On discrete GB-splines

ANZIAM J. 42 (E) ppC877–C899, 2000 C877

On discrete GB-splines

Boris I. Kvasov ^∗

(Received 7 August 2000)

Abstract

Explicit formulae and recurrence relations are obtained for dis- crete generalized b-splines (discrete gb-splines for short). Properties of discrete gb-splines and their series are studied. It is shown that the series of discrete gb-splines is a variation diminishing function and the systems of discrete gb-splines are weak Chebyshev systems.

∗

School of Mathematics, Suranaree University of Technology, Nakhon Ratchasima 30000, Thailand. mailto:[email protected]

0

See http://anziamj.austms.org.au/V42/CTAC99/Kvas for this article and ancillary

services, c Austral. Mathematical Soc. 2000. Published 27 Nov 2000.

Contents C878

Contents

1 Introduction C878

2 Discrete generalized splines C879

3 Construction of discrete GB-splines C882

4 Properties of discrete GB-splines C885

5 Local approximation by discrete GB-splines C887 6 Recurrence formulae for discrete GB-splines C890 7 Series of discrete GB-splines (uniform case) C892

References C898

1 Introduction

The tools of generalized splines and gb-splines are widely used in solving problems of shape-preserving approximation (e.g., see [7]). Recently, in [1]

a difference method for constructing shape-preserving hyperbolic tension splines as solutions of multipoint boundary value problems was developed.

Such an approach permits us to avoid the computation of hyperbolic func-

2 Discrete generalized splines C879 tions and has substantial other advantages. However, the extension of a mesh solution will be a discrete hyperbolic tension spline.

The contents of this paper is as follows. In Section 2 we give a definition of a discrete generalized spline. Next, we construct a minimum length local support basis (whose elements are denoted as discrete gb-splines) of the new spline; see Section 3. Properties of gb-splines are discussed in Section 4, while the local approximation by discrete gb-splines of a given continuous function from its samples is considered in Section 5. In Section 6 we derive recurrence formulae for calculations with discrete gb-splines. The properties of gb-spline series are summarized in Section 7.

2 Discrete generalized splines

Let a partition ∆ : a = x 0 < x 1 < · · · < x N = b of the interval [a, b] be given.

We will denote by S ₄ ^DG the space of continuous functions whose restriction

to a subinterval [ x i , x i+1 ], i = 0, . . . , N − 1 is spanned by the system of

four linearly independent functions {1, x, Φ i , Ψ i }. In addition, we assume

that each function in S ₄ ^DG is smooth in the sense that for given τ _i ^L

> 0

and τ _i ^R

> 0, j = i − 1, i, the values of its first and second central divided

differences with respect to the points x i − τ _i ^L

, x i , x i + τ _i ^R

and x i − τ _i ^L

,

x i , x i + τ _i ^R

coincide.

2 Discrete generalized splines C880 Given a continuous function S we introduce the difference operators

D 1 S(x) ≡ D i,1 S(x) = (λ ^R _i

S[x − τ _i ^L

, x] + λ ^L _i

S[x, x + τ _i ^R

])(1 − t) +( λ ^R _i+1

S[x − τ _i+1 ^L

, x] + λ ^L _i+1

S[x, x + τ _i+1 ^R

]) t, D 2 S(x) ≡ D i,2 S(x) = 2S[x − τ _i ^L

, x, x + τ _i ^R

](1 − t)

+2 S[x − τ _i+1 ^L

, x, x + τ _i+1 ^R

] t,

x ∈ [x _i , x _i+1 ) , i = 0, . . . , N − 1,

where λ ^R _j

= 1 − λ ^L _j

= τ _j ^R

/(τ _j ^L

+ τ _j ^R

), j = i, i + 1 and t = (x − x i ) /h i , h i = x i+1 − x i . The square parentheses denote the usual first and second divided differences of the function S with respect to the argument values x j − τ _j ^L

, x j , x j + τ _j ^R

, j = i, i + 1.

