Let Lp (1 ¬ p &lt

Radosława Kranz and Aleksandra Rzepka

On the approximation of conjugate functions from L^p by some special matrix means of conjugate

Fourier series

Abstract. The results corresponding to some theorems of W. Łenski and B. Szal are shown. From the presented pointwise results the estimates on norm approximation are derived. Some special cases as corollaries are also formulated.

2000 Mathematics Subject Classification: 42A24.

Key words and phrases: Degree of approximation, Fourier series, matrix means, con- jugate function.

1. Introduction. Let L^p (1 ¬ p < ∞) be the class of all 2π–periodic real–

valued functions integrable in the Lebesgue sense with p–th power over Q = [−π, π]

with the norm

kfk := kf(·)kLp = Z

Q

| f(t) |^pdt

!1/p

. Consider the trigonometric Fourier series

Sf (x) := ao(f)

2 +

X∞ ν=1

(aν(f) cos νx + bν(f) sin νx) and the conjugate one

Sf (x) :=e X∞ ν=1

(aν(f) sin νx − b^ν(f) cos νx)

with the partial sums eSkf . We know that if f ∈ L¹, then f (x) :=e −1

π Z π

0 ψx(t)1 2ctg t

2dt = lim

→0⁺f (x, ) ,e

where

f (x, ) :=e −1 π

Z π



ψx(t)1 2ctgt

2dt with

ψx(t) = f(x + t) − f(x − t), exists for almost all x and if ef ∈ L¹, then eSf (x) = S ef (x) [3].

Let A := (an,k) and B := (bn,k) be infinite lower triangular matrices of real numbers such that

an,k ­ 0 and b^n,k ­ 0 when k = 0, 1, 2, ...n, (1)

an,k = 0 and bn,k = 0 when k > n ,

(2)

Xn k=0

an,k = 1 and Xn k=0

bn,k= 1, where n = 0, 1, 2, ... .

Let the AB−transformation of ( eSkf ) be given by Te_n,A,Bf (x) :=

Xn r=0

Xr k=0

a_n,n−rbr,kSekf (x) (n = 0, 1, 2, ...) .

As a measure of approximation of function ef and ef

·,n+1^π

by eT_n,A,Bf in the space L^p(1 ¬ p < ∞) we will use the pointwise modulus of continuity of f defined, for β ­ 0, by the formula

e

wxf (δ)_p,β :=



 1 δ Zδ

0



|ψ^x(t)| sin^β t 2

^p dt





1 p

,

and

e

ωL^pf (δ)_β:= sup

0<t¬δ

ψ^·(t) sin^β t 2

L^p

.

The deviation Tn,A,Bf − f, where Tn,A,Bf defines the AB−transformation of (Skf ), was estimated by M. L. Mittal [2], with special matrix B. In the case of conjugate function, the deviation eT_n,A,Bf − ef was considered by W. Łenski. and B.

Szal [1] as follows:

Theorem A Let f ∈ L¹. If entries of our matrices satisfy the conditions an,n  1

n + 1, 1

s + 1 Xn r=0

an,r  a^n,r for 0 ¬ s ¬ n and

|a^n,rbr,r−l− a^n,r+1br+1,r+1−l|  an,r

(r + 1)² for 0 ¬ l ¬ r ¬ n − 1

then 

 e^{Tn,A,Bf(x) −}fe x, π

n+ 1  

Xn r=0

an,r+ Xr k=1

an,k

r+ 1

! "

1 r+ 1

Xr k=0

e wxf _π

k+ 1



1,0

#

+ 1 n+ 1

Xn k=0

e wxf _π

k+ 1



1,0

and under the additional condition 1

π

Z π/(n+1) 0

|ψ^x(t)|

t dt ewxf

 π n + 1



1,0

,

 eT_n,A,Bf (x)− ef (x)  Xn r=0

an,r+ Xr k=1

an,k

r + 1

! "

1 r + 1

Xr k=0

e wxf

 π k + 1



1,0

#

+ 1 n + 1

Xn k=0

e wxf

 π k + 1



1,0

for every natural n and all considered real x.

