GB-splines of arbitrary order
1Boris I. Ksasova;∗, Pairote Sattayathamb
a
Institute of Computational Technologies, Russian Academy of Sciences, Lavrentyev Avenue 6, 630090, Novosibirsk, Russia
b School of Mathematics, Suranaree University of Technology, University Avenue 111, 30000,
Nakhon Ratchasima, Thailand
Received 6 April 1998; received in revised form 23 October 1998
Abstract
Explicit formulae and recurrence relations for the calculation of generalized B-splines (GB-splines) of arbitrary order are given. We derive main properties of GB-splines and their series, i.e. partition of unity, shape-preserving properties, invariance with respect to ane transformations, etc. It is shown that such splines have the variation diminishing property and are Chebyshevian splines. c1999 Elsevier Science B.V. All rights reserved.
Keywords:Spline; GB-spline; Weak Chebyshevian system; Variation diminishing; Shape-preserving approximation
1. Introduction
Fitting curves and surfaces to functions and data requires the availability of methods which pre-serve the shape of the data. In practical calculations, we usually deal with data given with prescribed accuracy. Therefore, we need to develop methods for constructing fair-shape-preserving approxima-tions that satisfy given tolerances and inherit major geometric properties of the data such as positivity, monotonicity, convexity, presence of linear sections, etc. Such approximations, based on GB-splines [12] with automatic choice of tension parameters are suggested in [11].
Until recently, local support bases for computations with generalized splines have been available only for some special types of splines [3, 13, 15]. This limits the choice of methods when using generalized splines. In [7–9] local support basis functions for exponential splines were introduced and their application to interpolation problems was considered. A recurrence relation for rational B-splines with prescribed poles was obtained in [5]. In [10, 12] one of the authors constructed
∗Corresponding author. E-mail: boris@math.sut.ac.th.
1Supported by grant BRG
=16=2540 of the Thailand Research Fund.
GB-splines for tension generalized splines allowing the tension parameters to vary from interval to interval.
In this paper, we expand the main results in [12] to GB-splines of arbitrary order. These GB-splines are nonnegative functions with supports of minimal length which form a partition of unity. We get explicit formulae for such GB-splines and develop recurrence algorithms for their calculation. In the particular case of polynomial B-splines, we recover the well-known recurrence relation for such B-splines [1]. The main properties of GB-splines and their series such as shape-preserving properties, invariance with respect to ane transformations, etc., are investigated. It is shown that the GB-spline series is a variation diminishing function and the systems of GB-splines are weak Chebyshevian systems.
2. GB-splines of arbitrary degree
Let a partition :a=x0¡x1¡· · ·¡xN=b of the interval [a; b] be given to which we associate
a space of splines SG
n whose restriction to a subinterval [xi; xi+1]; i= 0; : : : ; N−1 is spanned by the
system on linearly independent functions {1; x; : : : ; xn−3;
i; n; i; n}, n¿2, and where every function
in SG
n has n−2 continuous derivatives.
Denition 2.1. A generalized spline of order n is a function S∈SG
n such that
(i) for any x∈[xi; xi+1], i= 0; : : : ; N −1, S(x) =Pi; n−2(x) +S(n
−2)(x
i)i; n(x) +S(n
−2)(x
i+1) i; n(x); (1)
where Pi; n−2 is a polynomial of order n−2, and
i; n(r)(xi+1) = i; n(r)(xi) = 0; r= 0; : : : ; n−2;
(n−2)
i; n (xi) = (n
−2)
i; n (xi+1) = 1;
(2)
(ii) S∈Cn−2[a; b].
The functionsi; n and i; n depend on tension parameters. In practice, we choose i; n(x) =i; n(pi; x);
i; n(x) = i; n(qi; x); 06pi; qi¡∞. In the limiting case when pi; qi→ ∞ we require that limpi→∞
i; n(pi; x) = 0, x∈[xi; xi+1] and limqi→∞ i; n(qi; x) = 0, x∈[xi; xi+1] so that the function S in formula
(1) turns into a polynomial of the ordern−2. Additionally, we require that ifpi=qi=0 for alliwe get
a conventional polynomial spline of degreenwithi; n(x)=−(x
−xi+1)n−1
(n−1)!hi ; i; n(x)= (x−xi)n−1
(n−1)!hi ; hi=xi+1−xi.
Consider now the problem of constructing a basis in the space SG
n consisting of functions with
local supports of minimal length. For this, it is convenient to extend the mesh by adding points
x−n+1¡· · ·¡x−1¡a; b¡xN+1¡· · ·¡xN+n−1. As dim(SnG) = (n)N −(n−1)(N −1) =N+n−1, it
is sucient to construct a system of linearly independent GB-splines Bj; n; j=−n+ 1; : : : ; N −1; in
SG
n such that Bj; n(x)¿0 if x∈(xj; xj+n) and Bj; n≡0 outside (xj; xj+n).
For n¿2 we require the fulllment of the normalization condition
N−1 X
j=−n+1
According to (1), on the interval [xj+l; xj+l+1], l= 0; : : : ; n−1, the GB-spline Bj; n has the form
Taking into account the continuity conditions for neighboring polynomials Pj; l−1; n−2 and Pj; l; n−2,
l= 1; : : : ; n−1, in (4), we have the relations
In particular, the following identity is valid:
n−1
Using the expansion of polynomials by powers of x we can rewrite (7) in the form
n−1
at a system of n−2 linear algebraic equations which denes the unknown quantities B(n−2)
j; n (xj+l);
To obtain the unique solution of this system we can use the normalization condition (3).
Substi-tuting formula (4) into the identity (3) written for x∈[xi; xi+1], we obtain
Since according to (3)
it follows from (6) that
i−1 X
j=i−n+2
Pj; i−j; n−2(x) =
i−1 X
j=i−n+2
i−j
X
l=1 B(n−2)
j; n (xj+l) n−3 X
r=0
z(jr+)l; n(x−xj+l)r=r!≡1:
This gives us the system of linear equations
i−1 X
j=i−n+2
i−j
X
l=1 B(n−2)
j; n (xj+l) n−3 X
r=
zj(r+)l; n(−xj+l)
r−
(r−)! =0; ; = 0; : : : ; n−3;
where 0; is the Kronecker symbol.
