Interpolational and extremal properties of L-spline functions

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Interpolational and extremal properties of L-spline functions

Citation for published version (APA):

Morsche, ter, H. G. (1982). Interpolational and extremal properties of L-spline functions. Technische Hogeschool

Eindhoven. https://doi.org/10.6100/IR76507

DOI:

10.6100/IR76507

Document status and date:

Published: 01/01/1982

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SNOil~Nn:l

3NildS-

3

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!? -SPLINE FUNCTIONS

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL EINDHOVEN, OP GEZAG VAN DE

RECTOR MAGNIFICUS, PROF. IR. J. ERKELENS, VOOR

EEN COMMISSIE AANGEWEZEN DOOR HET COLLEGE VAN DEKANEN IN HET OPENBAAR TE VERDEDIGEN OP

VRIJDAG 16 APRIL 1982 TE 16.00 UUR

DOOR

HENRICUS GERHARDUS TER MORSCHE

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Dit proefschrift is goedgekeurd door de promotoren

Prof.Dr.Ir. F. Schurer en

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Judith en Robert-Jan Met dank aan mijn ouders

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ar

_{m n}(p l and ar+(p _{m n}J 112

5.5. The Landau problem for second order differentlal operators 113 5.6. The Landau problem for third order differential operators 126

6. ON THE LANDAU PROBLEM FOR PERIODIC FUNCTIONS

6,1. Introduetion and summary 132

-6.2. A few preliminary lemmas and the sets F{pn,T) and f(pn,T} 134 6.3. Some properties of the sets F(pn1T} and f(pn1T) 137 6.4. Perfect Euler l-splines as extrema! functions 144

- 3

6.5. A parametrization of the set êlf(D ,1) 148

7. PERFECT l-SPJ,INES AND THE LANDAU PROBLEM ON THE HALF LINE

7.1. Introduetion and summary

7.2. Perfect l-splines and ~-approximate perfect l-splines 7.3. A representation theerem for the set P (p )

r,.!!_ n

7.4. An extrema! property of perfect l-splines 7.5. A characterization of ar+(p )

m n

REFERENCES LIST OF SYMBOLS SUBJECT INDEX AUTHOR INDEX SAMENVATTING CURRICULUM VITAE 151 153 161 167 178 184 192 194 198 200 204

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I

GENERAL INTROVUCTION

No doubt, the theory and application of spline functions (splines for short) has been a flourishing branch of approximation theory during the last few decades. Although Bernoulli and Euler already used very simple splines, i.e. polygons, for the approximate salution of differential equa-tions, it is in general agreed upon that a systematic investigation of splines began with the work done by I.J. Schoenberg during the secend World War. It took some time till it was widely recognized that splines have interesting extremal properties, and are a good tool for the numerical approximation of functions as well. Since 1960 a large number of papers have been published. The first bock on the subject by Ahlberg, Nilson and Walsh [1] dates from 1967 and in recent years a few more have appeared

(cf., for instance, Schumaker [56], which also contains an extensive bibliography) •

In their original farm splines are piecewise polynomials, in general of rather low degree, tied tagether with a certain degree of smoothness at the so-called knots. A natural generalization is obtained if the polynomials are replaced by functions in the kernel of a linear differential operator pn(D) of the ferm

where D is the ordinary differentiation operator and the coefficients a₁ (i O,l, •.• ,n-1) are real. The associated splines are then called ! -aplinea; the polynomial splines are obtained if p (D) = Dn.

n

There is an extensive literature on the problem of interpolation by means of polynomial splines. If the knots of the interpolating !-spline and the points of interpolation, the so-called nodes, are equally spaeed on ~' then, following Schoenbarg's terrninology, one speaks of cardinal t-apline interpolation. The interpolation problem may then be formulated as fellows. Let f be defined on ~, let h be the distance between consecutive knots

(the mesh distance) and let a E (O,h] be prescribed, then one is asked to determine an t-spline, corresponding to pn(D) and withknotsat O,±h,±2h, ••• ,

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that interpolates fat the nodes a,a±h,a±2h, •••• Questions concerning existence and uniqueness of such an .C-spline interpolant, and the problem of determining (best possible) error estimates naturally present itself. Cardinal polynomial spline interpolation has been investigated in detail by Schoenberg [51] in the cases a = h/2 and a = h. Results for arbitrary values of a € (O,h] in the case of periadie 'cardinal polynomial spline interpolation are contained in Ter Morsche [40]. Assuming that the charac-teristic polynomial pn of pn(D) has only real zeros, Michelli [37] has generalized Schoenherg's results for cardinal polynomial spline interpola-tion to cardinal .C-spline interpolainterpola-tion.

In the first part of this thesis we study (periodic) cardinal .C-spline interpolation without the assumption that pn has only real zeros. When extending the theory to arbitrary operators pn(D) one encounters the concept of

disconjugaay.

A differentlal operator pn(D) is said to be dis-conjugate on an interval (a,b) if pn(D) can be factorized in a sequence of first order differential operators o

1,o2, ••• ,on of the form Di= winw;

1 ,

where wi is a positive function defined on (a,b)1 apparently, this is true on any interval (a,b) if pn has only real zeros. In general, it will be assumed that pn(D) is disconjugate on an appropriate interval, as in the absence of this proparty some problems are substantially more difficult. Chapters 3 and 4 deal with existence and uniqueness of an .C-spline inter-polant, and error estimates are derived that are best possible. The esti-matas obtained in general are of the form

(x € JR) '

where K :?: 1 is an appropriate constant, 11•11 is the supramum norm on JR, sf is the .C-spline interpolant to f and f

0 is a so-called

perfect Euler

.e-epZine,

Perfect .C-splines are characterized by the fact that their "pn

(D)-derivative" has constant absolute value c and jumps from ± c to

+

c at the

knots. Perfect Euler 1:. -splines f

0 addi tionally satisfy the functional relation f

0 (x+ h)

= -

f0 (x) (x e: JR) and are such that their "pn (pl -derivative" is ± 1. We emphasize that our analysis of the (periodic) cardinal .C-spline interpolation problem makes essential use of specific relations between derivatives and finite differences of cardinal 1:.-splines. As a simple illustration we mention the relation

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III

which holds for polynomial cubic splines, xi being the equally spaeed knots and h being the mesh distance. The relations we derive in Chapter 3 are far-reaching generalizations of (1).

The secend part of this thesis deals with extremal properties of perfect !-splines in conneetion with so-called Landau probZems. This name has its crigin in a few interesting inequalities that Landau [30] derived in 1913

for twice differentiable functions. If f and f" are bounded on lR (on lR; '"' [0,"')) and i f we write llfll+ : llflllR+, then

0

llf'll ~

12

lilrllilf"il , ~ 2lllfll llf''ll ,

+ +

where the constants

12

and 2 are best possible. A generalization of the first inequality to higher derivatives is due to Kolmogorov [27]; the

+

corresponding problem on JR

0 has been solved by Schoenberg and Cavaretta [54].

