polynomial approximation in

Constr. Approx. (1992) 8:187-201 CONSTRUCTIVE

APPROXIMATION �9 1992 Springer-Verlag New York Inc.

Polynomial Approximation in Lt, (0 < p < 1)

Ronald A. DeVore, Dany Leviatan, and Xiang Ming Yu

Abstract. We prove that for f e Lp, 0 < p < 1, and k a positive integer, there exists an algebraic polynomial P. of degree _< n such that

tlf - P.II, < Cco~ f. p.

where e)~(f, t)p is the Ditzian-Totik modulus of smoothness o f f in Lp, and C is a constant depending only on k and p. Moreover, if f is nondecreasing and k _< 2, then the polynomial P, can also be taken to be nondecreasing.

I. Introduction

We are interested in the approximation of functions f ~ Lp(I), 0 < p < 1, I = [ - 1 , 1], by algebraic polynomials. Such approximation has previously been studied by other authors, most notably, Storozhenko, Krotov, and Oswald [-S-K-O] and Khodak [K]. Our main departure from these previous works is that we shall prove direct estimates for the error of polynomial approximation in terms of the Ditzian-Totik modulus of smoothness. This modulus measures smoothness differently at the endpoints of I than in the interior. Such dependence on the position of the point is crucial if we wish to characterize functions with a certain error of polynomial approximation (see [D-T]). We, however, do not in this paper discuss inverse estimates in terms of this modulus.

A second variant of our work is to consider the approximation of monotone functions by monotone algebraic polynomials in Lp, 0 < p < 1. We establish the same estimates as for the unconstrained case but only for the first- and second- order moduli. There is a result of Shvedov I-S] which says that such estimates cannot hold for smoothness order greater than 2.

The usual estimates for approximating f ~ Lp(I) by algebraic polynomials are described in terms of the ordinary kth order modulus-of smoothness of f. If J is

Date received: September 13, 1990. Date revised: April 5, 1991. Communicated by Vilmos Totik. AMS classification: 41A25, 41A20. Key words and phrases: Degree of approximation, Monotone approximation, Polynomials.

187

188 R.A. DeVore, D. Leviatan, and Xiang Ming Yu

an interval, we let

~Ok(f, t, J)p := sup I h(f, X, J)lV dx . O < h < _ t \ J d

Here, A k is the symmetric difference:

~ ( - 1 ) k-~ f if x_+ h e J, (1.1) A~(f, x, J):= 0

otherwise.

We reserve the notation I for the interval [ - 1 , 1] throughout this paper. In the case of this interval, we simply write ~Ok( f , t)p .'= O)k( f , t, I)p. A fundamental inequality (usually called a Jackson inequality) for algebraic polynomial approximation of f e Lv(I), 0 < p < o% says that for each positive integer k there exists an algebraic polynomial P, of degree _< n such that

(1.2) [[f - P, llp < Co~k(f, ~ ) ,

where C is a constant depending only on k and p (see IS-K-O]) . We shall improve upon (1.2) by incorporating the position of the point x ~ I

into the analysis. If ~p(x):= x/1 - x 2, the Ditzian-Totik modulus of smoothness of f e Lp(1) is defined by

]Ah~(x)(f, x, I)1 p dx , o)~(f, t)p := sup k O < h < t - 1

where

( - 1) k-~ f x - h~o(x) + ih~o(x) x +_ he(x) e I, (1.3) Ak~,~(f, I) X, 0

otherwise.

It is easy to see that ~ ( f , ")p is defined for each f E Lp(I) and satisfies ~ ( f , t)p < C[tfllp(I) for sufficiently small t > 0 (see (D-T, p. 21-1).

For 1 < p < ~ , Ditzian and Totik improved the estimate (1.2) by showing that e~k(f, ")p can be replaced by the smaller e~k~(f, ")p. They proved that for f ~ Lp(I), 1 < p < ~ , and positive integer k, there exists an algebraic polynomial P,(x) of degree < n such that

where C depends only on k. In this paper, we shall show that this conclusion is also valid for 0 _ N (with N a constant depending only on p and k), there exists an algebraic

Polynomial Approximation in Lv (0 < p < 1) 189

polynomial P, of deoree < n such that

(1.4) , , f - Pnllp <_ Cco~(f, ln)p,

where C depends only on k and p. Moreover, if f is nondecreasing on I, then for k < 2 the polynomial Pn in (1.4) can also be taken to be nondecreasiny.

Remark. (i) For 1 _ 3 for monotone approximation.

