Constructing Cubic Splines on the Sphere
Except where reference is made to the work of others, the work described in this
thesis is my own or was done in collaboration with my advisory committee. This
thesis does not include proprietary or classi ed information.
Mosavverul Hassan
Certi cate of Approval:
Narendra Kumar Govil
Professor
Department of Mathematics
University of Montana
Amnon J. Meir, Chair
Professor
Mathematics and Statistics
Bertram Zinner
Associate Professor
Mathematics and Statistics
George T. Flowers
Acting Dean
Graduate School
Constructing Cubic Splines on the Sphere
Mosavverul Hassan
A Thesis
Submitted to
the Graduate Faculty of
Auburn University
in Partial Ful llment of the
Requirements for the
Degree of
Master of Science
Auburn, Alabama
August 10, 2009
Constructing Cubic Splines on the Sphere
Mosavverul Hassan
Permission is granted to Auburn University to make copies of this thesis at its
discretion, upon the request of individuals or institutions and at
their expense. The author reserves all publication rights.
Signature of Author
Date of Graduation
iii
Thesis Abstract
Constructing Cubic Splines on the Sphere
Mosavverul Hassan
Master of Science, August 10, 2009
(M.Sc., I.I.T. Guwahati{India, 2006)
(B.Sc., Ranchi University, 2002)
47 Typed Pages
Directed by Amnon J. Meir
A method to approximate functions de ned on a sphere using tensor product cu-
bic B-splines is presented here. The method is based on decomposing the sphere into
six identical patches obtained by radially projecting the six faces of a circumscribed
cube onto the spherical surface. The theory of univariate splines has been general-
ized in di erent forms to functions of several variables. Among these extensions the
tensor product splines are the easiest to handle. Although the tensor product splines
are restricted to rectangular domains rendering their applicability limited they are
extremely e cient compared to other surface approximation techniques which are far
more complicated and hence computationally less attractive.
iv
Acknowledgments
I would like to acknowledge and thank my professor, Dr. A.J. Meir for his
continuous help and support he provided me throughout the thesis work. His vast
knowledge, experience and patience helped me explore and bring my work to its
conclusion. His constant encouragement motivated me to enrich myself with the
scienti c acumen necessary for the present work.
I would also like to express my gratitude to my committee members Dr. Narendra
Kumar Govil and Dr. Bertram Zinner for their advice and support.
v
Style manual or journal used Journal of Approximation Theory (together with the
style known as \aums"). Bibliograpy follows van Leunen?s A Handbook for Scholars.
Computer software used The document preparation package TEX (speci cally
LATEX) together with the departmental style- le aums.sty.
vi
Table of Contents
List of Figures viii
1 Introduction 1
1.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Spline approximation and its signi cance . . . . . . . . . . . . 2
1.2 Spline Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 B-spline Representation . . . . . . . . . . . . . . . . . . . . . . 5
1.2.2 Tensor Product Splines . . . . . . . . . . . . . . . . . . . . . . 9
1.2.3 Error Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 Radial Projection 16
2.1 Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3 Analysis 19
3.1 Univariate Cubic Spline Interpolation . . . . . . . . . . . . . . . . . . 19
3.1.1 Radial Projection: The One Dimensional Case . . . . . . . . . 21
3.1.2 Periodic Splines on a Square . . . . . . . . . . . . . . . . . . . 25
3.2 B-spline representation on a Square . . . . . . . . . . . . . . . . . . . . 27
3.3 Tensor Product Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4 Conclusion 35
Bibliography 36
Appendices 37
A Notations 38
A.0.1 One dimensional case . . . . . . . . . . . . . . . . . . . . . . . 38
A.0.2 Bivariate case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
vii
List of Figures
3.1 Approximation of the function f( ) = sin ;N = 28 . . . . . . . . . . . 24
3.2 Approximation of the function f( ) = sin ;N = 60 . . . . . . . . . . . 24
3.3 Approximation of the function f( ) = sin cos ;N = 28 . . . . . . . . 24
3.4 Approximation of the function f( ) = sin cos ;N = 60 . . . . . . . . 24
3.5 Approximation of the function f( ) = sin3 ;h = 2:5 10 1 . . . . . . 26
3.6 Approximation of the function f( ) = sin3 ;h = 6:25 10 2 . . . . . 26
3.7 Approximation of a function f =2C1[a;b];h = 1 . . . . . . . . . . . . . 26
3.8 Approximation of a function f =2C1[a;b];h = 1:5625 10 2 . . . . . . 26
3.9 Approximation of a function f( ) = sin 4 ;h = 6:25 10 2 . . . . . . 29
3.10 Approximation of a function f( ) = sin 4 ;h = 1:56 10 2 . . . . . . 29
3.11 Function f(x;y) = x6y6 . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.12 Approximation of a function f(x;y) = x6y6;h = 1 . . . . . . . . . . . 32
3.13 Approximation of a function f(x;y) = x6y6;h = 1:25 10 1 . . . . . . 33
3.14 Mesh on the sphere Sr . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.15 Mesh on the cube Bd . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
viii
Chapter 1
Introduction
1.1 Objective
Cubic splines on a spherical domain may be used for the approximation of func-
tions de ned on such domains which serve as a tool for modeling and analyzing of
physical processes. However evaluation of these functions explicitly is sometimes dif-
cult due to the limitations of the underlying processes. In such cases it becomes
pertinent to form an approximation of the de ned function. We also run into cases
where the evaluation of the exact function is computationally expensive forcing us
to use approximation techniques. The problem of functional approximation may be
broadly classi ed into two categories. The rst involving problems where the exact
function is unknown and approximation technique is based on function value at cer-
tain set of discrete points. The second class of problems is related to physical process
modeling. These usually involve operator equations. Our aim is to look into a suitable
set of approximations A and develop means to select an appropriate approximation.
