EAST JOURNAL ON APPROXIMATIONS

Volume , Number (2002), 1-12

THREE–CONVEX APPROXIMATION BY QUADRATIC

SPLINES WITH ARBITRARY FIXED KNOTS

A. V. Prymak

Department of Mathematical Analysis,

Faculty of Mechanics and Mathematics,

Kyiv National Taras Shevchenko University,

Kyiv, 01033, UKRAINE

E-mail: prymak@mail.univ.kiev.ua

A function fis said to be 3-convex on [a, b] if it has a convex derivative

on (a, b). For every 3-convex function an estimate of approximation by

3-convex quadratic splines with arbitrary ﬁxed knots is obtained in terms

of the modulus of smoothness of order 3 of the function.

1. Introduction

An approach to the construction of shape-preserving splines goes back to

the famous paper by DeVore [1], where he has constructed monotone splines

of order ≤rthat provide proper estimates for the approximation of monotone

functions. For other constructions of comonotone and coconvex splines see

the papers by Hu [3], Yu, Leviatan and Hu [4], Kopotun [6], Shevchuk [10].

Recently, the interest to 3-convex (and in general, to q-convex, q≥3) ap-

proximation was intensiﬁed.

Let us ﬁrst explain what do we mean by a generalized convexity (i.e., by

q-convexity). For a ﬁxed positive integer q,

[x0, . . . , xq;f] :=

q

X

i=0

f(xi)

Qq

j=0,j6=i(xi−xj)

2Convex approximation

is the q-th divided diﬀerence of the real function fat the points x0, . . . , xq. We

shall denote by ∆qthe set of all q-convex functions on [−1,1], that is, the set of

all functions such that for all choices of q+1 distinct points x0, . . . , xq∈[−1,1]

we have [x0, . . . , xq;f]≥0. Thus, ∆1and ∆2are the sets of non-decreasing

and convex functions on [−1,1], respectively, ∆3is the set of 3-convex func-

tions, or, in other words, the set of all functions, having convex ﬁrst derivative

on (−1,1). Remark that if f∈∆q,q≥2, then fis a continuous function on

(−1,1).

For the free knots shape-preserving spline approximation Kopotun and

Shadrin have proved [7] the estimate

E(q)

N,r (f)p≤B·EN,r (f)p

for f∈∆q∩Lp[−1,1], where

EN,r (f)p:= inf{kf−skLp[−1,1] :s∈SN,r },

E(q)

N,r (f)p:= inf{kf−skLp[−1,1] :s∈SN,r ∩∆q},

SN,r is the set of splines of degree r−1 with Nfree knots, and the constants

Aand Bdepend on qand ronly. This estimate in turn generalizes the results

by Leviatan and Shadrin [8] for q= 1,2, and Petrov [9] for q= 3. So, the

case of free knots spline shape-preserving approximation is reduced to the

approximation without constrains, and hence, Jackson-type estimates hold in

the shape-preserving case.

In the present paper we consider splines with arbitrary, but ﬁxed knots.

Let us denote by L(x) := L(x;f;x0, . . . , xn) the Lagrange polynomial of

degree ≤nthat interpolates the function fat the points x0, . . . , xn. As

usual, ωk(f;t;I) is the k-th modulus of smoothness of the function fon the

interval I, and kfkI:= max

t∈I|f(t)|is the uniform norm of the continuous

function fon the closed interval I.

Konovalov and Leviatan kindly informed the author that for each

f∈∆3∩C(2)[−1,1] they have constructed (see [5]) a 3-convex quadratic

spline swith equidistant knots, satisfying

(1) kf−sk[−1,1] ≤c

n2ω1µf00;1

n; [−1,1]¶.

Also, the spline provides a proper estimate for the simultaneous approxima-

tion for the ﬁrst and second derivatives.

They also raised a question: whether such a spline exists for arbitrary

ﬁxed knots, as well as, whether for a given f∈∆3∩C[−1,1] a spline s

A. V. Prymak 3

exists that provides ω3¡f;1

n; [−1,1]¢in the right-hand side of (1) instead of

c

n2ω1¡f00;1

n; [−1,1]¢.

