View of Approximation of derivatives by nonlinear operators

Rev. Anal. Num´er. Th´eor. Approx., vol. 31 (2002) no. 2, pp. 187–194 ictp.acad.ro/jnaat

APPROXIMATION OF DERIVATIVES BY NONLINEAR OPERATORS

RADU P ˘ALT ˘ANEA^∗

Abstract. There are obtained two theorems on simultaneous approximation, by using generalized convex operators.

MSC 2000. 41A28, 41A35, 26B25.

Keywords. Simultaneous approximation, generalized convexity of higher order, convex operators.

1. INTRODUCTION

In [6] Tiberiu Popoviciu obtained that Bernstein operators preserve conve- xity of higher orders. On the other hand the sequence of Bernstein operators has the property of the uniform approximation of the derivatives of higher order. This is a fact more general. Sendov and Popov obtained in [8] that, roughly speaking, if a sequence of linear positive operators that preserve con- vexity of higher orders has the property of the uniform approximation of con- tinuous functions then it has also the propriety of the uniform approximation of derivatives of higher orders on any compact subinterval strictly contained in the interval of definition of functions.

In this paper we shall obtain two theorems concerning the uniform approxi- mation of derivatives of higher orders by using sequences of nonlinear operators having the propriety of preservation of some type of generalized convexity of higher orders. As regard to [8] our scheme of the proof is simplified, but it requires a supplementary order of derivability.

2. CONVEX OPERATORS FOR APPROXIMATION OF VECTOR-VALUED FUNCTIONS

Let [a, b] be an interval of the real axis and letF be an Euclidean space with the scalar product h,i and the corresponding norm k · k. Denote respectively byF [a, b], Fthe space of functions defined on [a, b] and with values inF, by C [a, b], F the subspace of continuous funtions, endowed with the Chebysev norm k · k_[a,b] and for the integer m≥1 denote byC^m [a, b], F the subspace of m times continuously derivable functions. In the the caseF =R we omit to write F. If x₀, x₁, . . . , x_m+1, m≥ −1 are distinct points of [a, b] then, for a function f : [a, b]→F denote by [f;x0, x1, . . . , xm+1] the divided difference

∗Department of Mathematics, Transilvania University, 2200 Bra¸sov, Romania, e-mail:

[email protected].

of function f on the points xi,0 ≤ i ≤ m+ 1. We introduce the following definition:

Definition 1. A function f : [a, b] → F is c-nonconcave of order m ≥

−1, if for any choice of two sequence of distinct points x₀, x₁, . . . , x_m+1 and y0, y1, . . . , ym+1 of [a, b]we have:

(1) [f;x₀, x₁, . . . , x_m+1],[f;y₀, y₁, . . . , y_m+1]≥0.

In a similar mode, by replacing in (1) the inequality ”≥” by ”>”, or ”=” one can define the functions that are c-convex, respectively c-polynomial of order m. Denote by K_m [a, b], F the space of functions that are c-nonconcave of order m.

Remark. In the caseF =Ra function is c-nonconcave of orderm≥ −1 if and only if it is either usual nonconcave of order m or it is usual nonconvex

of order m(see [5]).

Lemma 2. If a, b, v ∈ F, kak = kbk = kvk = 1, ha, bi ≤ 0 then max{ha, vi,hb, vi} ≥ −√

2/2.

Proof. Letp := dimF. If p = 1 Lemma 2 is obvious. Letp ≥2. We have a 6= b. First consider the case a 6= −b. Let {₁, . . . , p} be an orthonormal basis of the space F such that ₁ = (a+b)/ka+bk and ₂ = (b−a)/kb−ak.

Represent v =λ1·1+. . .+λp·p, where (λ1)²+. . .+ (λp)² = 1. We have hv, ai=λ₁(1+ha, bi)/ka+bk+λ₂(hb, ai−1)/kb−ak;hv, bi=λ₁(ha, bi+1)/ka+

bk+λ2(1− ha, bi)/kb−ak. Hence max{ha, vi,hb, vi}=λ1(1 +ha, bi)/ka+bk+

|λ₂|(1− ha, bi)/kb−ak.By consideringaandbfixed one obtains the minimum value of max{ha, vi,hb, vi}in the case λ₁ =−1 and λ₂ =. . .=λ_p = 0 and it is equal to −(1 +ha, bi)/ka+bk = −(1/√

2)^p1 +ha, bi ≥ −√

2/2. The case

a=−bis immediate.

Theorem 3. If the sequence of functions (f_n)_n, f_n ∈C¹ [a, b], F is uni- formly convergent to the function f ∈ C¹ [a, b], F on [a, b] and if f_n⁰ ∈ K0 [a, b], F, n ∈ N, then, for any subinterval [c, d] ⊂ (a, b), the sequence (f_n⁰)_n is uniformly convergent on[c, d]to the function f⁰.

