Let Lp (1 ¬ p &lt

Włodzimierz Łenski, Bogdan Szal

Approximation of functions from L^p(ω)_β by linear operators of their Fourier series

Abstract. We show the results corresponding to theorems of S. Lal [Appl. Math.

Comput., 209 (2009) 346-350] on the rate of approximation of functions from the generalized integral Lipschitz classes by matrix summability means of their Fourier series as well as to the authors theorems [Acta Comment. Univ. Tartu. Math., 13 (2009), 11-24] also on such approximations.

2000 Mathematics Subject Classification: 42A24.

Key words and phrases: Rate of approximation, Summability of Fourier series, Lip- schitz classes.

1. Introduction. Let L^p (1 ¬ p < ∞) be the space of all 2π-periodic real- valued functions integrable in the Lebesgue sense with p-th power over Q = [−π, π]

equipped with the norm

(1) kfk := kf(·)kLp = Z

Q

| f(t) |^pdt

!1/p

when 1 ¬ p < ∞

and consider the trigonometric Fourier series

Sf (x) := ao(f)

2 +

X∞ ν=1

(aν(f) cos νx + bν(f) sin νx)

with the partial sums Skf .

Let A := (an,k) be an infinite lower triangular matrix of real numbers such that an,k ­ 0 when k = 0, 1, 2, ...n, a^n,k= 0 when k > n ,

Xn k=0

an,k = 1, for n = 0, 1, 2, ... ,

and let the A−transformation of (S^kf ) be given by

T_n,Af (x) :=

Xn k=0

an,kSkf (x) ( n = 0, 1, 2, ...) . Denote, for m = 0, 1, 2, .., n,

An,m= Xm k=0

an,k and An,m= Xn k=m

an,k.

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

A sequence c := (ck) of nonnegative numbers tending to zero is called the Rest Bounded Variation Sequence, or briefly c ∈ RBV S, if it has the property

X∞ k=m

|cⁿ− cⁿ⁺¹| ¬ K (c) c^m

for all positive integers m, where K (c) is a constant depending only on c.

A sequence c := (ck) of nonnegative numbers will be called the Head Bounded Variation Sequence, or briefly c ∈ HBV S, if it has the property

mX−1 k=0

|cⁿ− cⁿ⁺¹| ¬ K (c) c^m

for all positive integers 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 two another classes of sequences.

Follows by L. Leindler([3]) a sequence c := (ck) of nonnegative numbers ten- ding to zero is called the Mean Rest Bounded Variation Sequence, or briefly c ∈ M RBV S, if it has the property

(2)

X∞ k=m

|c^k− ck+1| ¬ K (c) 1 m + 1

Xm k­m/2

ck

for all positive integers m.

Analogously, a sequence c := (ck) of nonnegative numbers will be called the Mean Head Bounded Variation Sequence, or briefly c ∈ MHBV S, if it has the property

(3)

n−m−1X

k=0

|c^k− c^k+1| ¬ K (c) 1 m + 1

Xn k=n−m

ck,

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 [6])

RBV S MRBV S and HBV S MHBV S.

Consequently, we assume that the sequence (K (αn))^∞_n=0 is bounded, that is, that there exists a constant K such that

0 ¬ K (αⁿ) ¬ K

holds for all n, where K (αn) denote the sequence of constants appearing in the inequalities (2) or (3) for the sequences αn = (an,k)ⁿ_k=0, n = 0, 1, 2....

Now we can give the conditions to be used later on. We assume that for all n and 0 ¬ m < n

(4)

n−1

X

k=m

|a^n,k− a^n,k+1| ¬ K 1 m + 1

Xm k­m/2

an,k

and

(5)

n−m−1X

k=0

|a^n,k− a^n,k+1| ¬ K 1 m + 1

Xn k=n−m

an,k

hold if (an,k)ⁿ_k=0 belong to MRBV S or MHBV S, for n = 1, 2, ..., respectively.

As a measure of approximation of f by T_n,Af we use the generalized modulus of continuity of f in the space L^p defined for β ­ 0 by the formula

ωβf (δ)_Lp:= sup

0¬|t|¬δ



  sint

2  

βpZπ 0

|ϕ^x(t)|^pdx





1 p

,

where

ϕx(t) := f (x + t) + f (x − t) − 2f (x) . It is clear that for β ­ α ­ 0

ωβf (δ)_Lp ¬ ω^αf (δ)_Lp,

and ω0f (·)L^p= ωf (·)L^p is the classical modulus of continuity.