We devote Sections 4 and 5 to the proof of Theorem 1. In Section 4, we deal with the case of nonconstrained approximation. These estimates are later applied in Section 5 in order to settle the question of monotone approximation. However, the monotone case is much more involved and we start in Section 2 with a construction of continuous monotone piecewise linear functions, namely, splines of order 2, which yields "good" approximation to a monotone f ~ Lp(I). This result is interesting in its own right and plays a critical role in the construction of monotone polynomial approximants. We collect in Section 3 some well-known results about polynomials that are needed in the proof of Theorem 1.1. Throughout the paper 0 < p < 1 is fixed.

2. Monotone Piecewise Linear Approximants

Let a = 4o < 41 < "'" < 4, = b be such that adjacent I j = [~j_ 1, 4s], j = 1 . . . . . n, have comparable lengths, i.e.,

(2.1) [Ii+_ 1] ~ CO IIsl

with Co an absolute constant. We shall be interested in this section in the approximation of f ~ Lv[a, b] by the elements of 5 ~, where 5# denotes the class of piecewise linear functions on [a, b] for this partition. Each function S ~ 5 P is completely determined by its left- and right-hand values S(~ _+), j = 1 . . . . . n - 1, and the values S(4o), S(~,).

If f is a function in Lv(J), 0 < p < 1, J an interval, then a polynomial P of degree k is a near best L v approximation (with constant M) to f from among all polynomials of degree _< k if

(2.2) lit - P]lp(J) <- MEk(f, J)p,

where Ek(f, J)v is the error of best approximation to f on J in the L v (quasi-)norm from among all polynomials of degree _< k. Of course, if M = 1, then P is a best approximant.

We shall frequently make use of the following remark which was proved in [D-P] . If P is a near-best approximation to f with constant M on an interval J ,

190 R A. DeVore, D. Leviatan, and Xiang Ming Yu

then for on any larger interval J we have

(2.3) I]f - PIlv(Y) < C m E k ( f , Y)v,

with a constant that depends only on p, k, and the ratio [J[/ld[. That is, P is a near-best approximant on the larger interval ] with constant CM.

Given any measurable function f and an interval J, we can speak of a best La(J ) approximant l to f from linear functions (i.e., polynomials of degree < 1) in the following sense. There should exist a measurable function h with [h[ = 1 on J such that h(x) = sgn(f - / ) (x) , whenever x �9 J and If(x) - l(x)[ > 0 and h is orthogonal to all linear functions on d. Brown and Lucier [B-L] have shown that for each f �9 Lv(J), 0 0} = meas{x �9 J: f ( x ) - l(x) < 0}.

In particular, in this case, f - 1 has a (weak) sign change on J. If we suppose in addition that f is nondecreasing on J, then it is easy to see

that any l (which is a best L1 approximation to f in the above sense) must also be nondecreasing on J. Indeed, i f f = I on a set of positive measure this is obvious. On the other hand, if f # 1 a.e., then h ;= sgn(f - l) must have at least two sign changes (because it is orthogonal to all linear functions), say at the points 4 and ~/with ~ < t/. Then l(~) < l(q) and therefore l is nondecreasing.

Now we return to the question of piecewise polynomial approximation. For each of our intervals I t we let Ij be a best L 1 approximant to f on 1j in the sense given above. Then from our above remarks, 1j. is nondecreasing and from (2.3) we have that for any interval Tj _~ Ij of comparable length to I/i[

(2.5) l i T - ljllp(/'j) < m E ~ ( f , rj)p

with M depending only on p and on the ratio of the lengths of i" t and I s We define the piecewise linear function S �9 5 p by

(2,6) S(x) := lj(x), 4 i - t < x < 4t,

and prove the following.

Theorem 2.1, For any interval [a, b] and partition a = 4o < 41 < "'" < 4, = b there is a piecewise linear function S � 9 5 a with the following properties:

(i) S is nondecreasing; (ii) there is a constant Mo > 0 depending only on p and the constant Co

of (2.1) such that for j = 1 . . . . . n, 7j satisfies (2.5) for an interval r t with Ij c 7"i c ~..= I j _ 2 U I j_ 1 tJ Ij U I~+ ~ W Ij+ 2 (for the purposes o f this for- mula, I u .'= (25, k < O, and k > n).

(iii) S(40+) -> S(40+) and S(~n--) < S(~,--).

Proof. The proof is by induction on the number of intervals n. Let Mt be a constant such that (2.5) is valid for all j = 1, . . . , n and any T~ with I~ c I't c f t "

Polynomial Approximation in Lp (0 1 a sufficiently large but fixed constant depending only on p and the constant Co of (2.1). If n = 1, we can take K = 11 = S .

