(1)ON THE DEGREE OF STRONG APPROXIMATION OF INTEGRABLE FUNCTIONS WŁODZIMIERZ ŁENSKI UNIVERSITY OFZIELONAGÓRA FACULTY OFMATHEMATICS, COMPUTERSCIENCE ANDECONOMETRICS 65-516 ZIELONAGÓRA,UL

ON THE DEGREE OF STRONG APPROXIMATION OF INTEGRABLE FUNCTIONS

WŁODZIMIERZ ŁENSKI UNIVERSITY OFZIELONAGÓRA

FACULTY OFMATHEMATICS, COMPUTERSCIENCE ANDECONOMETRICS

65-516 ZIELONAGÓRA,UL. SZAFRANA4A

POLAND

W.Lenski@wmie.uz.zgora.pl

Received 20 December, 2006; accepted 25 July, 2007 Communicated by L. Leindler

ABSTRACT. We show the strong approximation version of some results of L. Leindler [3] con- nected with the theorems of P. Chandra [1, 2].

Key words and phrases: Degree of strong approximation, Special sequences.

2000 Mathematics Subject Classification. 42A24, 41A25.

1. INTRODUCTION

Let L^p (1 < p < ∞) [C] be the class of all 2π–periodic real–valued functions integrable in the Lebesgue sense withp–th power [continuous] overQ = [−π, π]and letX^p =L^p when 1< p <∞orX^p =C whenp=∞. Let us define the norm off ∈X^p as

kfk_Xp =kf(·)k_Xp =





 R

Q|f(x)|^pdx¹_p

when 1< p <∞, sup_x∈Q|f(x)| when p=∞, and consider its trigonometric Fourier series

Sf(x) = a_o(f)

2 +

∞

X

ν=0

(a_ν(f) cosνx+b_ν(f) sinνx) with the partial sumsS_kf.

LetA:= (a_n,k) (k, n= 0,1,2, . . .)be a lower triangular infinite matrix of real numbers and let theA−transforms of(S_kf)be given by

T_n,Af(x) :=

n

X

k=0

a_n,kS_kf(x)−f(x)

(n= 0,1,2, . . .)

323-06

and

H_n,A^q f(x) :=

( _n X

k=0

a_n,k|S_kf(x)−f(x)|^q )¹_q

(q >0, n= 0,1,2, . . .).

As a measure of approximation, by the above quantities we use the pointwise characteristic w_xf(δ)_L^p :=

1 δ

Z δ 0

|ϕ_x(t)|^pdt

1 p

, where

ϕ_x(t) := f(x+t) +f(x−t)−2f(x).

wxf(δ)L^p is constructed based on the definition of Lebesgue points(L^p−points),and the mod- ulus of continuity forf in the spaceX^pdefined by the formula

ωf(δ)_Xp := sup

0≤|h|≤δ

kϕ·(h)k_Xp.

We can observe that withpe≥p,forf ∈X^ep, by the Minkowski inequality kw·f(δ)_pk

Xpe ≤ωf(δ)_Xep.

The deviation T_n,Af was estimated by P. Chandra [1, 2] in the norm of f ∈ C and for monotonic sequences a_n = (a_n,k). These results were generalized by L. Leindler [3] who considered the sequences of bounded variation instead of monotonic ones. In this note we shall consider the strong meansH_n,A^q f and the integrable functions. We shall also give some results on norm approximation.

ByKwe shall designate either an absolute constant or a constant depending on some param- eters, not necessarily the same of each occurrence.

2. STATEMENT OF THE RESULTS

Let us consider a functionwx of modulus of continuity type on the interval[0,+∞),i.e., a nondecreasing continuous function having the following properties:wx(0) = 0, wx(δ1+δ2)≤ w_x(δ₁) +w_x(δ₂)for any0≤δ₁ ≤δ₂ ≤δ₁+δ₂ and let

L^p(w_x) ={f ∈L^p :w_xf(δ)_Lp ≤w_x(δ)}. We can now formulate our main results.

To start with, we formulate the results on pointwise approximation.

