# Three-convex approximation by quadratic splines with arbitrary fixed knots

Article (PDF Available) · January 2002with 27 Reads
EAST JOURNAL ON APPROXIMATIONS
Volume , Number (2002), 1-12
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, q3) 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(xixj)
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 fq,q2, 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)pB·EN,r (f)p
for fqLp[1,1], where
EN,r (f)p:= inf{kfskLp[1,1] :sSN,r },
E(q)
N,r (f)p:= inf{kfskLp[1,1] :sSN,r q},
SN,r is the set of splines of degree r1 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
tI|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
f3C(2)[1,1] they have constructed (see [5]) a 3-convex quadratic
spline swith equidistant knots, satisfying
(1) kfsk[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 f3C[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) kFSk[a,b]cΛω3,
where
Λ := max ©Λi¯¯i= 0, n 3ª,
Λi:= Ãxi+3 xi
,
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(ba)/n we have
kFSk[a,b]c ω3µf;ba
n; [a, b].
In particular, if FC(1)[a, b], then
kFSk[a,b]cµba
nω2µf0;ba
n; [a, b],
if FC(2)[a, b], then
kFSk[a,b]cµba
n2
ω1µf00;ba
n; [a, b],
and if FC(3)[a, b], then
kFSk[a,b]cµba
n3°
°
°F(3)°
°
°[a,b].
Corollary 2. For the interval [1,1] and the Chebyshev partition
xj=cos ³
n´,j= 0, n, we have
kFSk[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) := 1x2. In particular, if FC(3)(1,1), then
kFSk[1,1] c
n3°
°
°ϕ3F(3)°
°
°[1,1].
Remark. The spline Sconstructed in Theorem 1 satisﬁes
°
°F0S0°
°[a,b]cpΛω3Var,
|F00(x+) S00 (x+)| ≤ cVar, x [a, b),
where cis an absolute constant, and
Var := max ©Var(F00,(xi1, 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
where cis an absolute constant and
mi:= Zxi
xi1
(L(x;F0;xi1, 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 fC[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
xi1
(L(x;f;xi1, xi)f(x)) dx ¯¯¯i= 1, n¾.
A. V. Prymak 5
We shall write sAi,j to note that sis a convex piecewise linear function on
[xi, xj] with knots xi, . . . , xj, satisfying
(4) f0(xl1)s0(θ)f0(xl), l =i+ 1, j, θ (xl1, xl),
and
s(xi) = f(xi), s(xj) = f(xj).
Notice that s0(θ) = const for θ(xl1, xl) since swas supposed linear on the
subintervals (xl1, xl). For each k= 0, n we introduce an auxiliary function
sk. To do this, we set
(5) gk(t) := ½f0(xi1), t [xi1, xi), i =k+ 1, n
f0(xi), t (xi1, 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 sjsi
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 sjsiis 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[xm1, xm] for some
integer m,i+ 1 mj. Using this m, we ﬁnally set
si,j (x) := ½si,j (x), x [xm1, xm]
L(x;si,j ;xm1, xm), x [xm1, 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 [xm1, xm],
and
(7) si,j (x)L(x;f;xm1, xm), x [xm1, 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[xm1, xm].
Since si,j is linear on [xm1, xm], then si,j fis concave on [xm1, 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) ki,j k[xi,xj]≤ |i,j (xj)|+ M.
Lemma 2. Let i,1in1, be a ﬁxed integer. Then there exist an
integer j,i+ 1 jnand 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 kni,
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 kni, such that ∆i,i+k(xi+k)<2 M. Assume
A. V. Prymak 7
that kis the smallest number satisfying this inequality. Evidently k2. Set
j:= i+kand
˜si,j (x) := ½si,j1(x), x [xi, xj1)
L(x;f;xj1, xj), x [xj1, 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 λsi,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,j1(xj1)M. Thus, Lemma 1 implies ki,j1k[xi,xj1]M and
hence
(11) °
°
°
°Zx
xi
si,j (t)f(t)) dt°
°
°
°[xi,xj]2 M.
We obtain
?
i,j (xj) = λi,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]λki,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+)(xz1) + F0(z1),
l1(x) := L(x;F0;z1, z2),
l2(x) := F00(z2)(xz2) + F0(z2),
the inequality
(12) ¯¯¯¯Zzi+1
zi
(F0(x)li(x)) dx¯¯¯¯cΛω3(F; (z3z0)/3; [z0, z3]), i = 0,2
8Convex approximation
holds, where
Λ := Ãz3z0
,
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
z3z0
([z3, z2, z1;f][z2, z1, z0;f])
=1
z3z0µ[z3, z2;f][z2, z1;f]
z3z1[z2, z1;f][z1, z0;f]
z2z0
=α1I1+α2I2+α3I3,
where
Ii:= Zzi
zi1
f0(x)dx =f(zi)f(zi1), i = 1,3,
and
α1=1
(z3z0)(z3z1)(z3z2), α3=1
(z3z0)(z2z0)(z1z0),
α2=1
z3z0µ1
z2z0
+1
z3z1.
