View of Using wavelets for Szász-type operators

Rev. Anal. Num´er. Th´eor. Approx., vol. 24 (1995) nos. 1–2, pp. 131–145 ictp.acad.ro/jnaat

USING WAVELETS FOR SZ ´ASZ-TYPE OPERATORS

HEINZ H. GONSKA and DING-XUAN ZHOU (Duisburg)

1. INTRODUCTION

The Sz´asz-Mirakjan operators are defined onC[0,∞) as

Sn(f, x) =

∞

X

k=0

f ^k_n

Sn,k(x), (1.1)

S_n,k(x) =e^−nx(nx)_k!^k.

There has been an extensive study on the approximation by these operators. In 1978, Beker [1] extended a result of Berens and Lorentz [2] to the interval [0,∞) and showed that forf ∈C_B:=C[0,∞)∩L∞[0,∞), 0< α <2,

(1.2) ω₂(f, t) =O(t^α) ⇔ |S_n(f, x)−f(x)| ≤M ^x_n^α₂ .

Here M is a constant independent of n ∈ N and x ∈ [0,∞), ω₂(f, t) is the modulus of smoothness defined as

ω₂(f, t) = sup

0<h<t

4²_hf(·) _∞, (1.3)

4²_hf(x) =







f(x+h)−2f(x) +f(x−h), when h≤x,

0, otherwise.

By (1.2), we know that the second-order Lipschitz functions (i.e., the Lips- chitz functions with respect to the second-order modulus of smoothness) can be

characterized by the rate of convergence of Sz´asz-Mirakjan operators. Another interesting result was given by Totik [10] in 1983 who proved the following equivalence:

(1.4) ω²_ϕ(f, t)_∞=O(t^α) ⇔ kS_n(f)−fk_∞=O n^−α2

, 0< α <2.

Here ω_ϕ²(f, t)_∞ is the so-called Ditzian-Totik modulus of smoothness, which is defined for 1≤p≤ ∞as

ω_ϕ²(f, t)_p= sup

0<h<t

4²_hϕ(x)f(x) p, (1.5)

ϕ(x) =√ x.

The Sz´asz-Mirakjan operators can not be used forLp(1≤p≤ ∞)-approximation.

For this purpose, we must modify these operators. Two versions of this type are Sz´asz-Kantorovich operators:

(1.6) K_n(f, x) :=

∞

X

k=0

n Z ^k+1

n k n

f(t)dt S_n,k(x) and Sz´asz-Durrmeyer operators:

(1.7) D_n(f, x) :=

∞

X

k=0

n Z ∞

0

f(t)S_n,k(t)dt S_n,k(x).

These operators can be used for Lp-approximation on [0,∞).In fact, for Ln = K_n orD_n,0< α <2, 1≤p <∞, f ∈L_p[0,∞),we have (c.f., [5], [7])

(1.8) ω²_ϕ(f, t)_p =O(t^α) ⇔ kL_n(f)−fk_p =O n^−α2 .

Parallel to Sz´asz-Mirakjan operators, it is natural to consider characteriza- tions of Lipschitz functions by means of the above two versions of Sz´asz-type operators. Early in 1985, Mazhar and Totik [9] modified the Sz´asz-Durrmeyer operators and gave the same equivalence to (1.2). However, their modified op- erators have the disadvantage that they can not be used forLp-approximation.

In fact, for the original Sz´asz-Durrmeyer operators (1.7), Mazhar and Totik [9] posed the open problem to find an inverse theorem to the following direct estimate:

(1.9)

Dn(f, x)−f(x)

≤M ω1

f,

qx n+_n¹2

. Here ω₁(f, t) is the modulus of continuity:

(1.10) ω₁(f, t) = sup

0<h<t

f(·+h)−f(·) _∞.

In 1991, Guo and the second author [6] solved this problem and showed that for 0< α <1, f ∈C_B,

(1.11) ω₁(f, t) =O(t^α) ⇔

D_n(f, x)−f(x)

≤M ^x_n+_n¹2

^α₂ . This is the first characterization of Lipschitz functions by means of linear oper- ators which do not reproduce linear functions. Extensions to higher orders of Lipschitz functions and some other discussions can be found in a series of the second author’s (joint) papers [8], [11], [12], [14]. On the other hand, we also showed that for any 1< α <2,there exist no functions{Ψ_n,α(x)}_n∈Nsuch that (1.12) ω₂(f, t) =O(t^α) ⇔

D_n(f, x)−f(x)

≤MΨ_n,α(x).