We give an aﬃrmative answer to both of these questions: our main result

is

Theorem 1. For every 3-convex and continuous on [a, b]function Fand

each partition a=x0< x1<··· < xn=b, there exists a quadratic 3-convex

spline Swith knots x0, x1, . . . , xnsatisfying

(2) kF−Sk[a,b]≤cΛω3,

where

Λ := max ©Λi¯¯i= 0, n −3ª,

Λi:= Ãxi+3 −xi

min ©xi+j+1 −xi+j¯¯j= 0,2ª!3

,

ω3:= max ©ω3(f; (xi+3 −xi)/3; [xi, xi+3]) ¯¯i= 0, n −3ª,

and cis an absolute constant.

Corollary 1. For the equidistant knots xj=a+j(b−a)/n we have

kF−Sk[a,b]≤c ω3µf;b−a

n; [a, b]¶.

In particular, if F∈C(1)[a, b], then

kF−Sk[a,b]≤cµb−a

n¶ω2µf0;b−a

n; [a, b]¶,

if F∈C(2)[a, b], then

kF−Sk[a,b]≤cµb−a

n¶2

ω1µf00;b−a

n; [a, b]¶,

and if F∈C(3)[a, b], then

kF−Sk[a,b]≤cµb−a

n¶3°

°

°F(3)°

°

°[a,b].

Corollary 2. For the interval [−1,1] and the Chebyshev partition

xj=−cos ³jπ

n´,j= 0, n, we have

kF−Sk[−1,1] ≤c ω3

ϕµf;1

n¶,

4Convex approximation

where ω3

ϕis the Ditzian – Totik [2] modulus of smoothness of order 3 and

ϕ(x) := √1−x2. In particular, if F∈C(3)(−1,1), then

kF−Sk[−1,1] ≤c

n3°

°

°ϕ3F(3)°

°

°[−1,1].

Remark. The spline Sconstructed in Theorem 1 satisﬁes

°

°F0−S0°

°[a,b]≤cpΛω3Var,

|F00(x+) −S00 (x+)| ≤ cVar, x ∈[a, b),

where cis an absolute constant, and

Var := max ©Var(F00,(xi−1, xi))¯¯i= 1, nª.

To prove Theorem 1, we ﬁrst prove Theorem 10and then, for interior

intervals we reduce Theorem 1 to Theorem 10and provide some additional

arguments for subintervals near the end-points aand b.

Theorem 10.For every 3-convex function Fwith continuous derivative

on [a, b]and each partition a=x0< x1<··· < xn=b, there exists a

quadratic 3-convex spline Swith knots x0, x1, . . . , xnsatisfying

(3) kF−Sk[a,b]≤cmax ©mi¯¯i= 1, nª,

where cis an absolute constant and

mi:= Zxi

xi−1

(L(x;F0;xi−1, xi)−F0(x)) dx, i = 1, n.

In Section 2 we prove some lemmas and present an auxiliary construction

for the proof of Theorem 10. Section 3 contains the proofs of Theorem 10and

Theorem 1.

2. An auxiliary construction and lemmas

Let f∈C[a, b] be a given convex function. Then f0exists a.e. in (a, b) and

is monotone. For briefness, we shall write f0(x) instead of f0(x+), if x6=b,

and f0(b) instead of f0(b−). Let a=x0< x1<···< xn=bbe an arbitrary

ﬁxed partition of [a, b]. Set

M := max ½Zxi

xi−1

(L(x;f;xi−1, xi)−f(x)) dx ¯¯¯i= 1, n¾.

A. V. Prymak 5

We shall write s∈Ai,j to note that sis a convex piecewise linear function on

[xi, xj] with knots xi, . . . , xj, satisfying

(4) f0(xl−1)≤s0(θ)≤f0(xl), l =i+ 1, j, θ ∈(xl−1, xl),

and

s(xi) = f(xi), s(xj) = f(xj).

