On the Convergence of Generalized Singular Integrals

On the Convergence of Generalized Singular Integrals

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Anastassiou, George A and Gal, Sorin G (2000) On the Convergence of Generalized Singular Integrals. RGMIA research report collection, 3 (3).

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On the Convergence of Generalized Singular Integrals

by

George A. Anastassiou

Department of Mathematical Sciences University of Memphis

Memphis, TN 38152 U.S.A.

E-mail: anastasg@msci.memphis.edu and

Sorin G. Gal

Department of Mathematics University of Oradea Str. Armatei Romane 5

3700 Oradea, Romania E-mail: galso@math.uoradea.ro

Abstract

In this article we study the degree of L^p-approximation (1≤ p≤ +∞) to the unit, by univariate and multivariate variants of the Jackson-type generalizations of Picard, Gauss–Weierstrass and Poisson–Cauchy singular integrals.

Part A: Univariate Results

1. Introduction

Let f be a function fromR into itself. For r ∈ N, the rth L_p-modulus of smoothness overR (1≤p≤+∞) is defined by

ωr(f;δ)_X = sup

|h|≤δ

k∆^r_hfk_X,

where

∆^r_hf(x) =

r

X

i=0

(−1)ⁱ r i

!

f(x+ih), r∈N,

X=L^p(R) or X =L^p_2π(R), kfk_Lp(R)=

Z +∞

−∞

|f(x)|^pdx 1/p

,

kfk_L^p

2π(R)= Z _π

−π

|f(x)|^pdx 1/p

.

Next, forξ >0 we consider the Jackson-type generalizations of Picard, Poisson–Cauchy and Gauss–Weierstrass singular integrals introduced in [3] by

P_n,ξ(f;x) =− 1 2ξ

n+1

X

k=1

(−1)^k n+ 1 k

!Z +∞

−∞

f(x+kt)e^−|t|/ξdt, Qn,ξ(f;x) = 1

−²_ξtan^−1π_ξ

n+1

X

k=1

(−1)^k n+ 1 k

!Z π

−π

f(x+kt) t²+ξ² dt.

and

W_n,ξ(f;x) =− 1 2C(ξ)

n+1

X

k=1

(−1)^k n+ 1 k

!Z _π

−π

f(x+kt)e^−t2/ξ2dt, C(ξ) =

Z π 0

e^−t2/ξ2dt,

respectively (the above operators are introduced by generalizing the usual Picard, Poisson–

Cauchy and Gauss–Weierstrass singular integrals, by following the same idea which is used to define the Jackson’s generalized operator in classical approximation theory).

Here we consider only f such thatP_n,ξ(f;x),Q_n,ξ(f;x),W_n,ξ(f;x)∈R, for all x∈R.

2. L^p-approximation, 1≤p <+∞

The first main result of this section is

Theorem 2.1. Here take X = L¹(R) (for P_n,ξ), X = L¹_2π(R), (for W_n,ξ, Q_n,ξ), ξ >0, n∈N,f ∈X. Then

kf−P_n,ξk_X ≤

"_n+1 X

k=0

n+ 1 k

! k!

#

ω_n+1(f;ξ)_X, ξ >0; (1) kf −W_n,ξ(f)k_X ≤

1/

Z π 0

e^−u2du

Z +∞

0

(u+ 1)ⁿ⁺¹e^−u2du

·ωn+1(f;ξ)_L¹

2π(R), 0< ξ ≤1; (2)

kf−Q_n,ξ(f)k_X ≤K(n, ξ)ωn+1(f;ξ)_L¹

2π(R), ξ >0, (3)

where K(n, ξ) =^h1/tan^{−1 π}_ξ^iR₀^{π/ξ (u+1)}_u2+1ⁿ⁺¹du.

Remark. For fixed n∈N, by (1) and (2) it follows that

kf −P_n,ξ(f)k_X →0, kf−W_n,ξ(f)k_X →0 asξ →0.

On the other hand, because K(n, ξ) → +∞, as ξ → 0, by (3) we do not obtain, in general, the convergence kf−Q_n,ξ(f)k_X →0 asξ →0. However, in some particular cases the convergence holds, as can be seen by the following.

Corollary 2.1. If f⁽ⁿ⁺¹⁾∈L¹_2π(R) and f⁽ⁿ⁾ is absolutely continuous on R, then kf−Q_n,ξ(f)k_L1

2π(R)≤C_nξ, 0< ξ≤1 where Cn>0 is a constant independent of f and ξ.

The second main result of the section follows.