Since we want the function to be approximated by some member of the approxima-
tion set A we need to device a method to select this member. Usually this is done
by choosing an approximation member such that the error is within a certain factor
of the least error that can be achieved. Functions arising from physical processes are
generally smooth implying an obvious need for the approximants to be su ciently
smooth. Functional approximations on the sphere are of research interest since many
geophysical applications including oceanography, climate modeling and modeling of
earth?s gravitational potential involve large amount of data on the surface of the
1
earth (basic model is a sphere) or on the satellite orbit (approximately a spherical
manifold).
1.1.1 Spline approximation and its signi cance
It appears that an obvious choice for function approximation is the polynomial
pm2Pm because of its relative smoothness and easy manipulation on a digital com-
puter. However it turns out that interpolating polynomials do not always converge
to the function being interpolated [3]. The following theorem justi es this.
Theorem 1.1. Let [a;b] be xed and suppose that for each k > 1;tk1;tk2;:::;tkk is a
collection of points in [a;b]: Then there exists a function f2C[a;b] such that
jj(f Lmf)jj1 !1 as m!1 (1.1)
where Lmf is the unique polynomial of order k interpolating f at tk1;tk2;:::;tkk:
This leads to the approximation using smooth piecewise polynomials i.e splines.
1.2 Spline Theory
We address here the one dimensional case of approximating a given function
using univariate splines. Mathematically we may represent it as
f(xi) = sm(xi);
where xi denote the nodal points. That is to say that the constructed spline agrees
with the function values at a certain set of points termed here as the nodal points. We
would want this spline to possess a certain degree of smoothness. We then represent
the spline on any partition say as a linear combination of the basis elements of
the linear space to which it belongs. These basis elements are generally called the
B-splines.
2
De nition 1.1 (Piecewise Polynomials [3]). Let a = x0 < x1 < x2 < <
xn 1 < xn = b; and write = fxign0: The set partitions the interval [a;b] into
n subintervals, Ii = [xi;xi+1); i = 0;1;:::;n 2; and In 1 = [xn 1;xn]: Given a
positive integer m, let
PPm( ) = ff : there exists polynomials p0;p1;:::;pn 1 in Pm with f(x) = pi(x)
for x2Ii;i = 0;1;:::;n 1g;
where
Pm =fp(x) : p(x) =
mX
i=1
cixi 1; c1;c2;:::;cm; x realg:
We call PPm( ) the space of piecewise polynomials of order m with knots x1;x2;:::;
xn 1:
Switching from the approximation of a given function by a polynomial to approx-
imation using piecewise polynomials provides us with a degree of exibility. However
piecewise polynomials are not necessarily smooth. To maintain exibility and at
the same time allow a certain degree of global smoothness we now de ne a class of
functions known as polynomial splines.
De nition 1.2. Let be a partition of the interval [a;b] as in De nition 1.1, and
let m be a positive integer. Let
Sm( ) = PPm( )\Cm 2[a;b];
where PPm( ) is the space of piecewise polynomials de ned in (1.1). We call Sm( )
the space of polynomial splines of order m (degree m 1) with respect to .