Definition 1 A discrete generalized spline is a function S ∈ S ₄ ^DG such that 1. for any x ∈ [x i , x i+1 ], i = 0, . . . , N − 1

S(x) ≡ S _i ( x) = [S(x _i ) − Φ _i ( x _i ) M _i ](1 − t) + [ S(x _i+1 ) − Ψ _i ( x _i+1 ) M _i+1 ] t

+ Φ _i ( x)M _i + Ψ _i ( x)M _i+1 , (1) where M _j = D _i,2 S _i ( x _j ), j = i, i + 1, and the functions Φ _i and Ψ _i are subject to the constraints

Φ _i ( x _i+1 − τ _i+1 ^L

) = Φ _i ( x _i+1 ) = Φ _i ( x _i+1 + τ _i+1 ^R

) = 0 ,

Ψ _i ( x i − τ _i ^L

) = Ψ _i ( x i ) = Ψ _i ( x i + τ _i ^R

) = 0 , (2)

D i,2 Φ _i ( x i ) = 1 , D i,2 Ψ _i ( x i+1 ) = 1;

2 Discrete generalized splines C881 2. S satisfies the following smoothness conditions

S i−1 ( x i ) = S i ( x i ) ,

D i−1,1 S i−1 ( x i ) = D i,1 S i ( x i ) , i = 1, . . . , N − 1. (3) D i−1,2 S i−1 ( x i ) = D i,2 S i ( x i ) ,

This definition generalizes the notion of a discrete polynomial spline in [9]

and of a generalized spline in [5, 6]. The latter one can be obtained by setting τ _j ^L

= τ _j ^R

= 0, j = i, i + 1 for all i. If τ _i ^L

= τ _i ^L and τ _i ^R

= τ _i ^R , j = i − 1, i then according to smoothness conditions (3) the values of the functions S i−1

and S _i at the three consecutive points x _i − τ _i ^L , x _i , x _i + τ _i ^R coincide. Setting τ _j ^L

= τ _j ^R

= τ i , j = i, i + 1 we obtain D 1,i S(x) = S[x − τ i , x + τ i ] and D 2,i S(x) = S[x − τ i , x, x + τ i ], which is the case discussed in [1].

The functions Φ _i and Ψ _i depend on the tension parameters which influ- ence the behaviour of S fundamentally. We call them the defining functions.

In practice one takes Φ _i ( x) = Φ i ( p i , x), Ψ i ( x) = Ψ i ( q i , x), 0 ≤ p i , q i < ∞.

In the limiting case when p i , q i → ∞ we require that lim p

→∞ Φ _i ( p i , x) = 0,

x ∈ (x _i , x _i+1 ] and lim _q

_→∞ Ψ _i ( q _i , x) = 0, x ∈ [x _i , x _i+1 ) so that the function S

in formula (1) turns into a linear function. Additionally, we require that if

p i = q i = 0 for all i, then we get a discrete cubic spline. If τ _i ^L

= τ _i ^R

= τ i ,

j = i − 1, i for all i then this spline coincides with a discrete cubic spline

of [10]. The case τ _i = τ for all i was considered in [ 8].

3 Construction of discrete GB-splines C882

3 Construction of discrete GB-splines

Let us construct a basis for the space of discrete generalized splines S ₄ ^DG by using functions which have local supports of minimum length. Since dim( S ₄ ^DG ) = 4 N − 3(N − 1) = N + 3 we extend the grid ∆ by adding the points x j , j = −3, −2, −1, N+1, N+2, N+3, such that x −3 < x −2 < x −1 < a, b < x _N+1 < x _N+2 < x _N+3 .

We demand that the discrete gb-splines B _i , i = −3, . . . , N − 1 have the properties

B _i ( x) > 0, x ∈ (x i + τ _i ^R

, x i+4 − τ _i+4 ^L

) , (4) B _i ( x) ≡ 0, x /∈ (x i , x i+4 ) ,

N−1 X

j=−3

B _j ( x) ≡ 1, x ∈ [a, b]. (5)

According to (1), on the interval [ x j , x j+1 ], j = i, . . . , i + 3, the discrete gb -spline B _i has the form

B _i ( x) ≡ B i,j ( x) = P i,j ( x) + Φ j ( x)M j,B

+ Ψ _j ( x)M j+1,B

, (6) where P i,j is a polynomial of the first degree and M l,B

= D j,2 B _i ( x l ), l = j, j + 1 are constants to be determined. The smoothness conditions ( 3) together with the constraints (2) give the following relations