Theorem 2.1. Leta_n= (a_n,m)satisfy the following conditions:

(2.1) a_n,m ≥0,

n

X

k=0

a_n,k = 1 and

(2.2)

m−1

X

k=0

|a_n,k−a_n,k+1| ≤Ka_n,m, where

m= 0,1, . . . , n and n= 0,1,2, . . . X⁻¹

k=0 = 0 . Supposew_xis such that

(2.3)

u^pq

Z π u

(wx(t))^p t^1+pq dt

¹_p

=O(uH_x(u)) as u→0+,

whereH_x(u)≥0,1< p ≤qand (2.4)

Z t 0

H_x(u)du=O(tH_x(t)) as t→0 +. Iff ∈L^p(w_x),then

H_n,A^q0 f(x) = O(a_n,nH_x(a_n,n)) withq⁰ ∈(0, q]andqsuch that1< q(q−1)≤p≤q.

Theorem 2.2. Let(2.1),(2.2)and(2.3)hold. Iff ∈L^p(w_x)then H_n,A^q0 f(x) = O

w_x

π n+ 1

+O

a_n,nH_x π

n+ 1

and if, in addition,(2.4)holds then

H_n,A^q0 f(x) =O

a_n,nH_x π

n+ 1

withq⁰ ∈(0, q]andqsuch that1< q(q−1)≤p≤q.

Theorem 2.3. Let(2.1),(2.3),(2.4)and (2.5)

∞

X

k=m

|a_n,k−a_n,k+1| ≤Ka_n,m, where

m = 0,1, . . . , n and n = 0,1,2, . . . hold. Iff ∈L^p(w_x)then

H_n,A^q0 f(x) =O(an,0Hx(an,0)) withq⁰ ∈(0, q]andqsuch that 1< q(q−1)≤p≤q.

Theorem 2.4. Let us assume that(2.1),(2.3)and(2.5)hold. Iff ∈L^p(w_x),then H_n,A^q0 f(x) =O

w_x

π n+ 1

+O

a_n,0H_x

π n+ 1

. If, in addition,(2.4)holds then

H_n,A^q0 f(x) =O

a_n,0H_x π

n+ 1

withq⁰ ∈(0, q]andqsuch that 1< q(q−1)≤p≤q.

Consequently, we formulate the results on norm approximation.

Theorem 2.5. Leta_n = (a_n,m)satisfy the conditions(2.1)and(2.2). Supposeωf(·)_Xpeis such that

(2.6)

u^pq

Z π u

(ωf(t)_Xpe)^p t^1+pq dt

¹_p

=O(uH(u)) as u→0+

holds, with1 < p ≤ q andpe≥ p,whereH (≥0)instead of H_x satisfies the condition (2.4).

Iff ∈X^pethen

H_n,A^q0 f(·)

Xpe

=O(a_n,nH(a_n,n)) withq⁰ ≤q andp≤pesuch that 1< q(q−1)≤p≤q.

Theorem 2.6. Let(2.1), (2.2)and(2.6)hold. Iff ∈X^pethen

H_n,A^q0 f(·)

Xep

=O

ωf π

n+ 1

X^pe

+O

a_n,nH

π n+ 1

. If, in addition,H(≥0)instead ofHx satisfies the condition(2.4)then

H_n,A^q0 f(·)

Xep

=O

a_n,nH π

n+ 1

withq⁰ ≤q andp≤pesuch that 1< q(q−1)≤p≤q.

Theorem 2.7. Let (2.1), (2.4) with a function H (≥0) instead of H_x, (2.5) and (2.6) hold.

Iff ∈X^pethen

H_n,A^q0 f(·)

Xpe

=O(a_n,0H(a_n,0)) withq⁰ ≤q andp≤pesuch that 1< q(q−1)≤p≤q.

Theorem 2.8. Let(2.1),(2.5)and(2.6)hold. Iff ∈X^pethen

H_n,A^q0 f(·)

Xep

=O

ωf π

n+ 1

X^pe

+O

an,0H

π n+ 1

. If, in addition,H(≥0)instead ofH_xsatisfies the condition(2.4),then

H_n,A^q0 f(·)

Xpe

=O

a_n,0H π

n+ 1

withq⁰ ≤q andp≤pesuch that 1< q(q−1)≤p≤q.

3. AUXILIARY RESULTS

In tis section we denote byωa function of modulus of continuity type.