Polynomial splines spaces are nite dimensional linear spaces. The dimension of
this space of splines is dim(Sm( )) = n+m 1: The above de nition clearly implies
that any polynomial on of degree 6m 1 is a spline function of degree m 1 on
3
. In general a spline of degree m 1 is represented by di erent polynomials in each
interval Ii; i = 0;1;:::;n 1: This may give rise to dicontinuities in its (m 1) th
derivatives at the internal nodes x1;x2;:::;xn 1. The nodes for which this actually
happens are called active nodes.
Let sm2Sm( ) be a spline of degree m 1 de ned on the partition . Let us
denote the restriction of this spline function as smj[xi;xi+1] where
smj[xi;xi+1] =
m 1X
j=0
sji(x xi)j; if x2[xi;xi+1]
so we have mn coe cients to determine. Again we have the continuity conditions at
the internal nodes. Each internal node has m 1 continuity conditions which amounts
to (n 1)(m 1) conditions. We therefore have mn (n 1)(m 1) = m + n 1
coe cients to determine. Since we are talking about an interpolatory spline we have
smj[xi;xi+1](xi) = f(xi) fi for i = 0;:::;n
where the n+1 function values are known. We now have (m+n 1) (n+1) = m 2
coe cients still unaccounted for. To be more precise we still need m 2 conditions
to determine the spline completely. This leads to imposing further constraints i.e
conditions for periodicity or conditions for the spline to be natural.
Mathematically they are represented as
1. Periodic splines, if
slm(a) = slm(b); l = 0;:::;m 2: (1.2)
2. Natural splines, if for m = 2p 1; with l> 2
sp+jm (a) = sp+jm (b) = 0; j = 0;1:::;p 2 (1.3)
4
We will be dealing with cubic periodic splines throughout our work and unless
otherwise mentioned m = 4:
1.2.1 B-spline Representation
Before we de ne the B-Spline representation for a spline sm2Sm( ) and what
the B-splines themselves are we de ne the concept of divided di erence since B-splines
can be de ned in terms of the divided di erence.
De nition 1.3 (Divided Di erence [2]). The n-th divided di erence of a function f
at the points x0;x1;:::;xn (which are assumed to be distinct) is the leading coe cient
(i.e the coe cient of xn) of the unique polynomial pn+1(x) of degree n which satis es
pn+1(xi) = f(xi); i = 0;1;:::;n:
Mathematically the n-th divided di erence is denoted as
f[x0;x1;:::;xn] =
nX
i=0
f(xi)
!0n+1(xi); (1.4)
where
!n+1(x) =
nY
i=0
(x xi):
We now de ne the B-splines in terms of divided di erence
De nition 1.4 (Normalized B-splines). The normalized B-splines of degree m 1
relative to the distinct nodes xi;xi+1;:::;xi+m is de ned as
Bi;m(x) = (xi+m xi)g[xi;:::;xi+m]: (1.5)
where
g(t) = (t x)m 1+ =
8>
><
>>:
(t x)m 1 if x6t;
0 otherwise.
5
To nd an explicit expression for the normalized B-splines we state here the the
Uniqueness Theorem for an interpolating polynomial which forms a basis for for this
explicit expression.
Theorem 1.2. Given n + 1 distinct points x0;x1;:::;xn and n + 1 corresponding
values f(x0);f(x1);:::;f(xn) there exists a unique polynomial pn+12Pn+1 such that
pn+1(xi) = f(xi) for i = 0;1;:::;n:
The uniqueness of the interpolating polynomial provides for the comparison be-
tween the Lagrange?s form of the interpolating polynomial and the Newton?s divided
di erence formula. This comparison along with the notion that the divided di erence
is the coe cient of xn in the interpolating polynomial yields the explicit represen-
tation for the n-th divided di erence as de ned in equation (1.4). Using equation
(1.5) in the expression for normalized B-splines we arrive at the following explicit
representation for B-splines
Bi;m(x) = (xi+m xi)
mX
j=0
(xj+i x)m 1+
mY
l=0l6=j
(xj+i xl+i)
: (1.6)
We note here that the m-th order normalized B-spline have active nodes
xi;xi+1;:::;xm and vanish outside the interval [xi;xi+m]: The term normalized for
B-splines has been introduced since B-splines can have varied sizes depending on the
location of the nodal points, for example
Qi;1(x) =
8>
><
>>:
1
xi+1 xi; xi 6x
><
>>:
1 if xi 6x 0 for x2(xi;xi+m):
In view of Theorem 1.4 we can now de ne any spline sm(x) 2Sm( ) uniquely
as a linear combination of these basis elements i.e
sm(x) =
n 1X
m+1
ciBi;m(x): (1.13)
The real numbers ci are called the B-spline coe cients of sm: The nodes in equation
(1.11) are generally chosen as periodic or coincident. For periodicity we must have
x i = xn i b+a;
xn+i = xi +b a: (1.14)
Using equations (1.14) and (1.6) we have the following condition for periodicity
B i;m(x) = Bn i;m(x+b a); i = 1;:::;m 1 (1.15)
Since the B-splines vanish outside their local support i.e [xi;xi+m] the condition de-
ned in equation (1.15) will be satis ed if and only if
c i = cn i; i = 1;:::;m 1: (1.16)
8
1.2.2 Tensor Product Splines
In this section we introduce the Tensor Product Splines as an extension of the
univariate B-spline representation. We require here only a few concepts concern-
ing tensor products of vector spaces to de ne the tensor product polynomial spline.