A still further generalization reads as fellows. Let pn be a monic poly-nomial of degree n, let J c lR be a closed interval, and let m be a positive number. Further, let Fm(pn,J) denote the set of functions f with f(n-l) absolutely continuous on every compact subinterval of J, 11 fll J ~ m and lip (D) fll., ~ 1. The generalized Landau problem then amounts to determining

n v .

the best possible upper bound for llpk (D) fiiJ on Fm (pn ,J) , where pk (D) is a given linear differential operator of order k ~ n-1. The cases J =lR (full-line case) and J =lR~ (half-line case) are of particular interest. In what

+

fellows we assume that J =lR or J JR

0•

Our analysis of the generalized Landau problem is based on an investigation of the set

It is shown that rm(pn,J,O} is a compact, convex, subset of lRn having 0 as an interior point. Solving a Landau problem is then equivalent to maximizing a linear function on rm(p

0,J,O). By the support hyperplane theerem for convex sets, every boundary point of rm(pn,J,O) has the property that an appropriate linear function attains its maximum on rm(pn,J,O) at that point. Consequently, every f E F (p ,J) with the

- - ·(n-1)

T

m n

property that (f(O) ,f' (O), ••. ,f (0)) E 3f (p ,J,O), the boundary of

m n .

rm(pn,J,O), is an extremaZ jUnction fora Landau problem, i.e., forsome pk the function f maximizes llpk (D) f liJ on Fm (p

0 ,J) • It is therefore of

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general second order and for some specific third order differential opera-tors in the cases J =~ and J =~~. There is a separate chapter on Landau problems for periodic functions, where, among ether things, a generalization of a result of Northcott [46] is given. Landau problems on the half line are discussed in the last chapter, under the additional hypotheses that pn has only real zeros and that p (0)

=

0. The problem we pose is to minimize

n +

liP (D) fll+ with respect to all functions in F (p ,1R

0l satisfying the initial

n (i) m n

conditions f (0) =ai (i= O,l, ••• ,n-1) for prescribed ai. It is shown, by means of a so-called representation theorem, that a perfect l-spline with a specific oscillation property furnishes the solution.

Landau problems are of importance with respect to the optima! recovery of the derivatives of smooth functions (cf. Michelli and Rivlin [39,_p. 27]). The foregoing is intended to introduce the main themes of the thesis and to give a rough sketch of the problems that are dealt with. We re~rain from giving here a detailed summary of the contents of the seven chapters. Instead of this we refer to the first sections of the various chapters which are introductory and summarize the main results obtained. A few

remarks are in order with respect to Chapters 1 and 2. We have strived to make this thesis reasonably self-contained. To achieve this, preliminary material of a general kind is brought together in Chapter 11 most of it is standard in approximation theory, apart from a generalization of ours of the Budan-Fourier theerem to piecewise continuous functions. This useful theorem is, among ether things, applied to give a rather simple proof of the so-called

total positivity property

of a sequence of consecutive B-splines. Chapter 2 introduces the nonpolynomial B-splines and contains various properties, notably recurrence relations, of these very useful functions. One of these recurrence relations is used to obtain the knot distributions for which the supremum norm of a polynomial B-spline is minimal.

Examples are given in each chapter to illustrate and clarify our assertions. In order to facilitate the readability of this thesis, at the :end a list of symbols, an author index and a subject index are added.

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1. BASIC CONCEPTS AND PRELIMINARY RESULTS

1. 1.

IntJtoduc.tion and

J..wnmaJty

The purpose of this chapter is to collect concepts and preliminary results that will be needed in this thesis. In Sectien 1.2 some general notations are listed and a number of classes of functions are introduced. Sectien 1.3 gives the definition of an

of some particular

function, tagether with the definitions such as cardinal !-splines and perfect L-splines. Sectien 1.4 contains a variety of concepts in the theory of L-spline functions, e.g. the concept of disconjugacy of a differential operator, Taylor's formula and Peano's remainder formula, the classical Budan- Fourier theerem and a generalized version of it, various forms of a Chebyshev system, the notions of solvent and unisolvent families, and di-vided differences. Most of them are standard in approximation theory, apart from the generalized Budan- Fourier theorem. In order to obtain this gener-alization a specific devi'ce for counting zeros of piecewise continuous func-tions is used. The generalized Budan-Fourier theerem is applied to !-splines in Subsectien 1.4.6 in order to catint its zeros.

7.2. NotationJ.. and c.onventionJ..

1.2.1. Formula indication

Formulas are numbered independently from theorems, lemmas '· corollaries and definitions. When referring, for example, to the third formula of Sectien 2 in Chapter 1, we shall write (1.2.3).

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1.2.2. Notatiens

Here only those notations are explained that are used throughout this thesis. A more extensive list of symbols is given on pp. 192,193.

JN lR [a,b] (a,b) T x 0 V

av

öi,j f{x+) f(x-) [x] sgn

the set of posi ti ve integers, JN = { 1, 2, 3, .•• } •

thesetof nonnegative integers, JN

0 = {0,1,2, ••• } the set of integers.

the set of real numbers. the set of complex numbers.

the closed interval {x E lR

I

a s x s b}

the open interval {x E

EI

a< x< b}; similarly (a,b], [a,b).

the set [O,oo).

the Euclidean space of dimension n consisting of column vectors. the transpose of a vector x.

the interior of a set

v.

the boundary of a set V. the empty set,

the Kronecker symbol, i.e., öi .

.~ lim f{x +hl. h+O lim f(x-hl. hi-0 1 and ö. . "' 0 if i

>!

j. ~.J

the largest integer not exceeding x. the sign function, i.e.,

sgn(x)

[_:

(X > 0) 1

(x 0)

(x < 0)

:= is used in a definition if a new symbol occurs on the left-hand side.

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1.2.3. Functions

As we mainly deal with real-valued functions, the range of a function is assumed to be E, unless otherwise stated.

If f(x) is the value of a function f at x, then the function is denoted by f or by XI--? f(x) or by f(•).

If f is a function of several variables, say two, then the function of one variable obtained from f by fixing one of the two variables, say x, is denoted by YHf(x,y) or by f(x,•).

We praeeed by defining some classes of functions. In these definitions J

stands for an interval and n E ~. unless otherwise stated.

C(J)

c(n) (J)

PC(J)

AC(n) (J)

the set of continuous functions defined on J.

the set of functions f defined on J having continuous n-th derivatives, i.e., f(n) c C(J) where n c

~

0

•

the set of functions f defined on J that are continuous on J

with the possible exception of finitely many points in any bounded subinterval of J, and such that at every point x of discontinuity f(x+) and f(x-) both exist with f(x) f(x+). the set of functions f E c<n-l) (J) for which the (n-1)-st derivatives f(n-1) are absolutely continuous on every compact subinterval of J.

3

PC(n) (J) _{thesetof functions}_f _E_{AC(n) (J) for which the (n-1)-st}

deriva-L"'(J)

w<nl (J)

t . _Lvesf(n-l) _{are Lil egra s o}· t 1 f f _{unctLons Lil}· · _PC( ) _{J , L.e., or}· f

f E Pc(n) (J) there · t f t' PC(J) h th t

every eXLS s a unc LOD g E suc a

t

f(n-1) (t) _f<n-1₎

_tt

0J +

J

g(T)dT ( t E J t t0 E J) •

to

The n-th derivative is defined on J with the possible exception of finitely many points in any bounded subinterval of J.

At any point t E J we define f(n) (t) : g(t).

the set of measurable functions f which are essentially bounded on J, i.e., for every functlon f E L."(J) there exists a number

M > 0 such that

I

f(t)

I

< M (a.e.) .

