Saturation in C[a,b] of a special sequence of linear positive operators

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Saturation in C[a,b] of a special sequence of linear positive

operators

Citation for published version (APA):

Morsche, ter, H. G. (1978). Saturation in C[a,b] of a special sequence of linear positive operators. (Eindhoven University of Technology : Dept of Mathematics : memorandum; Vol. 7805). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1978

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.~-- ....

EINDHOVEN UNIVERSITY OF TECHNOLOGY

Department of Mathematics

Memorandum 1978-05

June 1978

Saturation in C[a,bJ of a special sequence of linear positive operators

by

H.G. ter Morsche

Eindhoven University of Technology Department of Mathematics

PO Box 513, Eindhoven The Netherlands

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by

H.G. ter Morsche

1. Summary and introduction

In this note we investigate the saturation problem for a sequence of ~inear 00

positive operators (L) 1 defined in C[a,bJ, which are related to

distribu-n distribu-n=

tion functions in the following way. (l()

Let (Y. (x». 1 be a sequence of random variables depending on a parameter

J. J.=

X E [a,bJ, mutually independent with a common distribution function Fl

,x

defined on JR, such that

x •

By X (x) we denote the

n 1 mean of Y1 (x) ,Y2(x) , •.• ,Yn(x),

X (x) :=-(Y 1(x)

n n + ••• + Y (x)}, and by F n n,x the distribution function of X n (x).

For expectations we use the notation E(X), where X is a random variable. (l()

Now the sequence (L) 1 is defined as follows:

n n=

( 1.1) L (f ; x) :

=

E ( f (X

»

=

n n

f

b f (T ) dF n,x (T )

a

where f E C[a,bJ.

A well-known example of such a sequence is the sequence of Bernstein opera-tors defined in C[O,lJ,

B (fiX) ==

n

In this case we have

\ n k n-k

F (t) == L (k)x (1 - x) •

n,x k.:Snt

The saturation order of the Bernstein operators is given by the sequence

x( 1 - x) (l()

( _n ~ _n=l' with trivial class the space of linear functions (cf. [2J, p. 102), and saturation class the space of functions with a

Lipschitz-con-tinuous derivative. We shall prove that the sequence (L ) has the same tri-n

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2

-order is given by the sequence,

2. Preliminary notes

We start with a definition of saturation of a sequence of operators (L ) de-_n fined in C[a,b].

Definition (2.1). A sequence of operators (L ) defined on [a,bJ is said to

n

be saturated on [a,bJ, if there exists a sequence of nonnegative functions

(tV (x» on [a,bJ, which tends to 0 uniformly on [a,b] I and a class T (L ) of

n n

functions such that

(2.2) f(x) - L (f;x)

=

0(, (x»,

n n (n -+ co) ,

uniformly on [a,b] if and only if f E T(L

n), and there exists a function fo E C[a,bJ, fO

i

T(L

n) for which (2.3)

uniformly on [a,bJ. We let S(L ) denote the set of functions for which (2.3)

n

holds. The set S(L ) is called the saturation class of (L ) and the set

n n

T(L ) is cailed the trivial class of (L ).

n n

We remark that the definition given above is almost identical to the defini-tion given in ([3J, p. 123); we don't assume that L is an operator from

n

C[a,b] into C[c,d] and that we only require that the functions, (x) are

n

nonnegative on [a,b] instead of positive on (a,b).

Now we return to the special sequence (L ) defined by (1.1). Since, n b b

f

dF 1 , a (t) = 1 and

f

tdF 1/a(t)

=

a I a a we have b b

o

~

f

_{(t - a)2dF1/a (t)}

~

_{(b - a)}

_f

(t - a)dF1,a(t)

=

0 I a a hence (2.4) (J 2 (a) = 0 •

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In a similar way we can prove

(2.5) o 2 (b)

=

0 .

From (2.4) and (2.5) it follows that

(2.6) L (f;a)

=

f(a), L (f;b)

=

f(b), (n

=

1,2, ••• j f E C[a,b]) ,

n n

as illustrated in the proof of the following lemma.

Lemma 2.7. If x E [a,b] is such that 02(x)

=

0, then L (fiX)

=

f(x), (n = 1,2, ••• ; f E C[a,b]) •

n

Proof. Let £ > O. Because of the continuity of f at X, there exists a 0 > 0 such that If(x)-f(t) I < E, provided Ix - tl <

o.

Therefore,

b 1Ln(f;X) - f(x)

I

~

f

If(x) - f(t)

IdFn,x(~)

= a

I

f (x) - f (t)

I

dF (t) +

f

n,x

I

f (x) - f (t)

I

dF n,x (t) < E + Ix-tl;::o b + 2M

f

dF (t)

~

£ + 2M

f

n,x 62 Ix-tl;::o a 2 (x - t) dF (t) . n,x

where M = max{

I

f(t) I, t E [a,b]}.