Assume then that the theorem is true for any number of intervals < n. We let Aj := { y : y = Is{x), x e l j } = [ej, flj], j = 1 . . . . . n, be the range of lj on Ij. If f # l ~ a.e. on I i, i = j, j + l , then by (2.4), f - l i changes sign at some point t/i s I ~ (the interior of I~), i = j, j + 1. Since

lj(r/j) < f ( r / j+ ) < f(t/j+ 1 - ) < lj+l(rlj+l), we have c~j < flj+ 1- Clearly this also holds if f = I i on a set of positive measure in I~ for one or both of the values i = j, j + 1.

We define m as the largest integer such that ~m= 1 Ai # ~ , and we consider two possibilities. If m = 1 we have fll < e2 and we define S = 11 on I t . By our induction hypothesis there is a function S for [C~, ~ ] . The composi te function g is nondecreasing by virtue of (iii):

s G - ) = & < ~ = s G + ) _< g G + ) .

Propert ies (ii) and (iii) also follow from our induct ion hypothesis and the definition of S.

The second possibility is m > 2. Let e e A t n ' " n A m . We first define S(4j - ) . := ~ , j = 1 , . . . , m - 1, S(C~-) . '= c~. We have yet to define S(4o) which will then completely define S on [Co, 4m). TO define S(r we consider two cases.

Case 1. /2(Co) < 11(4o) = S(Co). In this case we define S(Co)= S(Co)= 11(Co). Then on [Co, 413 we h a v e 12(X)_< S(x) < l l (x ) where we used that S(C1) = ~ >/2(C1). Hence, with I , = 11 u 12, we have

lib - i l l l p ( I t ) < 11/1 - 121[p(I1)

< C [ l l f - 1~ lip(It) + I l f - 1211p(l,)]

< Cl-II / - /1l ip(TO + I I / - 12lip(TO]

<_ C M 1 E I ( f , I1)t,,

where the last inequality uses (2.5). We can now replace I t by T 1 on the left of our inequality (because the norm of l on two intervals of comparable size are equivalent). Therefore,

I[f - 11 ]lp(~l) ~ C l - I I f - 11 IIp(I~) + II11 - - / l lip(f1)]

and (ii) follows for j = 1 and M o : = C 1 M 1 provided C 1 is sufficiently large but depending only on p and C 0.

Case 2. /2(4o) > 11(Co) = S(Co). Then 11 and 12 intersect at some point C with Co < C < r If C < ~1 we define il as the linear function which takes the values ct at C1 and the common value 11(C) = 12(4) at 4. If C = C1 we can take 11 .'= 11. Then 11(x ) < il(x) < 12(x), Co <

x <_ 4, lz(x) < fl(X) < ll(x), ~ < x < ~1" Hence again we have

I1il - l l l [ p ( I 1 ) <- II11 - 1211p(I1)

and arguing as in Case 1 we have p roper ty (ii) for {1- This completes the definition of S on [Go, I1].

We check next that p roper ty (ii) is valid for our definition of S on 1 i, j = 2 . . . . . m - 1. O n such an interval, S(x) - c~. Let t/i e_I~ be the point where li(t/j) = e. Then for t/i < x < J + := [t/j, ~ ] , we have

Illj - ~llpU+) < _<

~j we have li+l(x ) <_ ~ = lj(x) <_ li(x ). Hence, for

II lj - tj+l ll,(J +)

C[llf-/lll~(J+) + I I f - l i+lll ,(d+)].

Similarly for J _ := [~j_ a, t/i], we have

[ll~ - ~ l ] p ( J - ) < C E I 4 f - l j - ~ l l , ( J - ) + I [ f - l j l l p ( J - ) ] .

Hence

Hlj -/jllp(Ij) -< Cl-IIf - li-~ljp(I j) + IIf - lj]]p(Ij) + ]If - lj+tHp(Ij) ].

We can replace Ij by T j : = I j _ ~ w I j u I i + ~ on the left and right and obtain as before

I[f - 71[I.(T~) < c [ l l f - lillp(~) + Illj - [jllp(~)]

<- C [ l l f - lj_ l llp(Ij) n t- I l f - ljll,(~) + I l f - li+~ II,(r~)]

CM1EI(f, Ii) p,

since each of the 1i_ 1, 1 i. and 1i+ 1 are near best on r i. This shows that (ii) holds for i i. Since l , , - l,,_ 1 - e , p roper ty (ii) holds for j = m if we define L:= r.-l.

