View of On the \(L_{p}\)-saturation of the Ye-Zhou operator

Rev. Anal. Num´er. Th´eor. Approx., vol. 34 (2005) no. 1, pp. 55–62 ictp.acad.ro/jnaat

ON THE Lp-SATURATION OF THE YE-ZHOU OPERATOR

ZOLT ´AN FINTA^∗

Abstract. We solve the saturation problem for a class of Ye-Zhou operator Tn(f, x) =Pn(x)AnLn(f) with suitable sequence of matrices{An}n≥1.The solu- tion is based on the saturation theorem for the Kantorovich operator established by V. Maier and S. D. Riemenschneider.

MSC 2000. 41A36, 41A40.

Keywords. Ye-Zhou operator, Kantorovich operator, saturation theorem.

1. INTRODUCTION

Forf ∈C[0,1] thenth Bernstein polynomial is defined by Bn(f, x) =

n

X

k=0

pn,k(x)f^k_n≡

n

X

k=0 n k

x^k(1−x)^n−kf_n^k. D. X. Zhou showed in [6] for the Kantorovich operator

(1) Kn(f, x) =

n

X

k=0

pn,k(x)(n+ 1)

Z (k+1)/(n+1) k/(n+1)

f(t) dt that if 0< α <1 then ω₂(f, t) =O(t^α) if and only if

|K_n(f, x)−f(x)| ≤M(x(1−x)/n+ 1/n²)^α/2.

He also showed [6] that if 1 < α < 2 then there exist no functions {ϕ_n,α(x)}_n≥1 such that

ω₂(f, t) =O(t^α)⇔ |K_n(f, x)−f(x)| ≤M ϕ_n,α(x).

Thus we cannot characterize the second orders of Lipschitz functions by the rate of convergence for the Kantorovich operators. To overcome this difficulty, M. D. Ye and D. X. Zhou [5] introduced a new method of linear approximation by means of matrices: let P_n(x) ≡ (p_n,0(x), . . . , p_n,n(x)) and

∗“Babe¸s-Bolyai” University, Faculty of Mathematics and Computer Science, 400084 Cluj- Napoca, Romania, e-mail: [email protected].

Ln(f) ≡ (Ln,0(f), . . . , Ln,n(f))^T, where {p_n,k(x)}ⁿ_k=0 is a bases of the linear space span{1, x, . . . , xⁿ}and

L_n,k(f) = (n+ 1)

Z (k+1)/(n+1) k/(n+1)

f(t) dt

are functionals onC[0,1],respectively. Then, for any (n+ 1)×(n+ 1) matrix A we get the so-called Ye-Zhou operator defined by

(2) Tn(f, x) =Pn(x)·A·Ln(f).

The aim of the present paper is to solve the saturation problem for the class of operator {T_n(f, x)}_n≥1 for a suitable sequence of matrices {A_n}_n≥1. By reason of (2), the saturation problem of the Ye-Zhou operator will be in connection with the saturation condition of the Kantorovich operator (see [2], [3], [4]).

2. THE CONSTRUCTION OF THE OPERATOR

LetAn= (ai,j)ⁿ_i,j=0 be an (n+ 1)×(n+ 1) matrix with restriction ai,j = 0 for|i−j| ≥2.We denotea_i,i=a_i, a_i,i+1=b_i, ai,i−1 =c_i and setc₀ =b_n= 0.

Now, we define the matrix An by

0≤ai, bi, ci≤1 for i= 0,1, . . . , n, (3)

a_i+b_i+c_i = 1 for i= 0,1, . . . , n, (4)

a_i+bi−1+c_i+1 = 1 for i= 0,1, . . . , n (b−1 =c_n+1 = 0), (5)

|ib_i−(i+ 1)c_i| ≤ ¹₂ for i= 1,2, . . . , n−1, (6)

bi ≤ ^M_n2 and ci≤ _n^M₂ for i= 0,1, . . . , n, (7)

with some absolute constant M >0.

The existence of a matrixAn with the properties (3)–(7) is guaranteed by the following numerical example:

a₀ = 1−_2n¹3, b₀= _2n¹3,

ai = 1−bi−ci, bi = ⁱ⁺¹_2n3, ci = _2nⁱ₃, for i= 1,2, . . . , n−1, an= 1−_2n¹₂, cn= _2n¹₂.

Then the explicit expression of the Ye-Zhou operator will be the following:

(8) Tn(f, x) =

n

X

i=0

pn,i(x)(ciLn,i−1(f) +aiLn,i(f) +biLn,i+1(f)), whereai, bi, ci (i= 0,1, . . . , n) satisfy the conditions (3)–(7).