The deviation T_n,Af− f with lower triangular infinite matrix was estimated in the norm of L^p by S. Lal [1, Theorem 2, p. 347] as follows:

Theorem 1.1 If f ∈ L^pβ(ω) (see Remark 2.9) and ω (t)

t is a decreasing function of t

(6)

(Z π/(n+1) 0

t|ϕ^x(t)|

ω (t)

p

sin^βptdt )1/p

= O

(n + 1)⁻¹ ,

(Z π π/(n+1)

t^−γ|ϕ^x(t)|

ω (t)

p

sin^βptdt )1/p

= O ((n + 1)^γ) (0 < γ < 1 p), then

1 n + 1

Xn ν=0

1 Pν

Xν k=0

pν−kSkf − f L^p

= O



(n + 1)^β+1pω

 1 n + 1



,

where Pn = Pⁿ

ν=0pν with nonnegative and nonincreasing sequence (pν).

In this paper we show (see Remark 2.10) that the condition (6) is not proper for the above estimate. It is clear that instead of this condition we have to take

(Z π/(n+1) 0

|ϕ^x(t)|

ω (t)

p

sin^βptdt )1/p

= Ox

(n + 1)^−1/p .

In this remark we also show the correct estimate of the integrals considered in the proof of the above theorem under additionally assumption that t^1−γω (t) is a nonincreasing function of t. Moreover, in the next remark we note that from our results the theorem of S. Lal follows.

In our theorems we shall consider the pointwise deviation introduced at the be- gin. We shall formulate the general and precise conditions for the functions and moduli of continuity. Consequently, we also give some results on norm approxima- tion. We shall use the matrix which rows belong to the classes of sequences MRBV S or MHBV S. Thus we essentially extend our earlier results (see [4]) for the classes of sequences RBV S and HBV S .

We shall write I1 I²if there exists a positive constant K, sometimes depended on some parameters, such that I1¬ KI².

2. Statement of the results. Let us consider a function ω of modulus of continuity type on the interval [0, 2π], i.e. a nondecreasing continuous function having the following properties: ω (0) = 0, ω (δ₁+ δ₂) ¬ ω (δ1) + ω (δ₂) for any 0 ¬ δ1 ¬ δ2 ¬ δ1+ δ₂ ¬ 2π. It is easy to conclude that the function δ⁻¹ω (δ) is almost nonincreasing function of δ. Let

L^p(ω)_β= {f ∈ L^p: ωβf (δ)_Lp¬ ω (δ)} ,

where ω is also the function of modulus of continuity type. It is clear that for β > α­ 0

L^p(ω)_α⊂ L^p(ω)_β .

We can now formulate our main results on the degrees of pointwise summability.

Theorem 2.1 Let f ∈ L^p (1 < p < ∞), and let ω satisfy

(7)

(Z _n+2^2π

0

|ϕ^x(t)|

ω (t)

p

sin^βp t 2dt

)1/p

= Ox

(n + 1)^−1/p

with β < 1 − ¹p and

(8) (Z π

n+22π

t^−γ|ϕ^x(t)|

ω (t)

p

sin^βp t 2dt

)1/p

= Ox((n + 1)^γ)

with 0 < γ < β +¹_p. If (an,k)ⁿ_k=0 ∈ MRBV S be such that A^n,τ = O 

n+1τ

, where

τ = [π/t] , then

T_n,Af (x)− f (x) = Ox



(n + 1)^β+1pω

 π n + 1



,

for considered x.

Theorem 2.2 Let f ∈ L^p (1 < p < ∞), and let ω satisfy (7) with β < 1 −¹p and (8) with 0 < γ < β +¹_p. If (an,k)ⁿ_k=0∈ MHBV S be such that A^n,n−2τ = O 

n+1τ

, where τ = [π/t] , then

T_n,Af (x)− f (x) = Ox



(n + 1)^β+1pω

 π n + 1



,

for considered x.

Remark 2.3 We can observe that taking ank = _n+1¹ Pn ν=kpν−k

Pν we obtain the mean considered by S. Lal [1] and if (pν) is monotonic with respect to ν then (an,k)ⁿ_k=0∈ MRBV S.

Corollary 2.4 Under the assumptions of Theorem 2.1 on a function f, if (pν)is nonincreasing sequence such that

(9) Pτ

Xn ν=τ

P_ν⁻¹= O (τ) ,

then from Theorem 2.1 we obtain the corrected form of the result of S. Lal.