In summary , we have defined a m o n o t o n e piecewise linear function S on [G0, ~m] and S has p roper ty (ii) and S(~0 + ) > S(~0 +) . By our induct ion hypothesis there is an S on [~m, 4,] which also satisfies (i)-(iii) for that interval. By the definition of m we mus t have ~ < S(~m +) < S(~,, +). Hence the composi te function

is m o n o t o n e nondecreas ing and satisfies (iii). P roper ty (ii) also follows f rom the induction hypothesis. �9

We now wish to alter the definition of S to make it continuous. We first note that in Theo rem 2.1 the intervals /) can be taken as [i by enlarging the constant M o if necessary.

Theorem 2.2. Under the hypothesis of Theorem 2.1, there is a nondecreasing pieeewise linear function S*~ 5 p satisfying (i) and (ii) of Theorem 2.1 with [i replaced by I* := kJl-al Ii+ 3 I~ and the additional property that S* is continuous.

Proof. We let S*(~o) = S(~o) and S*(~,) = S(~,). Then, for 1 _< k < n, we set

~S(~k--) if slope [k --< slope ik+ 1,

S*(~-k) = [ .g(r if s lope/k > s lopeik+t ,

and define S* on I k to be the linear function 1" connecting S*(~-2) and s*(~) .

Evidently S* is continuous and increasing. Therefore, we have only to show that l* is a near-best Lp approximant for I*. First assume 1 < k < n. We consider four cases.

Case 1. S*(~k- j ) = S(~-k-~--) and S*(~k) = S(~k--)" In this case slope [k-2 --< slope [~ _< slope 1~+1, thus lk-l(X) --< l*(x) <_ [~(x), x e [ ~ _ 2, ~k]" AS in the proof of Theorem 2.1, we obtain

HI* -- i~l[(l~) <_ C [ l l f - ik-~llp(l~) + }If - - ] g l l p ( I k ) ] �9

We can replace Ik by I* on the left and obtain as before

l i t - * * Ik II,(l~) --< C [ l l f - - lkll,(I~') + Ill* - lkll .(I*)] < C[ l I f - lkllp(I*) + IJf -- l~ll,(!k) + }}f -- lk- 2 }},(I~)]

C M 0 [ ~ l ( f ~ I~)p -J~ E2(f~ Ik-1)p] ~ CMoEI(f~ It)..

Case 2. S*(~k-O = S(~k-2--) and S*(~k) = S(~k+)" In this case slope [k-2 <-slope [k and slope Ik+ 1 < slope lk, thus ]~_ l (x )< l~(x) <_ ]k+l(x), x e [~k-a, ~k], and again we get as before that l~ is a near-best Lv approximant for I~'.

Case 3. S*(~k-1) = S(~k-2+) and S*(r = S(~k--). In this case, I~(x) = [k(X), X e [~,_ 2, ~k], SO obviously l~' is a near-best Lp approximant for I~'.

Case 4. S* ( {k -O = S(~k 1+) and S*(~k) = S(~k+). In this case slope 1~+ 2 < slope l~ < slope 7k_1, thus Ik(X) <_ l*(X) <_ [k+l(X), X ~ [~k- 1, ~k], SO that as in Case 1 we get that l* is a near-best L, approximant for I*.

If k = l, we have two cases. The first is S*(~0-'= S ($2- ) and l*:= 12 so there is nothing new to prove. In the second case, S*(~2):= S(~2+) and slope 7 z < slope i 2. Therefore, [2(x) <_ l~(x) <_/2(x), x e 11, and the proof is completed as before. The remaining k = n is dealt with in the same way. This completes the proof of (ii) and the theorem. �9

For our next result and for later use, we introduce the following averaged modulus of smoothness on an interval J:

' = I s(f , x , J)l" d x ds .

Then w k is equivalent to co k in the sense that (see [P~P] or [D-L])

(2.8) C-lwk(f, t, J)p ~ e)k(f, t, J)p <_ Cwk(f, t, J)., t > O,

with the constant C > 1 depending only on k and p.

We also recall Whitney's theorem which is known to hold for all 0 O.

Corollary 2.3. I f 0 __ 2, there is a continuous piecewise linear nondecreasin9 spline S with n equally spaced knots (the usual notation S E 5e~) such that

(2.10) Ilf - Sl[, _< Ccoz(f, 1/n)p

with a constant C depending only on p.

Proof, We choose ~ = (2j - n)/n, j = 0 . . . . . n, and apply Theorem 2.2 and (2.9) to the continuous piecewise linear nondecreasing S* and find

(2.11) II / -- S*[Ip(I3) -< GEl(f , I*)p < Cw2(f, 14/n, I*)~,.