We can eliminate the unknowns analogously as has been done in [10, 12]. Having computed the
unknowns B(n−2)
j; n (xj+l); l= 1; : : : ; n−1, we nd the coecients of the polynomials Pj; l; n−2; l=
1; : : : ; n−2, in (4) by using formulae (6). In this way, the computation of the coecients of the
polynomials Pj; l; n−2 can be realized starting from either the left or right endpoint of the support
interval.
3. Recurrence algorithm for the calculation of GB-splines
Let us dene the function
Bj;2(x) =
(n−2)
j; n (x); xj6x6xj+1; (n−2)
j+1; n(x); xj+16x6xj+2;
0; x6∈(xj; xj+2);
(9)
where the functions (n−2)
j; n and
(n−2)
j+1; n are assumed to be positive and monotone on (xj; xj+1) and
(xj+1; xj+2) respectively.
We will consider the sequence of GB-splines dened by the recurrence formula
Bj; k(x) =
Z x
xj
Bj; k−1() cj; k−1
d−
Z x
xj+1
Bj+1; k−1() cj+1; k−1
d; k= 3; : : : ; n; (10)
where
cj; k−1= Z xj+k−1
xj
Bj; k−1() d:
In practical calculations, an alternate representation of formula (10),
Bj; k(x) =−
Z xj+k−1
x
Bj; k−1() cj; k−1
d+
Z xj+k
x
Bj+1; k()
cj+1; k−1
d; k= 3; : : : ; n
is useful.
By dierentiating formula (10) we obtain
B′
Theorem 3.1. The recurrence formulae (9)and (10) dene the sequence of GB-splines of the form
Proof. For k= 2 the formula (12) takes the form
Bj;2(x) =
Using mathematical induction, we assume that the assertion of the theorem is fullled for some
k′
By virtue of (11) and because GB-splines have local supports,
Therefore,
We must show its validity for l6k−2. According to (10), and by the induction assumption for
x∈[xj+l; xj+l+1], we have
Using the formula of dierentiation (11) we obtain
B(k−2)
This permits us to transform expression (16) into the form
+
We have now proved formula (12) with
Pj; l; k−2(x) =
Taking into account the conditions for continuity we obtain the validity of the formula (17) for
l=k−1. However, according to (15), Pj; k−1; k−2≡0. So from (17) forl=k−1 we obtain the identity
(14). By subtracting this identity from (17) we arrive at the second formula in (13). This proves the theorem.
To use formulae (12) and (13) for calculations we rst need to nd the quantities B(k−2)
j; k (xj+1);
etc. Therefore, to nd the required values of the derivatives of the GB-splines in the interior nodes of
their support intervals, it is necessary to know the quantities cj; k, i.e., the integrals of the GB-splines
Bj; k; k= 2; : : : ; n−1.
Theorem 3.2. The integrals cj; k=
Proof. For k= 2 according to (9) we obtain
which corresponds to the formula (19). Let us suppose by induction that the formula (19) holds for allk′
pression (20) into the form
cj; k=
This proves the theorem.
Proof. We can write these identities
cj; k≡Fj; k(x) = k−1 X
l=1 B(k−2)
j; k (xj+1)
n−3 X
r=n−k−1
zj(+r)l; n(x−xj+1)
r−n+k+1
(r−n+k+ 1)!; k= 2; : : : ; n−1: (22)
For k= 2, formula (22) does not depend onx and coincides with (19). According to Theorem 3.2,
the polynomial Fj; k −cj; k; 26k6n−1, of order k−1 takes zero values at the points xj+; =
1; : : : ; k−1. Therefore, by the Fundamental Theorem of Algebra it must be identically equal to zero.
Using the expansion of polynomials in (22) by powers of x we obtain
cj; k = k−1 X
l=1 B(k−2)
j; k (xj+1)
k−2 X
=0 x
!
n−3 X
r=n−k−1+
zj(r+)l; n (−xj+1)
r−n+k+1−
(r−n+k+ 1−)!; k= 2; : : : ; n−1:
The right-hand side is a polynomial of order k−1 for xedj; k while the left-hand side is constant.
It follows that the coecient of x equals c
j; k when = 0 and equals zero otherwise. From this we
obtain the equalities (21). This proves the theorem.
To construct the GB-spline Bj; k; k= 3; : : : ; n, we can formulate an algorithm by applying formulae
(18) and (19), and requiring the calculation of the following quantities for GB-splines:
Bj;2(xj+1) · · · B(k −2)
j; k (xj+) · · · B(n
−2)
j; n (xj+)
..
. ... ... ···
(= 1; : : : ; n−1)
Bj+n−k;2(xj+n−k+1) · · · B(k −2)
j+n−k; k(xj+n−k+)
..
. ···
(= 1; : : : ; k−1)
Bj+n−2;2(xj+n)
and the integrals of GB-splines
cj;2 · · · cj; k · · · cj; n−1 cj+1;2 · · · cj+1; k · · · cj+1; n−1
..
. ... ... ···
cj+n−k;2 · · · cj+n−k; k
..
. ···
cj+n−2;2:
Algorithm 1. (a) Form the diagonal matrix A={aij}; i; j= 1; : : : ; n−1, with diagonal elements
al+1; l+1=Bj+l;1(xj+l+2) = 1; l= 0; : : : ; n−2. Attach to the matrix A at the left an additional column
with elements al+1;0=cj+l;2; l= 0; : : : ; n−2 calculated by formula (19).