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1.2.4. The supremum norm of a function If a function f is essentially

I

fl on J is denoted by .llf!IJ. + If J = JR or J

=

JR 0 then the

bounded on J, then the essential supremum of

notatien for the supremum norm is shortened

by writing llfll := llfiiJR, llfll+ :=

1.2.5. Interpolation of Hermite data Let!= {t

0,t1, ••• ,tm--1lT € JRm with t0

~

t1

~

•••

~

tm-l' If f i s a func-tion that is sufficiently often differentiable, then f{t) = f(t

0,t1, ••• ,t 1>

T

- -

-

m-denotes a vector.;:.= (a

0,et1, ••• ,etm_1l € JRm defined as fellows.: if t.

1 < t. t. 1 ••• = t.+k 1 < tj k' then et . . := f(i) (t.)

J- J J+ J - + J+~ J

(i O,l, ••• ,k-1), in particular, if the tj are distinct then aj := f(tj)

(j 0,1, ••• ,n-1). The sequence _{,tj+l'''''tj+k- 1 with} tj-1 < tj

=

tj+1

=

of length k.

tj+k- 1 < tj+k will be called a

aoinaident bloak

We say that a function f interpolates the data (y

0,y1, ••• ,ym-l) at the points t₀,t₁, ••• ,tm-l when !<t₀,t₁, ••• ,tm_₁l = (y₀,y₁, ••• ,ym-l)r. This kind of interpolation is called He~ite

interpolation.

1.2.6. Determinants

T

m

Let!= (t₀_{,t1, ... ,tm-l)} EJR with t₀s t₁s ... s tm-land let

~

₀

,~₁, ••• ,~m-

₁

be m functions that are sufficiently aften differentiable, then

: = det

<.Po I • • • I

_{.Pm-- 1) ,}

where det {.Po

I • • • I

.Pm_₁) is the determinant of the m x m matrix, the j-th column of which is given by ~j(t

₀

,t

₁

, ••• ,tm_

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Usually, spline functions (throughout the thesis the term "spline" will be used as a synonym for "spline function") are defined as functions with a certain degree of smoothness consisting of piecewise polynomials tied tagether at the so-called knots. More generally, the polynomials are re-placed by functions satisfying a given linear homogeneaus differentlal

equation! f 0 (cf. Michelli [37]) • This leads to the de fini tion of an

! -spline.

The space of all polynomials with real coefficients and of degree at most n E ~ is denoted by "n· A pelynemial pn in "n is called manie if pn has degree n and if its leading coefficient is equal te ene.

Let pn E: Tin be monic, i.e., pn {z) z n + a z n-1 +

...

+ ao, then the linear differentlal operator f 1-4 f (n) _{+ an-lf}n-1 (n-1) +

...

+ is denoted by Dn + a Dn-1 +

...

₊_a

0r or by p (D) for short, with Df := f' and

Dkf n-1

n Df(k-1) (k ;:> 2).

The kernel of the operator pn(D), denoted by Ker(pn) or Ker(pn(D)), is defined as fellows.

DEFINITION 1.3.1.

:= {f E AC(n) (lR) I p (D)f(t)

n 0 (a.e.)} •

DEFINITION 1.3.2. Lets bedefinedon [a,b]. We say that sis an l-spZine

funation of order n, if there exists a manie poZynomiaZ pn E "n and a

finite sequenee of points x₁,x₂, ••• ,~ with a= x

0 < x1 $ x2 •.. $ xk <

< ~+

1

band x₁ < xi+n (i= l, ••• ,k-n), sueh that

i) _{on every nonempty interval Cx1},xi+l) (i

coincides with an element of Ker(p

0),

O,l, ... ,k) the function s

5

ii) if ₁< x₁ = xi+l = . . . ₁< xi+j' i.e., if x₁has muZtipZicity

j, then s has a continuous (n-1-j)-th derivative in a neighbourhood of j

If pn (z) zn then the !-spline s consists of piecewise polynomials and i t

will be called a poZynomiaZ spline of order n (or of degree n-1).

The points x

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If a knot has multiplicity one, then it is called a simple knot. Occasional-ly, the distinct points among the knots x

1,x2, .•• ,~ are denoted by

y_{1,y2, ••• ,yk and their multiplicities by v1,v2, ••• ,vk, respectively.}

We also deal with .C-splines defined on

E

or on

E;.

In these cases the number of knots is allowed to be infinite;, however, a finite accumulation of knots is not permitted.

The following definition concerns an !-spline function that is fundamental in the theory of splines.

DEFINITION 1.3.3. Let~ E Ker(pn) be the jUnation satisfYing

, (n-2)

~(0)

=

~ (0)

= •••

=

~ (0)

=

0 1 ~ (n-1) (O) 1 •

Then the funation ~+ E PC(n-l) (lR) is defined by

( (!1 (t)

~+(t) := 0

(t ~ 0) ,

(t < 0)

We call q~ the fundamentaL funation corresponding to the operator pn (D) and (!1+ the Green's funation corresponding to pn(D). The dependenee of the func-tions ~ and ~+ on pn is not expressed in the notation.

The function ~+ can be used to represent .C-splines. For instance, if s is

an .C-spline with knots located at y₁,y₂, ••• ,yk with y₁< y₂< ••• < yk and with multiplicities v

1,v2, ••• ,vk, respectively, then scan be written as

(1.3.1)

k V·-1

s

ct>

= g

ct>

+

I

~

w. j

<

j >

ct - y

i> ,

1=1 j=O J.,

where g E Ker(p ) and the real numbers w. j are uniquely determined (cf.

n J.,

Karlin [21, p. 517]).

For numerical purposes the representation by means of ~+ can be bad (cf. De Boor [5, p. 104]). The so-called B-spline functions constitute a basis for .C-splines that is appropriate for both practical and theoretica! pur:-poses. Chapter 2 is devoted to these functions.

We shall give special attention to .C-splines with equally spaeed simple knots. With respect to this class the following definition is in order.

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7

DEFINITION 1.3.4. Anf.-spline function sconesponding to the opemtor pn(D) is called a cardinal .C -spline function of order n E .N i f there exists a number h > 0 such that

i) pn(D)s(t) =0 (ih< t< (i+1)h; iE:iZ),

ii) sE c<n-2) (JR) if n;:: 2 and sE PC(JR) i f n 1 .

The number h in the de fini ti on of a cardinal .,C -spline is called the mesh distance of the knot distribution.

Thesetof all cardinal .C-splines corresponding to the operator pn(D) and with mesh distance h is, in genera!, denoted by

( 1. 3. 2)

occasionally, the notatien S(pn(D) ,hl will also be used.

Usually, the term cardinal is reserved for l-splines with knots at the integers, but by a simple transformation of scale t

=

hT the knots are transformed to the integers, while the differential operator pn(D) is replaced by pn(hD).

The cardinal pclynomial splines are studied in detail by Schoenberg [51] in his monograph. Parts of the theory are extended by the author of this thesis in[40] and, independently, by Michelli [37].

Finally, the so-called perfect l-splines will be defined.Thesesplines often occur as "extremal functions" with respect to certain extremal problems in the space W(n) (J) in case the supremum norm is used (cf. Chapters 5, 6, 7).