Theorem 2.2. Let us consider X = L^p(R) (for P_n,ξ), X = L^p_2π(R) (for W_n,ξ, Q_n,ξ), 0< ξ≤1, n∈N, 1< p <+∞, ¹_p +¹_q = 1, f ∈X. Then

kf −P_n,ξ(f)k_X ≤(2/q)^1/qkgk_Lp(R+)ω_n+1(f;ξ)_X, where g(u) = (u+ 1)ⁿ⁺¹e^−u/2;

kf −W_n,ξ(f)k_X ≤ rπ

2q

1/q 1 Rπ

0 e^−u2dukhk_Lp(R+)ω_n+1(f;ξ)_X, where h(u) = (u+ 1)ⁿ⁺¹e^−u2/2;

kf−Q_n,ξ(f)k_X ≤K_p(n, ξ)ω_n+1(f;ξ)_L^p

2π(R), where K_p(n, ξ) =

1 tan^{−1 π}_ξ

Rπ/ξ

0 (u+ 1)^(n+1)p_u2¹+1du 1/p

.

Remark. Theorem 2.2 shows us that

kf−P_n,ξ(f)k_X ≤C1ωn+1(f;ξ)X, kf−W_n,ξ(f)k_X ≤C2ωn+1(f;ξ)X

whereC1, C2 >0 are independent off,nand ξ, whileKp(n, ξ) in the third estimation (in Theorem 2.2) tends to +∞ with ξ → 0. In this case, as in Corollary 2.1 we can improve the estimation of kf −Q_n,ξ(f)k_X.

3. Uniform Approximation by Qn,ξ Operator

We present

Theorem 3.1. For 0< ξ≤1, n∈N, f ∈X=C_2π(R), we get the estimation kf −Q_n,ξ(f)k_X ≤K(n, ξ)ωn+1(f;ξ)_X,

where K(n, ξ) is given by Theorem 2.1.

Corollary 3.2. If f⁽ⁿ⁺¹⁾∈C_2π(R) =X, then

kf −Q_n,ξ(f)k_X ≤Cnξ, 0< ξ ≤1, where Cn>0 is independent of f and ξ.

Part B: Multivariate Results

4. Introduction

Let f be a function defined onR^m with values inR.

Let x= (x1, . . . , xm), h= (h1, . . . , hm)∈R^m. Let us denote

∆^r_hf(x) =

r

X

i=0

(−1)^r−i r i

!

f(x+ih), r∈N.

We define the rth L^p-modulus of smoothness overR^m, 1≤p≤+∞, by ω_r(f;δ)_p = sup{k∆^r_hf(·)k_Lp(R^m);|h| ≤δ},

where|h|= (|h₁|,|h₂|, . . . ,|h_m|), δ= (δ1, . . . , δm),|h| ≤δ means|h_i| ≤δi,i=i, m and kfk_Lp(R^m)=

Z _+∞

−∞

· · · Z +∞

−∞

|f(x₁, . . . , x_m)|^pdx₁. . . dx_m 1/p

,

if 1≤p <+∞,

kfk_L^∞_(Rm)= sup{|f(x₁, . . . , xm)|;xi ∈R, i=i, m}, ifp= +∞.

Whenf ∈L^p_2π(R^m) ={f :R^m →R;fis 2π-periodic in each variable andkfk_L^p

2π(R^m)<

+∞}, therth L^p-modulus of smoothness is defined as above, where kfk_L^p

2π(R^m)= Z _π

−π

· · · Z π

−π

|f(x1, . . . , xm)|^pdx1. . . dxm

1/p

,

if 1≤p <+∞, kfk_L^p

2π(R^m)= sup{|f(x₁, . . . , x_m)|;x_i ∈[−π, π], i= 1, m}, ifp= +∞.

Next, for ξ= (ξ1, . . . , ξm)>0 (i.e., ξi >0,i= 1, m), we consider the multivariate vari- ants of the Jackson-type generalizations of Picard, Poisson–Cauchy and Gauss–Weierstrass singular integrals introduced in [2] by

P_n,ξ(f;x) =− 1 Qm i=1

(2ξ_i)

n+1

X

k=1

(−1)^k n+ 1 k

!Z +∞

−∞

· · · Z +∞

−∞

f(x1+kt1, . . . , xm+ktm)

m

Y

i=1

e^−|ti|/ξidt1. . . dtm,

Q_n,ξ(f;x) =− 1 Qm

i=1

h2

ξitan⁻¹_ξ^π

i

i

n+1

X

k=1

(−1)^k n+ 1 k

!Z π

−π

· · · Z π

−π

f(x1+kt1, . . . , xm+ktm)

m

Q

i=1

(t²_i +ξ_i²)

dt1. . . dtm,

and

W_n,ξ(f;x) =− 1

m

Q

i=1

(2C(ξi))

n+1

X

k=1

(−1)^k n+ 1 k

!Z π

−π

· · · Z π

−π

f(x1+kt1, . . . , xm+ktm)

m

Y

i=1

e^−t2i/ξ2i dt1. . . , dtm,

respectively, where C(ξi) =^R₀^πe^−t2i/ξ2idti,i= 1, m,x= (x1, . . . , xm)∈R^m. We denote

Erf(x_i) = 2

√π Z xi

0

e^−t2idt_i, x_i ∈R.