(b) For k= 3; : : : ; n, using formula (18) we nd the elements a; l+1=B(k
−2)
j+l−(k−2); k(xj+); =l+
3−k; : : : ; l+ 1; l=n−2; : : : ; k −2, and place them on the main diagonal and on the rst k −2
upper o-diagonals of the matrix A. At every step k (for k¡n−1) we also calculate the elements
al+1; k−2=cj+l−(k−2); k; l=n−2; : : : ; k−2, using formula (19) and place them into the (k −2)nd
As a result the matrix A is transformed to the form
The (k−1)st column of the upper triangular part of the matrixAcontains the quantitiesB(k−2)
j; k (xj+);
= 1; : : : ; k−1; k = 2; : : : ; n. This permits us to construct the GB-splines Bj; k; k = 2; : : : ; n, using
formulae (12) and (13). The integrals for these GB-splines are located along the main diagonal of
the matrix A.
The supports of the GB-splines Bj; k; k = 2; : : : ; n, begin at the point xj. We can also consider an
alternative version of the above algorithm in which the GB-splines Bj+k; n−k; k=n−2; : : : ;0, whose
supports end at the point xj+n are calculated.
Algorithm 2. (a) Form the diagonal matrix Aof dimension (n−1)×(n−1) with diagonal elements
al+1; l+1 =Bj+l;2(xj+l+1) = 1; l= 0; : : : ; n−2. Attach to A the additional (n)th row with elements
As a result the matrix A takes the form
A=
The elements of the (n−k+ 1)th row in the upper triangular part of the matrix A permit us to
construct the GB-splines Bj+n−k; k; k= 2; : : : ; n, using formulae (12) and (13). The integrals of these
GB-splines are located along the main diagonal of the matrix A.
4. Another representation for GB-splines
By summing over the jumps we arrive at the representation
Bj; k(x) = k−1 X
l=1
j+l; k(x)B(k
−2)
j; k (xj+1); k= 3; : : : ; n (23)
with
j+l; k(x) =
(n−k)
j+l−1; n(x)(x−xj+l−1) + "
(n−k)
j+l; n (x)−
(n−k)
j+l−1; n(x) + n−3 X
r=n−k
z(jr+)l; n(x−xj+1)
r−n+k
(r−n+k)!
#
×(x−xj+1)− (n−k)
j+l; n(x)(x−xj+l+1); (24)
(x−y) =
(
1; x¿y;
0; x¡y:
As (x−y) = 1−(y−x) we obtain
Bj; k(x) =− k−1 X
l=1
ˆ
j+l; k(x)B(k
−2)
j; k (xj+1) +Rj; k(x); (25)
where ˆj+l; k is derived from j+l; k by replacing (x−xj+m) with (xj+m−x); m=l−1; l; l+ 1.
Now according to (14)
Rj; k(x) = k−1 X
l=1 B(k−2)
j; k (xj+1)
n−3 X
r=n−k
zj(+r)l; n(x−xj+1)
r−n+k
(r−n+k)! ≡0; k= 3; : : : ; n
and it follows from formulae (23) and (25) that Bj; k(x)≡0 if x =∈(xj; xj+k). Any of these formulae
can be used to dene the GB-spline of order k;36k6n.
We will transform the expression for the function j+l; k in (24). Using the Taylor expansion of
the functions (n−k)
j+l; n ;
(n−k)
j+l; n with the remainder in integral form and the properties (2) we have
j+l; k(x) =
" Z x
xj+l−1
(x−)k−2
(k−2)!
(n−1)
j+l−1; n()d
#
(x−xj+l−1)
+
" Z x
xj+1
(x−)k−2
(k−2)!
(n−1)
j+l; n() d
Z x
xj+1
(x−)k−2
(k−2)!
(n−1)
j+l−1; n() d
#
×(x−xj+1)− "
Z x
xj+l+1
(x−)k−2
(k−2)!
(n−1)
j+l; n () d
#
(x−xj+l+1):
From here we obtain for polynomial splines with
j; n(x) =−
(x−xj+1)n−1
(n−1)!hj
; j; n(x) =
(x−xj)n−1
that
m; k(x) = (xm+1−xm−1)gk[x; xm−1; xm; xm+1]; m=j+l; gk(x; y) = (−1)k(x−y)k
−1
+ =(k−1)!; z+= max(0; z)
and thus (23) is transformed to
Bj; k(x) = (xj+k−xj)gk[x; xj; : : : ; xj+k]; k= 3; : : : ; n:
The recurrence formula (10) takes the form [14]
Bj; k(x) =
x−xj
xj+k−1−xj
Bj; k−1(x) +
xj+k−x
xj+k−xj+1
Bj+1; k−1(x)
with
cj; k =
Z xj+k
xj
Bj; k(x) dx=
xj+k−xj
k :
5. Properties of GB-splines
Let us formulate some properties of GB-splines which are similar to those of polynomial B-splines [14].
Theorem 5.1. The functions Bj; k; k= 2; : : : ; n; have the following properties:
(i) Bj; k(x)¿0 if x∈(xj; xj+k) and Bj; k(x)≡0 if x6∈(xj; xj+k);
(ii) the splines Bj; k have k−2 continuous derivatives;
(iii) for k¿3 and x∈[a; b]; PN−1
j=−k+1Bj; k(x) = 1;
(iv) for x∈[xj; xj+1]; (r)
j; n(x) = n−r−1
Y
k=2 cj; k
!
Bj; n−r(x); j; n(r)(x) = n−r−1
Y
k=2
(−cj−k+1; k)Bj−n+1+r; n−r(x)
j= 0; : : : ; N −1; r= 0; : : : ; n−2; where cj; k =
Z xj+k
xj
Bj; k() d:
Proof. The functions Bj+;2 given by formula (9) are positive if x∈(xj+; xj++2), while Bj+;2(x) = 0
if x6∈(xj+; xj++2), and are monotone on the intervals [xj++l; xj++l+1]; l= 0;1; = 0; : : : ; k−2.
Let us suppose by induction that Bj+; k−1(x)¿0 if x∈(xj+; xj++k−1); Bj+; k−1≡0 if x6∈(xj+;
xj++k−1) and B(k −3)
j+; k−1(x) is monotone on the intervals [xj++l; xj++l+1]; l = 0; : : : ; k − 2 with
(−1)l+1B(k−3)
j+; k−1(xj++l)¿0, l = 1; : : : ; k −2; = 0;1. Then by formula (11) the function B(k
−2)
j; k
is monotone in [xj+l; xj+l+1]; l= 0; : : : ; k−1, and in addition (−1)l+1B(k −2)
j; k (xj+l)¿0; l= 1; : : : ; k−1.
