On the role of the exponential eigen splines in translation invariant periodic spline spaces

On the role of the exponential eigen splines in translation

invariant periodic spline spaces

Citation for published version (APA):

Morsche, ter, H. G. (1989). On the role of the exponential eigen splines in translation invariant periodic spline spaces. (RANA : reports on applied and numerical analysis; Vol. 8923). Technische Universiteit Eindhoven.

Document status and date: Published: 01/01/1989

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Eindhoven University of Technology

Department of Mathematics and Computing Science

RANA89-23 October 1989

ON THE ROLE OF THE EXPONENTIAL EIGEN SPLINES IN TRANSLATION INVARIANT PERIODIC SPLINE SPACES

by

H.G.

ter Morsche

Reports on Applied and Numerical Analysis

Department of Mathematics and Computing Science Eindhoven University of Technology

P.O. Box 513 5600 MB Eindhoven The Netherlands

ON THE ROLE OF THE EXPONENTIAL EIGEN SPLINES IN

TRANSLATION INVARIANT PERIODIC SPLINE SPACES

1. Introduction

H.G. ter Morsche Department of Mathematics

University of Technology Eindhoven. The Netherlands

As pointed out by 1.1. Schoenberg in his beautiful work on cardinal spline functions (cf. SCHOEN-BERG [4]) the so-called exponential eigen splines playa fundamental rOle in cardinal spline spaces. The reason is the simple obselVation that the associated uniform knot distribution is invariant with respect to translation over the mesh size. The eigen functions for the corresponding shift operator

are

just the exponential eigen splines. In the multivariate setting translation invariance may concern several direc-tions and distances depending on the grid shape. In stead of saying that the grid-partition is translation invariant, we say that the corresponding spline space is translation invariant.

To be more precise and more general a linear space V (over C) of complex- valued functions defined on

JR" is called translation invariant with respect to

n

independent vectors VI , V2, ••. , v" in IR" if and only if

Til)

e V (je V;j=l, 2, ... , n).

Here Tj is the shift operator:

(1.1) Tj!(x) :=!(x+Vj).

Now a function

s

in V is called an (exponential) eigen function corresponding to an n-tuple z = (z I, ...• Z/I) e

e"

of complex numbers if

TjCs)

=

Z j S (j

=

1, 2, " ' , n) .

In this note our attention is focussed on the role of the exponential eigen functions in subspaces of V consisting of (multi) periodic functions! for which

!(x+Njvj)=!(x) (XE /R";j=I,2, ···,n),

where N =

CN

1> N 2. " ' , Nil.) E IN".

A function! satisfying this periodicity condition is called an N-periodic function. The linear subspace of V consisting of the N-periodic functions in V is denoted by VN. An exponential eigen function

s

based on the n-tuple (z 1, Z2 • •••• z,,) is N-periodic in case z7J

=

_{1 (j}

=

_{1, ...• n). Therefore it has the}

S(X+Vj) =e2fcil1jlNj S(X) (xe 1R";j=l • ..•• n). where J.1

=

Q.tl. . ..• J.111) e roN with

roN ={I1E

Z'll1j

E

(0, I, ...

,Nj

-I)} .

The next section deals with some basic properties of the N-periodic eigen functions. Especially if VN is supplied with a semi inner product. which is also translation invariant in some sense. In fact. it will be shown in that section that VN has an orthogonal base consisting of N-periodic eigen functions. which can be used to compute the dimension of VN. This is illustrated by an example with respect to bivariate periodic splines on a three direction mesh. Section 3 is devoted to an application of the obtained proper-ties to least semi nonn interpolation by means of periodic splines on a rectangular mesh.

2. Some basic properties of the N-periodic exponential eigen functions

By Ell (J.1e roN) we denote the space of the N-periodic eigen functions corresponding to the n-tuple of "frequencies" (J.1t/N 1> J.121N 2 •...• J.1"INII ). Hence. seEp. if and only if

s(x +Vj) == e2fciIL/Nj 8 (X) (x e 1R";j

=

1.

2 .... , n) .

The first basic property of the N-periodic eigen functions is contained in the following lemma. In this lemma a direct sum of spaces is denoted by l:e.

Lemma 2.1. Let V be a translation invariant space of functions defined on JR" with

respect

to the n independent vectors VI • V2. . ..• v" and let N == (N 1. N 2 ••••• N,,) e IN". Then

VN

=

L

$EJI.'

p.E(J)N

Proof. The Off (Discrete Fourier Transfonn) of an N-periodic sequence (av ) , ve

Z".

of complex

numbers

(a

v+N.e ,

=a

_v (j == 1, ...•

n»

is given by J J

The sequence (av ) can be recovered by means of the inverse DFT:

• 111 VI P" V8

1 "t" A 21tl(N;'""+ .•. +T.)

a

v

₌

_{N N N} ~ _aile

1 2 ' • • "p.E<q...

Now let

f

e VN• We apply the DFT to the N -periodic sequence

a"

=f(x-J.11VI-J.12VZ- ... -J.1"v,,).