Theorem 2.1. Let A_n and T_n be defined as above. Then we have (i) T_n is a positive operator on L_p[0,1];

(ii) T_n(f, x) =^R₀¹L_n(x, t)f(t) dt, where the kernel L_n(x, t) =

n

X

i=0

p_n,i(x)(n+ 1) c_iχ_Ii−1(t) +a_iχ_Ii(t) +b_iχ_Ii+1(t) satisfies

Z 1 0

L_n(x, t) dx= Z 1

0

L_n(x, t) dt= 1

withIi= (i/(n+ 1),(i+ 1)/(n+ 1))and χIi the characteristic function onI_i, respectively.

Proof. (i) it is a direct consequence of (8) and (3);

(ii) a simple calculation shows that Z 1

0

p_n,i(x) dx= _n+1¹ for i= 0,1, . . . , n.

Then, by (5), Z ₁

0

L_n(x, t) dx=

n

X

i=0

c_iχ_Ii−1(t) +a_iχ_Ii(t) +b_iχ_Ii+1(t)

=ci+1+ai+bi−1 = 1 fort∈I_i, i= 0,1, . . . , n, and

Z 1 0

L_n(x, t) dt=

n

X

i=0

p_n,i(x)(n+ 1)_n+1^ci +_n+1^ai +_n+1^bi

=

n

X

i=0

p_n,i(x)(c_i+a_i+b_i)

= 1

in view of (4).

Theorem 2.2. For each f ∈Lp[0,1],1≤p≤ ∞, we have

n→∞lim kT_n(f)−fk_p = 0.

Proof. In view of (8), (4) and (1) we obtain Tn(f, x)−Kn(f, x) =

=

n

X

i=0

pn,i(x){c_i[Ln,i−1(f)−Ln,i(f)] +bi[Ln,i+1(f)−Ln,i(f)]}.

Ifp=∞ then for 1≤i≤none has

|L_n,i(f)−Ln,i−1(f)| ≤

≤ (n+ 1)

Z (i+1)/(n+1) i/(n+1)

|f(t)|dt+ (n+ 1)

Z i/(n+1) (i−1)/(n+1)

|f(t)|dt

≤ 2kfk_∞

and for 0≤i≤n−1

|L_n,i+1(f)−L_n,i(f)| ≤

≤ (n+ 1)

Z (i+2)/(n+1) (i+1)/(n+1)

|f(t)|dt+ (n+ 1)

Z (i+1)/(n+1) i/(n+1)

|f(t)|dt

≤ 2 kfk_∞.

Thus we have forx∈[0,1]

|T_n(f, x)−Kn(f, x)| ≤

n

X

i=0

pn,i(x)(ci+bi) 2 kfk_∞≤ ^4M_n2 kfk_∞ in view of (7). Hence

(9) kT_n(f)−K_n(f)k_∞≤ ^4M_n2 kfk_∞. Ifp= 1 then for 1≤i≤none has

|L_n,i(f)−Ln,i−1(f)| ≤

≤ (n+ 1)

Z (i+1)/(n+1) i/(n+1)

|f(t)|dt+ (n+ 1)

Z i/(n+1) (i−1)/(n+1)

|f(t)|dt

≤ (n+ 1)kfk₁ and for 0≤i≤n−1,

|L_n,i+1(f)−L_n,i(f)| ≤

≤ (n+ 1)

Z (i+2)/(n+1) (i+1)/(n+1)

|f(t)|dt+ (n+ 1)

Z (i+1)/(n+1) i/(n+1)

|f(t)|dt

≤ (n+ 1)kfk₁.

Again, by (7), we have for x∈[0,1]

|T_n(f, x)−Kn(f, x)| ≤

n

X

i=0

pn,i(x)(ci+bi)(n+ 1)kfk₁≤ ^4M_n kfk₁. In conclusion

(10) kT_n(f)−Kn(f)k₁≤ ^4M_n kfk₁.

By the interpolation theorem of Riesz-Thorin we get for 1≤p≤ ∞that kT_n(f)−K_n(f)k_p≤ ^4M_n kfk_p

in view of (9) and (10). Hence we obtain the assertion of the theorem, because

n→∞lim kK_n(f)−fk_p = 0, 1≤p≤ ∞.

3. THE SATURATION RESULT

The matrix A_n and the corresponding operatorsT_n are determined in Sec- tion 2. We can now state our main result:

Theorem 3.1. Let {T_n}_n≥1 be defined as above.