DEFINITION 1.3.5. A function s, defined on the interval J, is called a perfect l-spline fUnction corresponding to the operator pn(D) i f sis an l-spline of order n + 1 corresponding to the operator Dpn (D) with the

property that a number c exists such that for all t

E

J, knots being ex-cluded,

IP<D)s(tll _n c .

It follows from this definition that the "pn (D) -derivative" of s has con-stant absolute value c and jumps from ± c to

+

c at the knots. Note that a perfect .C-spline correspcnding to the operator pn (D) i·s an l-spline of order n + 1 corresponding to the operator Dpn {D) •

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1. 4. Same bM-<..c. c.oi'U!ep.U bt .the .theoi!.!J oó

.c

-.&p.Une 6unc..:üo~

In order to make this thesis reasonably self-contained, in this section we will collect some results that will be frequently used. In general, proofs are omitted in case a reference is available or if it concerns! proofs that are standard in the theory of .C-spline functions.

1.4.1. Disconjugacy of a differential operator

DEFINITION 1.4.1.

The differentiaZ operator

pn{D)

is said to be disaonjugate

on an intervaZ

J

if there exist

n

striatZy positive jUnations

wiE c<n+1-i) (J) (i 1,2, ••• ,n)

suah that

pn(D)

aan be faatorized as

(1.4.1)

where the differentiaZ operator

Di

is defined by

(1.4.2) {i 1, 2, ... ,n)

The interval J is called an interval of disconjugacy. The following characterization was proved by Pólya in 1922.

LEMMA 1.4.2 (Pólya[48]).

The differentiaZ operator

pn(D)

is disconjugate on

an open intervaZ

{a,b)

if and onZy if no nontriviaZ funetion

f E Ker(pn)

has more than

n-1

zeros in

(a,b),

aounting muZtipZiaities.

If pn has only real zeros, i.e., if pn (D) == (D-À

1I) ••• (D- Àni} with

Ài E ~. then one has

so p (D) is disconjugate on the whole real line. If p has at ;least one

n n

nonreal zero, then an interval of disconjugacy is finite. As an example let us consider the operator p

2 (D) = D

2 _{+I, One easily verifies that for}

all a E :Rand f E AC(2) (lR)

1 2 f(t)

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Hence intervals of the form (a,a+n) are intervals of disconjugacy for p2 (D) •

Since pn(D) has constant real coefficients, the maximal length ~ of an interval of disconjugacy does not depend on the location of that interval on the real line. It turns out (cf. Troch [62, Theorem 1]) that the maximum intervals on which each nontrivial function in Ker(pn) has at most n- 1 zeros counting multiplicities, are half open intervals [a,a+~) or (a,a+t]. For second order differential operators the maximal length ~ is simply the distance between two consecutive zeros of the corresponding fundamental function (!). For third order differential operators the function (j) also

determines the number ~. This is a consequence of Lemma 2 in Troch [62] 1

which for third order differential operatorscan be statedas follows.

9

LEMMA 1.4.3 (Troch[62]).

Let

p

3 E n3

be a monic poZynomial having a nonreal

zero, and let

x

0 E R be

a nearest point to zero for which

(!) (x0)

x

0 ~

o.

Then

~

=

lx

0

1.

0

and

If the order of the differential operator exceeds three1 the zeros of its fundamental function !Jl do not in general determine the number t as in the case of second and third order differential operators. This is shown by the following example.

4

EXAMPLE. I f p

4 (D)

=

D -I, then !Jl (t)

=

~ (sinh t - sin t) and thus (j) (t) ~ 0 when t f. 0. As t~sin tE Ker(p

4), by Lemma 1.4.2 the number ~ for p4(D) is finite.

It is well known that a trigonometrie polynomial of degree n, i.e., a func-tion belonging to _{Ker(p 2n+l} with}

( 1. 4.3) _{P2n+l (D)} D(D +I) (D +4I) ••• (D +n I) 2 2 2 2 1

cannot have more than 2n zeros in an interval [a,b) with b-a~· 2n1 unless

it is identically zero. As shown by the function t~ sin(nt) 1 the number t

for the operator (1.4.3) is equal to 2n.

For more detailed information concerning the property of disconjugacy the reader is referred to the hook of Coppel [14].

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1.4.2. Taylor's formula and Peano's remainder formula for pn(D) For the differential operator pn(D) Taylor's formula can be statedas fellows.

LEMMA 1.4.4. Let f E AC(n) ([a,b]) and tet.g E Ker{pn) be the unique funation

• .., • (i) { ) (i) { ) (. 0 1 1) Th

sat~sJy~ng g a

=

f a ~

=

1 1••·,n- • en

'{1.4.4) f(t) g(t) +

r

qJ+(t-T)pn(D)f(T)dT a

(a ~ t ~ b) ,

UJhere the funation rp+ is given by Defin:ition 1.3.3.

We preeeed by giving Peano's remainder formula for pn(D).

LEMMA 1.4.5. Let À be a linea!' funationa'l ckfined on AC(n} ([a,b]) of the

fo!'m À(f) (f E AC(n) ([a,b])) 1 UJhere m E :N• n € :NJ aR. 1 j E JR. and _{a s x 1}< x2 < • • • < x s b. m If À(g) 0 for all g E Ker(p

>.

then one has

n b

{1.4.5) À {f)

I

K(T)pn(D)f(T)dT 1

a

UJhere the kemel K is ckfined by K(T) := À ( t -<p + (t- T)).

A proef of Lemma 1.4.5 if pn(D) on is contained in Davis [ 161 p. 70 J •

In the case of a general differential operator pn(D) a proof of Peano's remainder formula can be given along similar lines using Taylor's formula of Lemma 1.4.4. We note that Lemma 1.4.5 does not state Peano's remainder formula in its full generality.

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1.4.3. The theerem of Budan-Fourier

The classical form of the Budan-Fourier theerem gives an upper bound for the number of zeros in an open interval (a,b) of a polynomial of degree n

{i) n

by means of the number of sign changes in the sequences (p (a))i=O and (p (i) (b))

~:O

(cf. Karlin [ 21, p. 317]) • It reads as follows:

THEOREM 1.4.6 (Budan-Fourier). Let p E 'lfn be a poZynomiaZ of exaat

n EN. The total number of zeros of pin (a,b), aounting muZtipZiaities, is bounded by

(n) + (n)

(p(a) , ... ,p (a)) - S (p(b) , ... ,p (b)) , where and s+ are defined by Definition 1.4. 7.

11

An extension to polynomial splines with simple knots is given by De Boor and Schoenberg [6, p. 6] and to polynornial splines with knots of a~bitrary

rnultiplicity by Melkman [36].

The version of the Budan-Fourier theerem we present here is a generaliza-tion of the classical theerem to the set of functions PC(n) ([a,b]). Before we can forrnulate our result we have to agree on the way of counting zeros. This gives rise to the following definitions.

DEFINITION 1.4.7. Let a

1,a2, ••• ,an be a sequenae of reaZ numbers. Then

denotes the number of sign changes in a

1,a2, ••• ,an by deZeting zero entries. The maximum number of sign changes that aan be obtained by re-pZacing the zero entries in a₁,a₂, ••. , by +1 ar -1 is denoted by

DEFINITION 1.4.8. Let f be a function defined on (a,b). Then

We shall call S-{a

1,a2, ••• ,an) the nurnber of strong sign changes and S+(a

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One has the following equality (cf. De Boor and Schoenberg [6, p. 3]):

(1.4.6)

The next step is to define the way of counting zeros in an open interval (a,b) of functions in PC(n) ([a,b]) (n = 0,1, ••• ). This will be done as follows. First the procedure is described for the class PC([a,b]); one then extends it to the class PC(n) ([a,b]).