P i,j ( x j ) = P i,j−1 ( x j ) + z j M j,B

,

D j,1 P i,j ( x j ) = D j−1,1 P i,j−1 ( x j ) + c j−1,2 M j,B

,

3 Construction of discrete GB-splines C883

where

z j ≡ z j ( x j ) = Ψ _j−1 ( x j ) − Φ j ( x j ) , c j−1,2 = D j−1,1 Ψ _j−1 ( x j ) − D j,1 Φ _j ( x j ) . Thus in (6)

P i,j ( x) = P i,j−1 ( x) + [z j + c j−1,2 ( x − x j )] M j,B

. (7)

By repeated use of this formula we get

P i,j ( x) = ^{X j}

l=i+1

[ z l + c l−1,2 ( x − x l )] M l,B

= − ^{X i+3}

l=j+1

[ z l + c l−1,2 ( x − x l )] M l,B

.

As B _i vanishes outside the interval ( x i , x i+4 ), we have from (7) that P i,j ≡ 0 for j = i, i + 3. In particular, the following identity is valid

i+3 X

j=i+1

[ z j + c j−1,2 ( x − x j )] M j,B

≡ 0,

from which one obtains the equalities

X i+3 j=i+1

c _j−1,2 y _j ^r M _j,B

= 0 , r = 0, 1, y _j = x _j − z _j

c j−1,2 . (8)

3 Construction of discrete GB-splines C884 Thus the formula for the discrete gb-spline B _i takes the form

B _i ( x) =

 

 

 

 

 

 



 

 

 

 

 

 

Ψ _i ( x)M _i+1,B

, x ∈ [x _i , x _i+1 ) ,

( x − y i+1 ) c i,2 M i+1,B

+ Φ _i+1 ( x)M i+1,B

+ Ψ _i+1 ( x)M i+2,B

, x ∈ [x i+1 , x i+2 ) , ( y _i+3 − x)c _i+2,2 M _i+3,B

+ Φ _i+2 ( x)M _i+2,B

+ Ψ _i+2 ( x)M i+3,B

, x ∈ [x i+2 , x i+3 ) ,

Φ _i+3 ( x)M i+3,B

, x ∈ [x i+3 , x i+4 ) ,

0 , otherwise .

(9)

Substituting formula (9) into the normalization condition (5) written for x ∈ [x _i , x _i+1 ], we obtain

X i j=i−3

B _j ( x) = Φ i ( x) ^{X i−1}

j=i−3

M i,B

+ Ψ _i ( x) ^{X i}

j=i−2

M i+1,B

+( y i+1 − x)c i,2 M i+1,B

+ ( x − y i ) c i−1,2 M i,B

≡ 1.

As according to (5)

X i−1 j=i−3

M _i,B

=

X i j=i−2

M _i+1,B

= 0 (10)

the following identity is valid

( y i+1 − x)c i,2 M i+1,B

+ ( x − y i ) c i−1,2 M i,B

≡ 1.

4 Properties of discrete GB-splines C885 From here one gets the equalities

y _i+1 ^r c _i,2 M _i+1,B

− y _i ^r c _i−1,2 M _i,B

≡ δ _1,r , r = 0, 1,

where δ _1,r is the Kronecker symbol. Solving this system of equations and using (8) or (10), we obtain

M _j,B

= y _i+3 − y _i+1

c j−1,2 ω ⁰ _i+1 ( y j ) , j = i + 1, i + 2, i + 3, ω _i+1 ( x) = (x − y _i+1 )( x − y _i+2 )( x − y _i+3 )

or with the notation c j,3 = y j+2 − y j+1 , j = i, i + 1, M i+1,B

= 1

c i,2 c i,3 , M i+2,B

= − 1

c i+1,2

1

c i,3 + 1 c i+1,3

!

, (11)

M _i+3,B

= 1

c i+2,2 c i+1,3 .

4 Properties of discrete GB-splines

The functions B _j , j = −3, . . . , N − 1 possess many of the properties inherent

in usual discrete polynomial b-splines. To provide inequality (4), in what

follows we need to impose additional conditions on the functions Φ _j and Ψ _j .