In this paper we shall consider the pointwise deviations eT_n,A,Bf (·) − ef (·,n+1^π ) and eT_n,A,Bf (·) − ef (·). In the theorems we formulate the general and precise con- ditions for the matrices (an,r)ⁿ_r=0 and (br,k)ⁿ_k=0 and for the modulus of continuity type. Finally, we also give some results on the norm approximation.

We shall write K1  K², if there exists a positive constant C, sometimes de- pending on some parameters, such that K1¬ CK².

2. Statement of the results. Now we formulate our main results.

Theorem 2.1 Let f ∈ L^p (1 ¬ p < ∞). If the entries of matrices (a^n,r)ⁿ_r=0 and (br,k)ⁿ_k=0 satisfy the conditions (1), (2), and

(3)

Xn r=0

an,r n−rX

k=0

|bn−r,k− bn−r,k+1|  1 n + 1, then, for 0 ¬ β ¬ 1 when p = 1 and 0 ¬ β < 2 −¹p when p > 1,

Ten,A,Bf (x)− ef

 x, π

n + 1



 (n + 1)^β (

e wxf

 π n + 1



p,β

+ 1

n + 1 Xn k=0

e wxf

 π k + 1



1,β

)

holds for almost all considered x .

Let us consider the classes L^p( ewx)β =n

f ∈ L^p: ewxf (δ)_p,β¬ ewx(δ)o , and

L^p(eω)^β=n

f ∈ L^p: eω^Lpf (δ)_β¬ eω (δ)o ,

with β ­ 0 and 1 ¬ p < ∞, where ewx and eω are the functions of modulus of continuity type.

Theorem 2.2 Let f ∈ L^p( ewx)β (1 < p < ∞). If the entries of matrices (a^n,r)ⁿ_r=0 and (br,k)ⁿ_k=0 satisfy the conditions (1), (2), (3) and ewx satisfies

(4) (Z _n+1^π

0

"

e wx(t) t sin^{β t}₂

#^q dt

)¹q

 (n + 1)^β+p1wex

 π n + 1

 ,

(5)

(Z _n+1^π

0

|ψ^x(t)|

e wx(t)

^p sin^βp t

2dt )¹_p

 (n + 1)^−p1, for β ­ 0, then

 eTn,A,Bf (x)− ef (x)  (n + 1)^β−1 Xn k=0

e wx

 π k + 1

 ,

holds for almost all considered x such that ef (x)exists.

Theorem 2.3 Let f ∈ L^p (1 ¬ p < ∞). If the entries of matrices (a^n,r)ⁿ_r=0 and (br,k)ⁿ_k=0 satisfy the conditions (1), (2) and (3), then

Ten,A,Bf (·) − ef



·, π n + 1



L^p

¬ (n + 1)^β−1 Xn k=0

e ωL^pf

 π k + 1



β

,

holds for almost all considered x such that ef (x) exists and for 0 ¬ β ¬ 1 when p = 1, and 0 ¬ β < 2 −¹p when p > 1 .

Theorem 2.4 Let f ∈ L^p(eω)^β (1 < p < ∞). If the entries of matrices (a^n,r)ⁿ_r=0 and (br,k)ⁿ_k=0 satisfy the conditions (1), (2), (3) and eω satisfies

(6) (Z _n+1^π

0

"

e ω(t) t sin^{β t}₂

#q

dt )¹q

 (n + 1)^β+1pωe

 π n + 1

 , for β ­ 0, then

eTn,A,Bf (·) − ef (·) _Lp (n + 1)^β−1 Xn k=0

e ω

 π k + 1

 , holds.

3. Corollaries.

Corollary 3.1 Under the assumptions of Theorem 2.1 we have 

Te_n,A,Bf (x)− ef

 x, π

n + 1

  (n + 1)^β−

1 p

( _n X

k=0

"

e wxf

 π k + 1



p,β

#p)¹p

.