We raise each of the inequalities (2.11) to the power p and then add them. Since a point x ~ I appears in at most seven of the I* we obtain

IIf - S*ll~ _< C ~ w2(f, 14/n, Ij*)p p <_ CWE(f, 14/n, I)~ < Co)2(f, 1/n, I)~. �9 j = l

3. Preparatory Results

The remainder of this paper is concerned with the approximation of a given function f ~ Lp(I), I := [ - 1 , 1], by algebraic polynomials. For this, we shall use a very common technique. We first approximate f by a piecewise polynomial S and then approximate S by a polynomial with the desired degree and monotonicity. For example, in the simplest case, S is a piecewise linear function

S(x) = ~ c~(x - ~j)+ j=o

of the type considered in the previous section. The improved estimates for algebraic polynomial approximation hinge on taking the breakpoints {j of the piecewise linear function S thicker near the endpoints +_ 1. To construct monotone approximants, we shall need other properties for the 4i- We begin by recalling a construction introduced in [D-Y] and used also in [-L-Y].

We shall discuss how to choose the points {j and how to approximate the truncated functions ~0j(x):= ( x - {j)+ by algebraic polynomials. We shall first approximate a corresponding periodic function by trigonometric polynomials, and then use the standard change of variables x = cos t to obtain algebraic polynomial approximants.

Let J , be a Jackson kernel

J,(t) = ) ~ , ~ ~ , J,(t) dt = 1. rr

Polynomial Approximation in L~ (0 < p < 1) 195

Here r is a fixed natural n u m b e r which is chosen large enough that certain condit ions (to be prescribed later) are satisfied. We begin by approx imat ing the characterist ic functions Xj(t):= Xt-tj, tjl, tj := j~ /n , j = 0 . . . . . n, by

f t t+~J Ti(t) := Xj * J, ( t ) = J,(u) du, j = 0 . . . . . n. -tj

Making the change of variable x = cos t we obtain algebraic polynomials r j (x) :=

T,_j(t) which are approx imat ions to the characterist ic functions )~txj, l~, x j : = cos t._~. Finally, we define

Rs(x ) := [ "x rj(u) du, j = 0 . . . . . n, .1- 1

which can be viewed as approx imat ions to the t runcated power functions qgj. No te that Ro(x ) = 1 + x and R~(x) =- 0 and, in general, Rj is a po lynomia l of degree < hr.

F o r the const ruct ion of m o n o t o n e algebraic po lynomia l approx imants , we need other proper t ies of the Rj. No te tha t rj - rj+ ~ _> 0, x e [ - - i, 1], and therefore Rj - Ri+ ~ is increasing in [ - 1, 1] for a l l j = 0, 1 . . . . . n - 1. We need to introduce some new points ~j which serve as a substi tute for the x~. They are defined by 1 - r = R~(1). I t follows that - 1 = ~o < ~1 < "'" < ~. = 1. We need the following further discription of the distr ibution of the 4j's in [ - 1, 1] which was proved in [ D - Y ] and says that the points ~ are dis tr ibuted like the points cos tn_ ~.

L e m m a 3.1. With q~(x):= ~ 1 - x 2 We have f o r any n > 10:

(i) Cl~o(x)n -~ < r - ~ j -a <C2~o(x) n - l , x e [ ( j - 3 , r J = 4 . . . . . n - 5; (ii) Ir - cos t , - j [ _< C(~j+I - ~j),j = 0 . . . . . n - 1;

(iii) C~(~+~ - ~) < Cj - 4j-1 < C2(~j+~ - ( j ) , j = 1, 2 . . . . . n - 1; (iv) 1 + x < C2(P(x)n -1 , - 1 < x <_ 47, and 1 - x < C2~o(x)n -1 , 4~-7 < x < 1; (V) Clq)(x)n -1 N 41 -t- 1, - -1 <_ x <_ 47;

Clq~(x)n -1 < 1 -- ~ , - 1 , ~,,-7 < x < 1;

where C, C1, and C2 are constants independent o f n and x.

We need est imates on how close r~ and Rj are to 0 j := Z[r 11 and ~0j:= (- - ~i)+, respectively. This is given by

L e m m a 3.2. For j : 1 . . . . . n -- 1 let di(x ) := 1 + [x - ~ j l / ( ~ j ~ 1 ~ - ~j). Then

(3.1) Ira(x) - Oj(x) l _< C[dj(x)] -r+ 1,

(3.2) I Rj (x) - q)j(x)l < C(~j + ~ - r - r + 2,

with the cons tan t C dependin9 only on r.