4 Properties of discrete GB-splines C886 The proofs of the following four assertions repeat those given in [5].

Lemma 2 If the conditions

0 < 2h ⁻¹ _j−1 Ψ _j−1 ( x j ) < D j−1,1 Ψ _j−1 ( x j ) ,

0 < 2h ⁻¹ _j Φ _j ( x _j ) < −D _j,1 Φ _j ( x _j ) , j = i + 1, i + 2, i + 3 (12) are satisfied, then in (11) c j,k > 0, j = i, . . . , i + 4 − k; k = 2, 3, and

( −1) ^j−i−1 M j,B

> 0, j = i + 1, i + 2, i + 3. (13)

Theorem 3 Let the conditions of Lemma 2 be satisfied, the functions Φ _j and Ψ _j be convex and D j,2 Φ _j and D j,2 Ψ _j be strictly monotone on the interval [ x j , x j+1 ] for all j. Then the functions B _j , j = −3, . . . , N − 1 have the following properties:

1. B _j ( x) > 0 for x ∈ (x _j + τ _j ^R

, x _j+4 − τ _j+4 ^L

), and B _j ( x) ≡ 0 if x /∈

( x j , x j+4 );

2. B _j satisfies the smoothness conditions (3);

3. ^{P N−1} _j=−3 y _j+2 ^r B _j ( x) ≡ x ^r , r = 0, 1 for x ∈ [a, b], Φ j ( x) = c j−1,2 c j−2,3 B _j−3 ( x),

Ψ _j ( x) = c j,2 c j,3 B _j ( x) for x ∈ [x j , x j+1 ], j = 0, . . . , N − 1.

5 Local approximation by discrete GB-splines C887 Lemma 4 The function B _i has support of minimum length.

Theorem 5 The functions B _i , i = −3, . . . , N − 1, are linearly independent and form a basis of the space S ₄ ^DG of discrete generalized splines.

5 Local approximation by discrete GB-splines

According to Theorem 5, any discrete generalized spline S ∈ S ₄ ^DG can be uniquely written in the form

S(x) = ^{N−1 X}

j=−3 b j B _j ( x) (14)

for some constant coefficients b j .

If the coefficients b _j in (14) are known, then by virtue of formula (9) we can write out an expression for the discrete generalized spline S on the interval [ x i , x i+1 ], which is convenient for calculations,

S(x) = b _i−2 + b ⁽¹⁾ _i−1 ( x − y _i ) + b ⁽²⁾ _i−1 Φ _i ( x) + b ⁽²⁾ _i Ψ _i ( x), (15) where

b ^(k) _k = b ^(k−1) _j − b ^(k−1) _j−1

c j,4−k , k = 1, 2; b ⁽⁰⁾ _j = b _j . (16)

5 Local approximation by discrete GB-splines C888 The representations (14) and (15) allow us to find a simple and effective way to approximate a given continuous function f from its samples.

Theorem 6 Let a continuous function f be given by its samples f(y j ), j =

−1, . . . , N + 1. Then for b j = f(y j+2 ), j = −3, . . . , N − 1, formula ( 14) is exact for polynomials of the first degree and provides a formula for local approximation.

Proof: It suffices to prove that the identities

N−1 X

j=−3

y _j+2 ^r B _j ( x) ≡ x ^r , r = 0, 1 (17)

hold for x ∈ [a, b]. Using formula ( 15) with the coefficients b j−2 = 1 and b j−2 = y j , j = i − 1, i, i + 1, i + 2, for an arbitrary interval [x i , x i+1 ], we find that identities (17) hold.

For b _j−2 = f(y _j ), formula (15) can be rewritten as

S(x) = f(y i ) + f[y i , y i+1 ]( x − y i ) + ( y i+1 − y i−1 ) f[y i−1 , y i , y i+1 ] c ⁻¹ _i−1,2 Φ _i ( x) +( y i+2 − y i ) f[y i , y i+1 , y i+2 ] c ⁻¹ _i,2 Ψ _i ( x), x ∈ [x i , x i+1 ] .

This is the formula of local approximation. The theorem is thus proved. ♠

5 Local approximation by discrete GB-splines C889 Corollary 7 Let a continuous function f be given by its samples f j = f(x j ), j = −2, . . . , N + 2. Then by setting

b _j−2 = f _j − 1 c j−1,2



Ψ _j−1 ( x _j ) f[x _j , x _j+1 ] − Φ _j ( x _j ) f[x _j−1 , x _j ] ^ (18) in (14), we obtain a formula of three-point local approximation, which is exact for polynomials of the first degree.

