2{∫02π/(n+2)[|ψx(t)|sinβt2]pdt}1/p{∫02π/(n+2)[tsinβ(t/2)]qdt}1/q≪(n+1)2-β-(1/p)w̃x(2πn+2){∫02π/(n+2)[t1-β]qdt}1/q≪w̃x(2πn+2)≪w̃x(πn+1)
for β<2-(1/p).

Using Lemma 3.3 and the Hölder inequality ((1/p)+(1/q)=1), we get|I2∘̃|≪1π∫2π/(n+2)π|ψx(t)|tAn,τdt≪1π(n+1)∫2π/(n+2)π|ψx(t)|t2dt≤1π(n+1)∫π/(n+1)π|ψx(t)|t2dt≤πβ-1n+1∫π/(n+1)π|ψx(t)|sinβ(t/2)t2+βdt≤1π3(n+1)∫0π|ψx(u)|sinβt2du+πβ-1(2+β)n+1{∫π/(n+1)π[t-1∫0t|ψx(u)|sinβ(u/2)duw̃x(t)]pdt}1/p{∫π/(n+1)π[w̃x(t)t2+β]qdt}1/q≪1π2(n+1){1π∫0π(|ψx(u)|sinβu2)pdu}1/p+πβ-1(2+β)n+1{∫π/(n+1)π[{t-1∫0t(|ψx(u)|sinβ(u/2))pdu}1/pw̃x(t)]pdt}1/p⋅{∫π/(n+1)π[w̃x(t)t2+β]qdt}1/q.
Since β+(1/p)>0, we have|I2∘̃|≪1n+1w̃x(π)+w̃x(πn+1){[t(-1-β)q+1(-1-β)q+1]t=π/(n+1)π}1/q≪(n+1)β+(1/p)w̃x(πn+1),
and our proof is complete.

4.4. Proof of Theorem <xref ref-type="statement" rid="thm4">2.4</xref>

The proof is the same as the proof of Theorem 2.3, the only difference is that, in place of An,τ, we have to write the quantity A¯n,n-2τ for which we suppose the same order.

4.5. Proof of Theorem <xref ref-type="statement" rid="thm5">2.5</xref>

We start with the obvious relationsT̃n,Af(x)-f̃(x)=1π∫02π/(n+2)ψx(t)∑k=0nan,kDk∘̃(t)dt+1π∫2π/(n+2)πψx(t)∑k=0nan,kDk∘̃(t)dt=I1∘̃+I2∘̃,|T̃n,Af(x)-f̃(x)|≤|I1∘̃|+|I2∘̃|.
Lemma 3.1 gives|I1∘̃|≪1π∫02π/(n+2)|ψx(t)|tdt≤πβ-1∫02π/(n+2)|ψx(t)|sinβ(t/2)t1+βdt=πβ-1[1t1+β∫0t|ψx(u)|sinβu2du]t=02π/(n+2)+πβ-1(1+β)∫02π/(n+2)t-1∫0t|ψx(u)|sinβ(u/2)dut1+βdt.
If β>0 and q>1, then, by the Hölder inequality ((1/p)+(1/q)=1),|I1∘̃|≤πβ-1(n+22π)1+β∫02π/(n+2)|ψx(u)|sinβu2du+πβ-1(1+β){∫02π/(n+2)[t-1∫0t|ψx(u)|sinβ(u/2)dut1/pw̃x(t)]pdt}1/p⋅{∫02π/(n+2)[w̃x(t)t(1/q)+β]qdt}1/q≤πβ-1(n+22π)β{n+22π∫02π/(n+2)(|ψx(u)|sinβu2)pdu}1/p+πβ-1(1+β){∫02π/(n+2)[{t-1∫0t(|ψx(u)|sinβ(u/2))pdu}1/pt1/pw̃x(t)]pdt}1/p⋅{∫02π/(n+2)[w̃x(t)t(1/q)+β]qdt}1/q≪w̃x(2πn+2)+{∫02π/(n+2)tβp-1dt}1/p{∫02π/(n+2)1t[w̃x(t)sinβ(t/2)]qdt}1/q≪w̃x(2πn+2)+(n+1)-β{∫02π/(n+2)1t[w̃x(t)sinβ(t/2)]qdt}1/q,
or if β=0, then|I1∘̃|≪w̃x(2πn+2)+∫02π/(n+2)w̃x(t)tdt.
Therefore, using (2.8), we have|I1∘̃|≪w̃x(πn+1).
The same estimation as in the proof of Theorem 2.3 gives|I2∘̃|≪(n+1)β+(1/p)w̃x(πn+1).
Collecting these estimates, we obtain the desired result.

4.6. Proof of Theorem <xref ref-type="statement" rid="thm6">2.6</xref>

For the proof, we use the analogical remark as these in the proofs of Theorems 2.2 and 2.4.

4.7. Proofs of Theorems from <xref ref-type="statement" rid="thm7">2.7</xref> to <xref ref-type="statement" rid="thm12">2.12</xref>

The proofs are similar to these above. We have only to use the generalized Minkowski inequality.

For example, in case of Theorem 2.7, we get‖Tn,Af(⋅)-f(⋅)‖Lp≤‖I1‖Lp+‖I2‖Lp≤‖w⋅,βf(2πn+2)Lp‖Lp+‖I2‖Lp≪‖w⋅,βf(πn+1)Lp‖Lp+1n+1‖w⋅,βf(π)Lp‖Lp+(n+1)β+(1/p)‖w⋅,βf(πn+1)Lp‖Lp≪1n+1w(π)+(n+1)β+(1/p)w(πn+1)=O((n+1)β+(1/p)w(πn+1)).

This completes the proof of Theorem 2.7.

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of conjugate Fourier series