Therefore, B(k−2)
j; k has exactly k−2 zeros in (xj; xj+k) and by Rolle’s theorem, Bj; k does not vanish
on (xj; xj+k). Taking formula (10) into account, we have Bj; k(x)¿0 if x∈(xj; xj+k) and Bj; k(x) = 0 if
Property (ii) is obvious by virtue of the recurrence formula (10) and by continuity of the func-tion Bj;2.
According to (12) for x∈[xj; xj+1],
Bj; n(x) =B(n
−2)
j; n (xj+1) j; n(x); Bj−n+1; n(x) =B(n
−2)
j−n+1; n(xj)j; n(x): (26)
Applying the dierentiation formula (11) we obtain
Bj; n(r)(x) =
r
Y
k=1 c−1
j; n−kBj; n−r(x);
Bj(−r)n+1; n(x) =
r
Y
k=1
(−c−1
j−n+1+k; n−k)Bj−n+1+r; n−r(x); x∈[xj; xj+1]; r= 1;2; : : : ; n−2
(27)
and in particular,
B(n−2)
j; n (xj+1) =
n−2 Y
k=1 c−1
j; n−k; B
(n−2)
j−n+1; n(xj) = n−2 Y
k=1
(−c−1
j−k; k+1): (28)
Therefore, if x∈[xj; xj+1] then according to (26)–(28) we have
(r)
j; n(x) = n−r−1
Y
k=2
cj; kBj; n−r(x); (j; nr)(x) = n−r−1
Y
k=2
(−cj−k+1; k)Bj−n+1+r; n−r(x); r= 0; : : : ; n−2:
Applying the recurrence formula (10) for k¿2, we have for x∈[a; b]:
N−1 X
j=−k+1
Bj; k(x) = N−1 X
j=−k+1 "
Z x
xj
Bj; k−1() cj; k−1
d−
Z x
xj+1
Bj+1; k−1() cj+1; k−1
d
#
=
Z x
x−k+1
B−k+1; k−1() c−k+1; k−1
d−
Z x
xN
BN; k−1() cN; k−1
d=
Z x0
x−k+1
B−k+1; k−1() c−k+1; k−1
d= 1;
i.e.,
N−1 X
j=−k+1
Bj; k+1(x)≡1; k = 3; : : : ; n if x∈[a; b]: (29)
This proves the theorem.
Corollary 5.2. The following identities are valid:
i−1 X
j=i−k+2
i−j
X
l=1 B(k−2)
j; k (xj+l) n−3 X
r=n−k
zj(r+)l; n(x−xj+l)
r−n+k
Proof. Substituting formula (12) into the identity (29) written for x∈[xi; xi+1], we obtain
i
X
j=i−k+1
Bj; k(x) =
(n−k)
i; n (x) i−1 X
j=i−k+1 B(k−2)
j; k (xi) +
(n−k)
i; n (x) i
X
j=i−k+2 B(k−2)
j; k (xi+1)
+
i−1 X
j=i−k+2
Pj; i−j; k−2(x)≡1:
According to (29)
i−1 X
j=i−k+1 B(k−2)
j; k (xi) = i
X
j=i−k+2 B(k−2)
j; k (xi+1) = 0:
Then using (13) one obtains
i−1 X
j=i−k+2
Pj; i−j; k−2(x) =
i−1 X
j=i−k+2
i−j
X
l=1 B(k−2)
j; k (xj+l) n−3 X
r=n−k
zj(+r)l; n(x−xj+l)
r−n+k
(r−n+k)! ≡1:
This proves the corollary.
Corollary 5.3. Let Fj; k; k = 2; : : : ; n−1; be polynomials as dened in (22). Then the following equalities are valid:
N−1 X
j=−k+1 Z b
a
Bj; k(x) dx= N−1 X
j=−k+1
Fj; k(x) =b−a; k = 2; : : : ; n−1:
Proof. Integrating the identity (29) on the interval [a; b] and using (22) we obtain
N−1 X
j=−k+1 Z b
a
Bj; k(x) dx= N−1 X
j=−k+1 Z xj+k+1
xj
Bj; k(x) dx
=
N−1 X
j=−k+1 cj; k =
N−1 X
j=−k+1
Fj; k(x) =b−a:
This proves the corollary.
Theorem 5.4. The GB-splines Bj; k; k= 2; : : : ; n; have supports of minimum length.
Proof. It follows from the explicit formula (9) that the support of the GB-spline Bj;2 cannot be
reduced. Let us suppose that the assertion of the theorem is fullled for some k′
=k−1¡n (k =
3; : : : ; n). Using mathematical induction we will prove its validity for k′
+ 1 =k6n.
By the properties of the functions j+l; n, j+l; n; 06l6k−1, and by formula (12), a GB-spline
Bj; k cannot be dierent from zero on only a part of the subinterval [xj+l; xj+1+l]; 06l6k−1. If we
suppose that Bj; k is zero on interval [xj+l; xj+1+l]; 06l6k−1, then due to the continuity of B
(k−2)
j; k ,
we have B(k−2)
j; k (xj+l) =B
(k−2)
Using formula (11) one can show, however, that (−1)l+1B(k−2)
j; k (xj+l)¿0; l= 1; : : : ; k −1. For
k = 2 we have Bj;2(xj+1) = 1. Suppose by induction that (−1)l+1B(k
′−2)
j; k′ (xj+l)¿0; l= 1; : : : ; k
′
−1,
for 16k′
6k−1. By formula (11) we get
(−1)l+1B(k−2)
j; k (xj+l) = (−1)l+1
B(k′−2)
j; k′ (xj+l)
cj; k′
−B
(k′−2)
j+1; k′ (xj+l)
cj+1; k′
= (−1)l+1B
(k′−2)
j; k′ (xj+l)
cj; k′
+ (−1)lB
(k′−2)
j+1; k′(xj+1+(l−1)) cj+1; k′
¿0; l= 1; : : : ; k−1:
We have obtained a contradiction. This proves the theorem.