Notice that s0(θ) = const for θ∈(xl−1, xl) since swas supposed linear on the

subintervals (xl−1, xl). For each k= 0, n we introduce an auxiliary function

sk. To do this, we set

(5) gk(t) := ½f0(xi−1), t ∈[xi−1, xi), i =k+ 1, n

f0(xi), t ∈(xi−1, xi], i = 1, k,

and then deﬁne

sk(x) := f(xk) + Zx

xk

gk(t)dt.

Evidently, skis a convex linear spline on [a, b] with knots x0, . . . , xnsatisfy-

ing (4), and

sk(x)≤f(x), x ∈[a, b], sk(xk) = f(xk).

Let 0 ≤i < j ≤nbe a ﬁxed pair of integers. Next we construct a function

si,j ∈Ai,j as follows. Since sj(xi)≤si(xi), sj(xj)≥si(xj), and sj−si

is a continuous function, then there exists a point θ∈(xi, xj) such that

si(θ) = sj(θ). In view of notation (5),

gi(t)≤gj(t), t ∈[a, b],

and hence sj−siis a non-decreasing function. This yields

max{si(x), sj(x)}=½si(x), x ≤θ

sj(x), x > θ,

and we set si,j(x) := max{si(x), sj(x)}. Clearly, θ∈[xm−1, xm] for some

integer m,i+ 1 ≤m≤j. Using this m, we ﬁnally set

si,j (x) := ½si,j (x), x ∈[xm−1, xm]

L(x;si,j ;xm−1, xm), x ∈[xm−1, xm].

It is easy to see that si,j ∈Ai,j .

This auxiliary construction of splines si,j plays an important role in the

proof of Theorem 10. It will be applied in the forthcoming Lemma 2.

6Convex approximation

Evidently,

(6) si,j (x)≤f(x), x ∈[xm−1, xm],

and

(7) si,j (x)≤L(x;f;xm−1, xm), x ∈[xm−1, xm].

Let us denote

∆i,j (x) := Zx

xi

(si,j (t)−f(t)) dt.

Lemma 1. The function ∆i,j is continuous on [xi, xj]. It has at most 3

intervals of monotonicity with at most one interval where the function is in-

creasing. Moreover, on this interval the oscillation of ∆i,j does not exceed M.

Proof. It is suﬃcient to show that there is at most one interval where

si,j (t)> f(t). Indeed, (6) implies that if si,j (t)> f (t), then t∈[xm−1, xm].

Since si,j is linear on [xm−1, xm], then si,j −fis concave on [xm−1, xm] and

hence, there is at most one interval where it is positive. The estimate of the

oscillation follows from (7) and the deﬁnition of M. 2

Remark. Lemma 1 yields the inequality

(8) k∆i,j k[xi,xj]≤ |∆i,j (xj)|+ M.

Lemma 2. Let i,1≤i≤n−1, be a ﬁxed integer. Then there exist an

integer j,i+ 1 ≤j≤nand a spline s?

i,j ∈Ai,j satisfying

(9) °

°∆?

i,j °

°[xi,xj]≤5 M,

and if j < n, then

(10) ∆?

i,j (xj)≤0

where ∆?

i,j (x) := Zx

xi

(s?

i,j (t)−f(t)) dt.

Proof. Consider the numbers ∆i,i+1(xi+1),∆i,i+2(xi+2), . . . , ∆i,n(xn). Lem-

ma 1 implies that any of them is bounded by M. If for some k, 1 ≤k≤n−i,

the inequalities −2 M ≤∆i,i+k(xi+k)≤0 hold, then (8) yields that if one

takes j:= i+kand s?

i,j := si,j , then (9) and (10) are true. If ∆i,i+k(xi+k)>0

for all k,k= 1, n −i, then we take j:= n,s?

i,j := si,j and (9) holds. Other-

wise, there exists k, 1 ≤k≤n−i, such that ∆i,i+k(xi+k)<−2 M. Assume

A. V. Prymak 7

that kis the smallest number satisfying this inequality. Evidently k≥2. Set

j:= i+kand

˜si,j (x) := ½si,j−1(x), x ∈[xi, xj−1)

L(x;f;xj−1, xj), x ∈[xj−1, xj].

It is easy to see that ˜si,j ∈Ai,j . Finally, we deﬁne s?

i,j as follows

s?

i,j := λsi,j + (1 −λ)˜si,j ,

where

λ:= 2 M

|∆i,j (xj)|.