Proof By Lemma 4.3  eT_n,A,Bf (x)− ef (x) ¬

 (n + 1)^β



 1 n + 1

Xn k=0

"

e wxf

 π k + 1



p,β

#p!p¹

+ 1

n + 1 Xn k=0

e wxf

 π k + 1



1,β





¬ (n + 1)^β−1p Xn k=0

"

e wxf

 π k + 1



p,β

#p!¹p

.

This completes the proof of Corollary 3.1. _

Corollary 3.2 Under the assumptions of Theorem 2.1, Corollary 3.1 and Lemma 4.3 we obtain another estimation for norm approximation

Ten,A,Bf (·) − ef



·, π n + 1

 _Lp ¬ (n + 1)^β−1p ( _n

X

k=0

e ωL^pf

 π k + 1



β

!p)¹p

.

If matrix A is defined by an,r= (r+1) ln(n+1)¹ when r = 0, 1, 2, ...n and an,r= 0 when r > n, and matrix B is defined by br,k = _r+1¹ when k = 0, 1, 2, ...r and br,k= 0 when k > r, then from Theorem 2.1 we obtain

Corollary 3.3 Let f ∈ L^p (1 ¬ p < ∞), then 

 eT_n,A,Bf (x)− ef (x)

=   

1 ln(n + 1)

Xn r=0

1 (r + 1)²

Xr k=0

Sekf (x)− ef (x)   

= Ox(n + 1)^β (

e wxf

 π n + 1



p,β

+ 1

n + 1 Xn k=0

e wxf

 π k + 1



1,β

) ,

for almost all considered x and 0 ¬ β < 1 −¹p,when p > 1, and β = 0, when p = 1.

Proof For the proof we will show that (an,r) and (br,k) satisfy the assumptions of Lemma 4.2. The conditions (1) and (2) are satisfied evidently. Since

Xn r=0

1 (r + 1) ln(n + 1)

n−r−1X

k=0

  1

n− r + 1− 1 n− r + 1

 + 1 n− r + 1

!

= 1 ln(n + 1)

Xn r=0

1 r + 1

1 n− r + 1

= 1

ln(n + 1) 1 n + 2

Xn r=0

 1

r + 1+ 1 n− r + 1



¬ 1

ln(n + 1) 1

n + 22 ln(n + 1)  1 n + 1

the proof of Corollary 3.3 is complete. _

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

Zygmund [3, Section 5 of Chapter II].

It is clear that

Sekf (x) =−1 π

Z π

−π

f (x + t) fDk(t) dt and

Te_n,A,Bf (x) =−1 π

Z π

−π

f (x + t) Xn r=0

Xr k=0

a_n,n−rbr,kDfk(t) dt, where

Dfk(t) = Xk ν=0

sin νt = cos₂^t− cos(k +¹2)t 2 sin₂^t . Hence

Te_n,A,Bf (x)− ef

 x, π

n + 1



= −1 π

Z _n+1^π

0 ψx(t) Xn r=0

Xr k=0

a_n,n−rbr,kDfk(t) dt

+1 π

Z π

n+1π

ψx(t) Xn r=0

Xr k=0

an,n−rbr,kDf_k^◦(t) dt, and

Te_n,A,Bf (x)− ef (x) = 1 π

Z π 0 ψx(t)

Xn r=0

Xr k=0

an,n−rbr,kDf^◦_k(t) dt, where

(7) Df^◦_k(t) = cos^(2k+1)t₂ 2 sin₂^t .

Now, we formulate some estimates for the conjugate Dirichlet kernels.

Lemma 4.1 (see [3]) If 0 < |t| ¬ π/2, then (8)  fD^◦_k(t) ¬ π

2 |t| and  fDk(t) ¬ π

|t|, and for any real t we have

(9)  fDk(t) ¬1

2k (k + 1)|t| and  fDk(t) ¬ k + 1.

Lemma 4.2 If (an,r)ⁿ_r=0 and (br,k)^r_k=0 satisfy (1), (2) and (3), then 

Xn r=0

an,n−r

Xr k=0

br,kDf_k^◦(t)    τ²

n + 1 for n = 0, 1, 2, 3, ..., where τ = [π/t] ¬ n/2.