Theorem 5.5. The GB-splines Bj; k; j=−k+ 1; : : : ; N−1; k= 2; : : : ; n; are linearly independent and form a basis for the space SG
k of generalized splines.
Proof. Let us assume to the contrary that there exist constantsbj; k; j=−k+1; : : : ; N−1; k=2; : : : ; n,
which are not all equal to zero and such that
b−k+1; kB−k+1; k(x) +· · ·+bN−1; kBN−1; k+1(x) = 0; x∈[a; b]: (30)
According to formula (9), and taking into account the properties of the functions i; n and i; n in
(2), we obtain from (30) for k = 2
N−1 X
j=−1
bj;2Bj;2(xi) =bi−1;2= 0; i= 0; : : : ; N:
Thus, bj;2= 0; j=−1; : : : ; N −1, and the functions Bj;2; j=−1; : : : ; N −1, are linearly independent.
Suppose by induction that the functions Bj; k′; j=−k
′
; : : : ; N−1 are linearly independent for some
k′
=k−1¡n (k= 3; : : : ; n). We will prove the assertion of theorem for k′
+ 1 =k6n. Dierentiating
the equality (30) and using the recurrence formula (10) we have
N−1 X
j=−k+1 bj; kB
′
j; k(x) = N−1 X
j=−k+1 bj; k
"
Bj; k−1(x) cj; k−1
−Bj+1; k−1(x)
cj+1; k−1 #
=b−k+1; k
B−k+1; k−1(x) c−k+1; k−1
+
N−1 X
j=−k+2
(bj; k−bj−1; k)
Bj; k−1(x) cj; k−1
−bN−1; k
BN; k−1(x) cN; k−1
= 0:
(31)
The supports of the GB-splines Bj; k−1; j=−k+ 1; N, however, are outside the interval [a; b]. By
the induction assumption, the GB-splines Bj; k−1; j=−k+ 2; : : : ; N −1 are linearly independent and
thus from (31) we get bj; k−bj−1; k= 0; j=−k+ 2; : : : ; N−1 or bj; k=c= const; j=−k+ 1; : : : ; N−1.
By Theorem 5.1, the GB-splines Bj; k; j=−k+ 1; : : : ; N−1 give a partition of unity on the interval
[a; b]. Using this property, and Eq. (29), we arrive at the equality
N−1 X
j=−k+1
bj; kBj; k(x) =c N−1 X
j=−k+1
Therefore, bj; k= 0; j=−k+ 1; : : : ; N−1 and the GB-splines Bj; k; j=−k+ 1; : : : ; N −1, k= 2; : : : ; n,
are linearly independent.
Since by Denition 2.1, dim(SG
k) = (k)N−(k−1)(N−1) =N+k−1, we see that the GB-splines
Bj; k∈SkG; j=−k+ 1; : : : ; N −1 form a basis of this space. This proves the theorem.
By virtue of this theorem, any spline S∈SG
k; k= 2; : : : ; n, can be uniquely written in the form
S(x) =
N−1 X
j=−k+1
bj; kBj; k(x) for x∈[a; b] (32)
for some constant coecients bj; k.
Corollary 5.6. Any splineS 6≡0in SG
k; k= 2; : : : ; n; with nite support of minimal length coincides with a GB-spline up to a constant multiplier.
Proof. By Theorem 5.4 the minimal support of a spline S∈SG
k , k= 2; : : : ; n dierent from identical
zero, is an interval (xi; xi+k); i= 0; : : : ; N −k. Using representation (32) we get
S(x) =bi−k+1; kBi−k+1; k +· · ·+bi+k−1; kBi+k−1; k(x):
As S≡0 for x6∈(xi; xi+k), when choosing sequentially x∈(xp; xp+1); p=i−k + 1; : : : ; i−1, we
obtain bp; k= 0. In the same manner,bp; k= 0 for p=i+k−1; : : : ; i+ 1. Therefore, S(x) =bi; kBi; k(x).
This proves the corollary.
6. Series of GB-splines
In practical applications such as approximation of functions, discrete data etc., one considers linear
combinations of GB-splines. According to Theorem 5.5, any generalized spline S∈SG
k ; k= 2; : : : ; n,
can be uniquely represented in the form
S(x) =
N−1 X
j=−k+1
bj; kBj; k(x) for x∈[a; b] (33)
with some constant coecients bj; k.
Let us study how the behaviour of a spline S depends on the coecients bj; k. Since GB-splines
are local, from (33) we obtain the inequalities
min
i−k+16j6i bj; k6S(x) = i
X
j=i−k+1
bj; kBj; k(x)6 max
i−k+16j6i bj; k; xi6x6xi+1; k= 2; : : : ; n: (34)
Hence it follows that the behaviour of the spline S on the interval [xi; xi+1] is determined by the
coecients bi−k+1; k; : : : ; bi; k. In particular, in order for a spline S to be zero at a point of the interval
The estimate (34) can be substantially improved on. Applying the dierentiation formula (11), we
obtain for r6k−2
S(r)(x) =
N−1 X
j=−k+r+1
bj; k(r)Bj; k−r(x); (35)
where
bj; k(l)=
bj; k; l= 0;
b(l−1)
j; k −b
(l−1)
j−1; k
cj;k−l
; l= 1;2; : : : ; r:
(36)
Lemma 6.1. If bj; k¿0 (60); j=−k+ 1; : : : ; N −1; k= 2; : : : ; n; then S(x)¿0(60) for all x.
The conclusion is obvious, because the GB-splines Bj; k are nonnegative.
Lemma 6.2. If bj; k¿bj−1; k (bj; k¡bj−1; k); j=−k+ 2; : : : ; N −1; k= 3; : : : ; n; then the function S is monotonically increasing (decreasing).