Note that λ∈(0,1). Then, since si,j ,˜si,j ∈Ai,j , the function s?

i,j also belongs

to Ai,j . Now we prove that s?

i,j satisﬁes (10). The minimality of kyields

0<∆i,j−1(xj−1)≤M. Thus, Lemma 1 implies k∆i,j−1k[xi,xj−1]≤M and

hence

(11) °

°

°

°Zx

xi

(˜si,j (t)−f(t)) dt°

°

°

°[xi,xj]≤2 M.

We obtain

∆?

i,j (xj) = λ∆i,j (xj) + (1 −λ)Zxj

xi

(˜si,j (t)−f(t)) dt

≤ −2 M + (1 −λ) 2 M <0,

so, the inequality (10) holds. Applying (8) and (11), we get

°

°∆?

i,j °

°[xi,xj]≤λk∆i,j k[xi,xj]+ (1 −λ)°

°

°

°Zx

xi

(˜si,j (t)−f(t)) dt°

°

°

°[xi,xj]

≤λ(|∆i,j (xj)|+ M) + 2 M = 2 M + λM + 2 M ≤5 M,

and thus (9) holds, which completes the proof. 2

Lemma 3. Let Fbe a 3-convex function on [z0, z3]and z0< z1< z2< z3

be some real numbers. With

l0(x) := F00(z1+)(x−z1) + F0(z1),

l1(x) := L(x;F0;z1, z2),

l2(x) := F00(z2−)(x−z2) + F0(z2),

the inequality

(12) ¯¯¯¯Zzi+1

zi

(F0(x)−li(x)) dx¯¯¯¯≤cΛω3(F; (z3−z0)/3; [z0, z3]), i = 0,2

8Convex approximation

holds, where

Λ := Ãz3−z0

min ©zj+1 −zj¯¯j= 0,2ª!3

,

and cis an absolute constant.

Proof. Note that

F0(x)≥li(x), x ∈(zi, zi+1), i = 0, i = 2,

and

F0(x)≤l1(x), x ∈[z1, z2].

Therefore the sign of Rzi+1

zi(F0(x)−li(x)) dx equals (−1)i.

If fis 3-convex on [z0, z3], then

[z0, z1, z2, z3;f] = 1

z3−z0

([z3, z2, z1;f]−[z2, z1, z0;f])

=1

z3−z0µ[z3, z2;f]−[z2, z1;f]

z3−z1−[z2, z1;f]−[z1, z0;f]

z2−z0¶

=α1I1+α2I2+α3I3,

where

Ii:= Zzi

zi−1

f0(x)dx =f(zi)−f(zi−1), i = 1,3,

and

α1=1

(z3−z0)(z3−z1)(z3−z2), α3=1

(z3−z0)(z2−z0)(z1−z0),

α2=−1

z3−z0µ1

z2−z0

+1

z3−z1¶.

Thus,

(13) 0 ≤[z0, z1, z2, z3;f] = α1I1+α2I2+α3I3.

We shall make use of the Whitney inequality. Let Pbe the quadratic poly-

nomial of best uniform approximation of Fon [z0, z3]. Then

kF−Pk[z0,z3]≤c0ω3(F; (z3−z0)/3; [z0, z3]).

where c0is an absolute constant. We have

[z0, z1, z2, z3;F] = [z0, z1, z2, z3;F−P] =

3

X

i=0

F(zi)−P(zi)

Q3

j=0,j6=i(zi−zj)

≤4c0

ω3(F; (z3−z0)/3; [z0, z3])

(min ©zj+1 −zj¯¯j= 0,2ª)3.

A. V. Prymak 9

Thus

(14) [z0, z1, z2, z3;F]≤4c0

ω3(F; (z3−z0)/3; [z0, z3])

(min ©zj+1 −zj¯¯j= 0,2ª)3.