As zeros in (a,b) of a function f € PC([a,b]) are considered: points x

0 € (a,b) where f(x0-)f{x0l s 0 and intervals [c,d] c {a,b) such that f i s identically zero in (c,d).

The function f is said to "vanish" at a point x

0 if x0 is a zero of f as defined above; so f may vanish at x

0 even when f(x0) ~ 0. The function f is said to "vanish identically" in an interval J if f vanishes at every point of J.

We shall only take into account those zeros a (a point x

0 or an interval [c,d] c (a,b)) having the property that f does not vanish identically in any interval containing {x

0} {or [c,d]) as a proper subset. These zeros will be called

strong zeros.

DEFINITION 1.4.9.

Let

f € PC([a,b])

and Zet a he astrong zero of fin

{a,b).

Then the multipliaity

M

0(f,a}

of a is determined as follows.

If

S-{f,U) = oo

in eaah neighbourhood

u

c (a,b)

of a, then

M

0(f,a) := ~.

If

s-

(f,U}

=

1

for aU sufficiently small neighbourhoods

u

c (a,bl

of

a~

then

M

0(f,a) := 1.

If

s-{f,U) = 0

for all suffiaiently smaU neighbourhoods

u c (a,bl

of a,

then

M

0(f,a) := 0.

Note that in this definition oo, 1 and 0 are the only possible values of S-(f,U). Clearly, fora sufficiently small neighbourhood U of a streng

zero a we have either

s-

(f,U) = 0 or

s-

(f,U)

=

1 or

s-

(f,U)

= "' •

According to the counting rule as given above and taking into account Definition 1.4.8, the total number of strengzerosin {a,b), counting multiplicities, of a function f € PC([a,b]) is equal to S-(f,(a,b)).

The following example illustrates the way of counting zeros for functions f € PC{[a,b]).

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13

~"---

_.

L_~

0 1 2--~ 5 6....---7 ! . , _ . . / 9

Figure 1.

The streng zeros are 2, 3, [4,5], 6, [7,8] with multiplicities 0, 1, 1, 1, 0,

respectively. Hence

s

(f,(0,9)) ~ 3.

Our next purpose is to define the multiplicity of a strong zero of a func-tion f E PC(n) ([a,b]) (n

~

1) with respect to a sequence of differential operators

o

₁

,n

₂, ••• ,Dn given by

(1.4. 7) D. ~ w. D -1

l. l. w. l.

(i 1, ..• , n) ,

where w. E C(n+1-i) ([a,b]) is a strictly positive function.

l.

[iJ

We define the operators D as fellows:

(1.4.8) D[O] := I , D[i]

:~Di

D ••• D

i-1 1 (i 1,2, ... , n) •

(n) ]

DEFINITION 1.4.10. Let f E PC ([a,b l (n ~ 1} and let a be astrong zero

of fin (a,b). The multipliaity of a with respect to the sequence of

dif-n

ferentiaZ operators n₁,n₂, •.• ,Dn, denoted by M((Dil₁,f,a), is defined as foZlCJWs. If D[i]f(a) 0 for i n M( (Di) ₁,f,a) := k o[i]f(a)

=

0 for i n M((Di)₁,f,a) := n.

0,1, ••• ,k-1 and D[k]f(a) # 0 with k

~

n-1, then

[n]

O,l, ••• ,n-1 and D f does not vanish at a, then

If D[i]f vanishes at a for i

REMARK 1. If a is astrong zero of f E PC(n) ([a,b]) with an odd multiplicity M((D₁l7,f,a) then f changes sign at a, and if M((Di)~,f,a) is even then f

does not change sign at a.

REMARK 2. If a is a streng zero of f E PC(n) ([a,b]) with

M((D

1 )~,f,a)

=

k ~ n-1, then, of course, a reduces to a single point x

0 and

1 (k-1} (k)

f(x

0J = f (x0) = ••• = f (x0) 0, f (x0) # 0. Moreover, x0 then is an isolated zero, i.e., there exists a neighbourhood of x₀such that f

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vanishes in U only at the point x 0• on the other hand, if x

0 e (a,b) satisfies f(x0)

=

f'(x0J

= .•.

=

f(k-1) (x

0_nJ

=

0 and f(k) _. (x0) f 0 with k ~ n-1, then it is easy to verify that M((Di)

1,f,a)

=

k for any sequence of differentlal operators D₁,D

2, ••• ,D0 given by (1.4.7).

REMARK 3. We show that there may exist two different sequences of

differ-~ n n

ential operators (Di)

1 and (Di) 1 such that for some function

f e PC(n) ([a,b]) and some x

0 e (a,b)

M((Di)~

1 f,x

0 J

E

{n,~+1},

whereas

n

M((Di)

1,f,x0)

= ""•

To this end we take n

=

1, D1

=

D and D1

=

w1 Dw 1 with w 1 given by

J

1 +

( I

x

I

~ 1, x f 0) I w 1 (x) :=

l1

(x

=

0) •

I f f E PC(l) ([-1,1]) satisfies f(O) = 0, then

and thus f<x>

=

w₁<x>

r

w;1_<t>n1f<tldt 0

-

Jx

-1 -f'(x) = D 1f(x} + wÎ(x) w1 (t)D1f(T)dT 0 2

Substituting a function f for which D₁f(x) =x cos (1/x), we may easily verify that M((D₁J,f,O)

=

2 and M((D) ,f,O) = ~.

REMARK 4. If a is a strong zero of f € PC(n) ([a,b]) (n

~

1) with

n { } - [n] - [n]

M((D

1J1,f,a) E n,n+l, then S (D f,U) = 0 or S (D f,U) = 1 in a neigh-bourhood U of a. From this it follows that a is an isolated zero.

The total number of strong zerosin (a,b) of f E PC(n) ([a,b]) (n E

~

0 ),

counting multiplicities according to Definitions 1.4.9 and 1.4,10, is denoted by

(1.4.9)

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15

Let f E PC(n) ([a,b]). Taking into account (1.4.8) one has

[ '] (n- . )

D J f E PC J ([a,b]), and the total number of streng zeros in (a,b) of D[j]f, counting multiplicities in accordance with Definitions 1.4.9 and 1.4.10, is denoted by

(1.4.10) (j 0, 1 , ••. , n) •

Note that if j n then ( 1. 4 .10) may be rep la eed by S - (D [nJ f, (a ,b)) . The following lemma may be considered as a generalization of Rolle's theorem.

LEMMA 1.4.11. Let f E PC(n) ([a,b]) (n ;" 1). Then

(1.4.11) (n EN) •

PROOF. Let a₁and a₂be two distinct streng zeros of f. Assume that a₁and

a₂are points x₁and x₂with x₁< x

2• Since 0 x2 f(x 2l

=

w1(x2J

J

---1 --- D[l]f(T}dT W l (T) 0 ,

and f does not vanish identically in (x₁,x₂), it follows that

- [1]

s

(D f,(x₁

,x

2JJ ;" 1 and therefore

If a

1 is a streng zero of multiplicity k ;" 2 of f, then it fellows immedi-ately from Definition 1.4.10 that a

1 is a zero of multiplicity k- 1 of

D[l]f. This proves our lemma, since the case that at least one of the strong zeros is an interval can be treated in a similar way.