Proof. According to formulae (35) and (36), we have
S′
(x) =
N−1 X
j=−k+2
bj; k(1)Bj; k−1(x); bj; k(1)=
bj; k−bj−1; k
cj; k−1 :
Because the GB-splines Bj; k−1; k= 3; : : : ; n, are nonnegative, the formula above and Lemma 1 imply
that S is monotonic. This proves the lemma.
Lemma 6.3. If bj; k(1)¿bj(1)−1; k (b
(1)
j; k¡b
(1)
j−1; k); j=−k+ 3; : : : ; N−1; k= 4; : : : ; n; then the function S is convex downwards (upwards).
Proof. By virtue of (35) and (36), we have
S′′
(x) =
N−1 X
j=−k+3
bj; k(2)Bj; k−2(x); b (2)
j; k =
bj; k(1)−bj(1)−1; k
cj; k−2
: (37)
Because the GB-splinesBj; k−2; k= 4; : : : ; n, are nonnegative, taking into account Lemma 1 we obtain
that S is convex. This proves the lemma.
Let Z[a; b](f) denote the number of isolated zeros of a function f on the interval [a; b].
Lemma 6.4. If the spline S(x) =PN−1
j=−kbj; kBj; k(x); k= 2; : : : ; n; is not identically zero on any subin-terval of [a; b]; then
Proof. According to (35) and (9), for x∈[xi; xi+1], we have S(k−2)
(x) =
N−1 X
j=−1 b(k−2)
j; k Bj;2(x) =b (k−2)
i−1; k
(n−2)
i; n (x) +b
(k−2)
i; k
(n−2)
i; n (x):
This function has at most one zero on [xi; xi+1], because the functions(n
−2)
i; n and
(n−2)
i; n are
monoto-nous and nonnegative on this subinterval. Hence Z[a; b](S(k−2))6N. Then, according to Rolle’s
theorem [14], we nd Z[a; b](S)6N+k−2. This proves the lemma.
Denote by suppBj; k ={x|Bj; k(x)6= 0}; k= 2; : : : ; n, the support of GB-spline Bj; k, i.e. the interval
(xj; xj+k).
Theorem 6.5. Assume that −k+1¡−k+2¡· · ·¡N−1; k= 2; : : : ; n. Then
D= det(Bj; k(i))= 06 ; i; j=−k+ 1; : : : ; N −1
if and only if
j∈suppBj; k; j=−k+ 1; : : : ; N + 1: (38)
If condition (38) is satised; then D¿0.
Proof. Let us prove the theorem by induction. It is clear that the theorem holds for a single basis
function. Assume that it also holds for l−1 basis functions. Let us show that if (38) is satised,
then D6= 0 for l basis functions.
Let l∈=suppBl; k. If l lies to the left (right) with respect to the support of Bl; k then the last
column (line) of the determinant D consists of zeros, i.e., D= 0. If l∈suppBl; k and D= 0, then
there exists a nonzero vector c= (c−k+1; k; : : : ; cl−k; k) such that
S(p) = l−k
X
j=−k+1
cj; kBj; k(p) = 0; p=−k + 1; : : : ; l−k;
i.e., the spline S has l isolated zeros. But this contradicts Lemma 6.4, which states that S can have
no more than l−1 isolated zeros. Hence c= 0 and D6= 0.
Now it only remains to prove that D¿0 if (38) is satised. Let us choosexp¡p¡xp+1 for all p.
Then the diagonal elements of D are positive and all the elements above the main diagonal are
zero, i.e., D¿0. It is clear that D depends continuously on p, p=−k+ 1; : : : ; l−k, and D 6= 0
for p∈suppBp; k. Hence the determinant D is positive if condition (38) is satised. This proves the
theorem.
The following three statements follow immediately from Theorem 6.5.
Corollary 6.6. The system of GB-splines {Bj; k}; j=−k + 1; : : : ; N −1; k= 2; : : : ; n; is a weak Chebyshevian system in the sense of [6]; i.e., for any −k+1¡−k+2¡· · ·¡N−1 we have D¿0;
and D¿0 if and only if condition (38) is satised. If the latter is satised; then the generalized spline S(x) =PN−1
Corollary 6.7. If the conditions of Theorem 6.5 are satised; the solution of the interpolation problem
S(i) =fi; i=−k+ 1; : : : ; N −1; fi∈R (39)
exists and is unique.
Let A={aij}; i= 1; : : : ; m; j= 1; : : : ; n; be a rectangular (m×n) matrix with m6n. The matrix
A is said to be totally nonnegative (totally positive) [4] if the minors of all orders of the matrix are
nonnegative (positive), i.e., for all 16l6m we have
det(aipjq)¿0 (¿0) for all 16i1¡· · ·¡il6m; 16j1¡· · ·¡jl6n:
Corollary 6.8. For arbitrary integers −k + 16−k+1¡· · ·¡l−k6N −1 and −k+1¡−k+2¡· · · ¡l−k; k= 2; : : : ; n; we have
Dl= det{Bj; k(i)}¿0; i; j=−k+ 1; : : : ; l−k;
and Dl¿0 if and only if
j∈suppBj; k; j=−k+ 1; : : : ; l−k;
i.e., the matrix {Bj; k(i)}; i; j=−k+ 1; : : : ; N −1 is totally nonnegative.
The last statement is proven by induction on the basis of Theorem 6.5 and the recurrence relations
for the minors of the matrices {Bj; k(i)}; k= 2; : : : ; n. The proof does not dier from that described
by Schumaker [14].
Since the supports of GB-splines are compact, the matrix of the system (39) is a banded matrix
and has 2k−1 nonzero diagonals in general. If the knots of the spline xi; i=−k+ 1; : : : ; N−1;are
placed in a suitable manner, then the number of nonzero diagonals of this matrix can be reduced
to k−1.
De Boor and Pinkus [2] proved that linear systems with totally nonnegative matrices can be solved by Gaussian elimination without choosing a pivot element. Thus, the system (39) can be solved eciently by the conventional Gauss method.
Denote by S−
(C) the number of sign changes (variations) in the sequence of components of the
vector C= (v1; : : : ; vn), with zero being neglected. Karlin [6] showed that if a matrix A is totally
nonnegative, then it decreases the variation, i.e.
