P
1-reproducing shape-preserving quasi-interpolant
using positive quadratic splines with
local support
C. Contia;∗, R. Morandia, C. Rabutb
aDipartimento di Energetica, Universita di Firenze, via C. Lombroso 6=17, I-50134 Florence, Italy b
Departement de Genie Mathemathique I.N.S.A., Complexe Scientique de Rangueil, 31077 Toulouse-Cedex 4, France
Received 20 January 1998; received in revised form 13 October 1998
Abstract
In this paper a strategy is presented to construct a shape-preserving quasi-interpolant function expressed as a linear combination of quadratic splines with local support where the coecients are given by the data. The quasi-interpolant is shown to be linear-reproducing, monotone and=or convex conforming to the data. c 1999 Elsevier Science B.V. All rights reserved.
Keywords:Quadratic spline; Quasi-interpolant; Shape-preserving
1. Introduction
Shape-preserving interpolation and=or approximation methods appear of great interest when applied to graphical problems and to reconstruction of functions and curves according to the data. While many shape-preserving interpolant methods with dierent approaches have been presented in the last 20 years (e.g. [5, 7, 8] and references quoted therein), little has been done for the denition of shape-preserving quasi-interpolant methods.
The problem of constructing a quasi-interpolant shape-preserving function tting a set of data
{(xi; f(xi))}ni=1, with x16· · ·6xn, in [x1; xn], expressed as
(x) =
n X
i=1
f(xi)Ci(x); x∈[x1; xn]; (1)
∗Corresponding author. E-mail: [email protected]
was solved by Beatson and Powell [3] and by Wu and Schaback [9], where the functions Ci were
multiquadric functions. However, the quasi-interpolant dened in [3] requires the derivatives of f at the endpoints x1 and xn. This problem was subsequently solved by Conti et al. [6] through the
use of natural cubic splines with no local support.
In this work we present a shape-preserving quasi-interpolant spline of the form (1) where {Ci}n i=1
are positive quadratic splines with local support. The quasi-interpolant dened here satises the
P1-reproduction property in [x1; xn], that is for any p∈P1, Pni=1p(xi)Ci(x) =p(x); ∀x∈[x1; xn] and
interpolates the rst and the last data point. Furthermore, satises shape-preserving properties, that is, if the data are monotone and=or convex, then is monotone and=or convex as well.
Note that the quadratic functions Ci dened in this paper have a larger support than the usual
quadratic B-splines but do guarantee the required properties of .
As in [6], we assume that the values {f(xi)}ni=1 come from a function belonging to the space
D−2L2(
R), the space of all the functions with second derivatives in L2(
R). Endowed with the scalar product
(f; g)2=
Z
R
f′′ (x)g′′
(x) dx+f(a1)g(a1) +f(a2)g(a2) (2)
(where a1; a2 are two distinct real points), D−2L2(R) is a Hilbert space. The proposed
quasi-interpolant is obtained by approximating the right-hand side of the identity
f(x) = (f; H(x;•))2; ∀x∈R;
where H is the reproducing kernel of the Hilbert space.
Concerning the error bound, in [3] it is proved that if f∈C2[x
1; xn] the error of the multiquadric
quasi-interpolant isO(h2logh) with h= maxi=1;:::;n−1{hi=xi+1−xi}. By means of theP1-reproduction
property of our quasi-interpolant spline function, we obtainO(h2) as the error bound for any function
f∈C2[x
1; xn]. This is the same error bound obtained in [6].
The paper is organized as follows: in Section 2 some known properties of the reproducing kernel of the Hilbert space D−2L2(R) are listed. In Section 3, the strategy to construct the shape preserving
quasi-interpolant spline function is described and a graph is shown to illustrate the behaviour of the basis functions Ci. In Section 4, the P1-reproduction of the quasi-interpolant is proved and
in Section 5 the shape-preserving properties of the quasi-interpolant are investigated. Finally, in Section 6, some graphs are shown to illustrate the behaviour of our approximant in comparison with the quasi-interpolants given in [6, 9].
2. Preliminaries
The reproducing kernel of the Hilbert space D−2L2(R) is the map H:R×R→R satisfying the
following conditions:
(i) ∀x∈R; H(x;•)∈D−2L2(
R); ∀y∈R; H(•; y)∈D−2L2(
R); (ii) ∀x∈R; ∀f∈D−2L2(
R); f(x) = (f; H(x;•))2:
where the scalar product (f; H(x;•))2, dened by (2), is
the denition of P1-reproduction.