Proof. Consider ad absurdum that there is a number λ > 0, a sequence (x_k)_k of points x_k∈[c, d] and a subsequence (n_k)_k of indices such that

kf⁰(xk)−f_n⁰_k(xk)k> λ, k∈N.

There is a numberδ₁ >0 such that

kf⁰(x)−f⁰(y)k< λ/4, if |x−y|< δ₁.

Putδ := min{δ₁, c−a, b−d} andρ:=λδ/8.Afterwards fixk such that kf−f_nkk_[a,b] < ρ.

Define g:=fn_k,y:=x_k and I1 :=

Z y

y−δ(g⁰(t)−g⁰(y)) dt, I2 :=

Z y+δ y

(g⁰(t)−g⁰(y)) dt.

Here it is used the Riemann integral for functions with values in an Euclidean space.

Since g⁰ ∈K₀([a, b], f) it follows for any pointsa≤t₁ < y < t₂≤b:

g⁰(t₁)−g⁰(y), g⁰(t₂)−g⁰(y)≤0.

By approximating I₁ andI₂ by Riemann sums we obtain hI₁, I2i ≤0.

First consider the case I1 6= 0 and I2 6= 0. Set α :=I1/kI₁k, β := I2/kI₂k and v := u/kuk, where u := δ(g⁰(y) −f⁰(y)). From Lemma 2 it follows max{hα, vi,hβ, vi} ≥ −√

2/2.Suppose, for a choice, thathβ, vi ≥ −√

2/2. We have

kg(y+δ)−f(y+δ)k

=

g(y) + Z y+δ

y

g⁰(t) dt − f(y) − Z y+δ

y

f⁰(t) dt

≥

Z y+δ y

(g⁰(t)−f⁰(t)) dt

− ρ

≥

Z y+δ y

(g⁰(t)−f⁰(y)) dt

−

Z y+δ y

(f⁰(y)−f⁰(t)) dt

− ρ

≥

Z y+δ y

(g⁰(t)−f⁰(y)) dt

− 3ρ

= kI₂+uk − 3ρ

≥kI₂k²+kuk²−√

2kI₂k · kuk^1/2 − 3ρ

≥ kuk/√ 2 − 3ρ

> λδ/√ 2 − 3ρ

> ρ.

One obtains a contradiction. In the case I2 = 0 one obtains as above that kg(y+δ)−f(y+δ)k ≥ kuk −3ρ > ρ. The caseI₁ = 0 is similar. Theorem is

proved.

Remark. In the case F =R the result in Theorem 3 is given in [8].

Lemma 4. Let [a, b] ⊂R and f : [a, b]→ F. For any xi,∈ R,1 ≤ i ≤n, and any t ∈ R such that x_i and x_i +t, 1 ≤ i ≤ n, are 2n distinct points of

the interval [a, b], by denoting ft(x) := (f(x+t)−f(x))/t, x ∈ [max{a, a− t},min{b, b−t}],we have

(2) [f_t;x₁, . . . , x_n] =

n

X

k=1

f;x₁+t, . . . , xk−1+t, x_k+t, x_k, x_k+1, . . . , x_n. Proof. Forp ≥1, u 6= 0, y_i,1≤i≤p, such that y1 6=yi, y1 6=yi−u, 2≤ i≤p, denote

Θ^p_u(y₁, . . . , y_p) :=

p

X

j=1 j

Y

i=2

(y₁−y_i)⁻¹·

p

Y

i=j

(y₁+u−y_i)⁻¹. (For j= 1 take^Qj_i=2 = 1). Using the relation

Θ^p+1_u (y1, . . . , yp+1) = (y1+u−yp+1)^−1hΘ^p_u(y1, . . . , yp) +

p+1

Y

i=2

(y1−yi)⁻¹ⁱ, one can proved by induction with regard to p that

Θ^p_u(y₁, . . . , y_p) =u·

p

Y

i=2

(y₁−y_i)

−1

.

We have

[f_t;x₁, . . . , x_n] =

n

X

k=1

f(x_k+t)−f(x_k) t

Y

1≤i≤n,i6=k

(x_k−x_i)⁻¹

=

n

X

k=1

f(x_k+t)·Θ^n−k+1_t (x_k, . . . , x_n)·

k−1

Y

i=1

(x_k−x_i)⁻¹ +

n

X

k=1

f(x_k)·Θ^k_−t(x_k+t, . . . , x₁+t)·

n

Y

i=k+1

(x_k−x_i)⁻¹

=

n

X

k=1

f;x1+t, . . . , xk−1+t, xk+t, xk, xk+1, . . . , xn

.