Thus, the second-order Lipschitz functions can not be characterized by means of Sz´asz-Durrmeyer operators when 1< α <2.This happens also for Kantorovich operators, see [14]. To overcome this difficulty, Ye and the second author [11]

introduced a technique of matrices and modified the Kantorovich operators so that they can be used for characterization of second-order Lipschitz functions as well as forL_p-approximation.

The puropose of this paper is to introduce a class of Sz´asz-type operators by means of Daubechies’ compactly supported wavelets (see [3], [4]). These operators have the following advantages: Firstly, they have the same moments of finitely many orders as we arbitrarily choose as Sz´asz-Mirakjan operators, hence they can be used to characterize the second-order Lipschiz functions.

Secondly, they can be used for Lp-approximation (1< p≤ ∞) and a similar result to (1.8) holds.

In the following sections we discuss these two aspects. We shall denote byM a constant which may be different at each occurrence.

2. CONSTRUCTION OF SZ ´ASZ-TYPE OPERATORS BY WAVELETS

We recall some facts about Daubechies’ compactly supported wavelets (see [3], [4]).

GivenN ∈N, Daubechies’ compactly supported scaling wavelet_Nφis defined by the following refinement equation

φ(·) = 2

∞

X

k=0

hkφ(2· −k), (2.1)

φ(0) = 1,

where{h_k}_k∈Z is the finitely sequence given by

∞

X

k=0

h_ke^−ikω = ^1+e₂^−iωN

L_N(ω) with

L_N(ω)

2 =P_N sin^{2 ω}₂

N

X

n=0

N−1+n n

sin^{2 ω}₂n

.

This function is compactly supported with suppNφ = [0,2N−1]. Moreover, there exists a positive constantβ >0 such that forN ≥2, Nφ∈C^βN(R) and for 1≤k≤βN,

(2.2)

Z

R

x^k _Nφ(x)dx= 0.

In particular, whenN = 1, 1φ=χ_[0,1] is the classical Haar basis.

In what follows, we assume thatφ∈L∞(R) has the following properties:

(i) suppφ⊂[0, C] with 0< C <∞.

(ii) Z

R

φ(x)dx= 1,and, for 1≤k≤K,(2.2) is satisfied where K∈N. Then, our Sz´asz-type operators are defined as

(2.3) L_n(f, x) :=

∞

X

k=0

n Z

R

f(t)φ(nt−k)dtS_n,k(x).

When K = 0, and φ is the Haar basis, these operators are exactly the Sz´asz- Kantorovich operators. Thus, we see that our operators are extensions of the Sz´asz-Kantorovich operators.

By the moment condition (ii),we have the following theorem.

Theorem2.1. Let Ln(f(t), x)be defined by(2.3). Then, for0≤k≤K,we have

(2.4) Ln t^k, x

=Sn t^k, x

, x∈[0,∞).

In particular,

(2.5) Ln(1, x) = 1;

(2.6) L_n(t, x) =x.

The moment condition (2.4) is the main improvement to Sz´asz-Kantorovich and Sz´asz-Durrmeyer operators.

3. CHARACTERIZATION OF SECOND-ORDER LIPSCHITZ FUNCTIONS

We need some preliminary results to state our main result in this section.

For our proof, we need Peetre’sK-functional given by (3.1) K₂(f, t) := inf

g∈C²[0,∞)∩CB

kf−gk_∞+t g⁰⁰

_∞ , t >0.

Since forg /∈L∞[0,∞), kf−gk_∞ =∞, this K-functional is equivalent to the modulus of smoothness:

(3.2) M⁻¹ω₂(f, t)≤K₂(f, t²)≤M ω₂(f, t), f ∈C_B, 0< t≤1.

Two types of Bernstein-Markov inequalities are necessary for our purpose, which we state as follows.