Proof: To prove the corollary, it is sufficient to take the monomials 1 and x as f. Then according to ( 18), we obtain b j−2 = 1 and b j−2 = y j and it only remains to make use of identities (17). This proves the corollary. ♠ Equation (15) permits us to write the coefficients of the spline S in its representation (14) of the form

b _j−2 =

( S(y _j ) − D _j−1,2 S(x _j−1 )Φ _j−1 ( y _j ) − D _j,2 S(x _j )Ψ _j−1 ( y _j ) , y _j < x _j , S(y j ) − D j,2 S(x j )Φ _j ( y j ) − D j+1,2 S(x j+1 )Ψ _j ( y j ) , y j ≥ x j . According to this formula we have b j−2 = S(y j ) + O(h ² _j ), h j = max( h j−1 , h j ).

Hence it follows that the control polygon (e.g., see [4]) converges quadratically

to the function f when b _j−2 = f(y _j ), or if the formula (18) is used.

6 Recurrence formulae for discrete GB-splines C890

6 Recurrence formulae for discrete GB-splines

Let us define functions

B _j,2 ( x) =

 



 

D _j,2 Ψ _j ( x), x ∈ [x _j , x _j+1 ) , D j+1,2 Φ _j+1 ( x), x ∈ [x j+1 , x j+2 ] ,

0 , otherwise , j = i, i + 1, i + 2. (19) We assume that the functions D j,2 Ψ _j and D j+1,2 Φ _j+1 are strictly monotone on [ x j , x j+1 ) and [ x j+1 , x j+2 ] respectively. The splines B _j,2 are a generalization of the “hat-functions” for polynomial b-splines. They are nonnegative and, furthermore, B _j,2 ( x j+l ) = δ 1,l , l = 0, 1, 2.

According to (9), (11) and (19) the function D ₂ B _i can be written as D 2 B _i ( x) = ^{i+3 X}

j=i+1

M j,B

B _j−1,2 ( x)

= 1 c _i,3

 B _i,2 ( x)

c _i,2 − B _i+1,2 ( x) c _i+1,2

 − 1 c _i+1,3

 B _i+1,2 ( x)

c _i+1,2 − B _i+2,2 ( x) c _i+2,2

 . (20)

The function D 1 B _i satisfies the relation D 1 B _i ( x) = B _i,3 ( x)

c i,3 − B _i+1,3 ( x)

c i+1,3 , (21)

6 Recurrence formulae for discrete GB-splines C891 where

B _j,3 ( x) =

 

 

 

 

 

 

 

 

 



D j,1 Ψ _j ( x)

c _j,2 , x ∈ [x _j , x _j+1 ) ,

1 + D ^j+1,1 Φ _j+1 ( x)

c _j,2 − D ^j+1,1 Ψ _j+1 ( x)

c _j+1,2 , x ∈ [x _j+1 , x _j+2 ) ,

−D ^j+2,1 Φ _j+2 ( x)

c _j+1,2 , x ∈ [x _j+2 , x _j+3 ) ,

0 , otherwise .

(22)

Using formula (22) it is easy to show that functions B _j,3 , j = −2, . . . , N−1 satisfy the first and second smoothness conditions in (3), have supports of minimum length, are linearly independent and form a partition of unity,

N−1 X

j=1

B _j,3 ( x) ≡ 1, x ∈ [a, b].

Applying formulae (20) and (21) to the representation (14) we also obtain

D 1 S(x) = ^{N−1 X}

j=−2 b ⁽¹⁾ _j B _j,3 ( x), D 2 S(x) = ^{N−1 X}

j=−1 b ⁽²⁾ _j B _j,2 ( x), (23)

where b ^(k) _j , k = 1, 2 are defined in ( 16).

7 Series of discrete GB-splines (uniform case) C892

7 Series of discrete GB-splines (uniform case)

Let us suppose that each step size h _i = x _i+1 − x _i of the mesh ∆ : a = x ₀ <

x 1 < · · · < x N = b is an integer multiple of the same tabulation step, τ, of some detailed uniform refinement on [ a, b].