S−
(AC)6S(C):
By virtue of Corollary 6.8, the totally nonnegative matrix {Bj; k(i)}; i; j=−k+ 1; : : : ; N −1; k =
2; : : : ; n, formed by the GB-splines decreases the variation.
For a bounded real function f, let S−
(f) be the number of sign changes of the function f on
the real axis R without taking into account the zeros
S−
(f) = sup
p
S−
Theorem 6.9. The generalized spline S(x) =PN−1
j=−k+1bj; kBj; k(x); k= 2; : : : ; n; is a variation dimin-ishing function; i.e., the number of sign changes of S does not exceed the one in the sequence of its coecients
S−
R
N−1 X
j=−k+1 bj; kBj; k
6S −
(b); b= (b−k+1; k; : : : ; bN−1; k):
Proof. We use the approach proposed by Schumaker [14]. Let S−
(b) =d−1. Let us divide the
coecients bj; k into d groups:
b−k+1; k; : : : bk2; k; bk2+1; k; : : : ; bk3; k; : : : ; bkd+1; k; : : : ; bN−1; k:
In each group at least one coecient is not zero, and all the nonzero coecients have the same sign.
Putting k1=−k and kd+1=N−1, we dene the function
˜
Bj; k(x) = kj+1
X
i=kj+1
|bi; k|Bi; k(x); j= 1; : : : ; d:
Then for arbitrary 1¡2¡· · ·¡d we have
det( ˜Bj; k(i))di; j=1=
k2
X
1=k1+1
· · ·
kd+1
X
d=kd+1
|b1; k| · · · |bd; k|det(Bj; k(i))¿0;
i= 1; : : : ; d; j=1; : : : ; d; k= 2; : : : ; n
by virtue of Corollary 6.8 and because at least one coecient bi; k is not zero in each group. It is
clear that we can choose 1¡2¡· · ·¡d such that det( ˜Bj; k(i))¿0. Hence the functions ˜Bj; k are
linearly independent.
Assume that=±1 is the sign of the rst group of the coecients bi; k. Let us take ˜bi; k=(−1)i
−1 ,
i= 1;2; : : : ; d. Then
˜
S(x) =
d
X
i=1
˜
bi; kB˜i; k(x)≡S(x) = N−1 X
j=−k+1
bj; kBj; k(x):
Applying Lemma 6.4, we obtain
Z
N−1 X
j=−k+1 bj; kBj; k
=Z
d
X
i=1
˜
bi; kBi; k
!
6d−1 =S−
(b−k+1; k; : : : ; bN−1; k); k= 2; : : : ; n:
This proves the theorem.
Theorem 6.10. Assume that the inequalities (−1)iS(
i)¿0; i= 1;2; : : : ; d; are valid for the gener-alized spline S(x) =PN−1
j=−k+1bj; kBj; k(x)at some 1¡2¡· · ·¡d. Then there exist −k+ 16j1¡j2 ¡· · ·¡jd6N −1 such that
(−1)ibji; kBji; k(i)¿0; i= 1; : : : ; d:
The proof of this statement does not dier from the proof of the corresponding theorem for poly-nomial B-splines [14].
7. Invariance of generalized splines with respect to ane transformations
In some applications of spline approximation we encounter ane transformations of the
inde-pendent variable: ˆx =px +q, where p6= 0 and q are constant. It is well-known that the usual
Lagrange–Newton, Chebyshev, etc., polynomials are invariant with respect to such transformations. Let us show that generalized splines also have this property.
Let ˆSnG be a set of generalized splines on the mesh ˆ={xˆi|xˆi=pxi+q; i= 0; : : : ; N} which is
obtained from the linear space SnG by ane transformation of the variable x.
Theorem 7.1. An approximating generalized spline S∈SnG is invariant with respect to ane trans-formations of the real axis R= (−∞;∞).
Proof. The function Bj;2 in (9) can be written in the form
Bj;2(x) =
(n−2)
j; n
x−xj
hj
!
; xj6x6xj+1;
’(n−2)
j+1; n
x−xj+1 hj+1
!
; xj+16x6xj+2;
0 otherwise;
where
j; n
x−xj
hj
! hn−2
j = j; n(x); ’j+1; n
x−xj+1 hj+1
! hn−2
j+1 =j+1; n(x):
Using the change of the variable ˆx=px+q we get
(n−2)
j; n
x−xj
hj
!
= (n−2)
j; n
ˆ
x−xˆj
ˆ
xj+1−xˆj
! ;
’(n−2)
j+1; n
x−xj+1 hj+1
!
=’(n−2)
j+1; n
ˆ
x−xˆj+1
ˆ
xj+2−xˆj+1 !
Therefore,Bj;2(x)= ˆBj;2( ˆx). Let the equalityBj; l(x)= ˆBj; l( ˆx) be fullled forl=k−1¡n(k=3; : : : ; n).
By virtue of the recurrence relation (10) we have
Bj; k(x) =
Z x
xj
Bj; k−1() cj; k−1
d−
Z x
xj+1
Bj+1; k−1() cj+1; k−1
d; k= 3; : : : ; n;
with
cj; k−1= Z xj+k−1
xj
Bj; k−1() d:
Using the substitution ˆ=p+q we obtain by induction
cj; k−1= Z xˆj+k−1
ˆ
xj
ˆ
Bj; k−1( ˆ)
1
pd ˆ=
1
pcˆj; k−1; Bj; k(x) =
Z xˆ ˆ
xj
ˆ
Bj; k−1( ˆ)
ˆ
cj; k−1
d ˆ−
Z xˆ ˆ
xj+1 ˆ
Bj+1; k−1( ˆ)
ˆ
cj+1; k−1
d ˆ= ˆBj; k( ˆx); k= 3; : : : ; n:
If now S∈SnG and ˆS∈SˆnG are approximating splines on the meshes and ˆ, respectively,
con-nected by an ane transformation ˆx=px+q, then by the uniqueness of the spline representation
as a linear combination of GB-splines we obtain
S(x) =X
j
bjBj; n(x) =
X
j
bjBˆj; n( ˆx) = ˆS( ˆx): (40)
Therefore, the approximating generalized splineS is invariant with respect to ane transformations
of its variable. This proves the theorem.