Lemma3.1. Let f ∈C_B.Then we have

kL_n(f)k_∞≤Mkfk_∞; (3.3)

L⁰_n(f)

∞≤M nkfk_∞; (3.4)

L⁰⁰_n(f)

_∞≤M n²kfk_∞; (3.5)

ϕ²L⁰⁰_n(f)

_∞≤M nkfk_∞. (3.6)

Proof of Lemma 3.1. We observe that

(3.7) L_n(f, x) =

∞

X

k=0

Z C 0

f ^t+k_n

φ(t)dtS_n,k(x). Therefore, forf ∈C_B, x∈[0,∞),

L_n(f, x) ≤

∞

X

k=0

Z C 0

|φ(t)|dtS_n,k(x)kfk_∞≤Ckφk_∞kfk_∞. Hence (3.3) holds.

To prove (3.4) and (3.5), we observe that

(3.8) S_n,k⁰ (x) =n S_n,k−1(x)−S_n,k(x) . Here we setSn,−1(x)≡0.Then for f ∈C_B, x∈[0,∞),

L⁰_n(f, x) =

n

∞

X

k=0

Z C 0

f ^t+k_n

φ(t)dt Sn,k−1(x)−Sn,k(x)

≤2nCkφk_∞kfk_∞,

and L⁰⁰_n(f, x)

=

n²

∞

X

k=0

Z C 0

f ^t+k_n

φ(t)dt Sn,k−2(x)−2Sn,k−1(x) +S_n,k(x)

≤4n²Ckφk_∞kfk_∞. Hence (3.4) and (3.5) hold.

Finally, using another expression for the derivative, namely (3.9) S_n,k⁰ (x) = ^k−nx_x S_n,k(x),

we obtain (3.6):

ϕ²(x)L⁰⁰_n(f, x) =

x

∞

X

k=0

Z C 0

f ^t+k_n

φ(t)dtSn,k(x)

_(k−nx)2

x² −_x^k₂

≤Ckφk_∞kfk_∞x¹

∞

X

k=0

(k−nx)²+k

Sn,k(x)

≤2nCkφk_∞kfk_∞.

Here we used the moments of Sz´asz-Mirakjan operators:

Sn(t, x) =x, (3.10)

Sn (t−x)², x

= ^x_n. (3.11)

The proof of Lemma 3.1 is complete.

Lemma 3.2. Let f ∈C²[0,∞)∩C_B.Then we have L⁰_n(f)

_∞≤M f⁰

_∞; (3.12)

L⁰⁰_n(f)

∞≤M f⁰⁰

∞; (3.13)

ϕ²L⁰⁰_n(f)

∞≤M ϕ²f⁰⁰

∞. (3.14)

Proof of Lemma 3.2. By (3.7) and (3.8), for x∈[0,∞),we have L⁰_n(f, x)

=

∞

X

k=0

n Z C

0

f ^t+k+1_n

−f ^t+k_n

φ(t)dtS_n,k(x)

≤Ckφk_∞ f⁰

_∞,

and L⁰⁰_n(f, x)

=

∞

X

k=0

n² Z C

0

f ^t+k+2_n

−2f ^t+k+1_n

+f ^t+k_n

φ(t)dtS_n,k(x)

≤Ckφk_∞ f⁰⁰

_∞.

To prove (3.14), we need the following inequality derived from a result of Becker [1]:

(3.15)

Z _h

0

Z _h

0 1

x+u+vdu dv≤ (x+2h)(1−x−2h)^6h2

with 0< h≤ ¹₈, 0≤x≤1−2h.

Then, forn≥8, x∈(0,∞),using (3.15) with x= 0, h= ¹_n,we have ϕ²(x)L⁰⁰_n(f, x)

=

xn²

∞

X

k=0

Z C 0

Z ¹

n

0

Z ¹

n

0

f^{00 t+k}_n +u+v

dudv φ(t)dt S_n,k(x)

≤xn²

∞

X

k=0

Z C 0

Z ¹

n

0

Z ¹

n

0 1

t+k

n +u+vdu dv dt S_n,k(x)kφk_∞ ϕ²f⁰⁰

_∞

≤xkφk_∞ ϕ²f⁰⁰

_∞ (

C

∞

X

k=1 n

kS_n,k(x) + 12CnS_n,0(x) )

≤Ckφk_∞ ϕ²f⁰⁰

_∞ (_∞

X

k=1

2S_n,k+1(x) + 12S_n,1(x) )

≤12Ckφk_∞ ϕ²f⁰⁰

_∞.

In the casen <8,(3.14) is easily derived from (3.6).

