ON SATURATION ORDER OF FUNCTIONS OF SOME VARIABLES BY SINGULAR INTEGRALS

ISSN: 1311-1728 (printed version); ISSN: 1314-8060 (on-line version)

doi:_{http://dx.doi.org/10.12732/ijam.v31i3.4}

ON SATURATION ORDER OF FUNCTIONS OF SOME VARIABLES BY SINGULAR INTEGRALS

Ali M. Musayev

Azerbaijan State Oil and Industry University, Azadlig av. 20, AZ 1601, Baku, AZERBAIJAN

Abstract: In the paper we consider approximation of functions f(x) _∈

Lp_{(Rn), byα-singular integrals, determine approximation order and saturation} class.

AMS Subject Classification: 41A35, 42A38, 42B20

Key Words: singular integrals, one 0-dimensional kernels, Fourier transfor-mation, Fejer operator

1. Introduction

Let Rn be n dimensional Euclidean space and λ = (λ1, λ2, ..., λn),

t = (t1, t2, ..., tn), Kλ(t) = n Q l=1

Kl, λl(tl), where Kl, λl(tl) (tl ∈ R1, λl > 0,

1_≤l_≤n) are one-dimensional kernels satisfying the following conditions:

1 √

2π

+∞ Z

−∞

Kl, λl(tl)dtl = 1, kKl, λl(tl)kL(R₁₎≤Ml<∞,

lim λl→∞

Z

|tl|≤δ

|Kl, λl(tl)|dtl = 0 (1)

for every 1_≤l_≤n.

Consider for every 1_≤l_≤n, the α-singular integral of the general form

Q(α)_{l, λ}

l(f, x) =

1 (√2π)n

R

Rn

( ∞ P s₁, ..., sn=1

_n Q l=1

(₋1)sl−1

α sl

×f(x1−s1t1, ..., xn−sntn)} _n

Q l=1

Kl, λl(tl)

dt1...dtn,

(2)

whereα >0 is any real number, and one-dimensional kernels satisfy conditions (1).

Note that if f(x) _∈Lp(Rn) (1_≤ p <_∞) and the kernel Kl, λl(tl) satisfies

conditions (1), then singular integral (2) exists almost everywhere on Rn and the following relations are valid:

a) Q

(α) l, λl(f, x)

_Lp

(Rn) ≤

C2kf(x)kLp_(R n)·

n Q l=1k

Kl, λl(tl)kL(R1);

b) lim

λ1→ ∞ · · ·

λn→ ∞ Q

(α)

l, λl(f, x)−f(x)

Lp_(Rn)= 0;

c) lim λµ→∞

Q

(α)

l, λl(f, x)−f(x)

Lp_(R n)

= Q

(α)

l, λl(f, x)−f(x)

Lp_(R n)

for every 1_≤µ_≤nand lim λµ→∞

means thatλµ→ ∞for every 1≤µ≤n(µ6=l);

d)

f(x)−Q (α) l, λl(f, x)

Lp_(R n)≤

n−1 P µ=1

( ∞ P S1, ..., Sn=1

" n Q l=µ+1 α sl # × " n Q l=µ+1

Kl, λl(tl)

#

L(R1)

 

 +

f(x)−Q (α)

n, λn(f, x)

Lp_(R n)

.

In the sequel, we will assume

G(α)_{l, λ}

l(ul) =

∞ X

Sl=1

(₋1)sl−1

α sl

K_{l, λ}∧ _l(ulsl)

at λl>0 for every 1≤l ≤n, where Kl, λ∧ l(ul) is Fourier transformation of the

functions Kl, λl(tl).

Denote byF the set of all infinitely differentiable functions with a compact support. Introduce the class of functions

MFl(ψ) ≡

ψ(x)_∈F, ηl(ul)ψ∧(u) =r∧_ψ(u)

Theorem 1. Letf(x)_∈Lp_{(Rn) (1≤p <∞)and one-dimensional kernels}

Kl, λl(tl) (tl ∈R1, λl >0, l= 1, n) of singular integrals (2) be such that the function

β_λ(α)

l (ul) =

1₋G(α)_{l, λ}

l(ul)

τl(λl)ηl(ul) (τl(λl)>0, λliml→0

τl(λl) = 0)

be Fourier-Stieltjes transformation of some function

µ(α)_λ (t)_∈N BV(_−∞; +_∞)

(i.e. √1 2

∞ R −∞

dµ(α)_λ

l (t) = 1 and

δ R −∞

+ +_R∞

δ dµ

(α) λl (t)

→ 0asλl→ ∞, 1≤l≤n).

Then: I. if

f(x)−Q (α) l, λl(f, x)

_Lp_(R

n)

=O

n X

l=1

τl(λl) !