It was shown above that Bj; k(x) = ˆBj; k( ˆx); k= 2; : : : ; n. By dierentiation of this equality we obtain
d
dx[Bj; k(x)] =
d
dx[ ˆBj; k( ˆx)] =
d
d ˆx[ ˆBj; k( ˆx)]
d ˆx
dx =pBˆ
′
j; k( ˆx)]; k= 3; : : : ; n:
Dierentiating now the equality (40), we can write down
S′
(x) =X
j
bjB
′
j; n(x) =
X
j
bjpBˆ
′
j; n( ˆx) =pSˆ
′
( ˆx):
By repeated dierentiation of the last and next to last equalities we arrive at
S(r)(x) =X
j
bjB(j; nr)(x) =
X
j
bjprBˆ(j; nr)( ˆx) =prSˆ(r)( ˆx); r= 1; : : : ; n−2:
8. Local approximation by GB-splines
Using the locality of GB-splines one can reduce the representation of a spline S as a linear
combination of GB-splines (33) for k=n to the form
S(x) =
i
X
j=i−n+1
Theorem 8.1. The restriction (41) of the spline S to the interval [xi; xi+1] can be written in the
Proof. We use induction on n. According to formula (9), the representation (42) holds for n= 2:
S(x) =bi−1;2Bi−1;2(x) +bi;2Bi;2(x) =bi−1;2i;2(x) +bi;2 i;2(x); x∈[xi; xi+1]: (43)
where according to (6)
Pi; l−2(x) =
By applying the formula of dierentiation (11) one gets
+ i; l(x)
bi−l+2; l
B(l−3)
i−l+2; l−1(xi+1) ci−l+2; l−1
+
i
X
j=i−l+3
bj; l−bj−1; l
cj; l−1
B(l−3)
j; l−1(xi+1)−bi; l
B(l−3)
i+1; l−1(xi+1) ci+1; l−1
:
(45)
BecauseB(l−3)
i−l+l′+1; l−1(xi+l′) =B
(l−3)
i+l′; l−1(xi+l′) = 0,l
′
= 0;1, expression (45) can be rewritten in the form
I=i; l(x) i−1 X
j=i−l+2
b(1)j; lB(l−3)
j; l−1(xi) + i; l(x) i
X
j=i−l+3
b(1)j; lB(l−3)
j; l−1(xi+1)
with
b(1)j; l=bj; l−bj−1; l
cj; l−1
; j=i−l+ 2; : : : ; i:
By the assumption of induction however,
i−1+l′
X
j=i−(l−2)+l′
bj; l−1B(l −3)
j; l−1(xi+l′) =b
(l−3)
i−1+l′; l−1; l ′
= 0;1:
Therefore one gets
I=b(l−2)
i−1; li; l(x) +b (l−2)
i; l i; l(x)
and using (44), we can write down (43) in the form
S(x) =Pi; l−2(x) +b (l−2)
i−1; li; l(x) +b
(l−2)
i; l i; l(x):
This is formula (42) in the assertion of the theorem for n=l. This proves the theorem.
Theorem 8.2. In formula (42) the polynomial Pi; n−2 can be written in the form
Pi; n−2(x) =
n−3 X
k=0
b(i−k)n+2+k; nQ (n−3−k)
i; n−2 (x);
where
Q(i; nk)−2(x) =
Qi; n−2(x)=ci−2;1; k= 0; Q(k−1)
i−1; n−2(x)−Q (k−1)
i; n−2(x) ci−k−1; k+2
; k= 1;2; : : : ; n−3;
Qj; n−2(x) =
n−3 X
l=0
z(j; nr)(x−xj)
r
r! ; j=i−n+ 3; : : : ; i; Q
(n−3)
i; n−2(x)≡1:
9. Examples
Let us give examples of the dening functions i; n(x) and i; n(x) in (1) which are most commonly
used. In the examples given below they depend on the parameters
i; n(x) =’i; n(t)hn
−2
i = n(pi;1−t)(−hi)n
−2;
i; n(x) = i; n(t)hn
−2
i = n(qi; t)hn
−2
i ;
where t= (x−xi)=hi and 0 6pi; qi¡∞.
By Denition 2.1 the function i; n(t) satises the conditions
(r)
i; n(0) = 0; r= 0; : : : ; n−2;
(n−2)
i; n (1) = 1: (46)
(1) Rational splines:
i; n(t) =
tn−1
(n−1)!
C1; n(qi)
1 +qi(1−t)
; C−1 1; n=
n−2 X
j=0 Cnj−2
qij j+ 1;
where
Cnj−2=
n−2
j
is the usual binomial coecient.
(2) Exponential splines:
i; n(t) =
tn−1
(n−1)!e
−qi(1−t)C
2; n(qi); C
−1 2; n(qi) =
n−2 X
j=0 Cnj−2
qij
(j+ 1)!:
(3) Hyperbolic splines:
i; n(t) =
i; n(t)
qn−2
i sinh(qi)
;
where
i; n(t) =
sinh(qit)−P
m−2
j=0
(tqi)2j+1
(2j+1)! if n= 2m;
cosh(qit)−Pm
−2
j=0 (tqi)2j
(2j)! if n= 2m−1:
(4) Splines with additional nodes:
i; n(t) =
1 +qi
(n−1)!
t− qi
1 +qi
n−1 +
:
If we take i= (1 +pi)−1 and i= 1−(1 +qi)−1, then the pointsxi1=xi+ihi andxi2=xi+ihi
x the positions of two additional nodes of the spline on the interval [xi; xi+1]. By moving them, we
can go from a spline of the order n to a spline of the order n−2.
The constants Ck; n(qi), k= 1;2, in the expressions for the function i; n(t) above are calculated
from the condition (n−2)
Acknowledgements
Appreciation is rendered to the Thailand Research Fund for the nancial support which has made this research possible.
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