Mathematically we denote the tensor product of two vector spaces U and V as U V:
Let us x a eld say C and let U and V be vector spaces de ned over this eld. Then
we can have
De nition 1.6 (Bilinear Mapping). A function f from U V to the vector space
P is said to be bilinear if it is linear in each of the two variables when the other is
kept xed.
The space of all such bilinear functions forms a vector space over the eld C
under addition and scalar multiplication.
If U is a linear space of functions de ned on some set X and V; a linear space of
functions de ned on some set Y into R then for each u2U and v2V the rule
w(x;y) = u(x)v(y); 8(x;y)2X Y (1.17)
de nes a function on X Y called the tensor product of u with v [1] and is denoted
as u v: Further the set of all nite linear combinations of the functions on X Y
of the form u v is called the tensor product of U with V. Hence
U V =
( nX
i=1
i(ui vi) : i2R;ui2U;vi2V; (1.18)
i = 1;:::;n; for some n
)
(1.19)
It can be veri ed that the tensor product de ned above is bilinear, i.e., the map
U V !U V : (u;v)7 !u v (1.20)
9
is linear in each argument.
( 1u1 + 2u2) v = 1(u1 v) + 2(u2 v)
u ( 1v1 + 2v2) = 1(u v1) + 2(u v2): (1.21)
Remark 1.1. The tensor product discussed above forms a linear space of functions
de ned on X Y and its dimension is given by the following proposition.
Proposition 1.1 (Tensor Product Splines). If U and V are some vector spaces
de ned over a eld C, then
dim(U V) = (dimU)(dimV)
As mentioned earlier the tensor product we use here is simply an extension from
the univariate case to the bivariate case and hence we will be considering two sets of
partitions one along the horizontal and one along the vertical.
De nition 1.7 (Tensor product Splines). Consider the strictly increasing se-
quences
a = x0 0 in x and l 1 > 0 in y with respect to the partition along
the horizontal and the vertical as de ned in equations (1.22)-(1.23) if the following
conditions are satis ed
10
1. On each subrectangle Ri;j = [xi;xi+1] [yi;yi+1];s(x;y) is given by a polynomial
of degree m 1 in x and l 1 in y:
sjRi;j 2Pm Pl; i = 0;1;:::;n 1; j = 0;1;:::;p 1:
2. The function s(x;y) and all its partial derivatives
@i+js(x;y)
@xi@yj 2C(R); i = 0;1;:::;m 2; j = 0;1;:::;l 2:
For our particular casem = l = 4 and the dimension of the vector space S( 1; 2)
of all functions satisfying the above two conditions is dim(S( 1; 2)) = (m + n
1) (l + p 1) where i; i = 1;2 are the partitions along the horizontal and the
vertical and are de ned as 1 = fxign0 and 2 = fyjgp0 respectively. We can de ne
an extended partition as in De nition 1.5 and because of the tensor product nature of
the vector space it is possible to work with the one dimensional B-spline basis. This
gives us the unique tensor product representation of a spline s(x;y) in terms of its
basis functions
s(x;y) =
n 1X
i= m+1
p 1X
j= l+1
ci;jBi;m(x)Bj;l(y) (1.24)
where Bi;m(x) and Bj;l(y) are the normalized B-splines de ned on the partitions
1 and 2 respectively. For the approximating function to be periodic in both
x(horizontal) and y(vertical) direction we have
@is(a;y)
@xi =
@is(b;y)
@xi ; i = 0;:::;m 2; c6y 6d
@js(x;c)
@yj =
@is(x;d)
@yj ; j = 0;:::;l 2; a6x6b: (1.25)
11
1.2.3 Error Estimates
We present here a priori error bounds for the interpolation procedures introduced
in Sections 1.2.1 and 1.2.2. The derivation of the interpolation error, f Sf and its
derivatives in L2 norm and the L1 norm are given. Here Sf denotes the approximation
to the function f. It is observed that if the function f is su ciently smooth, Sf is a
fourth order approximation to f in both L2 and L1.