Here we obtain analogous results as in Part A.

5. L^p-approximation 1≤p≤+∞

The first main result here is

Theorem 5.1. Let X = L¹(R^m) (for P_n,ξ), X = L¹_2π(R^m) (for W_n,ξ, Q_n,ξ), ξ ∈ R^m, ξ >0, n∈N, f ∈X. Then

kf−P_n,ξ(f)k_X ≤

"_n+1 X

k=0

n+ 1 k

! k!

#m

ω_n+1(f;ξ)_X, ξ >0;

kf−W_n,ξ(f)k_X ≤

" R+∞

0 (u+ 1)ⁿ⁺¹e^−u2du Rπ

0 e^−u2du

#m

ωn+1(f;ξ)_L¹

2π(R^m), 0≤ξ≤1;

and

kf−Q_n,ξ(f)k_X ≤

"_m Y

i=1

K(n, ξi)

#

ωn+1(f;ξ)_X, ξ >0, where

K(n, ξ_i) =

"

Z _π/ξi

0

(u+ 1)ⁿ⁺¹ u²+ 1 du

#

/tan⁻¹ π ξi

, i=i, m.

The next main result has as follows:

Theorem 5.2. Let X = L^p(R^m) (for P_n,ξ), X = L^p_2π(R^m) (for W_n,ξ, Q_n,ξ), ξ ∈ R^m, 0< ξ_i ≤1, i= 1, m, n∈N, 1< p <+∞, _p¹+¹_q = 1,f ∈X. Then

kf−P_n,ξ(f)k_X ≤ 2

q m/q

kgk^m_Lp(R+)ω_n+1(f;ξ)_X, where g(u) = (u+ 1)ⁿ⁺¹e^−u/2,u∈R₊;

kf −W_n,ξ(f)k_X ≤

"

rπ 2q

1/q

khk_Lp(R+)

Z π 0

e^−u2du

#m

ω_n+1(f;ξ)_X, 0< ξ ≤1,

where h(u) = (u+ 1)ⁿ⁺¹e^−u2/2; and kf−Q_n,ξ(f)k_X ≤

"_m Y

i=1

K_p(n, ξ_i)

#

ω_n+1(f;ξ)_X, 0≤ξ≤1,

where Kp(n, ξi) =^hR₀^π/ξi[(u+ 1)^(n+1)p/(u²+ 1)]du/tan⁻¹_ξ^π

i

i1/p

. The last result has to do with the uniform convergence.

Theorem 5.3. Let X = L^∞(R^m) (for P_n,ξ), X = L^∞_2π(R^m) (for W_n,ξ, Q_n,ξ), ξ ∈ R^m, 0< ξ≤1, n∈N, f ∈X. Then

kf−P_n,ξ(f)k_X ≤

"_m X

k=0

n+ 1 k

! k!

#m

ω_n+1(f;ξ)_X, ξ >0;

kf−W_n,ξ(f)k_X ≤

" R+∞

0 (u+ 1)ⁿ⁺¹e^−u2du Rπ

0 e^−u2du

#m

ωn+1(f;ξ)_X, 0< ξ ≤1;

and

kf−Q_n,ξ(f)k_X ≤

"_m Y

i=1

K(n, ξ_i)

#

ω_n+1(f;ξ)_X, ξ >0, where K(n, ξ_i) is given in Theorem 5.1.

Remark. Theorems 5.1–5.3 show us that while the generalized operators P_n,ξ and W_n,ξ give very good estimates (such that if ξ → 0, i.e., ξ_i → 0, i = 1, m, then P_n,ξ(f) → f, W_n,ξ(f)→f), for the generalized operatorQ_n,ξ(f) in general this does not happen, because ifξ_i →0, we haveK_p(n, ξ_i)→+∞, for all 1≤p <+∞.

However, under some smoothness conditions for f, (as for example if

∂^|k|f

∂x^k₁¹···∂x^km_m

≤M onR^m, for all |k| ∈ {0,1, . . . , n+ 1}, where|k|=k1+· · ·+km,ki∈N∪ {0},i= 1, m) and reasoning as in the univariate case, Part A, we easily get that Q_n,ξ(f)→f, asξ →0.

We intend to publish the above results with full proofs and more discussion elsewhere.

References

[1] Anastassiou, G. A. and S. G. Gal, General theory of global smoothness preservation by singular integrals, univariate case,J. Comp. Anal. Appl. 1, No. 3 (1999), 289–317.

[2] Anastassiou, G. A. and S. G. Gal, Global smoothness preservation by multivariate singular integrals,Bull. Austal. Math. Soc. 61(2000), 489–506.

[3] Gal, S. G., Degree of approximation of continuous functions by some singular inte- grals, Rev. Anal. Num´er. Th´eor. Approx. (Cluj), Tome XXVII, No. 2 (1998), 251–261.