Proof Since Kn(t) :=

Xn r=0

an,n−r

Xr k=0

br,kDf_k^◦(t) = Xn r=0

an,r n−r

X

k=0

bn−r,kDf^◦_k(t) and using Abel’s transformation and from (7) we get

= Xn r=0

an,r

2 sin₂^t

"_n−r−1 X

k=0

(bn−r,k− bⁿ−r,k+1) Xk

l=0

cos(2l + 1) t 2 +bn−r,n−r

n−rX

l=0

cos(2l + 1) t 2

#

= 1

2 sin₂^t Xn r=0

an,r n−r

X

k=0

(bn−r,k− bⁿ−r,k+1) Xk l=0

cos(2l + 1) t

2 .

Then

|Kⁿ(t)| ¬ 1 sin^t₂

Xn r=0

an,r n−r

X

k=0

|bⁿ−r,k− bⁿ−r,k+1|   

Xk l=0

cos(2l + 1) t 2

  . A simple calculation for the last sum gives

Xk l=0

cos(2l + 1) t 2

  ¬ 1

sin^t₂. So

|Kⁿ(t)|  π² t²

Xn r=0

an,r nX−r k=0

|bⁿ−r,k− bⁿ−r,k+1| and from (3) we obtain

|Kⁿ(t)|  τ² n + 1.

The desired estimate is now evident. _

Lemma 4.3 Let f ∈ L^p (1 ¬ p < ∞) and β ­ 0, then

e wxf

 π n + 1



p,β

 ( 1

n + 1 Xn k=0

"

e wxf

 π k + 1



p,β

#p)¹p

holds for every natural n and real x.

Proof By our assumption we get

e wxf

 π n + 1



p,β

=

(n + 1 π

Z _n+1^π

0

ψ^x(u) sin^βu 2  ^pdu

Xn k=0

2 (k + 1) (n + 1) (n + 2)

)_p¹

= ( 2

n + 1 Xn k=0

(k + 1) (n + 1) π (n + 2)

Z _n+1^π

0

ψ^x(u) sin^βu 2  ^pdu

)¹_p

 ( 1

n + 1 Xn k=0

(k + 1) π

Z _k+1^π

0

ψ^x(u) sin^β u 2  ^pdu

)¹_p

=

( 1 n + 1

Xn k=0

"

e wxf

 π k + 1



p,β

#^p)¹p

,

and this completes the proof. _

4.1. Proof of Theorem 2.1. We start with Te_n,A,Bf (x)− ef

 x, π

n + 1



= −1 π

Z _n+1^π

0

ψx(t) Xn r=0

Xr k=0

an,n−rbr,kDfk(t) dt

+1 π

Z π

n+1π

ψx(t) Xn r=0

Xr k=0

an,n−rbr,kDf_k^◦(t) dt

= Ie1(x) + eI₂^◦(x) and Te_n,A,Bf (x)− ef

 x, π

n + 1

 6 

 eI1(x) + eI₂^◦(x) . In case p = 1 and 0 ¬ β ¬ 1 from Lemma 4.1 and (9), we obtain

 eI1(x) ¬ 1 2π

Xn r=0

Xr k=0

a_n,n−rbr,kk (k + 1) Z _n+1^π

0 |ψ^x(t)| tdt, and using the inequality sin₂^t ­ π^t (0 ¬ t ¬ π), we get

 eI1(x) ¬ n (n + 1) π

Z _n+1^π

0 |ψ^x(t)|

 sin^β t

2

 t^1−βdt

= n

 π n + 1

1−β

e wxf

 π n + 1



1,β  (n + 1)^βwexf

 π n + 1



1,β

.

In case p > 1 and 0 ¬ β < 2−¹p, the H¨older inequality for integrals, with ¹_p+¹_q = 1, Lemma 4.1 and (9) give

 eI1(x) ¬ n (n + 1) 2π

Z _n+1^π

0 |ψ^x(t)| tdt

¬ n(n + 1)^1−1p 2

(n + 1 π

Z _n+1^π

0



|ψ^x(t)| sin^β t 2

p

dt )¹p

(Z _n+1^π

0

"

t sin^{β t}₂

#q

dt )¹q

 (n + 1)^2−1pwexf

 π n + 1



p,β

(Z _n+1^π

0 t^(1−β)qdt )¹_q

 (n + 1)^2−1pwexf

 π n + 1



p,β

(n + 1)^β−1−1q

= (n + 1)^βwexf

 π n + 1



p,β

.