Proof. We first let d~t ) := max{ l , nit - tjl} and approx ima te Z~(t) by Tj. It was shown in [ D - Y , L e m m a 3] tha t for 0 _< t _< n and a = ]t - t~] we have

Iz (t) - Y j ( O t = E z j ( t ) - x / t - u ) ] J ~ du ,,1_ g

<_ J.(u) du <_ J.(u) du ul>_a n

<_ C(an)- 2r + 2,

where we used the wel l -known inequalities for the momen t s of the Jackson kernel (see [L, p. 57]):

f ~ 'ul'J"(u) d u < - c n - ~ ' s = O 2 r - 2 " ~ . . . . .

Since b~(t) - Tj{t)] _< 1, 0 < t < n, we have p roved that

tz~( t ) - Tj(t)l < C[,ti(t)] -2~+2

To obtain inequalities (3.1) and (3.2), we make the subst i tut ion x = cos t and proceed as in [ D - Y , L e m m a 5] to obta in for x = cos t, 0 _< t _< n,

1 5 ( x ) - o~(x)l _< c[3._j(t)] -2'+2

If we integrate this inequality with respect to x, we obtain (see [D-Y, L e m m a 5] or [L-Y, L e m m a 6] for details)

I Rj(x) - ~0j(x)l ~ C - -

It follows f rom L e m m a 3.1 that

sin t ._ j [d._j( t)]-z~+~ ?l

sin t . _ j (3.3) -< C(r - {~).

n

Hence, in order to complete the p roof of (3.1) and (3.2), it suffices to prove that for x = cos t, 0_< t < n,

(3,4) dj(x) < C[d,,_j(0] z.

We consider two cases. If It - t .-~l _< n/n, then d._j(t) _> 1 while dj(x) <_ C and (3.4) is obvious. Consider then the second case: in/n < It - t._j[ < (i + 1)n/n for some 1 < i < n - 1. Then, for some r with I~ - t._jl < (i + 1)n/n, we have

rcos t - cos t._~t = lsin ~llt - t ._jl < Citsin 6 - j i l t - t n - j l ~ C i 2 ( ~ j + l - - ~j),

where the last inequality uses (3.3). It follows that

Ix - ~jl < Ix - cos t._~l + Icos t ._ j - ~a[ < Ci2(~i+1 - ~j),

Therefore dj(x) < Ci 2 while d._j(t) > ni and (3.4) follows. �9

We shall also need some wel l -known inequalities for algebraic polynomials . The

first of these est imates is the n o r m of a po lynomia l on a large interval in terms of its no rm on a smaller interval. I t follows f rom the wel l -known extremal p roper ty of Chebyshev polynomials .

Lemma 3.3. I f P is a polynomial of degree < k, then for x ~ [ - 1 , 1] we have

I x - - ~sl "~k max [P(u)[. ]P(x)l < C 1 § ~ j + l - - 4j/] {i_<u<~j+,

The second result we need follows f rom the fact that any two (quasi-)norms are equivalent on the space of polynomials of a fixed degree.

Lemma 3.4. I f 0 < p < oe and k = 0, 1 . . . . . then for any polynomial P of degree < k we have

C p v max ]P(x)l" < I1 [ILpta,bl, a<_x<_b - - b - a

where C depends only on k and p.

4. Proof of Theorem 1.1--The Nonconstrained Case

We fix n = 1, 2 . . . . and let ~s be the points of Section 3. Further , we denote by I j : = [gs-1, ~ ] and as in Theo rem 2.2, I* := [4 j -4 , 42+3] ~ I, j = 1 . . . . , n - 1, where 4s.'= 4o = - 1,j < 0, and 4s:= 4, = 1,j > n. F o r f e Lp(I) we denote by pj( f ) an algebraic po lynomia l of degree < k - 1 which is a near-best approx imat ion to f on the interval I*. We should r emark at the outset that if we wish only to discuss uncons t ra ined approx imat ion , we could work with the intervals I s in place o f /* . However , we wish to give a p roof which applies in the case of m o n o t o n e approx ima t ion as well.

F o r j = 1 . . . . . n - 1 we obta in f rom Whitney 's theorem (2.9) and the equivalence of OOk and w k (see (2.8)),

(4.1) ] ] / - - Ps[lpP(Is) -< CC~ ]Is], IS)~

<_ cw,(I, I* J, <- c,i l- -fl I A ts, x, i>,, dx dh

= C II*~ [ ho(x)(f, x, I*)1" dh dx,

where we changed the order of integrat ion and made a simple change of variables to arrive at the last integral.