Lemma 2.8. The function 02(x) is bounded on [a,b].

Proof.

b

2

o

~ 0 (x)

=

f

_{(x -} _t)_{2 dF 1,}_x_(t)

a

From lemma (2.8) it follows that

2

+ x

tend uniformly to x2 on [a,b] and since

2

~ (b - a) •

L (1;x)

=

1, L (t;x)

=

x for all x E [a,b],

n n

=

£

o

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4

-we can apply Korovkin's theorem to the convergence of the sequence L (fiX)

n

with the following result:

Theorem 2.9. Let f E C[a,b], then L (fiX) + f(x), (n +~) uniformly on [a,b].

n .

We will end this section with a qualitative result regarding the. convergence of the sequence L (fiY) if the function f E C[a,b] is twice continuously

dif-n

ferentiable in some neighbourhood of a point y E (a,b).

Lemma 2.10. Let f E C[a,b] have a continuous second derivative in a

neigh-bourhood of some point y E (a,b). Then

L (f iY)

n

fn (v) 2 02 ( )

=

f (Y) + "" 0 (y) + 0 ( X) ,

2n y n (n + ~) •

Proof. First we compute L «t - y) 4 iY). From the first section of this note n

there follows that

4 4

Ln«t - Y) iY) = E«Xn(y) - Y) ) •

4

Setting ~4(Y)

=

E«Y1 (y) - y) ), then a short calculation shows that

(2.11) Hence, (2.12) 4 L

«t -

y) iY) n b 3 (n - 1) 4 1

--'---=3""":" 0 (Y) +"3 J..I₄(y)

n n

f

( t - _y)4_dF (t)

n,y (n + 00) •

a

If 02(y)

=

0, then L (fiY)

=

fey), so in this case lemma 2.10 is trivial. n

We now assume that 02(y) ~ O.

Let E > 0, then there exists a 0 > 0 such that the function R(y,t), defined by

f(t)

=

fey) + f' (y) (t - y) +

~f"(y)

(t _ y)2 + R(y,t) (t _ y)2 , satisfies the inequality

I

R(y, t)

I

< E:, provided Iy - t I < 0.< Thus,

=

a b

f

(f(t) - f(y»dF (t)

=

n,y a b

f

fl (y) ( t -y)dF )t) +

~

n,y a b

f

fll(y) (t_y)2dF (t) + n,y

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b b +

_f

R(y,t) (t- y) 2dF (t) n,y fn (y) 2 = 2n cr (y) +

I

R(y,t) (t - y) 2dF n,y (t) • a

The following estimation proofs the lemma.

a b

f

R(y,t) (t- y) 2dF (t)

I ::;

n,y + M

f

I

t-yl ~o 2 cr (y) 1 = E: - + 0 ( - 2 2) n n 0 Here M := maxiIR{y,t)

I,

a ::; t::; b}.

3. The saturation of the sequence (L )

n

f

I

t-yl <0 a IR(y,t)

I

(t - y) 2dF (t) + n,y

We start with the definition of a special subset of C[a,b] denoted by Lip(1,M), (M ~ 0).

Definition 3.1. f E Lip(1,M) if and only if

If{X) - fey)

I ::;

Mix - yl for all x,y E [a,b] • Now we state the main theorem of this note.

o

Theorem 3.2. The sequence of operators (L ) defined by (1.1) are saturated

n

with order cr2(x)/n and trivial class (L) the set of linear functions on

n [a,bJ. If f E C[a,bJ then 2

I

f (x) - L (f; x)

I ::;

Ncr (x) I n 2n

if and only if f' E Lip(l,M).

Remark. In fact this theorem is more or less a direct consequence of theorem 5.4 in ([3J, p. 136) I we prefer to give the whole proof here.

In the proof of theorem 3.2 we need a characterization of those ~functions

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6

-Lemma 3.3. Let f E c[a,b] then the following assertions are equivalent.

i) The function f has a continuous derivative f' with f' E Lip(l,M) •

ii) For all x E (a,b) and h > 0 with x - h,x + h E [a,b] the following

inequa-lity holds

~I

f (x + h) - 2f (x) + f (x - h)

I

~

M •

h

Proof. It is obvious that i) implies ii) •

In order to prove i) from ii) we first show that in each subinterval

(c,d) c [a,b] there exists a point where f is differentiable. Let! be the linear function such that t(c)

=

fCc) and ted)

=

f(d). For the function

!p (x) := f (x) - 11, (x) I we have <p (c) = <p (d)

=

0 I and since t (x + h) - 211, (x) +

+ 11, (x - h) = 0 for all x, we have

(3.4)

~21

<p (x + h) - 2q> (x) + <p (x - h)

I

~

M •

h

The function <p attains an extreme value at an interior point ~ E (c,d) and

i t follows from (3.4) that <p is qifferentiable in ~ with <p'(~)

=

O. Hence,

f is differentiable in ~.