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
kFPk[z0,z3]c0ω3(F; (z3z0)/3; [z0, z3]).
where c0is an absolute constant. We have
[z0, z1, z2, z3;F] = [z0, z1, z2, z3;FP] =
3
X
i=0
F(zi)P(zi)
Q3
j=0,j6=i(zizj)
4c0
ω3(F; (z3z0)/3; [z0, z3])
A. V. Prymak 9
Thus
(14) [z0, z1, z2, z3;F]4c0
ω3(F; (z3z0)/3; [z0, z3])
Let us set
I?
i:= Zzi
zi1
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?
iα1Zz1
z0
(F0(x)l0(x)) dx
= [z0, z1, z2, z3;F]α1Zz1
z0
(F0(x)l0(x)) dx
4c0
ω3(F; (z3z0)/3; [z0, z3])
z0(F0(x)l0(x)) dx
(z3z0)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
xi1
(L(t;f;xi1, xi)f(t)) dt M, x [xi1, xi], i = 1, n.
We set S(x) := Rx
x0s(t)dt, where sA0,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;xi1, xi)Ai1,i,i= 1, n. Besides, if 0 i < j < k n,
then the assumptions hAi,j and hAj,k imply hAi,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
sA0,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 jn, 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, . . . , yn2:= xn1. The ﬁrst
derivative of Fexists on (x0, xn) and is continuous. Hence, it is continuous
on [y0, yn2] and we can apply Theorem 10to the function Fand the segment
[y0, yn2] with the partition y0, y1, . . . , yn2. We get some quadratic spline S1
with knots y0, . . . , yn2satisfying
kFS1k[y0,yn2]=kFS1k[x1,xn1]
cmax (Zxi
xi1
(L(x;F0;xi1, xi)F0(x)) dx¯¯¯¯¯
i= 2, n 1),
(18) F0(x1) = S0
1(x1), F 0(xn1) = S0
1(xn1),
(19) F00(x1+) = S00
1(x1+), F 00 (xn1) = S00
1(xn1).
Put
s(x) :=
F00(x1+)(xx1) + F0(x1), x [x0, x1)
S0
1(x), x [x1, xn1]
F00(xn1)(xxn1) + F0(xn1), x (xn1, 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; (x3x0)/3; [x0, x3]),
12 Convex approximation
and
Zxn
xn1
(F0(x)s(x)) dx cΛn3ω3(F; (xnxn3)/3; [xn3, 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.
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[3] Yingkang Hu, Convex approximation by quadratic splines, J. Approx. The-
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For any 3-monotone on $[a,b]$ function $f$ (its third divided differences are nonnegative for all choices of four distinct points, or equivalently, $f$ has a convex derivative on $(a,b)$) we construct a cubic 3-monotone (like $f$) spline $s$ with $n\in \Bbb N$ "almost" equidistant knots $a_j$ such that $$\left\Vert f-s \right\Vert_{[a_j,a_{j-1}]} \le c\, \omega_4 \left(f,(b-a)/n,[a_{j+4},a_{j-5}]\cap [a,b]\right), \quad j=1,...,n,$$ where $c$ is an absolute constant, $\omega_4 \left(f,t,[\cdot,\cdot]\right)$ is the $4$-th modulus of smoothness of $f$, and $||\cdot ||_{[\cdot,\cdot]}$ is the max-norm.
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Let f ∈ C[−1, 1] be 3-convex function on [−1, 1], that is the function , having convex derivative on (−1, 1). Then, an estimate for uniform approximation of f by 3-convex polynomials is obtained involving the Ditzian-Totik modulus of smoothness of order 3 in the right-hand side.
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For every 3-convex piecewise-polynomial function s of degree ≤4 with n equidistant knots on [0, 1] we construct a 3-convex spline s1 (s1 ∈ C(3)) of degree ≤4 with the same knots that satisfies the inequality $$\left\| {S - S_1 } \right\|_{C_{[0,1]} } \leqslant c\omega _5 (s;1/n),$$ where c is an absolute constant and ω5 is the modulus of smoothness of the fifth order.