The proof of Lemma 3.2 is complete.

With the above preparations, we can state our characterization theorem as follows.

Theorem3.3. Let 0< α <2, f ∈CB, Ln(f, x) be given by (2.3). Then

(3.16) ω2(f, t) =O(t^α) if and only if

(3.17)

L_n(f, x)−f(x)

≤M ^x_n+_n¹2

^α₂ .

Proof of Theorem 3.3. Necessity. Suppose that (3.16) holds. By (3.1) , we know that for 0< t <1,

K2(f, t)≤M t^α2. Letg∈C²[0,∞)∩CB.We use the Taylor expansion (3.18) g(t) =g(x) +g⁰(x) (t−x) +

Z t x

(t−u)g⁰⁰(u)du.

By Theorem 2.1 forx∈(0,∞),we have

|L_n(g, x)−g(x)|=

L_n Z t

x

(t−u)g⁰⁰(u)du, x

=

∞

X

k=0

Z C 0

Z ^t+k

n

x

t+k n −u

g⁰⁰(u)duφ(t)dtSn,k(x)

≤ g⁰⁰

∞

∞

X

k=0

Z C 0

^t+k_n −x

2kφk_∞dt Sn,k(x)

≤2Ckφk_∞ g⁰⁰

∞

∞

X

k=0

n k n−x2

+ ^C_n2²

o

Sn,k(x)

≤2Ckφk_∞ 1 +C² _x

n+_n¹2 g⁰⁰ _∞. Taking the infimum over g∈C²[0,∞)∩C_B,by (3.3) we have

Ln(f, x)−f(x)

≤ inf

g∈C²[0,∞)∩C_B

n

kL_n(f−g)k_∞+kf−gk_∞+

Ln(g, x)−g(x) o

≤ inf

g∈C²[0,∞)∩C_B

n

(1 +M)kf −gk_∞+M ^x_n+_n¹2 g⁰⁰ _∞

o

≤M K2 f,^x_n+_n¹2

≤M ^x_n+_n¹2

^α₂ .

Hence (3.17) holds. The proof of necessity is complete.

Sufficiency. Suppose that (3.17) holds. For d > 0, by (3.2) we choose f_d ∈ C²[0,∞)∩CB such that

kf −fdk_∞≤M ω2(f, d),

f⁰⁰

∞≤M d⁻²ω2(f, d).

Let 0< t≤ ¹₈, x > t, n∈N.Then, by (3.15) whenx≤ ¹₂,Lemmas 3.1 and 3.2, we have

∆²_tf(x) ≤

≤

∆²_tL_n(f) (x) +

∆²_t(f−L_nf) (x)

≤ Z ^t

2

−¹₂

Z ^τ

2

−^t₂

L⁰⁰_n(f−f_d, x+u+v)

dudv+ Z ^t

2

−₂^t

Z ^t

2

−^t₂

L⁰⁰_n(f_d, x+u+v) dudv

+|f(x−t)−Ln(f, x−t)|+ 2|f(x)−Ln(f, x)|+|f(x+t)−Ln(f, x+t)|

≤min (

M n²kf−f_dk_∞t², M nkf −f_dk_∞ Z ^t

2 t 2

Z ^t

2

−^t

2

1

x+u+vdudv )

+M f_d⁰⁰

∞t²+ 4M ^x+t_n +_n¹2

^α₂

≤M nkf−f_dk_∞minn

nt²,^12t_x+t²o +M

f_d⁰⁰

_∞t²+ 8M

max_x+t

n ,_n¹2

^α₂

≤M ω²(f, d)t²

max₁

n²,^x+t_n −1

+M d⁻²ω²(f, d)t²+M

max_x+t

n ,_n¹2

^α

2.

Letd= max

nqx+t n ,_n¹

o .Then

∆²_tf(x) ≤

≤M

maxnq

x+t n ,¹_noα

+M t²

maxnq

x+t

n ,_n¹o−2

ω₂

f,maxnq

x+t n ,_n¹o

. Now for anyδ∈ 0,¹₈

,we choosen∈Nsuch that

δ

2 <max

qx+t n ,¹_n

≤δ.

Under this choice, we have ∆²_tf(x)

≤Mn

δ^α+^h2ω_δ²2^(f,δ)

o ,

which implies

ω2(f, h)≤M n

δ^α+^h2ω_δ^2(f,δ)2

o .