Theorem 1.5 (Variational Problem). Let and f ff0;f1;:::;fn;f0a;f0bg be
given and V fw 2 PC22(I) j w(xi) = f(i); 0 6 i 6 n and Dw(xi) = f0i; i =
0 and ng. The variational problem of nding the functions p2V which minimize
jjD2wjj22 over all w2V has the unique solution Sf:
The function p2V is a solution of the variational problem if and only if
D2p;D2
2
= 0
for all 2V0 fw2PC22(I) jw(xi) = 0;0 6 i 6 n;and Dw(xi) = 0; i = 0 and ng:
By de nition we have
jjD2p+ D2 jj22 = (D2p;D2p)2 + 2(D2p;D2 )2 + (D2 ;D2 )2: (1.26)
By the orthogonality condition we have
jjD2p+ D2 jj22 = (D2p;D2p)2 + (D2 ;D2 )2; (1.27)
and this would gives us the following corollary
Corollary 1.1 (First integral relation). If f2PC22(I); then
jjD2fjj22 =jjD2Sfjj22 +jjD2Sf D2fjj22: (1.28)
12
Theorem 1.6 (Preliminary Result). If f2PC22(I); then
jjD2(f Sf )jj2 6 jjD2fjj2 (1.29)
jjD(f Sf )jj2 6 2 1hjjD2fjj2 (1.30)
jj(f Sf )jj2 6 2 2h2jjD2fjj2 (1.31)
Inequality (1.29) follows directly from Corollary 1.1. We can see from the above
theorem that Sf is a second-order approximation to f. Intuitively we may assume
that a smoother function will result in a higher order of convergence and hence we
have the following theorem.
Theorem 1.7. If f2PC14 ; then
jjf Sfjj16 5584h4jjD4fjj1: (1.32)
Moreover if f2C5(I) and is a uniform partition, then
jjf Sfjj16h4( 1384jjD4fjj1 + 1240hjjD5fjj1) (1.33)
Proof. Since f Sf = f Sh + Sh Sf where Sh is the cubic Hermite interpolate of f,
we have
jjf Sfjj16jjf Shjj1 +jjSh Sfjj1: (1.34)
Now
jjf Shjj16 1384h4jjD4fjj1 (1.35)
13
and
Sh Sf =
nX
i=1
e0ih0i(x); hence
jjSh Sfjj1 = jj
nX
i=1
e0ih0i(x)jj1
6 jje0jj1
n nX
i=1
h0i(x)
o
; (1.36)
where
jje0jj1 max16i6nje0ij6 124h3jjD4fjj1: (1.37)
Now, since
jh0ij+jh0i+1j6 h4 for all x2[xi;xi+1] and 0 6i6n; (1.38)
we have from equation (1.37)
jj
nX
i=1
e0ih0i(x)jj16 196h4jjD4fjj1; (1.39)
which gives us the required result
jjf Sfjj16 1384h4jjD4fjj1 + 196h4jjD4fjj1 = 5584h4jjD4fjj1: (1.40)
We proceed now to the error bounds of the bivariate interpolation procedure and
nd that as in case of the the univariate splines the approximation Sf for a su ciently
smooth function is fourth order accurate in both the L2 and the L1-norm.
Theorem 1.8. If f2PC24(U); then
jjf Sfjj2 6 4 4
h4jjD4xfjj2 +h2k2jjD2xD2yfjj2 +k4jjD4yfjj2
: (1.41)
14
The above theorem gives us the error bound in L2-norm. For the error bound in
the L1-norm we have the following theorem
Theorem 1.9. If f2PC14 (U); then
jjf Sfjj16 5384h4jjD4xfjj1 + 49h2k2jjD2xD2yfjj1 + 5384k4jjD4yfjj1; (1.42)
which clearly shows that the approximating spline is fourth order accurate.
15
Chapter 2
Radial Projection
The radial projection method which we describe here (see [4], p. 24) is a method
to radially project the points on the surface of the cube onto the sphere. Since the
tensor product B-splines are restricted to rectangular domains all calculations will
essentially be done on the cube.
The terminology used here will run as follows: The surface of the cube will be
termed as the box Bd centered at the origin and of side length 2d: We will denote the
sphere centered at the origin with radius r as Sr: Mathematically we may represent
the box and the sphere as follows
Bd = fxjx(x1;x2;:::;xn)2Rn;jjxjj1 = dg
Sr = faja(a1;a2;:::;an)2Rn;jjajj2 = rg
The radial projection from the box to the sphere is de ned as a mapping
P : Bd !Sr
given by
P(x) = r xjjxjj = a;
where as the inverse mapping from the sphere to the box is given by
P 1(a) = d ajjajj
1
= x:
16
2.1 Characteristics
The radial projection P is a one-one mapping from the box Bd to the sphere
Sr: We mention below some related properties. The following lemma shows that the
mapping P and its inverse P 1 are both Lipschitz continuous.