Lemma 3.1. If(2.3)with0 < p ≤ q and(2.4)with functionsω andH (≥0)instead of w_x andHx,respectively, hold then

(3.1)

Z u 0

ω(t)

t dt=O(uH(u)) (u→0+). Proof. Integrating by parts in the above integral we obtain

Z u 0

ω(t) t dt=

Z u 0

td dt

Z π t

ω(s) s² ds

dt

=

−t Z π

t

ω(s) s² ds

u 0

+ Z u

0

Z π t

ω(s) s² ds

dt

≤u Z π

u

ω(s) s² ds+

Z u 0

Z π t

ω(s) s² ds

dt

=u Z π

u

ω(s)

s^1+pq+1−pqds+ Z u

0

Z π t

ω(s) s^1+pq+1−pqds

dt

≤u^pq Z π

u

ω(s) s^1+p/qds+

Z u 0

1 t

t^pq

Z π t

(ω(s))^p s^1+p/q ds

¹_p dt, since1− ^p_q ≥0.Using our assumptions we have

Z u 0

ω(t)

t dt=O(uH(u)) + Z u

0

1

tO(tH(t))dt=O(uH(u))

and thus the proof is completed.

Lemma 3.2 ([4, Theorem 5.20 II, Ch. XII]). Suppose that 1 < q(q−1) ≤ p ≤ q and ξ = 1/p+ 1/q−1.If

t^−ξg(t)

∈L^p then (3.2)

(|a_o(g)|^q

2 +

∞

X

k=0

(|a_k(g)|^q+|b_k(g)|^q) )¹_q

≤K Z π

−π

t^−ξg(t)

pdt ¹_p

.

4. PROOFS OF THERESULTS

SinceH_n,A^q f is the monotonic function ofqwe shall consider, in all our proofs, the quantity H_n,A^q f instead ofH_n,A^q0 f.

Proof of Theorem 2.1. Let

H_n,A^q f(x) = ( _n

X

k=0

a_n,k 1 π

Z π 0

ϕ_x(t)sin k+¹₂ t 2 sin¹₂t dt

q)¹_q

≤ ( _n

X

k=0

a_n,k 1 π

Z an,n

0

ϕ_x(t)sin k+ ¹₂ t 2 sin¹₂t dt

q)¹_q

+ ( _n

X

k=0

an,k

1 π

Z π an,n

ϕx(t)sin k+¹₂ t 2 sin¹₂t dt

q)¹_q

=I(a_n,n) +J(a_n,n) and, by(2.1),integrating by parts, we obtain,

I(a_n,n)≤ Z an,n

0

|ϕ_x(t)|

2t dt

= Z an,n

0

1 2t

d dt

Z t 0

|ϕ_x(s)|ds

dt

= 1

2a_n,n Z an,n

0

|ϕ_x(t)|dt+ Z an,n

0

w_xf(t)₁ 2t dt

= 1 2

wxf(an,n)₁+ Z an,n

0

w_xf(t)₁ t dt

=K a_n,n Z π

an,n

w_xf(t)₁ t² dt+

Z an,n

0

w_xf(t)₁ t dt

!

≤K a_n,n Z π

an,n

(w_xf(t)_L1)^p t² dt

!¹_p +K

Z an,n

0

w_xf(t)_L1

t dt

≤K a^p/q_n,n Z π

an,n

(w_xf(t)_Lp)^p t^1+p/q dt

!¹_p +K

Z an,n

0

w_xf(t)_Lp

t dt

. Since f ∈L^p(w_x)and(2.4)holds, Lemma 3.1 and(2.3)give

I(a_n,n) = O(a_n,nH_x(a_n,n)).