By the Berens-Lorentz Lemma (see [2], [5]), we have ω₂(f, h) =O(h^α) (h→0). Hence (3.16) holds and the sufficiency is true.

The proof of Theorem 3.3 is complete.

4. APPROXIMATION THEOREMS INLp

In this section, we give approximation theorems in Lp (1< p≤ ∞) for our Sz´asz-type operators.

To state the equivalence result, we need the Ditzian-Totik K-functional de- fined by

(4.1) K_ϕ,2(f, t)_p = inf

g,g⁰∈A.C._loc,ϕ²g⁰⁰∈Lp

n

kf−gk_p+t ϕ²g⁰⁰

p

o

, t >0.

ThisK-functional is equivalent to the Ditzian-Totik modulus of smoothness:

(4.2) M⁻¹ω²_ϕ(f, t)_p ≤Kϕ,2 f, t²

≤M ω²_ϕ(f, t), where 1≤p≤ ∞, 0< t <1, f ∈L_p[0,∞).

Some Bernstein-Markov-type inequalities are also needed here.

Lemma 4.1. Let 1≤p≤ ∞, f ∈Lp[0,∞).Then we have kL_n(f)k_p ≤Mkfk_p;

(4.3)

ϕ²L⁰⁰_n(f)

p ≤M nfkfk_p. (4.4)

Proof of Lemma 4.1. By the Riesz-Thorin Theorem and Lemma 3.1, we need only to consider the casep= 1.

We note that for 0≤k <∞,

Z ∞ 0

Sn,k(x)dx= 1.

LetC≤C₀ ∈N.For f ∈L₁[0,∞),we have kL_n(f)k₁≤

Z ∞ 0

∞

X

k=0

Z C 0

f ^t+k_n

kφk_∞dt Sn,k(x)dx

≤ kφk_∞

∞

X

k=0 1 n

Z C 0

f ^t+k_n dt

≤ kφk_∞C₀kfk₁. Hence (4.3) holds.

By (3.9), we also have

ϕ²L⁰⁰_n(f) 1 ≤

Z ∞ 0

∞

X

k=0

Z C 0

f ^t+k_n

kφk_∞dt x Sn,k(x)

_(k−nx)2

x² +_x^k2

dx

=

∞

X

k=0

Z C 0

f ^t+k_n

kφk_∞dt

k n Z ∞

0

Sn,k−1(x)dx−2k+

+n(k+ 1) Z ∞

0

Sn,k+1(x)dx+n Z ∞

0

Sn,k−1(x)dx

≤2kφk_∞

∞

X

k=0

Z C0

0

f ^t+k_n dt

≤2C₀kφk_∞nkfk₁. Hence (4.4) also holds.

The proof of Lemma 4.1 is complete.

Lemma4.2. Let 1≤p≤ ∞, f, f⁰ ∈A.C._loc, ϕ²f⁰⁰∈L_p. Then we have

(4.5)

ϕ²L⁰⁰_n(f)

p≤M ϕ²f⁰⁰

p. Proof of Lemma 4.2. We prove only the casep= 1 again.

By (3.8) and (3.15), forn≥8,we have

ϕ²L⁰⁰_n(f) 1 ≤

≤

∞

X

k=0

Z C 0

Z ¹

n

0

Z ¹

n

0

f^{00 t+k}_n +u+v

dudvdt n(k+ 1) Z ∞

0

Sn,k+1(x)dxkφk_∞

≤

∞

X

k=1 n

k(k+ 1) Z C

0

Z ¹

n

0

Z ¹

n

0

ϕ²f^{00 t+k}₂ +u+v

dudvdtkφk_∞+

+ Z C

0

Z _n

0

Z _n

0 1 u+v

ϕ²f⁰⁰ _n^t +u+v

du dv dtkφk_∞

≤2nkφk_∞ Z ¹

n

0

Z ¹

n

0

( _∞ X

k=1

Z _C

0

ϕ²f^{00 t+k}_n +u+v dt

) du dv

+kφk_∞ Z ¹

n

0

Z ¹

n

0 n u+v

Z ∞ 0

ϕ²f⁰⁰(y)

dy du dv

≤2n²kφk_∞ Z ¹

n

0

Z ¹

n

0

C0

ϕ²f⁰⁰

1du dv+nkφk_∞ ϕ²f⁰⁰

1 12

n

≤M ϕ²f⁰⁰

1.