(i) Forf ∈Lp[0,1],1≤p≤ ∞,we have kT_n(f)−fk_p=O(n⁻¹) if and only if

f(x) =k+ Z x

a h(u)

ϕ²(u) du a.e. x∈[0,1],

wherea∈(0,1), ϕ(x) =^px(1−x),k is a constant andh(0) =h(1) = 0. For 1 < p ≤ ∞, h is absolutely continuous with h⁰ ∈ Lp[0,1]; for p= 1,h is of bounded variation on [0,1].

(ii) Moreover, ifkT_n(f)−fk_p =o(n⁻¹) then f is a.e. constant.

Proof. The proof is based on the ideas of the saturation theorems for Kan- torovich polynomials established by Maier [2], [3] and Riemenschneider [4]

(see also [1, pp. 315–321]). Therefore we shall prove only the essential steps regarding the operators Tn.To be more precise, a Maier-type inequality (see [2, p. 225] or [1, p. 315, Lemma 6.1]) and a property in connection with the bilinear functional (16) will be established (see [1, p. 320, Lemma 6.6]).

Lemma 3.2. Let g₁(x) = lnx, 0 < x ≤ 1. Then there exists an absolute constant C >0 such that

(11) |T_n(g1, x)−g1(x)| ≤

∞

X

j=n+1 (1−x)^j

j +C

n

X

k=0

h 1

(k+1)² + ¹_nⁱpn,k(x).

Proof. LetS0 = 0 andSk =^Pk_j=11/j (k= 1,2, . . .).Then (12)

n

X

k=0

(S_n−S_k) p_n,k(x) =

n

X

k=1 1

k(1−x)^k (see also [2, p. 225]). Because

g₁(x) = lnx= ln(1−(1−x)) =−

∞

X

k=1 1

k(1−x)^k, x∈(0,1],

and bn=c0 = 0,then, by (8), (4) and (12), we obtain Tn(g1, x)−g1(x) =

=

n

X

k=0

pn,k(x) (ckLn,k−1(g1) +akLn,k(g1) +Ln,k+1(g1)) +

n

X

k=0

(Sn−S_k)p_n,k(x) +

∞

X

j=n+1 (1−x)^j

j

=

n

X

k=1

p_n,k(x)c_k{L_n,k−1(g₁) + (S_n−S_k)}

+

n

X

k=0

p_n,k(x)a_k{L_n,k(g₁) + (S_n−S_k)}

+

n−1

X

k=0

p_n,k(x)b_k{L_n,k+1(g₁) + (Sn−S_k)}+

∞

X

j=n+1 (1−x)^j

j

=

n

X

k=1

p_n,k(x)c_k{L_n,k−1(g1) + (Sn−Sk−1)}+

n

X

k=1

p_n,k(x)c_k(Sk−1−S_k) +

n

X

k=0

p_n,k(x)a_k{L_n,k(g₁) + (Sn−S_k)}

+

n−1

X

k=0

pn,k(x)bk{L_n,k+1(g1) + (Sn−Sk+1)}

+

n−1

X

k=0

p_n,k(x)b_k(S_k+1−S_k) +

∞

X

j=n+1 (1−x)^j

j . (13)

On the other hand, by [2, p. 225], one has

|L_n,k(g₁) + (S_n−S_k)|=

(n+ 1)

Z (k+1)/(n+1) k/(n+1)

lnt dt+ (S_n−S_k)

≤ _6k⁵₂ +_n+1¹ , (14)

wherek= 1,2, . . . , n−1 and, by (6) and (3),

n

X

k=1

p_n,k(x)c_k(Sk−1−S_k) +

n−1

X

k=0

p_n,k(x)b_k(S_k+1−S_k)

=

=

n

X

k=1

−^c_k^k p_n,k(x) +

n−1

X

k=0 bk

k+1 p_n,k(x)

=

−^c_nⁿ pn,n(x) +

n−1

X

k=1

−^c_k^k +_k+1^bk p_n,k(x) +b₀ pn,0(x)

≤ pn,0(x) +

n−1

X

k=1

|kb_k−(k+1)c_k|

k(k+1) pn,k(x) + ¹_n pn,n(x)

≤ p_n,0(x) +

n−1

X

k=1 1

2k(k+1) p_n,k(x) +_n¹ p_n,n(x)

≤ p_n,0(x) +

n−1

X

k=1 1

(k+1)² p_n,k(x) +_n¹ p_n,n(x)

≤

n−1

X

k=0 1

(k+1)² p_n,k(x) + ¹_n pn,n(x).