The Abel transformation shows that (J(a_n,n))^q=

n−1

X

k=0

(a_n,k−a_n,k+1)

k

X

ν=0

1 π

Z π an,n

ϕ_x(t)sin ν+ ¹₂ t 2 sin¹₂t dt

q

+a_n,n

n

X

ν=0

1 π

Z π an,n

ϕ_x(t)sin ν+ ¹₂ t 2 sin¹₂t dt

q

, whence, by(2.2),

(J(a_n,n))^q ≤(K + 1)a_n,n

n

X

ν=0

1 π

Z π an,n

ϕ_x(t)sin ν+¹₂ t 2 sin¹₂t dt

q

. Using inequality(3.2),we obtain

J(a_n,n)≤K(a_n,n)^1q (Z π

an,n

|ϕx(t)|^p t^1+p/q dt

)¹_p . Integrating by parts, we have

J(a_n,n)≤K(a_n,n)^1q (

1

t^p/q (w_xf(t)_Lp)^p π

t=an,n

+

1 + p q

Z π an,n

(w_xf(t)_Lp)^p t^1+p/q dt

)¹_p

≤K(an,n)^1q (

(wxf(π)_Lp) + Z π

an,n

(w_xf(t)_Lp)^p t^1+p/q dt

)¹_p . Since f ∈L^p(wx),by(2.3),

J(a_n,n)≤K(a_n,n)^1q (

(w_x(π)) + Z π

an,n

(w_x(t))^p t^1+p/q dt

)_p¹

≤K (

(a_n,n)^pq Z π

an,n

(w_x(t))^p t^1+p/q dt

)¹_p

=O(a_n,nH_x(a_n,n)).

Thus our result is proved.

Proof of Theorem 2.2. Let, as before, H_n,A^q f(x)≤I

π n+ 1

+J

π n+ 1

and

I π

n+ 1

≤ n+ 1 π

Z _n+1^π

0

|ϕx(t)|dt=wx

π n+ 1

L¹

. In the estimate ofJ _n+1^π

we again use the Abel transformation and(2.2). Thus

J π

n+ 1 q

≤(K+ 1)a_n,n

n

X

ν=0

1 π

Z π

π n+1

ϕ_x(t)sin ν+ ¹₂ t 2 sin¹₂t dt

q

and, by inequality(3.2),

J π

n+ 1

≤K(a_n,n)^1q (Z π

π n+1

|ϕ_x(t)|^p t^1+p/q dt

)¹_p . Integrating(2.3)by parts, with the assumptionf ∈L^p(wx),we obtain

J π

n+ 1

≤K((n+ 1)a_n,n)^1q (

π n+ 1

^p_q Z π

π n+1

(w_x(t))^p t^1+p/q dt

)¹_p

=O

((n+ 1)a_n,n)^1q π n+ 1H_x

π n+ 1

as in the previous proof, with _n+1^π instead ofa_n,n.

Finally, arguing as in [3, p.110], we can see that, forj = 0,1, . . . , n−1,

|a_n,j −a_n,n| ≤

n−1

X

k=j

(a_n,k−a_n,k+1)

≤

n−1

X

k=0

|a_n,k−a_n,k+1| ≤Ka_n,n , whence

a_n,j ≤(K + 1)a_n,n and therefore

(K+ 1) (n+ 1)a_n,n≥

n

X

j=0

a_n,j = 1.

This inequality implies that J

π n+ 1

=O

an,nHx

π n+ 1

and the proof of the first part of our statement is complete.

To prove of the second part of our assertion we have to estimate the termI _n+1^π

once more.

Proceeding analogously to the proof of Theorem 2.1, witha_n,nreplaced by _n+1^π , we obtain I

π n+ 1

=O π

n+ 1H_x π

n+ 1

.

By the inequality from the first part of our proof, the relation (n+ 1)⁻¹ = O(a_n,n) holds,

whence the second statement follows.

Proof of Theorem 2.3. As usual, let

H_n,A^q f(x)≤I(an,0) +J(an,0).

Since f ∈L^p(w_x),by the same method as in the proof of Theorem 2.1, Lemma 3.1 and(2.3) yield

I(a_n,0) =O(a_n,0H_x(a_n,0)).