Now we estimate the term eI₂^◦. Using Lemma 4.2 and the inequality sin₂^t ­ π^t

(0 ¬ t ¬ π), we obtain 

 eI₂^◦(x) ¬ 1 π

Z π

π n+1

|ψ^x(t)|

Xn r=0

Xr k=0

a_n,n−rbr,kDf_k^◦(t)  

dt¬ π n + 1

Z π

π n+1

|ψ^x(t)|

t² dt

¬ (n + 1)^β−1 Z π

π n+1

|ψ^x(t)| sin^{β t}2

t² dt, for β ­ 0. Integrating by parts we get

 eI₂^◦(x)  (n + 1)^β−1 (1

t² Z t

0 |ψ^x(s)| sin^βs 2ds

^π

n+1π

+ 2 Z π

n+1π

1 t³

Z t

0 |ψ^x(s)| sin^βs 2ds

 dt

)

¬ (n + 1)^β−1 ( 1

π² Z π

0 |ψ^x(s)| sin^βs 2ds + 2

Z π

n+1π

1

t²wexf (t)_1,βdt )

 (n + 1)^β−1

 e

wxf (π)_1,β+Z n+1 1 wex

 π u



1,βdu



¬ (n + 1)^β−1 (

e

wxf (π)_1,β+ Xn m=1

Z m+1

m wexf π u



1,βdu )

 (n + 1)^β−1 (

e

wxf (π)_1,β+ Xn m=0

e wxf

 π

m + 1



1,β

)

¬ (n + 1)^β−1 Xn m=0

e wxf

 π

m + 1



1,β

.

Collecting these estimates the desired result is proved. _ 4.2. Proof of Theorem 2.2. Let us usually

Te_n,A,Bf (x)− ef (x) = −1 π

Z _n+1^π

0 ψx(t) Xn r=0

Xr k=0

a_n,n−rbr,kDf_k^◦(t) dt

+1 π

Z π

π n+1

ψx(t) Xn r=0

Xr k=0

a_n,n−rbr,kDf^◦_k(t) dt

= Ie₁^◦(x) + eI₂^◦(x)

and  eT_n,A,Bf (x)− ef (x) 6 eI₁^◦(x) + eI₂^◦(x) .

From Lemma 4.1 and (8), using the H¨older inequality for integrals, with ¹_p+¹_q = 1, and from (4) and (5) we obtain

 eI₁^◦(x) ¬ 1 π

Z _n+1^π

0 |ψ^x(t)|

Xn r=0

Xr k=0

an,n−rbr,k

 fD_k^◦(t) dt

Z _n+1^π

0

|ψ^x(t)|

t dt

(Z _n+1^π

0

|ψ^x(t)|

e wx(t)

p

sin^βp t 2dt

)p¹

(Z _n+1^π

0

"

e wx(t) t sin^{β t}₂

#^q dt

)¹q

 (n + 1)^−1p(n + 1)^β+1pwex

 π n + 1



= (n + 1)^βwex

 π n + 1



¬ (n + 1)^β−1 Xn k=0

e wx

 π n + 1



for 1 < p < ∞. Using Lemma 4.2 for eI₂^◦ we have 

 eI₂^◦(x) ¬ 1 π

Z π

n+1π

|ψ^x(t)|

Xn r=0

Xr k=0

an,n−rbr,kDf_k^◦(t)  

dt¬ π² n + 1

Z π

n+1π

|ψ^x(t)|

t² dt.

Analogously to the second part of the proof of Theorem 2.1, we obtain 

 eI₂^◦(x)  (n + 1)^β−1 Xn m=0

e wxf

 π

m + 1



1,β

and  eI₂^◦(x)  (n + 1)^β−1 Xn m=0

e wx

 π

m + 1

 .