Let us set

I?

i:= Zzi

zi−1

F0(x)dx, i = 1,3,

and deﬁne

f0(x) := ½l0(x), x ∈[z0, z1)

F0(x), x ∈[z1, z3],

f1(x) := ½l1(x), x ∈[z1, z2]

F0(x), x /∈[z1, z2],

f2(x) := ½F0(x), x ∈[z0, z2)

l2(x), x ∈[z2, z3].

It is easy to see that the functions f0,f1and f2are convex on [z0, z3]. This

implies that the functions Fj(x) := Rx

z0fj(t)dt,j= 0,1,2, are 3-convex on

[z0, z3]. We shall prove (12) in the case i= 0 only. The cases i= 1 and i= 2

are similar. Recall that

sign αi= sign µZzi+1

zi

(F0(x)−li(x)) dx¶= (−1)i.

Taking into account (13) and (14), we obtain

0≤[z0, z1, z2, z3;F0]

=

3

X

i=1

αiI?

i−α1Zz1

z0

(F0(x)−l0(x)) dx

= [z0, z1, z2, z3;F]−α1Zz1

z0

(F0(x)−l0(x)) dx

≤4c0

ω3(F; (z3−z0)/3; [z0, z3])

(min ©zj+1 −zj¯¯j= 0,2ª)3−Rz1

z0(F0(x)−l0(x)) dx

(z3−z0)3,

hence (12) holds for i= 0. 2

3. Positive results

Proof of Theorem 10.Assume that F(0) = 0, set f:= F0, and apply the

arguments of Section 2 to this function. Then

M = max ©mi¯¯i= 1, nª,

10 Convex approximation

and

(15) 0 ≤Zx

xi−1

(L(t;f;xi−1, xi)−f(t)) dt ≤M, x ∈[xi−1, xi], i = 1, n.

We set S(x) := Rx

x0s(t)dt, where s∈A0,n is a linear convex spline, constructed

consequently as follows.

On the ﬁrst step we put

s(x) := L(x;f;x0, x1), x ∈[x0, x1].

Remark that L(x;f;xi−1, xi)∈Ai−1,i,i= 1, n. Besides, if 0 ≤i < j < k ≤n,

then the assumptions h∈Ai,j and h∈Aj,k imply h∈Ai,k. On each step we

assume that sis already deﬁned on [x0, xi], and for x∈[x0, xi] it satisﬁes the

conditions

(16) ¯¯¯¯Zx

x0

(s(t)−f(t)) dt¯¯¯¯≤10 M,

and

(17) ¯¯¯¯Zxi

x0

(s(t)−f(t)) dt¯¯¯¯≤5 M.

Then we extend sfurther on [xi, xj], for some j,i < j ≤n, such that (16)

remains true for x∈[x0, xj] and the inequality (17), with xireplaced by xj,

holds whenever j < n. If j=n, then we need (16) only, for x∈[x0, xn].

Since there is a ﬁnite number of intervals, then we ﬁnish our procedure in a

ﬁnite number of steps. Further, taking into account (16) and the fact that

s∈A0,n, we obtain (3).

Let us describe our procedure in detail. Suppose (16) and (17) hold for

some i, 1 ≤i < n. If Rxi

x0(s(t)−f(t)) dt ≤0, then we take j:= i+ 1 and put

s(x) := L(x;f;xi, xj), x ∈[xi, xj].

The inequality (15) yields (16) for x∈[x0, xj], and also (17), with xjinstead

of xi. Otherwise (that is, if Rxi

x0(s(t)−f(t)) dt > 0), we apply Lemma 2. It

gives us some integer j,i+ 1 ≤j≤n, and a spline s?

i,j , satisfying (10) if

j < n, and (9). We put s(x) := s?

i,j (x), x∈[xi, xj]. So, if j=n, then (9)

implies (16) for x∈[x0, xn] and the procedure is ﬁnished. Otherwise, (9),

(17), and (16) for x∈[x0, xi] imply (16) for x∈[x0, xj]. The inequality (10)

gives Rxi

x0(s(t)−f(t)) dt ≥Rxj

x0(s(t)−f(t)) dt. Now, taking into account that

Rxi

x0(s(t)−f(t)) dt > 0 and the estimate (9), we obtain (17) with xjinstead

of xi. Theorem 10is proven. 2

A. V. Prymak 11

Remark. The spline Sconstructed in the proof of Theorem 10satisﬁes

F(x0) = S(x0),

F0(x0) = S0(x0), F 0(xn) = S0(xn),

F00(x0+) = S00 (x0+), F 00 (xn−) = S00(xn−).