Lemma 2.1. The radial projection P and its inverse P 1 satisfy the inequalities
jjP(x) P(y)jj 6 2rjjxjj
jjx yjj
; (2.1)
jjP 1(a) P 1(b)jj1 6 2djjajj
1
jja bjj1
: (2.2)
Proof.
jjP(x) P(y)jj = jj rjjxjjx rjjyjjyjj
= rjjxjjjjyjj
n
jj jjyjjx jjxjjy jj
o
= rjjxjjjjyjj
n
jj
jjyjj(x y) y(jjyjj jjxjj)
jj
o
6 rjjxjjjjyjj
n
jjyjjjjx yjj+jjyjj
jjyjj jjxjj
o
6 2rjjxjj
n
jjx yjj
o
In a similiar way we can prove inequality (2.2)
Corollary 2.1. The radial projection P and its inverse P 1; are globally Lipscitz
continuous mappings, that is,
jjP(x) P(y)jj6 2rdjjx yjj; (2.3)
and
jjP 1(a) P 1(b)jj16 2dnr jja bjj1: (2.4)
17
This follows directly from the lemma 2.1 by observing that for anyx2Rn;jjxjj16
x 6pnjjxjj1 and that for any a2 Sr; rpn 6jjajj1 6 r: For any x2 Bd; we have
x2Bd; d6x6dpn:
18
Chapter 3
Analysis
In this chapter we discuss the construction of Tensor Product Splines as a natural
extension of the B-spline representation of a spline. We also estimate the di erence
(f sm(x)) where f is the function de ned at the nodal points and sm(x) is the
approximating spline. Univariate spline representation is analyzed in Section 3.1.
In Section 3.2 we analyze the B-spline representation of a spline and in Section 3.3
extend the construction to Bivariate splines and analyze it.
3.1 Univariate Cubic Spline Interpolation
Let us consider a partition of an interval [a;b] as de ned in De nition 1.1
with a x0 = xn b and the corresponding function evaluations at the nodal points
fi; i = 0;1;:::;n 1: Our aim here will be to develop an e cient method to construct
a periodic cubic spline interpolating the function values at the distinct nodal points
[5]. Since the degree of the spline is m 1 = 3 the spline must be twice continuously
di erentiable i.e the second order derivative must be continuous. We introduce here
the following notations
fi = s4(xi); m = s04(xi); and M = s004(xi); i = 0;1;:::;n 1:
Due to the periodic consideration we have fn+j = fj and Mn+j = Mj for j = 0;1:
Since s4;i 12P4; s004;i 1 is linear and
s004;i 1(x) = Mi 1xi xh
i
+Mix xi 1h
i
for x2[xi 1;xi] (3.1)
19
where hi = xi xi 1; i = 1;:::;n: Integrating (3.1) twice we get
s4;i 1(x) = Mi 1(xi x)
3
6hi +Mi
(x xi 1)3
6hi +Ci 1(x xi 1) +
~Ci 1 (3.2)
and the constants Ci 1 and ~Ci 1 are determined by imposing the end point values
s4(xi 1) = fi 1 and s4(xi) = fi:This gives us, for i = 1;:::;n
~Ci 1 = fi 1 Mi 1h2i
6 ; Ci 1 =
fi fi 1
hi
hi
6 (Mi Mi 1): (3.3)
Imposing the continuity of the rst derivatives at xi; we get
s04(x i ) = hi6 Mi 1 + hi3 Mi + fi fi 1h
i
= hi+13 Mi hi+16 Mi+1 + fi+1 fih
i+1
= s04(x+i );
where
s04(x i ) = limt!0s04(x t):
This gives us the following linear system also know as the M-continuity system
iMi 1 + 2Mi + iMi+1 = di; i = 1;:::;n (3.4)
where
i = hih
i +hi+1
; i = hi+1h
i +hi+1
di = 6h
i +hi+1
(fi+1 fih
i+1
fi fi 1h
i
) i = 1;:::;n:
It is clear from (3.2) that the only unknows are M0;M1;:::;Mn 1: So our task of
nding a periodic cubic spline representation interpolating the given function values
20
now reduces to solving the linear system (3.4) of n equations and n unknowns. This
construction of the spline produces a system tridiagonal in nature. In matrix notation
it is represented as
0
BB
BB
BB
BB
BB
@
2 n 1 0 ::: n 1
n 2 2 n 2 ...