COROLLARY 1.4.12. Let f E PC(n) ([a,b]) (n

~

1), then

(1.4.12) Z((D n - [n]

1J 1,f, (a,bll

ss

(D f, (a,b)) +n PROOF. obviously, the inequality

(1.4.13) Z((D.). n [ j ]

1,0 f,(a,b)) ~ J+

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is a direct consequence of Lemma 1,4.11. Repeated application of (1.4.13) yields (1.4.12).

In the proef of our version of the Budan-Fourier theerem the following lemma is needed.

LEMMA 1.4,13.

Let

f E PC(n) ([a,b]),

If

D[n]f(a)

~

0,

then

(1.4.14) S (f(a),D - [1] f(a), ... ,D [n] f(a)) =

+ [1] [n] lim S (f(x),D f{x), ••• ,D f{x)) x.j.a

If

o[n]f{b-)

~·

0

then

(1.4.15} S (f(b) ,D + [1] f(b), ... ,D [n] f(b-}) + [ 1] [n]

=

lim S {f(x),D f(x), ••• ,D f(x-)) xtb

PROOF. Since D[n]f(a)

~

0 there exists a number E

1 > 0 such that

[i] - [n]

0

D f(x) ~ 0 (i= O,l, ••• ,n; a< x< a+E

1). Hences (f(x), ••• ,o f(x))

=

S+(f(x), •.• ,D[n]f(x)). If Ó[j+l]f(a}

=

D[j+2]f(a)

= ••• =

D[j+r-l]f(a)

=0,

while D[j]f(a)D[j+r]f(a) I O, then sgn(D[j+k]f(x))

=

sgn{D[j+r]f(a))

[ '] [ ·]

(k = l, ••• ,r) and sgn(D J f(x))

=

sgn(D J f(a)), As a consequence we have for x E (a,a+E

1)

(1.4.16) s -(D [j] f (a}, ••• ,D [j+r] f (a)) = S (D + [ j ] f (x) , .•• ,D [j+r] f (x)) .

[n]

Application of (1.4.16) to consecutive segmentsof (f(a), ••.

,o

f(a)) yields (1.4.14),

If D[n]f(b-)

~

0 then there exists a number E

2 > 0 such that D[i]f(x)

~

0

. [j+l] (L

=

0,1, ••• ,n; b-E 2 <x< b). Furthermore, if D f(b) = ••• = D[j+r-l]f(b)

=

0 and D[j]f(b)D[j+r]f(b) I 0, then sgn(D[j+k]f(x))

=

sgn((-1)r+k D[j+r]f(b-)) and sgn(D[j]f(x)) = sgn(D[j]f(b)) (k = 1,2, ••• ,r; b-E 2 <x< b). Hence + [ j ] [j+r] S (D f(x), ••• ,D f(x)) , which implies (1.4.15).

0

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17

We remark that a particular case of the previous lemma, i.e., when

D[i] =Di (i= 1,2, ••• ,n), can be found in De Boor and Schoenberg [6,p. 6], in Karlin and Michelli [24], and in Schumaker [56, p. 163].

THEOREM 1.4.14 (Generalized Budan-Fourier theorem).

Let

f E PC(n) ([a,b])

and Zet

Di

and

D[i]

he given

by (1.4.7)

and

(1.4.8),

respectively.

If

D[n]f(a)D[n]f(b-) I O,

then

( 1.4 .17)

- [n] - [1] [n]

s: S (D f, (a,b)) + S (f(a) ,D f(a), ... ,D f(a)) +

+ [1] [n]

-S (f(b),D f(b), ••. ,D f(b-)) PROOF. First we prove (1.4.17) under the additional condition that

(i= 0,1, ••• ,n-1) We assert that for j = O,l, ••• ,n

(1.4.19) Z((D.). n [ j ] 1,D f,(a,b)) S: ~ J+ n [j+l] ::;; Z({Di)j+ 2,D f,(a,b)) +

Varying n in (1.4.19) one can see that it suffices to prove (1.4.19) for

n [1] n

j=O. I f Z{(Di)₂,D f,(a,b)) ~ 1 and Z((Di)₁,f,(a,b)) 0 then (1.4.19)

holds trivially. Now let us assume that

n

Z( {Di) l

,f,

(a,b)) 0 .

Then sgn ( f (a)) sgn(f(b)) and sgn(D [1] f(a))

=

sgn(D [1] f(b-)) and again (1.4.19) holds trivially. So, the case Z((Di)~,f,(a,b)) ~ 1 remains. Since f is continuous we can find points c and d in (a,b) such that

n [1]

f(a)/f(c} > 1 and f(b}/f(d} > 1. If Z((Di}

2,D f,(a,c)) 0, then f(c) > f(a) in case D[1]f(a) > 0 and f(c) < f(a) in case D[l]f(a) < 0,

Hence S+(f(a) ,D[1]f(a}) = 1. A similar reasoning applied

~o

the interval

n [ 1]

if Z( (Di)

2,D , f, (d,b)) = 0, then (d,b) leads to the following assertion:

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and

These inequalities combined with (1.4.11) give

We thus obtain (1.4.19). Repeated application of (1.4.19) yields

n - [ n] n-l [ ] [ 1]

Z((Di)

1,f,(a,b)) s; S (D f,(a,b)) +

I

S-(D j f(a),D j+ f(a)) + j=O

In view of this and taking into account condition (1.4.18), inequality (1.4.17) follows from the identity

If condition (1.4.18) is not satisfied, then because of the assumption D[n]f(a)D[n]f{b-) # 0 it follows that for all sufficiently small e: > 0 one bas D[l]f(a+E)D[i]f(b-e:-) # 0 (i= O,l, ••• ,n). Therefore, (1.4.17) holds on the interval (a+e:,b-e:). Since Z((Di)~

1

f,(a,b)) = Z((Di)~

1

f,(a+e:,b-e:)) for sufficiently small e: > 0, Theorem 1.4.14 is obtained by letting e: tend to zero and using Lemma 1.4.13.

REMARK. Theorem 1.4.14 implies the classica! Budan-Fourier theorem 1.4.6; this can beseen as fellows. Since p(n) (a) I 0 one has that

S-(p(n) ,(a,b)) 0. Furthermore, the multiplicity of a zero of p with

n

respect to the operators (Di)

1 is as usual determined by consecutive derivatives. Hence {1.4.17) yields Theorem 1.4.6,

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1.4.4. Chebyshev systems

DEFINITION 1.4.15.

Let

J

be

an

intePval and let

~

₀

,~

₁

, ••• ,~n-l

ben

func-tions in

C(J),

The set

{~

₀

,~

₁

, ••• ,~n-l}

is called a Chebyshev system on

J

if (cf. Beetion

1.2.6

for notation)

> 0

tor each choice of

tr~ n

pointstiE

J

with

t

0 < t1 < ••• < tn_1• If the determinant in the foregoing definition is nonnegative for all t₀< t₁< ..• < tn_₁and if the functions ~

₀

,~₁, ••. ,~n-

₁

are linearly in-dependent, then {~

₀

,~

₁

, ••• ,~n-

₁

} is called a

weak Chebyshev system on

J.