Denition 2.1. LetX be the Hilbert space D−2L2(R), letE be an approximation operator E:X →X
and P1 the space of degree 1 polynomials. Then the approximation operator E is P1-reproducing if
and only if ∀p∈P1 E(p) =p.
3. Quasi-interpolant spline function
In this section a method is discussed to obtain a spline function quasi-interpolating the values
{(xi; f(xi))}i∈Z where f∈C2[x1; xn]. The approximation is done in two steps. First we derive a
Outside the interval [x1; xn], the function f∗ is dened by
We now want to make some piecewise polynomial approximation of f. For this purpose, we ap-proximate the right-hand side of (9) by approximating the integral and=or the second partial derivative ofH(x;•). In order to obtain a quadratic spline function, we substitute the second partial derivatives of H(x;•) in the right-hand side of (9), by some discrete derivative applied to @H(x; y)=@y. From now on, for simplicity, the partial derivative
@H(x; y)
where the coecients {i
In each interval [xi; xi+1]; i= 1; : : : ; n−1, let us substitute @2H(x; y)=@y2 with D12; iHy(x; xi). We thus
obtain the following approximation of f:
(x) =
Then (13) can be written as
the function in (18) can be expressed as
or, in a more concise form,
(x) =fC(x)−pf1C(x) +p
f
1(x) (22)
where fC is the quasi-interpolant function
fC(x) =
We can now state the following theorem
Theorem 3.1. The functions {Ci}i∈Z dened in (19) are positive functions with local support on [xi−2; xi+2]; i∈Z.
Fig. 1. A function Ci.
Fig. 1 shows the graph of a Ci function (the symbol ‘+’ denotes the xi positions).
4. Shape-preserving properties
In this section we prove some properties of the approximant and the quasi-interpolant function fC dened in the previous section. We rst prove that (x) =fC(x) in [x1; xn]. In addition, fC
interpolates f at x1 and xn.
Theorem 4.1. Let and fC be the functions dened by (18) and (23) respectively. Then (x) =
fC(x); ∀x∈[x1; xn].
Proof. In fact, if we consider the value of fC at x1 we get
fC(x1) =f(x0)C0(x1) +f(x1)C1(x1) +f(x2)C2(x1)
=f(x0)·14 +f(x1)·12 +f(x2)·14
In an analogous way we can obtain fC(xn) =f(xn). As a consequence, p
To investigate the shape-preserving properties of fC, we begin by giving the following denition.
Denition 4.2. A set of real values {(xi; f(xi))}i=1;:::; n, x16· · ·6xn; is called monotone increasing
(decreasing) if f(xi)6f(xi+1); i= 1; : : : ; n−1 (f(xi)¿f(xi+1); i= 1; : : : ; n−1) and convex
(con-cave) if i−16i6i+1; i= 1; : : : ; n−2 (i−1¿i¿i+1; i= 1; : : : ; n−2), where i= (f(xi+1)−
f(xi))=(xi+1−xi); i= 1; : : : ; n−1.
We can now prove the following theorem.
Theorem 4.3. If f is a monotone function; the quasi-interpolant spline fC dened by (23) is
monotone.
Proof. Let Dei; i= 0; : : : ; n+ 2, be the following functions:
e
Di(x) =D12; i−1vy(x−xi−1)−D12; ivy(x−xi): (28)
The function fC dened in (23) can be also written as
fC(x) =
Furthermore, the following relations are true:
Fig. 2. A function De′
i.
After dierentiation, we get
f′
C(x) = n+1
X
i= 1
f(xi)−f(xi−1)
hi−1
e
D′
i(x): (32)
We now prove that each piecewise linear function De′
i is a positive function with support on
[xi−2; xi+1]. To do so, it is sucient to evaluate this function at xi−1 and xi. Thus we obtain
e
D′
i(xi−1) =
hi−1
hi−2+hi−1
¿0; De′
i(xi) =
hi−1
hi−1+hi
¿0: (33)
This concludes the proof.
Fig. 2 shows a De′
i function (the symbol ‘+’ denotes the xi positions).
The next theorem deals with convexity preservation.