0 ... ... ... 0
... 0
1 2 1
n 0 0 n 2
1
CC
CC
CC
CC
CC
A
0
BB
BB
BB
BB
BB
@
Mn 1
Mn 2
...
M1
M0
1
CC
CC
CC
CC
CC
A
=
0
BB
BB
BB
BB
BB
@
dn 1
dn 2
...
d1
d0
1
CC
CC
CC
CC
CC
A
(3.5)
and can be easily solved on a computer using existing techniques.
3.1.1 Radial Projection: The One Dimensional Case
In the above discussion we have considered the spline to be periodic in nature as
we want to apply the construction on a circle C with radius r = 1 such that the rst
and last nodal point coincide i.e x0 = xn. The radial projection of the nodal points
on the square onto a unit circle involves basic geometry. Let the four corners, of the
square under consideration be P1(1;1);P2( 1;1);P3( 1; 1);P4(1; 1) in this order.
Let P(x;y) be any point on the side of the square, say the side joining P1 and P2: Let
P0(x0;y0) be the radial projection of the point P(x;y) on the circle. Then we have
the relation
tan = y
0
x0 =
y
x (3.6)
where is the angle between the X-axis and the vector joining the point P to the
origin.
Since the point P0 lies on the circle we have
y02 +x02 = 1: (3.7)
21
Solving for x0 and y0 we have
x0 = xpx2 +y2
y0 = ypx2 +y2
Taking points systematically anticlockwise on the square starting at P1 and then
projecting them onto the circle we have points on the circle which we now label
as xi; i = 0;1;:::;n 1: This provides us with the setting required to apply the
univariate spline constructed in Section 3.1. However we are still required to nd the
interval lengths hi; i = 0;1;:::;n 1 between the nodal points. To do this we make
use of the inner product of two vectors. Let ~xi and ~xi 1 be two vectors then
cos i = <~xi;~xi 1 >jj~x
ijjjj~xi 1jj
; (3.8)
where i as the angle between two vectors labelled here as ~xi and ~xi 1: Hence the arc
length or the interval length hi = r i:
Table 3.1: The following table shows the error and the observed rate of convergence
for various step sizes for the function f( ) = sin( ).
Step Size Error Order of Convergence
2:243994752564133e 01 1:521985704066554e 04
1:047197551196593e 01 7:326907390286181e 06 3:980413480161499e+ 00
5:067084925144792e 02 4:029381312631907e 07 3:995561255766796e+ 00
2:493327502848618e 02 2:703779530789965e 08 3:809570269779505e+ 00
1:236847501412775e 02 1:903539716356657e 09 3:785053304137925e+ 00
The experimentally observed order of convergence is de ned as
p =
ln (Erj+1Erj )
ln (hj+1hj )
;
22
where Erj denotes the error at the jth re nement and is de ned as
Erj =jjhx(f s4(x))jj2 ;
and hx represents the step size for the intermediate points taken to test the spline
function developed.
23
0 1 2 3 4 5 6 7
!1
0
1
Plot of the function
0 1 2 3 4 5 6 7
!1
0
1
Plot of the Spline Function
0 1 2 3 4 5 6 7
!1
0
1
x 10
!3
Plot of the difference
Figure 3.1: Approximation of the function
f( ) = sin ;N = 28
0 1 2 3 4 5 6 7
!1
0
1
Plot of the function
0 1 2 3 4 5 6 7
!1
0
1
Plot of the Spline Function
0 1 2 3 4 5 6 7
!1
0
1
2
x 10
!4
Plot of the difference
Figure 3.2: Approximation of the function
f( ) = sin ;N = 60
0 1 2 3 4 5 6 7
!0.5
0
0.5
Plot of the function
0 1 2 3 4 5 6 7
!0.5
0
0.5
1
Plot of the Spline Function
0 1 2 3 4 5 6 7
!1
0
1
x 10
!3
Plot of the difference
Figure 3.3: Approximation of the function
f( ) = sin cos ;N = 28
0 1 2 3 4 5 6 7
!0.5
0
0.5
Plot of the function
0 1 2 3 4 5 6 7
!0.5
0
0.5
1
Plot of the Spline Function
0 1 2 3 4 5 6 7
!1
0
1
2
x 10
!4
Plot of the difference
Figure 3.4: Approximation of the function
f( ) = sin cos ;N = 60
24
3.1.2 Periodic Splines on a Square
De ning periodic splines on a square is based again on the construction given in
Section 3.1. However in this case the nodal points on the square are not projected on
a circle. Hence with equally spaced nodes on the edges of the square i.e hi = hi+1 we
have from equation (3.4)
i = 12; i = 12
di = 3h2
i
(fi+1 2fi +fi 1); i = 1;:::;n:
Table 3.2: The following table shows the error and the observed rate of convergence
for various step sizes for the function f( ) = sin3( ).