19

DEFINITION 1.4.16.

The Chebyshev system

~

₀

,~

₁

, •.• ,~n-

₁

is called

an

ex-tended Chebyshev system of orderpon

J

if

~- "c(p-l) (J)

(i

0,1, ••• ,n-1)

~

and

if

> 0

for all

_{t 0}:s: _{t 1 s ••• s tn_ 1 (tiE}J)~

with coincident bloeks of Length

not exceeding

p

(cf. Section

1.2.5).

If the determinant in Definition 1.4.16 is nonnegative for every choice of

:S: t₁:S: ••• s tn_

1 with coincident blocks of length not exceeding pand

if the functions ~

₀

,~₁, ••• ,~n-

₁

are linearly independent, then these

func-tions are called an

extendedweak Chebyshev system of orderpon

J.

In what follows we give examples of Chebyshev systems together with a number of lemmas that will be used in the sequel of this thesis.

EXAMPLE 1. If ~ is the fundamental function corresponding to the operator

pn(D), then the functions

~~~·,

•••

,~(n-l)

are linearly independent on every

interval J, but they only form an extended Chebyshev system of order n on

intervals where p (D} is disconjugate. This assertien follows from Theorem _n _, 4.3 in Karlin and Studden [25, p. 24]. An important consequence of the

(n-1)

fact that ~.~·, ••• ,~ form an extended Chebyshev system on intervals

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LEMMA 1.4.17.

Let

(a,bl

be an interval of disaonjugaay

fo~

the

ope~ato~

pn(Dl,

let

a< x₁~ ••• ~ xn

<band let

a₁,a₂, ••• ,an

be

~eal numbe~s.

Then

the~

ereiste a unique jUnation

f é Ker(pnl

auah that

.

T

!(xl ,x2, ••• ,xnl = (al ,a2, •. • ,an) ,

i.e.~ f

ia the Hermite interpolate of the data

a₁,a₂, ••• ,an

(af. Seation

1.2.5). PROOF. Since (n-1) ql x n > 0 ,

the null function is the unique function interpolating zero data. This guarantees the existence and the uniqueness of a function f E Ker(pn) with

the required properties.

0

LEMMA 1.4.18.

Let

(a,b)

be an

inte~val

of disaonjugaay

fo~

the

ope~to~

pn(D)

and let

n

distinat points

x_{1 <}x_{2 < •••}< xn

be given in

(a,b).

Let

q1

be the fundamental jUnation of

pn (Dl

and de fine the funations

'Pi (i= 1, ••• ,n)

by

(1.4.20) epi (t) := cp(t-x

1l (t € lR) •

Then the jUnations

~P

₁

, ••• ,cpn

form an extended Chebyshev syetem of

o~de~ n

on every intewal of disconjugaay

fo~

the

ope~ator pn (D).

PROOF. Only a sketch of the proof will be given. First we show that the functions cp

1, ••• ,qln forma basis forthespace Ker(pn). If not, then there

would exist constants a

1, •.• ,an with (a1, ••• ,an) ~ (0,0, ••• ,0) such that

and we would have cp 0 x -x n n-1 (n-1) ql x -x n 1 (j

=

O,l, ••• ,n-1) , 0 ,

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which contradiets the fact that

~~~·,

•••

,~(n-

1

)

is an extended Chebyshev system on the interval (O,b-a) (cf. Example 1, p. 19).