It can easily be shown that the function sp. defined by sll(x)=a" is an N-periodic eigen function corresponding to the frequencies Q.tt/N 1> •••• IlnINn ). Hence. 8" e E", Note that 8" can be identically

zero. By the inverse DFT. we have 1

f(x)

=

N N , .. N l: SIl(X).

1 Z " liE lDtf

Apparently, every

f

e VN is a linear combination of N-periodic eigen functions. There remains to prove that E J1 ( l E v consists only of the null function if J.1, veroN and J.l :F. v.

e"bqJ./Nj

I

_(x)

=

I

_{(x +Vj)}

=

_e21N_/Nj

I

_{(x) (x E JRIl; j = I, ...• n) .}

Since P-j :¢:'Vj (mod Nj) for some j,

we

conclude that

I

(x) =0. I]

It is an immediate consequence of the previous lemma that (2.1) dim VN

=

L

dim Ell'

pem"

This formula

can

be very helpful for computing the dimension of VN•

As an example we consider the space S~ of bivariate polynomial splines of degree at most k and global smoothness

p

on the so-called three direction mesh (type-l triangulation) obtained by drawing in the

Xl, X2 plane the mesh lines x 1 = VI • x2

=

v2 • Xl

=

X2

+

V3 (VI. V2. V3 E Z). The space V

=

S~ is trans-lation invariant with respect to the unit vectors el • ez. It is shown in TER MORSCHE [3] as a part of a much more general result that for p

=

21 and k

=

31

+

1 the following formula holds.

{

I (t=O) ,

dim Ell

=

1

+

1 (t= 1) •

31

+

1 (t=3).

where tis the number of ones in the sequence (eZtUIlIINI, eZtUIL2IN2. e2lti(JL\IN r+I1,;N2_{» (J.1e roN)'}

From (2.1) we then have

(2.2) dim VN=N1 Nz +l(Nl +Nz+gcd(NloNz

»

(p=2l) ,

In KREBS [1] a similar technique is applied for the computation of the dimension of continuous periodic spline spaces with respect to a uniform hexagon partition of the xl. Xz plane.

Let us now return to the general case of the translation invariant space of functions defined on JR /I and assume that the subspace VN is supplied with a semi inner product

<,

>

which is translation invariant in the sense:

<Tjl. Tj g>

=

</.

g> (f. g E VN;j=1 • ...• n).

Here Tj is defined in (1.1).

It is a direct consequence of the definition of the eigen function that

<I.

g> =0 (fe Ell' ge Ev. p., ve O)N.jJ.:¢:.V).

This property and Lemma 2.1 finally lead to the following theorem

Theorem 2.2. Let V be a translation invariant space of functions defined on JRn. with respect to the

n

independent vectors VI • V2 ••••• Vn.. Let the subspace VN (N e INn.) be supplied with a translation invariant inner product

<.

>.

Then the follOwing holds:

To every

I

e VN there exist N-periodic eigen functions S p. E Ell (J.1 e roN) such that

{

/=

L

Sp.'

JlEaw

<i.I>=

L

<S1l'S",>,

3. Minimum norm interpolation

The interpolation problem we will consider in this note is of cardinal type, which can be formulated in this context as follows.

Let

a

E [0, 1)n and let (yv) be an N-periodic sequence of data (complex numbers). The question is to

find a function

f

in aN-periodic space VN which is translation invariant with respect to the n indepen-dent vectors VI , Vz, .••• Vn , such that

(3.1) f(a+J.ll VI +J.lz Vz + ••• +J.ln Vlt.)

=

YJ.I. (J.LEZIt.).

Due to the periodicity, we are faced with a total number of N I N 2 .•. N n interpolation data. It happens frequently in the more dimensional situation that dim VN

>

N 1 N

z ...

N_n (cf. (2.2». In general, we like to have an interpolation problem that is unisolvent, i.e., it has a unique solution for every N-periodic sequence of data. Therefore, some additional conditions are needed. In cardinal spline interpolation. (cf. STOCKLER [5], TER MORSCHE [2]) one selects a specific N 1 Nz ... N_n dimensional subspace of VN generated by translates of a fixed function in V. Then the extra condition is that the interpolating function must belong to that subspace.

We propose here to use as a selection for the interpolating functions the solution having a minimal given semi norm among all the possible candidates. Thereby. we assume that the semi norm H-II stems from a translation invariant inner product. In fact that means again that the interpolating function is an element of a specific N I N 2 ••• Nn dimensional subspace. First. we have to verify that there exists at least one

function in VN satisfying (3.1). As shown in the next lemma this is controlled by the N-periodic eigen splines.

Lemma 3.1. Let a E [0, 1)1t. be given. Then for every N-periodic sequence of data there exists at least

one function

f

in VN satisfying (3.1) if and only if there corresponds to every J.l E roN a function

SJ.I. E EJ.I. such that

(3.2) S fl(a) ;f: 0 .