Let

n--1

(4.2) L,( f , x) = Pl( f , x) + ~ [pj+ a(f, x) -- pj(f, x)]Os(x), j = l

and

n--1 (4.3) P. ( f , x) = Pl(f~ X) -~- 2 [Pj+l(f~ X) -- pj(f, x)Jrj(x),

j= l

where Oj and rj are defined as in Section 3. We shall see that the polynomial P , has the desired approximat ion properties.

We first prove

Theorem 4.1. I f f e Lp(I), 0 10. F o r j = 5 . . . . . n - 4 we have by Lemma 3.1(i) that C~n- 1 < l I* [/q~(x) < C2n-1 for x e I*. Since L, ( f ) = pj(f) on I j, from (4.1) we obtain

(4.5) flj l f (x) - L,(f ,x)lP dx = flj l f (x) - p j f , x)lP dx

<_ Cn [ ho(x)(f, X, I*)[ p dx dh.

This same inequality also holds for j = 1, 2, 3, 4 and j = n - 3, n - 2, n - 1, n. For example, for j = 1, 2, 3, 4 we have A~etx)( f , x, I*) = 0 if x < - 1 + khq~(x)/2, that is, if h > 2 (x+ 1)/kqffx). This means that the inner integral on the right side of (4.1) can be taken over 0 < h < 2(x + 1)/krp(x) < Cn -1 by Lemma 3.1(iv). Also, by Lemma 3.1(% q)(x)/[I*l< Cn. Hence, we again obtain (4.5). The other values o f j are handled in the same way.

We now sum the inequalities (4.5) to find

(4.6)

;, ;?f, If(x) L,(f , x)l p dx < Cn I x, I)Pl dx dh <_ Cco'~ f , P, - _ a ~ ( x ) ( f , n /p

where we have used the fact that a point x e I appears in at most seven of the intervals I*, j = 1 . . . . . n. �9

We are ready to prove Theorem 1.1 in the nonconstra ined case.

Proof of Theorem 1.1 (Par t 1). In view of Theorem 4.1 and the fact that P,(f , x), which is defined in (4.3), is a polynomial of degree < (n - 1)r + k, we only need to prove that

; (7 (4.7) -1 IL,(f, x) - P,( f , x)l p dx < Cco'~ f , n p"

By L e m m a s 3.2 and 3.3 we have

(4.8) I ' : = I L , ( f , x ) - P , ( f , x)] p d x -1

= (Pi+*( f , x ) - - p , ( f , x))(Oi(x) - r,(x)) d x 1 i = 1

n - 1 t ' 1 Z [Pi+ l ( f~ X) -- Pi(f~ x)l p [Oi(x) -- ri(x)[ p d x

i = 1 �9 - 1

f(,x ~ c n ~ l 1 - -~i l ~(k- 1)p m a x [Pi+ x(f, x) -- Pi ( f , x) l p 1 +

i = 1 X~I,+I - 1 ~ i + 1 - - ~i/]

x IOi(x) - r,(x)l ~' dx

max IPi+l(f, x ) - p , ( f , x)[ p 1 + dx , i = 1 XEIi+I 1 ~ i + 1 - - ~i /]

where I i := [~ i -1 , ~i']" N o w we choose r so that rp - kp > 2 and it is readily seen that

f, (4.9) [d~(x)]-2 d x <_ C ( ~ + 1 - ~) , i = 1 . . . . . n - 1. - 1

Hence by L e m m a 3.4, (4.5), and (4.8) we obtain

n--1

(4.10) I ' < C ~ max I P i + l ( f , x ) - P i ( f , x ) l ' l Q + l - QI i=l x~Ii+l

n-1 I~t+l < C ~ I P ~ + l ( f , x ) - p i ( f , x ) l p d x

i= 1 d~i

< C I f (x) - p~(f, x)F d x i=1 J{i-1

< Co~ L n .