21

By a nonsingular linear transformation the basis

~~~·,

•••

,~(n-

1

)

can be transformed into the basis ~

₁

, •.• ,~n· It then suffices to note that non-singular linear transformations transfarm Chebyshev systems into Chebyshev systems.

A variant of the above lemma is obtained by dropping the assumption that x₁, ..• ,xn are distinct. If there is a coincident bleek xi-l < xi

= xi+j-l < xi+j' then the functions ~i'~i+

₁

, ..• ,~i+j-

₁

in (1.4.20} only have to be replaced by

(1.4.21) (k = 0 t 1 t • • • 1 j-1 1 t E :R) •

Then ~

₁

, ••• ,~n again form an extended Chebyshev system of ordernon every interval of disconjugacy.

EXAMPLE 2. Let m distinct points x

1 < x2 < ••. < xm be given and let [x

1,xm] be an interval of disconjugacy for the operator pn(D}. The func-tions ~i (i = 1, ... ,m) are defined by ~i (t} := ~+(t-xi) (t E :R). Then on every interval of disconjugacy the functions ~

₁

, ..• ,~m farm an extended weak Chebyshev system of order n-1.

This assertien is a consequence of a result of Karlin, which for this par-ticular case takes on the following form.

IJ

LEMMA 1.4.19 (Karlin [21, p. 503]).

Let

(a,b)

be an interval of discon.jugacy

for the operator

pn(D).

Let

x

0 5 x1 5 ••• 5 xm

be points located in

(a,b)

with

x. <x.+ (i= O,l, ••• ,m-nJ.

Furthermore, let

t₀,t

1, ••• ,t

satisfy the

~ ~ n m

conditions

ii) < ti+n (i= 0,1, ••• ,m-n) ,

iii)

if

ti-l< ti= ti+l

= •••

_{ti+j- 1 < ti+j}

and

~-

₁

< ~

=

_{xk+ 1 =}

~H-l < xkH

for

i,j ,k,R.

in

:N

and if

~ ti•

then

R.+j 5 n+l.

Finally. let the functions

~

₀

, ••• ,~m

be defined by

~i+R.(~)

if

xi-1 < xi

=

xi+l = ••.

=

xi+R.'

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(1.4.22).

::1

~

0 •

Moreover, for n ~ 2 strict inequaUty in (1.4.22) holde if and only i f

where xj :="' if j > m.

For n = 1 strict inequality holds if and only if xi :S. ti < xi+l

(i= O,l, ••• ,m),

If the computation of (1.4.22) requires the (n-1}-st or n-th derivative of q1+ at zero we define fP!n-l) (0) := fP (n-1) (0) and IP!nl (0) := rp (n) (0).

1.4.5. Solventand unisolvent families of functions

DEFINITION 1.4.20. Let

n

c En, let J be an interval and let F₀(J) be a set of functions f(~;·l indexed by ~ E

o.

The set F

0(J) is called a solvent family of order p if

i) f (~;.) €

c

(p-l) (J) (~ € Q) ,

iil for any sequence of n data a.

1,a.2, ••• ,a.n and any sequence of n points

x₀ :S. x₁:S. ••• :S. xn-₁with coincident bloaks of Zength not exaeeding p

there exists a

point~

E 0 such that

!!~;x

1 ,x

2 ,

••• ,xn) = (a.

1,a2, ••• ,an)T.

If ~ E Q in ii) is uniqualy determined, then F

0(J) is said to be a uni-solvent family of order p.

The above definition is contained in Pinkus [47, p. 89]. An e~ample of a unisolvent family of order p is the linear span of an extended Chebyshev system of order p; an example of a solvent family consisting of a partic-ular set of t-splines shall be given in Chapter 7, p. 154.

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23

1.4.6. The number of zerosof an !-spline function

Rolle's theerem is one of the basic tools in the theory of polynomial interpolation. When counting zeros of !-splines we have to deal with situa-tions where the given !-spline may be identically zero on a subinterval without being the null function. Just for that reason zeros are counted according to the rules as given in Definitions 1.4.9 and 1.4.10. We reeall that the multiplicity of a zero of a function depends on the choice of the smoothness class in which the function is considered. Given an arbitrary !-spline s of order n defined on a finite interval [a,b], one always has

s E PC([a,b]). Between consecutive knots the function s coincides with a

function in Ker(pn) and since the number of knots in (a,b) is finite, i t fellows that the total number of strong zeros of s in (a,b) is also finite. In what fellows we give in particular attention to !-spline functions with simple knots. To begin with, Corollary1.4.12 yields the following

LEMMA 1.4.21. Lets be an l-spZine function on [a,b] of order n with simpZe

knots. If the corresponding operator p (D) is disconjugate on [a,b], then

n

(cf. pp. 11-14 for notation)

[n-1]

D₁and D

=

Dn-l ••• D₁(cf. (1.4.8)),

[n-lJ (n 1)

PROOF. We note that D s E PC - ([a,b]) and thus Corollary 1.4.12 may

be applied with n replaced by n-1. D

. [ n-1] [ n-1]

The funct~on D s satisfies the equation pn(D)s

=

DnD s

=

0 between each two consecutive knots in (a,b), hence (cf. (1.4.2)) D[n-l]s(t)

- [n-1]

= c wn (t). As a consequence we have S (D s, (a,b)) "; k, where k is the number of knots in (a,b). This observation combined with Lemma 1.4.21 gives the following

COROLLARY 1.4.22. Let s be an l-spZine on [a,b] of order n with simpZe

knots. If the corresponding operator pn(D) is disconjugate on [a,b], then

n-1 Z ((Di)

1 ,s, (a,b)) "; k + n- 1 ,

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The Budan-Fourier theerem 1.4.14 applied to t-splines yields

THEOREM 1.4.23.

Lets be an t-sptine funation on

[a,b]

of order

n

with

simpte knots. If the aorreaponding operator

pn(D)

is disconjugate on

[a,b]

• [n-1] [n-1]

and

~f D s(a)D s(b-l ~ 0~

then

- [n-1] - [1] [n-1]

:SS(D s,(a,b))+S(s(a),D s(a), ••• ,D s(a))+

+ [ 1] [n-1]

-s

(s(b),D s(b), ... ,D s(b-)),

[i]

where

D (i= 1,2, ••• ,n-1)

ia given

by {1.4.8).

COROLLARY 1.4.24.

Let

s

be an t-apUne on

[a,b]

of order

n

with simpte

knots. If the aarreeponding operator

pn(D)

ia diaaonjugate on

[a,b]

and if

D[n-l]s(a)D[n-l]s(b-l

~ 0~ then

- [1] [n-1]

Sk+S (s(a),D s(a), ••• ,D s(a)) +

+ [ 1] [n-1]

- s

(s(b),D s(b), ••• ,D s(b-)) ,

where

k

ia the nwrber of knots of

s

in

(a,b).

1.4.7. The operator p*(D)

n

The operator p*(D) is defined by n

If ~is the fundamental function oorreeponding to the operator pn(D), then n-1

evidently t a...-..t (-1) ~ (-t) is the fundamental function corresponding to

*

the operator p (D). Furthermore, we note that intervals of disconjugacy

n

*

for pn(D) arealso intervals of disconjugacy for pn(D), as

*

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25

1.4.8. Divided differences

Let n + 1 points x₀:S x₁:S ••• :S xn with x

0 < xn be given and let [x0,xn] be an interval of disconjugacy for the operator pn(D). Due to Lemma 1.4.17 the set V c Rn+l defined by

n+1 has dimension exactly n, and therefore there exists a vector ~ E R , uniquely determined up to a multiplicative constant, such that

(1.4.25)

To ensure the uniqueness of a a further condition is needed. Hence, we require that

(1.4.26) 1 ,

where gis a function satisfying pn(D)g{t) n~(tER).

DEFINITION 1.4.25. If

~

E Rn+1 is the unique vector satisfying (1.4.25} and (1.4.26), then the expression !<x₀,x₁, ••• ,xn) is caZZed the divided difference f with respect to the points x₀,x₁, ••• ,xn and the operator pn(D); it is denoted by f[pn;x

0,x1, ••• ,xn].

n

In the particular case that pn(D) = D we obtain the ordinary divided dif-ference common in numerical analysis and for this the notation

f[x

0,x1, ••• ,xn] will be used.

It fellows from (1.4.25) and (1.4.26) that

<p <p' Cjl (n-1) f xo xl x _n-1 x (1.4.27) f[pn;xO,xl, ..• ,xn] _(n-1) n <p <p' <p g xo xl x n-1 x n n If pn(D)

=

D and x

0 < x1 < ••• < xn, then formula (1.4.27) reduces to the

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where wis defined by W(t) := (t-x

0) ••• (t-xn) (t ":R).

T .

Writing ~

₌

(a

0,a1, ••• ,an) , we may compute the numbers ai from formula (1.4.27) by expanding the determinant in the numerator of (1.4.27) with respect to its last column. One then obtains

!p lP' lP (n-1}

(-1) n+i xo xl xi-1 xi+1 x

( 1. 4. 29) _ai n

(n-1} •

(j) q>' q> g

xo x1 x _n-1 x _n

According to Example 1 on p. 19, the functions q>,q> , ••• ,<p , (n-1) form an ex-tended Chebyshev system of ordernon [x

0,xn]. This implies that the deter-minant in the numerator of (1.4.29) has constant sign, hence

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2. THE B-SPLINE FUNCTIONS

2 • 1 •

1

rttttodu.c.tio

n a.nd

.6

wnma.Jty

The subject of this chapter is the class of the so-called basic t-spline functions, B-spline functions or B-splines for short. This name finds its origin in the fact that for some spaces of l-splines these functions form a basis; e.g. in the case of polynomial splines the well-known polynomial B-splines form a basis. For various properties of:polynomial B-splines the reader is referred to Lectures 1 and 2 of Schoenherg's monograph [51]. Fundamental properties of B-splines corresponding to a general operator pn(D) are studied in a paper by Schmidt, Lancaster and Watkins [50].

27

For the sake of completeness, in Sectien 2.2 we prove some fundamental properties of B-splines. Sectien 2. 3 is concerned wi th the se- called total positivity property of a consecutive sequence of B-splines. Applying our version of the Budan-Fourier theerem to l-splines (Theorem 1.4.23) we give a rather simple proof of this property. Using the Fourier transferm in Sectien 2.4, we derive various recurrence relations for B-splines. In the final Sectien 2.5 one of these recurrence relations is used to in-vestigate the behaviour of the supremum norm of the polynomial B-spline of order n as a function of its knot distribution. The optimal distribution of the knots, i.e., the distribution for which the supremum norm is minimal, is determined; this answers a question of Meinardus [34, p. 174].

2.2. Funda.mental-6 a.bout

B-.6ptine.6

2.2.1. Definition of a B-spline function

The B-spline will be defined by using the concept of divided differences as given in Definition 1.4.25. In order to obtain B-splines'corresponding to the operator pn(D) weneed divided differences with respect to the operator

Interpolational and extremal properties of L-spline functions