For θ ∈ IR, τ > 0 define

IR _θτ = {θ + iτ | i is an integer}

and let IR _θ0 = IR. For any a, b ∈ IR and τ > 0 let [ a, b] _τ = [ a, b] ∩ IR _aτ .

The functions B _j,2 , B _j,3 , and B _j with τ _j ^L

= τ _j ^R

= τ, j = i, i + 1 for all i are nonnegative on the discrete interval [ a, b] τ . This permits us to reprove the main results for discrete polynomial splines of [9] for series of discrete gener- alized splines. Even more, one can obtain the results of generalized splines of [5] from the corresponding statements for discrete generalized splines as a limiting case when τ → 0.

In particular, if in (14) and (23) we have the coefficients b ^(k) _j > 0, k = 0 , 1, 2, j = −3 + k, . . . , N − 1, then the spline S will be a positive, monoton- ically increasing and convex function on [ a, b] τ .

Let f be a function defined on the discrete set [a, b] τ . We say that f has

7 Series of discrete GB-splines (uniform case) C893 a zero at the point x ∈ [a, b] τ provided

f(x) = 0 or f(x − τ) · f(x) < 0.

When f vanishes at a set of consecutive points of [a, b] τ , say f is 0 at x, . . . , x + (r − 1)τ, but f(x − τ) · f(x + rτ) 6= 0, then we call the set X = {x, x + τ, . . . , x + (r − 1)τ} a multiple zero of f, and we define its multiplicity by

Z X ( f) =

 



 

r, if f(x − τ) · f(x + rτ) < 0 and r is odd, r, if f(x − τ) · f(x + rτ) > 0 and r is even, r + 1, otherwise.

This definition assures that f changes sign at a zero if and only if the zero is of odd multiplicity.

Let Z _[a,b]

( f) be the number of zeros of a function f on the discrete set [ a, b] τ , counted according to their multiplicity. Let us denote D ^L ₁ S(x) = S[x − τ, x].

Theorem 8 (Rolle’s Theorem For Discrete Generalized Splines.) For any S ∈ S ₄ ^DG ,

Z _[a,b]

( D ₁ ^L S) ≥ Z _[a,b]

( S) − 1. (24)

7 Series of discrete GB-splines (uniform case) C894 Proof: First, if S has a z-tuple zero on the set X = {x, . . . , x + (r − 1)τ}, it follows that D ^L ₁ S has a (z − 1)-tuple zero on the set X ⁰ = {x + τ, . . . , x + ( r − 1)τ}. Now if X ¹ and X ² are two consecutive zero sets of S, then it is trivially true that D ₁ ^L S must have a sign change at some point between X ¹ and X ² . Counting all of these zeros, we arrive at the assertion (24). This

completes the proof. ♠

Lemma 9 Let the function D i,2 Φ _i and D i,2 Ψ _i be strictly monotone on the interval [ x i , x i+1 ] for all i. Then for every S ∈ S ₄ ^DG which is not identically zero on any interval [ x _i , x _i+1 ] _τ , i = 0, . . . , N − 1,

Z _[a,b]

( S) ≤ N + 2.

Proof: According to (19) and (23), the function D 2 S has no more than one zero on [ x i , x i+1 ], because the functions D 2 Φ _i and D 2 Ψ _i are strictly monotone and nonnegative on this interval. Hence Z _[a,b]

( D ₂ S) ≤ N. Then according to the Rolle’s Theorem 8, we find Z _[a,b]

( S) ≤ N + 2. This completes the

proof. ♠

Denote by supp _τ B _i = {x ∈ IR a,τ |B i ( x) > 0} the discrete support of the spline B _i , i.e. the discrete set ( x _i + τ, x _i+4 − τ) _τ .

Theorem 10 Assume that ζ ₋₃ < ζ ₋₂ < · · · < ζ _N−1 are prescribed points on the discrete line IR _a,τ . Then

D = det(B i ( ζ j )) ≥ 0, i, j = −3, . . . , N − 1

7 Series of discrete GB-splines (uniform case) C895 and strict positivity holds if and only if

ζ i ∈ supp _τ B _i , i = −3, . . . , N − 1. (25)

The proof of this theorem is based on Lemma 9 and repeats that of Theorem 8.66 in [9, p.355]. The following statements follow immediately from Theorem 10.