Finally, we formulate the results on estimates of L^p norm of the deviation con- sidered above.

Theorem 2.5 Let f ∈ L^p(ω)_β (1 < p < ∞) with β < 1 − p¹. If (an,k)ⁿ_k=0 ∈ M RBV S be such that An,τ = O

τ n+1

, where τ = [π/t], then

T_n,Af (·) − f (·)

L^p= O



(n + 1)^β+1pω

 π n + 1



.

Theorem 2.6 Let f ∈ L^p(ω)_β (1 < p < ∞) with β < 1 − ¹p. If (an,k)ⁿ_k=0 ∈ M HBV S be such that A_n,n−2τ = O

n+1τ

, where τ = [π/t], then

T_n,Af (·) − f (·)

L^p= O



(n + 1)^β+1pω

 π n + 1



.

Remark 2.7 In the case p ­ 1 (specially if p = 1) we can suppose that the expres- sion t^−βω (t) nondecrease in t instead of the assumption β < 1−¹p.

Remark 2.8 Under additional assumptions β = 0 and ω (t) = O (t^α) (0 < α ¬ 1), the degrees of approximation in Theorem 2.1 and 2.2 are the following O

n^1p−α .

Remark 2.9 If we consider the modulus of continuity in the form

ωf (δ)_L^p

β := sup

0¬|t|¬δ



 Zπ

0

|ϕ^x(t)|^psinx 2  

βp

dx





1 p

,

then our theorems will be true in the class L^p_β(ω) =n

f ∈ L^p: ωf (δ)_L^p

β ¬ ω (δ)o and with the following norm

kfkLp

β := kf(·)kLp β

= Z

Q

| f(t) |^p  sin t

2  

βp

dt

!1/p

,

where 1 ¬ p < ∞.

Remark 2.10 In the paper [1, Proof of Theorem 2] sin t is considered instead of sin₂^t , for t ∈ (0, π). It generate some inconvenience since the inequality sin t ­ π²t does not hold for all t ∈ (0, π) and, among others, therefore the proofs in [1] are incorrect. But in such situation the problem can be solved by dividing the integrals Rπ

n+1π into two parts. One of them is (Z π

n+1π

 ω (t) t^2−γsin^βt

q

dt )1/q

=Z π−_n+1^π

n+1π

+Z π π−_n+1^π

.

Then , for t ∈

n+1π , π−n+1^π

we have sin t ­ sinn+1^π , whence , for γ < 1/p ,

Z π−_n+1^π

n+1π

 ω

n+1π



n+1π

(Z π−_n+1^π

n+1π

 1

t^1−γsin^βt

q

dt )1/q

¬ ω

n+1π



n+1π







Z π−_n+1^π

π n+1



 1

t^1−γ

sin_n+1^π β





q

dt







1/q

 (n + 1)^β+1ω

 π n + 1

 (Z π

n+1π

 1 t^1−γ

q

dt )1/q

 (n + 1)^β+1+1p−γω

 π

n + 1



= (n + 1)^1−γω

 π n + 1



(n + 1)^β+1p

and, since γ < 1, we have

Z π

π−_n+1^π  ω

n+1π



π n+1

(Z π π−_n+1^π

 1

t^1−γsin^βt

q

dt )1/q

 (n + 1) ω

 π

n + 1

 (Z π π−_n+1^π

 1 sin^βt

q

dt )1/q

= (n + 1) ω

 π n + 1

 (Z _n+1^π

0

 1

sin^β(π − t)

^q dt

)1/q

= (n + 1) ω

 π n + 1

 (Z _n+1^π

0

 1 sin^βt

^q dt

)1/q

¬ (n + 1) ω

 π

n + 1

 (Z _n+1^π

0

 1

2 πt

βq

dt )1/q

 (n + 1) ω

 π

n + 1



(n + 1)^β−1q

= (n + 1)^1−γω

 π

n + 1



(n + 1)^β−1q+γ

¬ (n + 1)^1−γω

 π

n + 1



(n + 1)^β+1p.

The another integral is (Z π

π n+1

 ω (t) t^1−γsin^βt

q

dt )1/q

=Z π−n+1^π π n+1

+Z π π−_n+1^π

.