By the Abel transformation (J(a_n,0))^q ≤

n−1

X

k=0

|a_n,k−a_n,k+1|

k

X

ν=0

1 π

Z π an,0

ϕ_x(t)sin ν+¹₂ t 2 sin¹₂t dt

q

+a_n,n

n

X

ν=0

1 π

Z π an,0

ϕ_x(t)sin ν+¹₂ t 2 sin¹₂t dt

q

≤

∞

X

k=0

|a_n,k−a_n,k+1|+a_n,n

! _∞ X

ν=0

1 π

Z π an,0

ϕ_x(t)sin ν+¹₂ t 2 sin¹₂t dt

q

. Arguing as in [3, p.110], by(2.5), we have

|a_n,n−a_n,0| ≤

n−1

X

k=0

|a_n,k−a_n,k+1| ≤

∞

X

k=0

|a_n,k−a_n,k+1| ≤Ka_n,0, whencea_n,n ≤(K+ 1)a_n,0and therefore

∞

X

k=0

|a_n,k−a_n,k+1|+a_n,n≤(2K+ 1)a_n,0 and

(J(a_n,0))^q ≤(2K+ 1)a_n,0

∞

X

ν=0

1 π

Z π an,0

ϕ_x(t)sin ν+ ¹₂ t 2 sin¹₂t dt

q

. Finally, by(3.2),

J(a_n,0)≤K(a_n,0)^1q (Z π

an,0

|ϕ_x(t)|^p t^1+p/q dt

)_p¹

and, by(2.3),

J(a_n,0) = O(a_n,0H_x(a_n,0)).

This completes of our proof.

Proof of Theorem 2.4. We start as usual with the simple transformation H_n,A^q f(x)≤I

π n+ 1

+J

π n+ 1

. Similarly, as in the previous proofs, by(2.1)we have

I π

n+ 1

≤wxf π

n+ 1

L¹

. We estimate the termJ in the following way

J

π n+ 1

q

≤

n−1

X

k=0

|a_n,k−a_n,k+1|

k

X

ν=0

1 π

Z π

π n+1

ϕ_x(t)sin ν+ ¹₂ t 2 sin¹₂t dt

q

+a_n,n

n

X

ν=0

1 π

Z π

π n+1

ϕ_x(t)sin ν+ ¹₂ t 2 sin ¹₂t dt

q

≤

∞

X

k=0

|a_n,k−a_n,k+1|+a_n,n

! _∞ X

ν=0

1 π

Z π

π n+1

ϕ_x(t)sin ν+ ¹₂ t 2 sin¹₂t dt

q

.

From the assumption(2.5), arguing as before, we can see that J

π n+ 1

≤K a_n,0

∞

X

ν=0

1 π

Z π

π n+1

ϕ_x(t)sin ν+¹₂ t 2 sin¹₂t dt

q!¹_q

and, by(3.2),

J π

n+ 1

≤K(a_n,0)^1q (Z π

π n+1

|ϕ_x(t)|^p t^1+p/q dt

)¹_p . From(2.5),it follows thata_n,k ≤(K + 1)a_n,0 for anyk ≤n,and therefore

(K+ 1) (n+ 1)a_n,0 ≥

n

X

k=0

a_n,k = 1, whence

J π

n+ 1

≤K((n+ 1)a_n,0)^1q (

π n+ 1

^p_q Z π

π n+1

|ϕ_x(t)|^p t^1+p/q dt

)¹_p .

≤K(n+ 1)a_n,0 (

π n+ 1

^p_q Z π

π n+1

|ϕ_x(t)|^p t^1+p/q dt

)¹_p . Sincef ∈L^p(wx), integrating by parts we obtain

J π

n+ 1

≤K(n+ 1)a_n,0 π n+ 1H_x

π n+ 1

≤Kan,0Hx

π n+ 1

and the proof of the first part of our statement is complete.

To prove the second part, we have to estimate the termI _n+1^π

once more.

Proceeding analogously to the proof of Theorem 2.1 we obtain I

π n+ 1

=O π

n+ 1H_x π

n+ 1

.

From the start of our proof we have(n+ 1)⁻¹ = O(an,0), whence the second assertion also

follows.

Proof of Theorem 2.5. We begin with the inequality H_n,A^q f(·)

Xpe ≤ kI(an,n)k

Xpe+kJ(an,n)k

Xep . By(2.1),Lemma 3.1 gives

kI(a_n,n)k

Xep ≤ Z an,n

0

kϕ(t)k

Xpe

2t dt

≤ Z an,n

0

ωf(t)_Xpe

2t dt

=O(a_n,nH(a_n,n)).