Here, the last inequali ty uses that each integral in the last sum can be es t imated as in (4.5) because of (4.1). This completes the p roof of Theo rem 1.1 in the noncons t ra ined case. �9

5. Proof of Theorem 1.1--The Monotone Case

Here we use the cont inuous piecewise linear function S* of Section 2. Fo r k = 2, we can take p j ( f ) in Section 4 to be the l* .'= a i x + bj of Theo rem 2.2. Let

n--1

(5.1) L * ( f , x ) = P l ( f , - - 1) + ~" aj+ x(tpj(x) - - (pj+ l(x)) j=O

and n 1

(5.2) P*( f , x) = P l ( f , -- 1) + ~ aj+ l(Rj(x) - Rj+,(x)) . j=O

We note that since R j -- Rj+ 1 is increasing for j = 0 . . . . . n - 1 and a~ > 0 for j = 1 . . . . . n, the polynomia l P,, (f, x) is nondecreasing in [ - 1, 1]. Also, because Pi+t( f , ~i) = Pi(f, ~i), i = 1 . . . . . n - 1, it follows that

n--1 L*( f , x) = P l ( f , - 1) + al(1 + x) + ~, (aj+l -- ai)q)j(x)

j = l

= p l ( f , x) + y~ % + 1 - aj)(x - ~j)Oj(x) j = l

n - 1 = p l ( L x) + ~ [pj+ l ( f , x) - Pi(f , x)lOj(x)

j = l

= L , ( L x).

We need to est imate

f l I f (x) - P* ( f , xll p dx - 1

< If(x) - g , ( f , x ) f dx + Ig*( f , x) - P*( f , x ) f dx. - I l

So in view of Theo rem 4.1, we have only to est imate the second term. T o this end,

n--1

I g * ( f , x) - P* ( f , x) l < ~ l a i+l - aill ~oi(x) - ei(x) l i=1

= ,~1 IPi+l(/, r Pi(f, ~+1)1 I~0~(x)- e~(x)l.

Hence, by Lemmas 3.2 and 3.3 we have, as in the derivat ion of (4.10),

f l IL*(f, x) - P*(f , x)[ p dx

< ~2 IP~+l(f, ~i+1) - - Pi(f, ~i+0f(~i+1 - - ~i) -p I~o~(x) -- R~(x)f dx /=1 1

n - 1 f l _< C ~ max I P,+ l ( f , x) - Pi(f, x)[ p [di(x)] --(r-- 2)p dx

i = l xeli+J - 1

n--1 _< C ~ max [Pi+l(f , x) - P i ( f , x ) f l ~ + x - ~l

i=1 xeli+l

n i p

provided rp -- 2p > 2. This completes the p roof of Theorem 1. �9

Acknowledgments. The first author was supported by NSF Grant DMS 8922154. The first two authors were supported by BSF Grant 89-00505.

References

[B-L]

[D-L]

[D-P]

[D-Y]

[D-T]

[K]

[L]

[L Y]

[Lo]

[P-P]

IS]

[S-K-O]

IV]

L. G. BROWN, B. J. LUOER (to appear): Best approximations in L 1 are near best in Lp, 0 < p < 1, Proc. Amer. Math. Soc. R. DEVORE and G. G. LORENTZ (1992) Constructive Approximation. New York: Springer- Verlag. R. DEVORE, V. POPOV (1987): Interpolation o f Besov spaces. Trans. Amer. Math. Sot., 305: 397414. R. A. DEVORE, X. M. YU (1985): Pointwise estimates for monotone polynomial approximation. Constr. Approx., 1:323-331. Z. DITZlAN, V. ToTIr (1987): Moduli of Smoothness. Series in Computational Mathe- matics. New York: Springer-Verlag. L. B. KHODAK (1981): Approximation of functions by algebraic polynomials in the metric Lpfor 0 < p < 1. Mat. Zametki, 30:649-655. D. LEVIATAN (1988): Monotone and comontone approximation revisited, J. Approx. Theory, 53:1-16. D. LEVIATAN, X. M. YU (1991): Shape preservin 9 approximation by polynomial in L p. Preprint. G. G. LORENTZ (1966): Approximation of Functions. New York: Holt, Rinehart and Winston. P. PE'rRUSHEV, V. POPOV (1987): Rational Approximation of Read Functions. Cambridge: Cambridge University Press. A. S. SHVEDOV (1979): Orders ofcoapproximation. English Trans. Math. Notes, 25:57-63; Mat. Zametki, 25:107-117. E. A. STOROZHENKO, V. G. KROTOV, P. OSWALD (1975): Jackson-type direct and converse theorems in Lp, 0 < p < 1, spaces. Mat. Sbornik, 98:395415. X. M. Yu (1987): Monotone polynomial approximation in Lp spaces. Acta Math. Sinica (New Series), 3:315-326.

R. A. DeVore Department of Mathematics University of South Carolina Columbia South Carolina 29208 U.S.A.

D. Leviatan Raymond and Beverly

Sackler Faculty of Exact Sciences

Tel Aviv University Tel Aviv 69978 Israel

Xiang Ming Yu Department of Mathematics University of South Carolina Columbia South Carolina 29208 U.S.A.

polynomial approximation in

Documents