Theorem5. Iff ∈C^k [a, b], F∩Km [a, b], F, k≥1, m−k+ 1≥0, then f^(k)∈Km−k [a, b], F.

Proof. Using Lemma 4 it is easy to obtain that if

f ∈C¹ [a, b], f∩Kn−1 [a, b], F, n≥1,

then it follows f⁰ ∈ ∩K_n−2 [a, b], F. Afterwards the theorem results by in-

duction.

Now consider the following definition.

Definition 6. An operator L :V → F [a, b], F, V ⊂ F [a, b], F is said to be k-convex,k≥ −1, if

(3) L(f)∈K_k [a, b], F, for anyf ∈V ∩K_k [a, b], F

and

(4) L(f)−L(g)∈Kk [a, b], F, for anyf, g∈V, f−g∈Kk [a, b], F. The main result of this section is the following one.

Theorem 7. Let (Ln)_n, Ln:C^k+1 [a, b], F→C^k+1 [a, b], F,k≥1, be a sequence of j-convex operators, for 1≤j ≤k. If we have

(5) lim

n→∞kL_n(f)−fk_[a,b]= 0, for allf ∈C^k+1 [a, b], F,

then for any f ∈C^k+1 [a, b], F, for any subinterval [c, d]⊂(a, b) and for any j, 1≤j≤k, one has

(6) lim

n→∞

(Ln(f))^(j)−f^(j)_[c,d]= 0.

Proof. Fix the interval [c, d] and let the subintervals [cj, dj],1≤j≤ksuch that [c₀, d₀] = [a, b], (c_j, d_j)⊃[c_j+1, d_j+1] and [c_k, d_k] = [c, d]. First note that for any functiong∈C^m [a, b], F, m≥0 and for any pointsy₀ < . . . < y_m of [a, b] we have (m!)k[g;y₀, . . . , y_m]k ≤ kg^(m)k_[a,b]. This one is a consequence of the Peano’s formula:

[g;yo, . . . , ym] = Z ym

y0

φ(t)·g^(m)(t) dt,

whereφ: [y₀, y_m]→Ris a continuous positive function independent ofg.

Fix now f ∈ C^k+1 [a, b], F. Denote ρ := max{|a|,|b|} We can choose by induction the numbers λj >0,2≤j≤k+ 1, such that:

(λ_j)² ≥2λ_j k+1

X

i=j+1 i j

ρ^i−j·λ_i+ (j!)⁻¹kf^(j)k_[a,b]

+ ^k+1

X

i=j+1 i j

ρ^i−j ·λ_i+ (j!)⁻¹kf^(j)k_[a,b]

²

, 2≤j ≤k+ 1, (for j=k+ 1 take ^Pk+1_i=j+1 = 0). Letv∈F withkvk= 1, and consider the function:

h(x) :=f(x) + k+1

X

j=2

λ_j·x^j

v, x∈[a, b].

We have h, h−f ∈Kj([a, b], F), (1≤j ≤k). Indeed, let 2≤j ≤k+ 1 and two sets of distinct points ofI: x₀, . . . , x_j andy₀, . . . , y_j. Using the inequalities above one obtains:

D[h;x₀, . . . , x_j],[h;y₀, . . . , y_j]^E=

=

*_k+1 P

i=j i j

λiξ_i^i−j

v+ [f;x0, . . . , xj], _k+1

P

i=j i j

λiη_i^i−j

v+ [f;y0, . . . , yj] +

≥0,

where ξi, ηi ∈[a, b],for j≤i≤k+ 1. Hence h∈Kj−1([a, b], F). In a similar mode we can see thath−f ∈Kj−1([a, b], F).

From Theorem 5 it follows (Ln(h))^(j),(Ln(h)−Ln(f))^(j) ∈ K0 [a, b], F, 1 ≤ j ≤ k, n ≥ 1, and from Theorem 3 it can deduce by induction, for 1≤j≤k:

n→∞lim k(L_n(h))^(j)−(h)^(j)k_[cj,dj]= 0 = lim

n→∞k(L_n(h)−L_n(f))^(j)−(h−f)^(j)k_[cj,dj].

From these limits it follows (6).

3. CONVEX OPERATORS FOR APPROXIMATION OF REAL-VALUED FUNCTIONS

Recall that forn≥1, a subsetZ ⊂C[a, b] is namedn-parameter family if for any distinct points xi ∈[a, b], 1≤i≤n, and any real numbersyi, 1≤i≤n there is an unique ψ ∈ Z such that ψ(x_i) = y_i, 1 ≤ i ≤n. Convexity with regard to an-parameter family was introduced by Tiberiu Popoviciu in [7] and was extensively studied by L. Tornheim in [9] and E. Popoviciu in [3] (and in others).

