Degree of L1 approximation to integrable functions by modified Bernstein polynomials

JOURNAL

OF APPROXlMATlON

Degree

THEORY

13, 66-72 (1975)

of L, Approximation Modified

Bernstein R.

Department

of Mathematics,

DEDICATEI)

to Integrable

by

Polynomials

ROJANIC

The Ohio State’ (/nicwsity,

TO PROFESSOR

OCCASION

Functions

G. <;. LORENTZ

OF HIS SIXTY-FIFTH

1. INTRODUCTION

Colmrbq

AND

Ohio 43210

ON Tilt.

BIRTHDAY

RESULTS

If f’is a function defined on [0, 11, the Bernstein polynomial

B,(f‘) off is

S. Bernstein [l] has proved that for every continuous function ,f on [0, I ],

A more precise version of this result due to T. Popoviciu [2] states that

where wf is the uniform modulus of continuity w,(h) =_ max{lf(x) A small modification

--f(J’)l:x,4’t[O,

11,I X--A

Gh1.

of Bernstein polynomials due to L. A. KantoroviC [3] 66

Copyright All rights

OfJ’defined by

,
MODIFIED

BERNSTEIN

makes it possible to approximate norm by the modified polynomials

POLYNOMIALS

67

Lebesgue integrable functions in the L,

The L, analog of Bernstein’s result was established by G. G. Lorentz [4] who has proved that for every Lebesgue integrable function f on [0, I],

As far as estimates of the degree of approximation to Lebesgue integrable functions by the polynomials P,(j) in the L, norm are concerned, very little is known. A result which gives the degree of approximation tof by P,(f) for a very special class of Lebesgue integrable functionsfis due to W. Hoeffding [5]. Hoeffding’s result may be stated as follows. Iffis a Lebesgue integrable function on [0, I], of bounded variation on every closed subinterval of (0, I), then

r ‘0

l 1P,(f; s) - f(x)1 c/s < (2/e)“” J(.f)n-‘“2,

where

J(f) = iI1 Sl’“(I - ,JC)1;2 1L/f(S)j. This result is useful when J(f) < co. In this paper we shall show that

c ‘0

l x1y 1 - X)1/B j PJf,

X) - f(.Y) du

can be estimated in terms of the L, modulus of continuity

We assume here and in the rest of the paper that the functionfis extended to (- x), co) by periodicity with period 1 (its value at the integers is immaterial). The L, norm with the weight function w(x) = ~“~(1 - x)lj2 seems to be a more convenient norm than the usual L, norm for the study of approximation properties of modified Bernstein polynomials.

68

BOJANIC AND SHISHA

Our result may be stated as follows. THEOREM.

Let f be a Lebesgue integrable on ,functiotz [0, I]. Then, for

?I > 2,

2. LEMMAS

The proof of our theorem is based on two lemmas. LEMMA 1. If f is a Lebesgue integrable Junction on [0, 11, then, for n 3 2 andfor x E (0, l), we have

x(1- x>(Pn-l(S, x>-“f(x)> = i. Proof.

np,&x)

(; - x) j;,7+--* (f(s $ t) -f(x))

dt.

We have P+l(J;

x) = I1 &(A-, t)f(t)

dt,

‘0

where n-1

K&-G

t> =

‘Z c

Pn--l.li(.~)X(k,n,(k+l),nl(f),

k==O

~(~,~,(~,.~),~l(t) being the characteristic partial summation we find that

n-1 ,c,Pn-d-4xkln.(R+l)/nl(t)

= Pn-13-1c4

function

xro,dt)

of (k/n, (k + 1)/n]. By

-

Pn-l.OW

X[o,ol(t)

Since

/7n-1,&1(x)- pnpl,*(x) = ((Z 1;)

(1 - .x) - (,, ; l)(x)x”‘(I

- x)+-l

= (3 (X - x) xl‘-l(l - +--1, we have x( 1 - X)(pn-l,&l(X)

- Pn-I,&))

= e/n - x> P%k(X)>

MODIFIED

BERNSTEIN

POLYNOMIALS

69

and it follows that

Hence,

and the proof of the lemma is complete, since

i. (t - zg2pn,Jc(.Y) = x(1 - x)/n. Our second lemma is a more precise version of a known inequality (see [6, p. 151). LEMMA

2. For n 13 2 and x E [0, I ] we have f I k/n - s I5 /T,,~,(.u) < I:=0

Proof.