Proof of Theorem 1. Take y0:= x1, y1:= x2, . . . , yn−2:= xn−1. The ﬁrst

derivative of Fexists on (x0, xn) and is continuous. Hence, it is continuous

on [y0, yn−2] and we can apply Theorem 10to the function Fand the segment

[y0, yn−2] with the partition y0, y1, . . . , yn−2. We get some quadratic spline S1

with knots y0, . . . , yn−2satisfying

kF−S1k[y0,yn−2]=kF−S1k[x1,xn−1]

≤cmax (Zxi

xi−1

(L(x;F0;xi−1, xi)−F0(x)) dx¯¯¯¯¯

i= 2, n −1),

(18) F0(x1) = S0

1(x1), F 0(xn−1) = S0

1(xn−1),

(19) F00(x1+) = S00

1(x1+), F 00 (xn−1−) = S00

1(xn−1−).

Put

s(x) :=

F00(x1+)(x−x1) + F0(x1), x ∈[x0, x1)

S0

1(x), x ∈[x1, xn−1]

F00(xn−1−)(x−xn−1) + F0(xn−1), x ∈(xn−1, xn],

and

S(x) := Zx

x1

s(t)dt +F(x1).

We shall prove that Ssatisﬁes (2). Indeed, note that the deﬁnition of S, the

convexity of S0

1, (18) and (19) imply that s0is a non-decreasing step function

on [x0, xn]. Hence, sis convex and consequently Sis 3-convex on [x0, xn]. In

order to prove (2) it is suﬃcient to show that, for i= 0, n −3,

Zxi+2

xi+1

(L(x;F0;xi+1, xi+2)−F0(x)) dx ≤cΛiω3(F; (xi+3 −xi)/3; [xi, xi+3]),

Zx1

x0

(F0(x)−s(x)) dx ≤cΛ0ω3(F; (x3−x0)/3; [x0, x3]),

12 Convex approximation

and

Zxn

xn−1

(F0(x)−s(x)) dx ≤cΛn−3ω3(F; (xn−xn−3)/3; [xn−3, xn]).

These inequalities are evident corollaries of Lemma 3, and thus Theorem 1 is

proven. 2

Acknowledgement. The author thanks Professor I. A. Shevchuk for the

useful discussions on the paper.

References

[1] Ronald A. DeVore, Monotone approximation by splines, SIAM J. Math.

Anal. 8, 5 (1977), 891–905.

[2] Z. Ditzian and V. Totik, Moduli of Smoothness, Springer Verlag, Berlin,

1987.

[3] Yingkang Hu, Convex approximation by quadratic splines, J. Approx. The-

ory 74, 1 (1993), 69–82.

[4] Y. Hu, D. Leviatan and X. M. Yu, Convex polynomial and spline approxi-

mation in C[−1,1], Constr. Approx. 10, 1 (1994), 31–64.

[5] V. N. Konovalov and D. Leviatan, Estimates on the approximation of 3-

monotone function by 3-monotone quadratic splines, East J. Approx. 7, 3 (2001),

333–349.

[6] K. A. Kopotun, Pointwise and uniform estimates for convex approximation of

functions by algebraic polynomials, Constr. Approx. 10, 2 (1994), 153–178.

[7] K. Kopotun and A. Shadrin, Shape-preserving approximation of k-monotone

functions by splines with free knots, to appear.

[8] D. Leviatan and A. Shadrin, On monotone and convex approximation by

splines with free knots, Annals of Numer. Math. 4, 1–4(1997), 415–434.

[9] P. P. Petrov, Three-convex approximation by free knot splines in C[a, b], Con-

str. Approx. 14, 2 (1998), 247–258.

[10] I. A. Shevchuk, One construction of cubic convex spline, Approximation and

Optimization, Proceedings of ICAOR, vol. 1, 357–368.

Received December 6, 2001