The proof of Lemma 4.2 is complete.

We estimate the approximation order first for smooth functions.

Theorem 4.3. Let 1< p≤ ∞, f⁰∈A.C._loc, f⁰, ϕ²f⁰⁰∈L_p.Then we have (4.6) kL_n(f)−fk_p≤ ^M_n

f⁰ p+

ϕ²f⁰⁰ p

.

Proof of Theorem 4.3. We denote the Hardy-Littlewood maximal function of a locally integrable function g by

(4.7) M(g) (x) = sup

t6=x

Z t x

|g(u)|du

|t−x| . Letx > _n¹, n∈N. Then, by (3.18) we have

|L_n(f, x)−f(x)|=

L_n Z t

x

(t−u)f⁰⁰(u)du, x

≤

∞

X

k=0

Z C 0

Z ^t+k

n

n

^t+k_n −u

f⁰⁰(u) du

kφk_∞dtS_n,k(x)

≤

∞

X

k=0

Z C 0

|^t+k_n ^−x|²

x dt S_n,k(x)kφk_∞M ϕ²f⁰⁰ (x)

≤ (

2

∞

X

k=0

(_n^k−x)²

x S_n,k(x) + 2_n^C2²x

)

Ckφk_∞M ϕ²f⁰⁰ (x)

≤ ²(^1+C2)^Ckφk∞

n M ϕ²f⁰⁰ (x). Here we used the fact foru∈[x, t] or [t, x],

|t−u|

u ≤ |t−x|

x .

For 0≤x < _n¹,we have

|L_n(f, x)−f(x)|=

L_n Z t

x

f⁰(u)du, x

≤

∞

X

k=0

Z C 0

Z ^t+k

n

x

f⁰(u) du

kφk_∞dtS_n,k(x)

≤

∞

X

k=0

Z C 0

^t+k_n −x

dtkφk_∞dtS_n,k(x)M(f⁰) (x)

≤







C²

n kφk_∞+Ckφk_∞ v u u t

∞

X

k=0 k n−x2

Sn,k(x)







M f⁰ (x)

≤ ^(1+C)Ckφk_n ^∞M(f⁰) (x).

Combining the above two cases, we have forx∈[0,∞),

|L_n(f, x)−f(x)| ≤ ^M_n M ϕ²f⁰⁰

(x) +M(f⁰) (x) , which implies for 1< p≤ ∞

kL_n(f)−fk_p≤ ^M_n

M ϕ²f⁰⁰ p+

M f⁰ p

≤ ^M_n ϕ²f⁰⁰

p+ f⁰

p

.

The proof of Theorem 4.3 is complete.

Finally, with all the above preliminary results, we state our main theorem in this section.

Theorem 4.4. Let f ∈ L_p[0,∞) for 1 < p < ∞ and f ∈ C_B for p = ∞.

Then, for 0< α <1, we have

(4.8) kL_n(f)−fk_p =O n^−α if and only if

(4.9) ω_ϕ²(f, t)_p=O t^2α

.

Proof of Theorem 4.4. When 1 < p < ∞, we use the following inequality (see [5])

(4.10)

f⁰

p ≤M

kfk_p+ ϕ²f⁰⁰

p

, f, f⁰ ∈A.C._loc.

Then the proof for this case can be completed by the standard method (c.f. [5]) using Lemmas 4.1, 4.2 and Theorem 4.3.

When p = ∞, we use the following K-functional introduced by the second author in [13]

K1,2(f, t) = inf

g∈C²[0,∞)∩C_B

kf−gk_∞+t g⁰

_∞+ ϕ²f⁰⁰

_∞ .

By Lemmas 3.1, 3.2, 4.1, 4.2 and Theorem 4.3, using the equivalence derived in [11], we know that (4.8) holds if and only if

K_1,2(f, t) =O(t^α), which is equivalent to (4.9).

The proof of Theorem 4.3 is complete.

It can thus be seen our Sz´asz-type operators have the advantages of both Sz´asz-Mirakjan operators and Sz´asz-Kantorovich or Durrmeyer operators.

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Received May 10, 1994 Department of Mathematics University of Duisburg

D-47057 Duisburg Germany