This lemma can easily be proven by using the DFf. To find a unique minimal norm solution we need a further assumption for the space VN in combination with the semi inner product This assumption can entirely be expressed in terms of the N-periodic eigen functions as follows.

{

For every J.l E roN there is a unique

S

Jl E E J.I.

(3.3) satisfying

IIslllI=min{lIslllIlsJ.l.E EJl,sJ.I.(a)=l}.

Note that assumption (3.3) implies that the trivial function S Jl

=

0 is the only function satisfying

USJ.l.1I =0. sJ.I.(a)=O.

Theorem 3.2. To every N-periodic sequence (yv) (VE Zit.) there corresponds a unique function/e VN

satisfying (3.1) and having minimal semi norm II/II if and only if (3.3) holds. Moreover, the unique solu-tion can be written as

1

f= -N-1-N-z---N-

I:

Yll sJ.I. •

It. p.emw

The proof of this theorem follows directly from Theorem 2.2 and properties of the DFf.

As an illustration of the previous theorem we present now a simple example dealing with bivariate poly-nomial splines on the rectangular mesh defined by the mesh lines Xl

=

VI • Xl

=

Va (VI. Va e Z).

The space

V

is the space of polynomial splines of degree

k

and global smoothness p on the rectangular

mesh. This space is translation invariant with respect to the unit vectors

el

and

e2.

Let /lm

be

the

m-th

iterate of the Laplacian/l

=

ail +ai2.

On the N-periodic subspace

VN

of

V

the following semi inner

pro-duct is defined.

where

g

is the complex conjugate of

g

and

2m

S P

+

1.

The inteIpOlation points are taken at the lattice points v

E

.z

2.

Hence ex

=

O.

In

order to solve the minimum nonn inteIpOlation problem. we need the exponential eigen functions in

VN.

For the rectangular mesh the exponential eigen splines can easily be represented by means of the

univariate exponential splines

t ~ HICA, t) (AE C, t E IR)

of degree

I,

which satisfy the functional

relation

HI(A, t

+

1)

=

A HI(A., t) (t E IR) ,

and

which have smoothness

I -

1 if A '#: 1 and smoothness

1 -

2 if A

=

1. It is well known that the

restric-tion of

H,(A,')

to the inteIVal [0, 1] corresponds to a generalized Euler-Frobenius polynomial in case

A'#: 1 or to a Bernoulli polynomial in case A

=

1 (cf. TER MORSCHE [3]). We nonnalize

H,(A,·)

by

requiring that

H~')(A •• ) E!

1 on (0, 1). For a given smoothness p, the smallest value

k

for which there

exist compactly supported splines in

V

is given by

k = 2p + 2.

From now on we assume that

k

=

2p + 2.

The eigen spaces

E" (J.L E roN)

may be written as a linear span in the following way:

<Hp+l (Z}o xt)H p+l(Z2' X2»

<H,(Z2.X2) Il=p+l, ••• ,2p+2>

(z

1

'#: 1,

Z 2

'#: 1) ,

(zl=I,Z2'#:1), Ep.= <H,(zt.Xt) Il=p+l, ... ,2p+2> (zl'#:I,Z2=1) , <Hi(l,xt),Hj(1,X2) I i,j=p+2,'" ,2p+2> (zl=I,Z2=1),

where z

1

=

e'bdIl1IN1

and

Z2

=

e211iJl<tIN2.

With help of these representations of

E" and

well-known

pro-perties of the univariate exponential eigen splines, we are able to solve

the

minimum nonn inteIpOlation

problem in

VN.

Due to Theorem 3.2 it suffices to give the functions

8",

For our inteIpOlation problem

we have obtained the following results.

If

p is even then (3.2) is always satisfied, and

--

-HP+l (zttxt)Hp+t (Z2. X2) (Zl '#:1,Z2'#:1) ,

-H2m-l (ZZ,X2) (ZI =1,Z2'#:1), (Zl '#:1.Z2=1) , (Zl =z2=1).

Here

HI (z}, x)

=

H, (z,x)1 H/(z, 0).

References

[11 F. Krebs (1988) Periodische Splines auf dem Regelmassigen sechseckgitter. Thesis, Dortmund. [21 H.G. ter Morsche (1986) Bivariate cubic periodic spline interpolation on a three direction mesh. EUT Report 86-Wsk-02, Eindhoven University of Technology, Eindhoven.

[3] H.G. ter Morsche (1988) On the dimension of bivariate periodic spline spaces: type-I triangulation. Reports on Applied and Numerical Analysis (RANA 88-05), Eindhoven University of Technology, Eindhoven

[4] U. Schoenberg (1973) Cardinal spline interpolation. Regional conference series in applied mathematics 12, Siam, Philadelphia

[5] J. StockIer (1988) Interpolation mit mehrdimensionalen Bernoulli-Splines und periodischen Box-Splines. Thesis, Duisburg.

H.G. ter Morsche Department of Mathematics University of Technology P.O. Box 513 5600 MB Eindhoven The Netherlands