Corollary 11 The system of discrete gb-splines {B _j }, j = −3, . . . , N − 1, associated with knots on IR _a,τ is a weak Chebyshev system according to the definition given in [9, p. 36], i.e. for any ζ −3 < ζ −2 < · · · < ζ N−1 in IR _a,τ we have D ≥ 0 and D > 0 if and only if condition ( 25) is satisfied. In the latter case the discrete generalized spline S(x) = ^{P N−1} _j=−3 b _j B _j ( x) has no more than N + 2 zeros.

Corollary 12 If the conditions of Theorem 5 are satisfied, then the solution of the interpolation problem

S(ζ i ) = f i , i = −3, . . . , N − 1, f i ∈ IR (26) exists and is unique.

Let A = {a ij }, i = 1, . . . , m, j = 1, . . . , n, be a rectangular m × n

matrix with m ≤ n. The matrix A is said to be totally nonnegative (totally

7 Series of discrete GB-splines (uniform case) C896 positive) (e.g., see [3]) if the minors of all order of the matrix are nonnegative (positive), i.e. for all 1 ≤ p ≤ m we have

det( a i

j

) ≥ 0 (> 0) for all 1 ≤ i 1 < · · · < i p ≤ m, 1 ≤ j ₁ < · · · < j _p ≤ n.

Corollary 13 For arbitrary integers −3 ≤ ν −3 < · · · < ν p−4 ≤ N − 1 and ζ ₋₃ < ζ ₋₂ < · · · < ζ _p−4 in IR _a,τ we have

D p = det {B ν

( ζ j ) } ≥ 0, i, j = −3, . . . , p − 4 and strict positivity holds if and only if

ζ i ∈ supp _τ B _ν

, i = −3, . . . , p − 4

i.e. the matrix {B j ( ζ i ) }, i, j = −3, . . . , N − 1 is totally nonnegative.

The last statement is proved by induction based on Theorem 5 and the recurrence relations for the minors of the matrix {B j ( ζ i ) }. The proof does not differ from that of Theorem 8.67 described by [9, p.356].

Since the supports of discrete gb-splines are finite, the matrix of sys-

tem (26) is banded and has seven nonzero diagonals in general. The matrix

is tridiagonal if ζ i = x i+2 , i = −3, . . . , N − 1.

7 Series of discrete GB-splines (uniform case) C897 An important particular case of the problem, in which S ⁰ ( x i ) = f _i ⁰ , i = 0 , N, can be obtained by passing to the limit as ζ −3 → ζ −2 , ζ N−1 → ζ N−2 .

De Boor and Pinkus [2] proved that linear systems with totally nonnega- tive matrices can be solved by Gaussian elimination without choosing a pivot element. Thus, the system (26) can be solved effectively by the conventional Gauss method.

Denote by S ⁻ (v) the number of sign changes (variations) in the sequence of components of the vector v = ( v ₁ , · · · , v _n ), with zeros being neglected.

Karlin [3] showed that if a matrix A is totally nonnegative then it decreases the variation, i.e.

S ⁻ ( Av) ≤ S ⁻ (v) .

By virtue of Corollary 4, the totally nonnegative matrix {B j ( ζ i ) }, i, j =

−3, . . . , N − 1, formed by discrete 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 IR, without taking into account the zeros

S ⁻ ( f) = sup

n S ⁻ [ f(ζ 1 ) , . . . , f(ζ n )] , ζ 1 < ζ 2 < · · · < ζ n .

Theorem 14 The discrete generalized spline S(x) = ^{P N−1} _j=−3 b j B _j ( x) is a

variation diminishing function, i.e. the number of sign changes of S does

References C898

not exceed that in the sequence of its coefficients:

S ⁻

 _N−1 X

j=−3 b j B _j



≤ S ⁻ (b) , b = (b −3 , . . . , b N−1 ) .

The proof of this statement does not differ from that of Theorem 8.68 for discrete polynomial b-splines in [9, p.356].

Acknowledgements: The author was supported by the Thailand Research Fund under grant number BRG/08/2543. He also wishes to express his grati- tude to the referees for their invaluable remarks and suggestions for improve- ments.

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