Definition 8. [7]. If Z ⊂C[a, b] is a n-parameter family, n ≥1, then a function f ∈C[a, b]is named Z-convex if for any points a≤x₁< . . . < x_n<

t ≤ b, it results f(t) > ψ(t), where ψ ∈ Z is the unique function such that ψ(x_i) =y_i, 1≤i≤n.

We consider the following definition.

Definition9. LetZ ⊂C[a, b]be an-parameter family,n≥1. An operator L:C[a, b]→ F[a, b] isZ-convex if the following conditions are verified (7) If f ∈C[a, b]is Z-convex then L(f) is usual convex of order n−1

If f −g is Z-convex , f, g∈C[a, b], (8)

then L(f)−L(g) is usual convex of order n−1.

The main result of this section is the following.

Theorem 10. Let k≥1 and for each 2≤j≤k+ 1let Z_j ⊂C^k+1[a, b] be a j-parameter family. Suppose that there are the numbers Mj >0 such that (9) kϕ^(j)k_[a,b]≤M_j, for anyϕ∈Z_j, 2≤j ≤k+ 1.

If (Ln)_n is a sequence of operatorsLn:C^k+1[a, b]→C^k+1[a, b] such that (10) L_n is Z_j-convex for any n≥1 and 2≤j≤k+ 1,

(11) lim

n→∞kL_n(f)−fk_[a,b]= 0, for anyf ∈C^k+1[a, b],

then for any f ∈C^k+1[a, b], any subinterval [c, d]⊂(a, b) and any j, 1≤j≤ k, one has

(12) lim

n→∞

(Ln(f))^(j)−f^(j)_[c,d]= 0.

Proof. First note that ifg ∈C^j[a, b], j ≥2 and g^(j)(x) > Mj , x ∈ [a, b], then g is Z_j-convex. Indeed, let a ≤ y₁ < . . . < y_j < t ≤b and ϕ ∈ Z_j the unique function such that ϕ(xi) =g(xi), 1≤i≤j. Since the function g−ϕ is usualj−1 convex it followsg(t)> ϕ(t).

Now fix f ∈ C^k+1[a, b] and [c, d] ⊂ (a, b). Denote ρ := max{|a|,|b|}. We can choose by induction the numbers λ_j, 2≤j≤k+ 1, such that

(j!)λ_j >

k+1

X

i=j+1

(i!)ρ^i−j·λ_i + kf^(j)k_[a,b] + M_j. (For j=k+ 1 take^Pk+1_i=j+1 = 0). Define the functionh by

h(x) :=f(x) +

k+1

X

j=2

λ_j·x^j, x∈[a, b].

Then hand h−f areZj-convex for 2≤j≤k+ 1 and consequentlyLn(h) and L_n(h)−L_n(f), n≥1, are usual convex of order j−1, for the same j.

Consider the intervals [cj, dj], 0 ≤ j ≤ k, such that [c0, d0] = [a, b], [cj+1, dj+1]⊂(cj, dj),[c_k, d_k] = [c, d]. Then by using the result in Theorem 3 in the case F =R, it follows by induction that

n→∞lim

(Ln(h))^(j)−h^(j)

[cj,dj]= 0 = lim

n→∞

(Ln(h)−L_n(f))^(j)−(h−f)^(j)

[cj,dj],

1≤j≤k, and consequently (12) is true.

REFERENCES

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[2] Korovkin P. P.,Linear operators and the theory of approximation, Fitmatgiz, Moscow, 1959 (in Russian).

[3] Moldovan (Popoviciu) E.,Sur une generalization des fonctions convexes, Mathemat- ica (Cluj),1(24), pp. 49–80, 1959.

[4] Nemeth A. B., Korovkin’s theorem for nonlinear 3-parameter families, Mathematica (Cluj),11(34), no. 1, pp. 135–136, 1969.

[5] P˘alt˘anea R.,Approximation operators and their connection to some particular allures, PhD Thesis, Babe¸s-Bolyai Univ., Cluj-Napoca, 1992 (in Romanian).

[6] Popoviciu T.,About the best approximation of functions by polynomials, Mathematical Monographs, Sec. Mat. Univ. Cluj,III, 1937.

[7] Popoviciu T.,Les fonctions convexes, Herman & Cie, Paris, 1945.

[8] Sendov B.andPopov V.,The convergence of derivatives of linear positive operators, C. R. Acad. Bulgare Sci.,22, pp. 507–509, 1969 (in Russian).

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Received by the editors: February 18, 1997.