(15)

Combining the relations (13), (14), (15) and (3) we arrive at (11).

Here we mention that we can deduce forT_nsimilar statements to the lemmas established in [1, pp. 316–318, Lemmas 6.2, 6.4 and 6.5] for Kn from the application of Theorem 1.

Furthermore, to prove the necessity in Theorem 3.1 (i), we have to employ the bilinear functionals

(16) A_n(f, ψ) = 2n

Z 1 0

[T_n(f, x)−f(x)]ψ(x) dx.

Lemma 3.3. For each fixed ψ ∈ C²[0,1] and 1 ≤ p ≤ ∞, the functionals (16) have bounded norm on Lp[0,1] :

(17) kA_n(·, ψ)k_p ≤ C_ψ.

Moreover, there exists the limit

(18) lim

n→∞ A_n(f, ψ) = Z 1

0

f(x)(ϕ²ψ⁰)⁰(x) dx.

Proof. We have A_n(f, ψ) = 2n

Z 1 0

[K_n(f, x)−f(x)]ψ(x) dx+2n Z 1

0

[T_n(f, x)−K_n(f, x)]ψ(x)dx.

Using [1, p. 320, Lemma 6.6] and (10) we get

|A_n(f, ψ)| ≤

2n Z 1

0

[K_n(f, x)−f(x)]ψ(x) dx

+ 2n

Z 1 0

[Tn(f, x)−Kn(f, x)]ψ(x) dx

≤ C⁰ kfk₁+ 8M kψk_∞ kfk₁

≤ C kfk_p,

withC depending onψ.This inequality implies (17).

By [1, p. 320, (6.14)] we have

(19) lim

n→∞ 2n Z ₁

0

[K_n(f, x)−f(x)]ψ(x) dx= Z ₁

0

f(x)(ϕ²ψ⁰)⁰(x) dx,

wheref ∈L_p[0,1].On the other hand, by (4), (7) andc₀ =b_n= 0, we obtain

2n[T_n(f, x)−K_n(f, x)]≤2n ⁿ

X

i=1

p_n,i(x)c_i(|L_n,i−1(f)|+|L_n,i(f)|) +

n−1

X

i=0

pn,i(x)bi(|L_n,i+1(f)|+|L_n,i(f)|)

≤^2M_n n

X

i=1

pn,i(x) (|L_n,i−1(f)|+|L_n,i(f)|) +

n−1

X

i=0

p_n,i(x) (|L_n,i+1(f)|+|L_n,i(f)|)

. (20)

Furthermore,|L_n,i(f)| ≤ kfk_∞ forf ∈C¹[0,1]. Hence, by (20), one has

|2n[T_n(f, x)−K_n(f, x)] | ≤ ^8M_n

n

X

i=0

p_n,i(x)· kfk∞= ^8M_n kfk∞. Thus

2n

Z 1 0

[Tn(f, x)−Kn(f, x)]ψ(x) dx

≤2n kT_n(f)−Kn(f)k_∞· kψk₁

≤ ^8M_n kfk_∞· kψk₁ which combined with (19) imply

n→∞lim 2n Z 1

0

[T_n(f, x)−f(x)]ψ(x) dx= Z 1

0

f(x)(ϕ²ψ⁰)⁰(x) dx=:A(f, ψ).

This A(f, ψ) is a linear functional on L_p[0,1]. By the Banach-Steinhaus theorem (see e.g. [1, p. 29]), lim_n→∞An(f, ψ) exists for allf ∈Lp[0,1] and is given by the integral^R₀¹f(x)(ϕ²ψ⁰)⁰(x) dx,that is (18).

REFERENCES

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[2] Maier, V.,TheL1-saturation class of the Kantorovich operator, J. Approx. Theory,22, pp. 223–232, 1978.

[3] Maier, V.,Lp-approximation by Kantorovich operators, Analysis Math.,4, pp. 289–295, 1978.

[4] Riemenschneider, S. D.,TheLp-saturation of the Bernstein-Kantorovich polynomials, J. Approx. Theory,23, pp. 158–162, 1978.

[5] Ye, M. D.andZhou, D. X.,A class of operators by means of three-diagonal matrices, J. Approx. Theory,78, pp. 239–259, 1994.

[6] Zhou, D. X.,On smoothness characterized by Bernstein type operators, J. Approx. The- ory,81, pp. 303–315, 1995.

Received by the editors: January 16, 2001.