As in the proof of Theorem 2.1, kJ(a_n,n)k

Xep ≤K(a_n,n)^1q

(Z π

an,n

|ϕ(t)|^p t^1+p/q dt

)¹_p

Xep

≤K(a_n,n)^1q (Z π

an,n

kϕ(t)k^p

Xep

t^1+p/q dt )¹_p

≤K(an,n)^1q (Z π

an,n

ωf(t)_Xpe

t^1+p/q dt )¹_p

, whence, by(2.6),

kJ(a_n,n)k

Xpe =O(a_n,nH(a_n,n))

holds and our result follows.

Proof of Theorem 2.6. It is clear that H_n,A^q f(·)

Xep

≤ I

π n+ 1

Xpe

+ J

π n+ 1

Xep

and immediately I

π n+ 1

Xpe

≤

w·f π

n+ 1

1

Xpe

≤ωf π

n+ 1

X^pe

and

J

π n+ 1

Xep

≤K(a_n,n)^1q (Z π

π n+1

kϕ(t)k^p

Xep

t^1+p/q dt )¹_p

≤K((n+ 1)a_n,n)^1q ( π

n+ 1 Z π

π n+1

ωf(t)_Xpe

t^1+p/q dt )_p¹

≤K((n+ 1)a_n,n)^1q π n+ 1H

π n+ 1

≤K(n+ 1)a_n,n π n+ 1H

π n+ 1

=O

a_n,nH π

n+ 1

.

Thus our first statement holds. The second one follows on using a similar process to that in the proof of Theorem 2.1. We have to only use the estimates obtained in the proof of Theorem 2.5, with _n+1^π instead ofa_n,n,and the relation

(n+ 1)⁻¹ =O(a_n,n).

Proof of Theorem 2.7. As in the proof of Theorem 2.5, we have

H_n,A^q f(·)

Xep ≤ kI(a_n,0)k

Xpe+kJ(a_n,0)k

Xpe

and

kI(an,0)k

Xpe =O(an,0H(an,0)) . Also, from the proof of Theorem 2.3,

kJ(an,0)k

Xpe ≤K(an,0)^1q

(Z π

an,0

|ϕ_·(t)|^p t^1+p/q dt

)¹_p

Xep

≤K(a_n,0)^1q (Z π

an,0

ωf(t)_Xpe

t^1+p/q dt )¹_p

and, by(2.6),

kJ(an,0)k

Xpe =O(an,0H(an,0)) .

Thus our result is proved.

Proof of Theorem 2.8. We recall, as in the previous proof, that H_n,A^q f(·)

Xep

≤ I

π n+ 1

Xpe

+ J

π n+ 1

Xep

and

I

π n+ 1

Xep

≤ωf π

n+ 1

X^ep

.

We apply a similar method as that used in the proof of Theorem 2.4 to obtain an estimate for the quantity

J _n+1^π

Xep

,

J

π n+ 1

Xep

≤K(n+ 1)a_n,0

( π n+ 1

^p_q Z π

π n+1

|ϕ_x(t)|^p t^1+p/q dt

)¹_p

Xpe

≤K(n+ 1)a_n,0 (

π n+ 1

^p_q Z π

π n+1

kϕ_·(t)k^p

Xep

t^1+p/q dt )¹_p

≤K(n+ 1)a_n,0 (

π n+ 1

^p_q Z π

π n+1

ωf(t)_Xpe

t^1+p/q dt )¹_p

and, by(2.6),

J

π n+ 1

Xep

≤K(n+ 1)a_n,0 π n+ 1H

π n+ 1

≤Kan,0H π

n+ 1

. Thus the proof of the first part of our statement is complete.

To prove of the second part, we follow the line of the proof of Theorem 2.6.

REFERENCES

[1] P. CHANDRA, On the degree of approximation of a class of functions by means of Fourier series, Acta Math. Hungar., 52 (1988), 199–205.

[2] P. CHANDRA, A note on the degree of approximation of continuous functions, Acta Math. Hungar., 62 (1993), 21–23.

[3] L. LEINDLER, On the degree of approximation of continuous functions, Acta Math. Hungar., 104 (2004), 105–113.

[4] A. ZYGMUND, Trigonometric Series, Cambridge, 2002.