(3)

asλ_{→ ∞}, then f(x) = 0 almost everywhere on Rn;

II. The following relations are equivalent: A)

f(x)−Q (α) l, λl(f, x)

Lp_(R n)

=O

n X

l=1

τl(λl) !

(4)

asλ_{→ ∞} (this means that λl→ ∞ for every1≤l≤nseparately);

B) there exist a bounded measure υ on Rn and the function

l(x) _∈ Lp_{(Rn) such that for every ψ(x) ∈ M}l

F(ψ) the following relation is

valid

Z

Rn

rψ(x)f(x)dx=   

 

R

Rn

ψ(x)dυ(x) for p= 1,

R

Rn

ψ(x)l(x) for 1< p <_∞. (5)

Proof. Let us consider the case 1< p <_∞. I. According to c) we have

f(x)−Q (α) l, λl(f, x)

Lp_(R n)

=O (τl(λl)) (1_≤l_≤n), (6)

asλl→ ∞. Then for any function ψ(x)∈F we have

lim λl→∞

Z

Rn

f(x)₋Q(α)_{l, λ}

l(f, x)

τl(λl)

As the singular integral (2) is a convolution type integral, we find

Z

Rn

f(x)₋Q(α)_{l, λ}

l(f, x)

τl(λl)

ψ(x)dx= Z

Rn

ϕ(x)₋Q(α)_{l, λ}

l(ψ, x)

τl(λl)

f(x)dx (8)

for everyψ(x)_∈F.

Furthermore, from ψ(x) _∈ Ml

F(ψ) and theorem on convolution of Fourier transformation, we have





f(x)₋Q(α)_{l, λ}

l(f, x)

τl(λl)



 ∧

(u) = 1−Q (α) l, λl(ul)

τl(λl)

ψ∧(u)

=hµ(α)_λ

l (tl)

i∨

(ul)r_ψ∧(u) = 

 1 √

2π

∞ Z

−∞

rψ(x₋tl)dµ(α)_λ

l (tl)



 ∧

(u).

Hence by the uniqueness of the Fourier transformation, we find:

ψ(x)₋Q(α)_{l, λ}

l(f, x)

τl(λl)

= √1 2π

∞ Z

−∞

rψ(x₋tl)dµ(α)_λ

l (tl).

From the last equality it follows that

ψ(x)₋Q(α)_{l, λ}

l(f, x)

τl(λl) −rψ(x) _Lp_(R

n)

≤ √1

2π krψ(x−tl)−rψ(x)kLp(Rn)dµ

(α)

λl (tl)→0

asλl→ ∞ (1≤l≤n). Hence we have

lim λl→∞

Z

Rn

ψ(x)₋Q(α)_{l, λ}

l(ψ, x)

τl(λl) f(x)d(x) = Z

Rn

rψ(x)f(x)dx (9)

forf(x)_∈Lp(Rn).

Therefore by (7) and (8), Z

Rn

forrψ(x)∈F. Hence we conclude, that f(x) = 0 almost everywhere on Rn. II. A)_⇒B). Taking into account c), from (4) we get

f(x)−Q (α) l, λl(f, x)

Lp_(R n)

=O (τl(λl)) (λl→ ∞).

Then by the theorem on weak compactness (see [4], p.16) there exist the function l(x) _∈ Lp_{(Rn) and sequence of numbers l}

i

lim li→∞

λli =∞

such that

lim li→∞

Z

Rn

f(x)₋Q(α)_{l, λ}

l(f, x)

τl(λl) ψ(x)d(x) = Z

Rn

ψ(x)l(x)dx (10)

for any functionψ(x)_∈F.

As the singular integral (2) is a convolution type integral, then taking ac-count (9), we find

lim λl→∞

Z

Rn

f(x)₋Q(α)_{l, λ}

l(f, x)

τl(λl) ψ(x)d(x)

= lim λl→∞

Z

Rn

ψ(x)₋Q(α)_{l, λ}

l(ψ, x)

τl(λl)

f(x)dx= Z

Rn

rψ(x)f(x)dx. (11)

Comparing (10) and (11), we have

Z

Rn

rψ(x)f(x)dx= Z

Rn

ψ(x)l(x)dx,

i.e. B) is valid.

Now prove B)_⇒A). As

Q(α)_l,λ

l(f) =

Z

Rn

ψ(x)₋Q(α)_{l, λ}

l(ψ, x)

τl(λl)

f(x)dx

= Z

Rn

f(x)₋Q(α)_{l, λ}

l(f, x)

τl(λl)

ψ(x)dx=Q(α)_l,λ

l(ψ),

then as in the proof of relation b)_⇒c) of the theorem in [7], we have

f(x)₋Q(α)_{l, λ}

l(f, x)

τl(λl)

= _√1 2π

∞ Z

−∞

or

f(x)₋Q(α)_{l, λ}

l(f, x)

τl(λl) −rψ(x)

Lp_(Rn)

≤ √1

2π kl(x)kLp(Rn)

∞ Z −∞ dµ (α) λl (tl)

≤M.