Let x/y be two arbitrary points in [a,b] with y

=

x + nh (n E:IN). Then

(3.5) _h(f1 _(x_{+ h) - f (x»} ₌ _{h (f (y) - f (y - h»}1

n

1

L

{f (x + kh) + h k=2

- 2f (x + kh - h) + f (x + kh - 2h» • It follows from ii) and (3.5) that

1 1

h(f(x+h) - f(x» = 'hefty) - fey-h»~ + R(x,y,h) I

where \R(X,y,h)

I

~ M\x-yl, uniformly in h.

Let £ > 0 and let y E [a,b] be such that Ix - yl < e and f is differentiable at y. Then for all h

1,h2 > 0 sufficiently small we have 'f(x + h

1) - f(x)' f(x + h2) - f(x)

I

h - h

I

< (M + 1)£ •

1 2

Hence, f has a right-hand derivative at x. Applying again ii) we conclude that f is differentiable at x. Moreover, according to (3.6), f' E Lip(l,M).O

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Proof of theorem 3.2. If the function f E C[a,b] has a continuous derivative f' E Lip{l,M) I then b If{x) - L (f;x) I

=

n

f

(f(x) - f(t»dF n,x (t)

I

== a b x

f

f' {T)dT)dF n,x (t)

I

=

(f' (T) - f ' (x» dT) dF nix (t)

I

:5 a t a b :5

~

f

(x - t) 2p n,x (t) ==

~n

02 (x) a t

Now let f E C[a,b] be such that IL (f;x) - f(x) I :5 M2 02(x) and f'

l

Lip(l,M).

n n

Then according to lemma 3.3 there exists a point Xo e(a,b) and a number

I

.

I

2

h > 0 such that f (x

O - h) - 2f (xO) + f (xO + h) > Mh • We assume that

(3.7) where Ml < -M I

otherwise we replace f by -f. The function ~(x) := f(x) - t(x), where ~ is the linear function with ~(xO ± h)

=

f(x

O ± h), satisfies the same relation (3.7), and in addition we have

(3.8) IL

(~iX)

- <v (x) I = IL (fiX) - f(x)

I

:5 M2 02(x),

n n n X E [a,b] .

Let a be a positive number with M < a < -M

1 and let C be such that the

quadra-tic function

(3.9) Q(x) := T(x - x-a 2 O) + C I

satisfies the inequality

(3.10) Q(x) > cp(x), (x E [xO-h,xO+h]) •

Now we have

Q(x

_o

± h)

= -

%

h2 + C ,

So, the function vex) := Q{x) - <vex) on [xO-h,x_O+h] attains its minimum

*

value m at a point y E (x

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...

8

-(3.11) Q (x) * = Q(x)

-

m, X E [a,bJ has the properties:

(3.12) Q * (x) :::::: qJ (x) , x E [xO-h,xO+hJ

,

* Q (y) q> (y)

.

Let

*

a' =min{x: X:S:; xO-h, Q (x) = q>(x)} and

*

b '

=

min{x: x ::::::

Xo

+h, Q (x) = q>{x)}

then a ::::; _{a '} < y < _{b '} :s:; band Q (x) :::::: q>(x) on [al,b'J.

*

Let hex) :=

{

a,

qJ(x) - Q*(X),. x

t

[a',b'J Then q> (x) :s:; Q (x) + h(x), (x

*

E [a,bJ). Hence

*

L (q>iY) - ql (Y) = L (q>iY) - Q (Y) ::::; L (Q + hiY) - Q (y)

=

n n n

2 2

=

L (Q*,y) - Q*(Y) + L (h;y) - hey)

= -

a

2 cr (Y) + 0 (cr (y» ,

n n n y n

according to lemma 2.10. This contradicts (3.8). To prove that the set

remark that

(L ) consists of all linear function, we have only to

n

. 2

I

f (x) - Ln (fiX)

I

=" o(cr

~x»,

(n + ~), uniformly in x ,

implies f' € Lip ( 1, E) for all E > O. Then f I is constant and so f is linear

.0

Remark. The parabola technique, applied in the proof of theorem (3.2) is in-troduced in [lJ byB. BajsVanski and R. Bojanic.

References

[lJ B. Baj~anski, and R. Bojanic, A note on approximation by Bernstein poly-nomials, Bulletin American Mathematical Society, 70 (675-677), 1964. [2J G.G. Lorentz, Approximation of Functions,: Holt, Rinehart and Winston,

New York, Chicago, San Fransisco, Toronto, London, 1966.

[3J Ronald A. De Vore, The approximation of Continuos Functions by Positive Linear Operators, Lecture Notes in Mathematics, 293, Springer-Verlag, Berlin, Heidelberg, New York, 1972.