Similarly, if the function t^γ−1ω (t) is a almost nonincreasing in t with γ < 1, then Z π−_n+1^π

n+1π

 ω

n+1π



 π n+1

1−γ

(Z π−_n+1^π

n+1π

 1 sin^βt

^q dt

)1/q

¬ ω

π n+1



 π n+1

1−γ







Z π−_n+1^π

n+1π



 1

sin_n+1^π ^β





q

dt







1/q

 (n + 1)^β+1−γω

 π n + 1



 (n + 1)^−γω

 π

n + 1



(n + 1)^β+1p and

Z π π−n+1^π

 ω

n+1π



 π n+1

1−γ

(Z π π−n+1^π

 1 sin^βt

^q dt

)1/q

 (n + 1)^1−γω

 π n + 1



(n + 1)^β−1q

¬ (n + 1)^−γω

 π n + 1



(n + 1)^β+1p.

Thus the desired estimate follows and Theorem 2 from [1] will be true (with proper assumptions).

Moreover, the condition (6) used in the estimate of the another integral R_n+1^π (from the proof of Theorem 2 of S. Lal) leads us to the divergent integral of the0

form Rn+1^π

0 t^{−(1+β)1−1/p} dt under the assumption β­ 0.

Remark 2.11 In the proof of the mentioned theorem of S. Lal [1] the condition

Pτ

Xn ν=τ

P_ν⁻¹= O (n + 1)

is used. It holds for every nonnegative sequences (pk) , but the proper condition should have the form (9).

3. Auxiliary results. We begin this section by some notations following A.

Zygmund [7].

It is clear that

Skf (x) = 1 π

Z π

−π

f (x + t) Dk(t) dt

and

T_n,Af (x) = 1 π

Z π

−π

f (x + t) Xn k=0

an,kDk(t) dt, where

Dk(t) =1 2 +

Xk ν=1

cos νt =

( _sin(2k+1)t

2 sin2^t₂ for t 6= 2πr, r = 0, 1, 2, ..., k +¹₂ otherwise.

Hence

T_n,Af (x)− f (x) = 1 π

Z π 0 ϕx(t)

Xn k=0

an,kDk(t) dt.

Next, we present the known estimates for the Dirichlet kernel.

Lemma 3.1 [7] If 0 < |t| ¬ π/2 then

|D^k(t)| ¬ π

|t|

and, for any real t, we have

|D^k(t)| ¬ k + 1 . More complicated estimates we give with proofs.

Lemma 3.2 (cf. [2, 4, 6])If (an,k)ⁿ_k=0∈ MHBV S, then 

Xn k=0

an,kDk(t)  

= O t⁻¹An,n−2τ
,

and if (an,k)ⁿ_k=0∈ MRBV S , then 

Xn k=0

an,kDk(t)  

= O t⁻¹An,τ
,

for ^2π_n ¬ t ¬ π (n = 2, 3, ...) , where τ = [π/t] . Proof Using partial summation

Xn k=0

an,ksin(2k + 1) t 2 sin t

2   

¬ tA^n,τ +1 2

nX−1 k=τ−1

|a^n,k− a^n,k+1|   

Xk l=τ−1

2 sin(2l + 1) t 2 sin t

2   

+1 2an,n

Xn l=τ−1

2 sin(2l + 1) t 2 sint

2   

¬ tA^n,τ +1 2

n−1

X

k=τ−1

|a^n,k− a^n,k+1|   

Xk l=τ−1

(cos lt − cos (l + 1) t)    +1

2an,n

Xn l=τ−1

(cos lt − cos (l + 1) t)   