That is, independently ofλl (1≤l≤n),

f(x)−Q (α) l, λl(f, x)

_Lp_(R

n)

=O (τl(λl)) (λl→ ∞).

Taking into account d) from the last equality we find

f(x)−Q (α)

λ (f, x)

Lp_(R n)

=O

n X

l=1

τl(λl) !

asλ_{→ ∞}, i.e. A) is valid.

The theorem has been proved for 1< p <_∞. For p= 1, it is proved in the same way.

Apply this theorem to Fejer’s specific linear operator,

σλ(f, x) = _n 1 Q l=1

(2πλl) Z

Rn

f(x₋t) n Y

l=1

sin1₂λltl 1 2tl

!2

dt (12)

in the case 1_≤p_≤2. In this case α= 1 and

Kλ = 1 n Q l=1

(2πλl) n Y

l=1

sin1₂λltl 1 2tl

!2 =

n Y

l=1

Kl, λl(tl).

Since

[Kl, λl(tl)]

∧_{(ul) =} (

1 √

2π

1₋|ul|

λl

for _|ul|< λl, 0 for _|u_{l| ≥}λl, then

G(α)_{l, λ}

l(ul) =

(

1₋|ul|

λl for |ul|< λl ,

0 for _|ul| ≥λl.

Therefore, the functionsτl(λl) = _λ1

1₋G(α)_{l, λ}

l(ul)

τl(λl)_|ul| =

1 for _|ul|< λl, λl

ul for |ul| ≥λl.

(13)

It is known [6] that

1 √ 2π ∞ Z −∞ r 2 π sintl

tl −

Citl

e−tlul_λl

dtl =

1 for _|ul|< λl, λl

ul for |ul| ≥λl,

(14)

whereCitl=− ∞ R tl

cosu u du. Introduce the function

lλl(tl) =λl

r 2

π

sintlλl

tl −

Ci(tlλl) . Since ∞ Z −∞

|lλl(tl)|dtl =λl

r 2 π ∞ Z −∞

sintlλl

tl −

Ci(tlλl) dtl = r 2 π ∞ Z −∞

sintl

tl −

Citl

dtl≤M1<∞,

thenlλl(tl)∈L(Rn).

On the other hand, if by

µλl(tl) =

tl

Z

−∞

lλl(ul)dul (1≤l≤u)

we denote the uniformly bounded measure on R1, then by ∞

Z

−∞ sintl

tl

dtl=π and ∞ Z

−∞

Citldtl= 0,

we getµλl(−∞) = 0, µλl(+∞) =

√

2π for all values ofλl and

[V arµλl(tl)]

∞ −∞=

∞ Z

−∞

|lλl(ul)|dul≤M2 <∞.

1₋C_{l, λ}(α)

l(ul)

τl(λl)ηl(ul)

is Fourier-Stieltes transformation of normalized function with bounded varia-tion.

Consequently, the conditions of the theorem are satisfied for Fejer’s singular integral.

Therefore we have the following

Corollary 2. Let f(x)_∈Lp_{(Rn) (1≤p≤2). Then for the relation} kσλ(f, x)₋f(x)_kLp =O

n X

l=1 1

λl !

to hold asλl→ ∞, it is necessary and sufficient that almost everywheref(x) = 0 onRn.

Corollary 3. Let f(x)_∈Lp_{(Rn) (1≤p≤2). For the relation} kσλ(f, x)−f(x)kLp_(R

n)=O

n X

l=1 1

λl !

to hold as λl→ ∞, it is necessary and sufficient that f(x)∈np(f), where

np(f) =   

 

f(x)_∈L(Rn)/f(x)_∈BS

(Rn), for p= 1, f(x)_∈Lp_{(Rn)/f(x)∈AC}

loc(R1), with respect to xl,

u∂f_∂x(x)

l ∈L

p_(R

n), (1≤p≤2, 1≤l≤n).

References

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[2] P. Butzer, R. Nessel, Fourier Analysis and Approximation, Vol. 1, New York and London, 1971, 553 p.

[4] H. Berens and P. Butzer, Uber die Darstelling holomorpher Funktionen-durch Laplace-und Laplace Stieltjes Integrale,Mat. Z., 81, 1963.

[5] Cynoymu (G. Sunouchi), Direct theorems in the theory of approximation,

Acta Math.,20, No 3-4 (1969), 409-420.

[6] A.M. Musayev, To the question of approximation of functions by the Mellin type operators in the spaceXσ1,σ2(E

+_{).Proc. of IMM of NAS Azerbaijan,} 28(2008), 69-73.