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This paper considers k-monotone estimation and the related asymptotic performance analysis over a suitable Hölder class for general k. A novel two stage k-monotone B-spline estimator is proposed: in the first stage, an unconstrained estimator with optimal asymptotic performance is considered; in the second stage, a k-monotone B-spline estimator is constructed (roughly) by projecting the unconstrained estimator onto a cone of k-monotone splines. To study the asymptotic performance of the second stage estimator under the sup-norm and other risks, a critical uniform Lipschitz property for the k-monotone B-spline estimator is established under the ℓ∞-norm. This property uniformly bounds the Lipschitz constants associated with the mapping from a (weighted) first stage input vector to the B-spline coefficients of the second stage k-monotone estimator, independent of the sample size and the number of knots. This result is then exploited to analyze the second stage estimator performance and develop convergence rates under the sup-norm, pointwise, and Lp-norm (with p ∈ [1, ∞)) risks. By employing recent results in k-monotone estimation minimax lower bound theory, we show that these convergence rates are optimal. © 2018, Institute of Mathematical Statistics. All rights reserved.
• Shape-preserving approximation of k-monotone functions by splines with free knots
• K Kopotun
K. Kopotun and A. Shadrin, Shape-preserving approximation of k-monotone functions by splines with free knots, to appear.
• One construction of cubic convex spline, Approximation and Optimization
• I A Shevchuk
I. A. Shevchuk, One construction of cubic convex spline, Approximation and Optimization, Proceedings of ICAOR, vol. 1, 357–368.
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Given a convex function ƒ without any smoothness requirements on its derivatives, we estimate its error of approximation by C1 convex quadratic splines in terms of ω3(ƒ, 1/n).
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We obtain Jackson-type estimates for the simultaneous approximation of a 3monotonetwice continuously dierentiable function x by means of quadratic splines withequidistant knots.x1.
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We prove Jackson type estimates for the approximation of monotone nondecreasing functions by monotone nondecreasing splines with equally spaced knots. Our results are of the same order as the Jackson type estimates for unconstrained approximation by splines with equally spaced knots.
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We prove that a convex functionf C[–1, 1] can be approximated by convex polynomialsp n of degreen at the rate of 3(f, 1/n). We show this by proving that the error in approximatingf by C2 convex cubic splines withn knots is bounded by 3(f, 1/n) and that such a spline approximant has anL third derivative which is bounded by n33(f, 1/n). Also we prove that iff C2[–1, 1], then it is approximable at the rate ofn –2 (f, 1/n) and the two estimates yield the desired result.
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Let wf k (f,d),f(x): = Ö{1 - x2 } ,\omega _\phi ^k (f,\delta ),\phi (x): = \sqrt {1 - x^2 } , , are the usual modulus and the Ditzian-Totik modulus of smoothness in the uniform metric, respectively. For a functionf | f(x) - pn (x) | \leqslant Cw3 (f,n - 1 Ö{1 - x2 } + n - 2 ),x Î [ - 1,1]; || f - pn ||¥ \leqslant Cwf 3 (f,n - 1 ); || f - pn ||p \leqslant Ct3 (f,n - 1 )p . \begin{gathered} \left| {f(x) - p_n (x)} \right| \leqslant C\omega _3 (f,n^{ - 1} \sqrt {1 - x^2 } + n^{ - 2} ),x \in [ - 1,1]; \hfill \\ \left\| {f - p_n } \right\|_\infty \leqslant C\omega _\phi ^3 (f,n^{ - 1} ); \hfill \\ \left\| {f - p_n } \right\|_p \leqslant C\tau _3 (f,n^{ - 1} )_p . \hfill \\ \end{gathered} As a consequence, for a functionf || f - pn* ||¥ \leqslant Cn - 1 w2 (f¢,n - 1 ),\left\| {f - p_n^* } \right\|_\infty \leqslant Cn^{ - 1} \omega _2 (f\prime ,n^{ - 1} ), wheren2 andC is an absolute constant.
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A Jackson-type estimate is obtained for the approximation of 3 -convex functions by 3 -convex splines with free knots. The order of approximation is the same as for the Jackson-type estimate for unconstrained approximation by splines with free knots. Shape-preserving free knot spline approximation of k -convex functions, k > 3 , is also considered.
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. We prove that the degree of shape preserving free knot spline approximation in L p [a; b], 0 ! p 1 is essentially the same as that of the non-constrained case. This is in sharp contrast to the well known phenomenon we have in shape preserving approximation by splines with equidistant knots and by polynomials. The results obtained are valid both for piecewise polynomials and for smooth splines with the highest smoothness. Supported by the Office of Naval Research Contract N0014-91-J1343 and the National Foundation Grant EHR 9108772. y Supported by the Russian Foundation of Fundamental Researches under Contract 95-01-00949a. AMS classification: 41A15, 41A25, 41A29. Keywords and phrases: Shape preserving approximation, Free knot splines, Degree of approximation. 0. Introduction Recent years have seen a growing interest in questions of shape preserving approximation. In particular there has been extensive activity in questions of estimating the degree of approximation of monotone ...