¬ tA^n,τ +1 2

n−1X

k=τ−1

|a^n,k− an,k+1| |cos (τ − 1) t − cos (k + 1) t|

+1

2an,n|cos (τ − 1) t − cos (n + 1) t|

¬ tA^n,τ +

n−1X

k=τ

|a^n,k− a^n,k+1| + a^n,n or 

Xn k=0

an,ksin(2k + 1) t 2 sin t

2   

¬ tA^n,n−τ+1 2

n−τ−1X

k=0

|a^n,k− a^n,k+1|   

Xk l=0

2 sin(2l + 1) t 2 sin t

2    +1

2an,n−τ

nX−τ l=0

2 sin(2l + 1) t 2 sint

2  

+ an,n−τ

¬ tA^n,n−τ+

n−τ−1X

k=0

|a^n,k− an,k+1| + 2a^n,n−τ. Because (an,k)ⁿ_k=0∈ MRBV S we have

an,s+1− a^n,r ¬ |a^n,m− an,s+1| ¬ Xs k=m

|a^n,k− an,k+1|

¬

n−1X

k=r

|a^n,k− a^n,k+1|  1 r + 1

Xr k­r/2

an,k (0 ¬ r ¬ m ¬ s < n) , whence

an,s+1 a^n,m+ 1 r + 1

Xr k­r/2

an,k (0 ¬ r ¬ m ¬ s < n) and therefore

Xn k=0

an,ksin(2k + 1) t 2 sin t

2   

 tA^n,τ + 1 τ + 1

Xτ k­τ/2

an,k+ 1 τ + 1

Xτ k­τ/2



an,k+ 1 τ + 1

Xτ k­τ/2

an,k





 tA^n,τ.

Analogously, the relation (an,k)ⁿ_k=0∈ MHBV S implies

an,s− a^n,m ¬ |a^n,m− a^n,s| ¬

s−1

X

k=m

|a^n,k− a^n,k+1|

¬

r−1X

k=0

|a^n,k− a^n,k+1|

 1

n− r + 1 Xn k=r

an,k (0 ¬ m < s ¬ r ¬ n)

and

an,s a^n,m+ 1 n− r + 1

Xn k=r

an,k (0 ¬ m < s ¬ r ¬ n) , whence

Xn k=0

an,ksin(2k + 1) t 2 sin t

2   

 tA^n,n−τ+ 1 τ + 1

Xn k=n−τ

an,k+ an,n−τ

1 τ

n−τ−1X

k=n−2τ

1

 tA^n,n−τ+1 τ

n−τ−1X

k=n−2τ

an,k+ 1 τ + 1

Xn k=n−τ

an,k

!

¬ 2tAn,n−τ+1 τ

n−τ−1X

k=n−2τ

an,k tAn,n−2τ

τ ¬n 2

.

Thus our proof is complete. _

4. Proofs of the results.

4.1. Proof of Theorem 2.1.

Let

T_n,Af (x)− f (x) = 1 π

Z _n+2^2π

0 ϕx(t) X∞ k=0

an,kDk(t) dt

+1 π

Z π

n+22π

ϕx(t) X∞ k=0

an,kDk(t) dt

= I1+ I2. Then T_n,Af (x)− f (x)

 ¬ |I1| + |I²| .

By the H¨older inequality ₁

p +¹_q = 1

, Lemma 3.1 and (7) , for β < 1−¹p,

|I¹| ¬ (n + 1) π

Z _n+2^2π

0 |ϕ^x(t)| dt

¬ (n + 1) π

(Z _n+2^2π

0

|ϕ^x(t)|

ω (t) sin^β t 2

^p dt

)¹_p(Z _n+2^2π

0

"

ω (t) sin^{β t}₂

#q

dt )¹q

 (n + 1)^1−1pω

 2π n + 2

 (Z _n+2^2π

0

1 t^β

^q dt

)¹_q

 (n + 1)^βω

 2π n + 2



¬ 2 (n + 1)^βω

 π n + 1

 .

Using the H¨older inequality₁

p+¹_q = 1

, Lemma 3.2 and (8)

|I²|  1 π

Z π

n+22π

|ϕ^x(t)|

t An,τdt 1 π (n + 1)

Z π

n+22π

|ϕ^x(t)|

t² dt

¬ 1

π (n + 1) (Z π

n+22π

t^−γ|ϕ^x(t)|

ω (t) sin^β t 2

p

dt

)_p¹(Z π

n+22π

"

ω (t) t^2−γsin^{β t}₂

#q

dt )¹q

 1

n + 1(n + 1)^γ+1ω

 2π n + 2

 (Z π

n+22π

 1

t^1−γ+β

q

dt )¹q

 (n + 1)^β+1pω

 π n + 1

 ,

for 0 < γ < β + ¹_p.

Collecting these estimates we obtain the desired result. 4.2. Proof of Theorem 2.2.

The proof is the same as the above, the only difference is that in place of An,τwe have to write the quantity A_n,n−τ for which we suppose the same estimate.

4.3. Proof of Remark 2.3.

Indeed,

an,k− an,k+1 = 1 n + 1

Xn ν=k

pν−k

Pν − Xn ν=k+1

pν−k−1

Pν

!

= 1

n + 1 p0

Pk

+ 1

n + 1 Xn ν=k+1

p_ν−k− pν−k−1

Pν