Theorem 4.4. If f is a convex function (concave); the quasi-interpolant spline fC dened in (23)
Proof. Let us show that if f is convex, or concave, then f′
C is a monotone function. For this
purpose, we use the following expression for f′
C:
f′
C= −
n X
i=1
i+1−i
2 D
1
2; i|x−xi| −
1
2 − n+1
2 : (34)
Letx∈[xi; xi+1] for somei∈ {1; : : : ; n−1}. At each pointx∈(xi; xi+1) the function fC′ is dierentiable,
so we can write
f′′
C(x) = (i+1−i)·i1+1−(i+2−i+1)·i−+11; (35)
where the coecients i+1
−1; i1+1 are dened by (12).
This quantity is constant so that f′
C is monotone in (xi; xi+1). We now go on to analyse the
quantities
f′
C(xi)−f
′
C(x); f
′
C(x)−f
′
C(xi+1): (36)
After a little algebra we obtain
f′
C(xi)−f
′
C(x) = (i+1−i)·i1(xi−x) + (i+2−i+1)·−i+11(x−xi+1);
f′
C(x)−f
′
C(xi+1) = (i+1−i)·i1(x−xi+1) + (i+2−i+1)·−i+11(xi+1−x):
(37)
The constant sign of the previous quantities concludes the proof.
5. Examples
In this section some graphs are shown to illustrate the behaviour of the shape-preserving spline function fC in comparison with the quasi-interpolant dened in [6, 9]. For each example the graph
of the rst derivatives f′
C is also shown to visualize the shape preservation properties. In all the
gures, the data are denoted by the symbol ‘+’. It should be noted that, according to the support of
Fig. 4. Test 1: quasi-interpolant dened in [9, 6].
Fig. 5. Test 2: function and approximationfC.
the basis functions involved in the denition of f′
C (see (32) or (34)), if the data are only locally
convex the method does not guarantee that the quasi-interpolant will have the same convexity as the data in the neighbour of change of the shape of the data.
The rst test relates to data given in [1]. In particular, Fig. 3 shows the shape-preserving quasi-interpolant fC function with the fC′ behaviour. Note that fC satises the endpoint interpolation
property and that the local support of the functions {Ci}n+1
i=0 guarantees a local linear reproduction.
For comparison, for the same data set, Fig. 4 shows the quasi-interpolant dened by the method given in [9] for c= 1 (:); c= 0:5 (−−); c= 0:1(−) respectively, and the cubic quasi-interpolant dened in [6].
The second test deals with functional data obtained by evaluating the function
f(x) = 1 + 2
x+1
Fig. 6. Test 2: derivative offC and error behaviour.
at seven uniformly distributed knots. Fig. 5 shows the function and the shape-preserving quasi-interpolant fC while Fig. 9 shows the fC′ graph. For this test a picture is also given showing the
error behaviour as the number of the data points increases (7;14;21; 40;80 uniformly distributed data points).
References
[1] H. Akima, A new method of interpolation and smooth curve tting based on local procedures, J. Assoc. Comput. Mach. 17 (1970) 589 – 602.
[2] M. Atteia, Hilbertian kernels and spline functions, Studies in Comp. Math. IV (1994).
[3] R.K. Beatson, M.J.D. Powell, Univariate multiquadric approximation: quasi-interpolation to scattered data, Construc. Approx. 8 (1992) 275 –288.
[4] C. Conti, Funzioni Spline Quasi-Interpolanti: Approssimazione di Curve e Superci, Ph.D. Thesis, 1997. [5] C. Conti, R. Morandi, PiecewiseC1 shape-preserving Hermite interpolation, Computing 56 (1996) 323 – 341. [6] C. Conti, R. Morandi, C. Rabut, Shape-Preserving Quasi-Interpolant Univariate Cubic Spline, in: M. Daehlen, T. Lyche,
L.L. Schumaker (Eds.), Mathematical Methods for Curves and Surfaces, Venderbilt University Press, Nashville, TN, 1998, pp. 55–62.
[7] P. Costantini, An algorithm for computing shape-preserving interpolating spline of arbitrary degree, J. Comput. Appl. Math. 22 (1988) 89 –136.
[8] F.N. Fritsch, R.E. Carlson, Monotone piecewise cubic interpolation, SIAM J. Numer. Anal. 17 (1980) 238 – 246. [9] Z. Wu, R. Schaback, Shape preserving properties and convergence of univariate multiquadratic quasi-interpolation,
![Fig. 4. Test 1: quasi-interpolant de ned in [9, 6].](https://thumb-ap.123doks.com/thumbv2/123dok/3107893.1377079/11.612.25.479.55.436/fig-test-quasi-interpolant-de-ned-in.webp)