.u(l

- x)/n5/*.

We have

and the result follows, since j”

(X - x)* #o,,&(X) = s(1 ,;

s, (3x(1 - x) + l ~ 6x(1 29) n

x(1 - x) 112

and

25x( 1 - x) - 130x2(1 - x)” n 1 - 6x(1 - x) - 36x2(1 - x)” + 168x3(1 - x)” 172 )

= xu n; -y) (15x71 - x)” + I

for x E [0, I].

70

BOJANIC

AND

3. PROOF OF

SHISHA

THEOREM

THE

Let x E (0, 1). By Lemma I we have

where 6 E (0, 1) and

Clearly

where

Hence, it follows that

Next I

-<

r

:<

we shall estimate the coefficients [l/S]. We have first

S,.(IT.8; X) for

r c= 0 and

,I S”(ll, s: s) =

c

/7/~n,,L(.Y)

~ k//7

~~- s

~

_’ ,7V~,yl/“(

Sr(t7,

S;

r

::.: x)

/7pn,,;(S)

[ k//l

~

s

1

I;=0

j/;,:n-~l$s

Next, for 1 :

c

1 -

[l/S], :-I

n(r

we -I-

(2)

.y)W.

have, by Lemma 2, I)-%F rs.-.li

C ?,-.I’/

; k/l1 <~r-~l~s

-

.K is p,,,.(x)

MODIFIED

BERNSTEIN

71

POLYNOMIALS

From (I), (2) and (3) it follows that xy

1 - .xp <

n*:2

1Pn-*(f, x) - f(x)i !

.I o 1f(.Y

+

f)

-f(X);

Integrating this inequality

dt

and taking into accoun

we find that

Choosing here 6 = ?I-~!~, we find that

Since the L, modulus of continuity every 0 < I71 ;< 11, ,

is a subadditive function,

we have, for

(see [7], p. 112). In particular we have, for k “T 1, w,(k/rz*~“),l :: 2kWf(n+Q Hence, (.l .P(l *0

- x)1/% I Pnpl(.l; x) - f(x)l

& < 4w,(n-1/“)L1 c Tk-2 I,=1

, 2.rr2 and the theorem is proved.

72

BOJANIC AND SHISHA REFERENCES

1. S. BERNSTUN, Demonstration du theoreme de Weierstrass, fonde sur le calcul de pro-

babilitts, Cow?mun.sot. Mar/r. Kharkow 13 (1912-13), l-2. 2. T. POPOVICIU,Sur I’approximation des fonctions convexes d’ordre superieur, Mutlzemntica (Chj) 10 (1935), 49-54. 3. L. V. KANTOKOVIC, Sur certains developpements suivant les polynbmes de la forme de S. Bernstein I, II, C. R. Acad. Sci. USSR 20 (1930), 563-568, 595-600. 4. G. G. LOKE~TZ,Zur Theorie der Polynome von S. Bernstein, Mar. Sb. 2 (1937). 543-556. 5. W. HOEFFDING, The L, norm of the Approximation Error for Bernstein-Type Polynomials, J. Approximation Theory 4 (1971). 347-356. 6. G. G. LORENTZ,Bernstein Polynomials, University of Toronto Press, Toronto, 1953. 7. A. F. TIMAN, “Theory of Approximation of Functions of a Real Variable” (Russian), Moscow, 1960.