Step Size Error Order of Convergence
1:000000000000000e+ 00 2:993908237928666e 02
5:000000000000000e 01 3:738564612692465e 03 3:001473631942863e+ 00
2:500000000000000e 01 1:044464521796866e 04 5:161649073884993e+ 00
1:250000000000000e 01 4:952085825195140e 06 4:398583359488427e+ 00
6:250000000000000e 02 2:868804616210702e 07 4:109514698407482e+ 00
3:125000000000000e 02 1:758372110067624e 08 4:028137400984083e+ 00
1:562500000000000e 02 1:093598905987019e 09 4:007084798744635e+ 00
Table 3.3: The following table shows the error and the observed rate of convergence
for various step sizes for a function f =2C1[a;b]
Step Size Error Order of Convergence
1:000000000000000e+ 00 8:728715607973290e 03
5:000000000000000e 01 8:728715607973281e 03 1:441541926716714e 15
2:500000000000000e 01 2:822924332662890e 03 1:628578924802774e+ 00
1:250000000000000e 01 9:724452119338235e 04 1:537501583136462e+ 00
6:250000000000000e 02 3:437050954186293e 04 1:500445730420506e+ 00
3:125000000000000e 02 1:215162841762690e 04 1:500021580057984e+ 00
1:562500000000000e 02 4:295235806309275e 05 1:500340417834547e+ 00
25
0 1 2 3 4 5 6 7 8
!1
0
1
Number of Points
Function f
0 1 2 3 4 5 6 7 8
!1
0
1
Number of Points
Spline
0 1 2 3 4 5 6 7 8
!2
0
2
x 10
!3
Number of Points
Function f
!
spline
Figure 3.5: Approximation of the function
f( ) = sin3 ;h = 2:5 10 1
0 1 2 3 4 5 6 7 8
!1
0
1
Number of Points
Function f
0 1 2 3 4 5 6 7 8
!1
0
1
Number of Points
Spline
0 1 2 3 4 5 6 7 8
!1
0
1
x 10
!5
Number of Points
Function f
!
spline
Figure 3.6: Approximation of the function
f( ) = sin3 ;h = 6:25 10 2
It is evident from the Figure 3.8 below that the function f =2C1[a;b] and hence
the observed order of convergence. In view of the approximation power of splines
[3] we may expect that the order of approximation attainable will increase with the
smoothness of the class of functions F being approximated. However this is true only
up to a limit. In fact if F\Pm =;; then the maximal order of convergence possible
for the class F is m; no matter how smooth F is assumed. In our case m = 4: This
is known as the saturation result.
0 1 2 3 4 5 6 7 8
0
0.5
1
Number of Points
Function f
0 1 2 3 4 5 6 7 8
0
0.5
1
Number of Points
Spline
0 1 2 3 4 5 6 7 8
!0.1
0
0.1
Number of Points
Function f
!
spline
Figure 3.7: Approximation of a function
f =2C1[a;b];h = 1
0 1 2 3 4 5 6 7 8
0
0.5
1
Number of Points
Function f
0 1 2 3 4 5 6 7 8
0
0.5
1
Number of Points
Spline
0 1 2 3 4 5 6 7 8
!4
!2
0
2
4
x 10
!3
Number of Points
Function f
!
spline
Figure 3.8: Approximation of a function
f =2C1[a;b];h = 1:5625 10 2
26
3.2 B-spline representation on a Square
Our aim here is to construct a cubic spline which is represented as a linear
combination of the B splines Bi;m(x) as in Section 1.4. We consider the extended
partition =fxign+m 1 m+1 of the interval [a;b] as de ned in the De nition 1.5. Using
the periodicity condition available to us through the equations (1.14),(1.15) and (1.16)
we have a system of linear equations available to us which we can represent in a matrix
form. We denote the matrix of all basis functions as B: Using the properties of the
B-spline and noting that the spline function must agree with the function values at
the nodal points xi; i = 0;1;:::;n 1 we arrive at the following matrix representation
of the linear system,
BC = F;
which implies C = B 1F:
B =
0
BB
BB
BB
BB
BB
BB
BB
BB
BB
BB
BB
BB
BB
B@
0 ::: ::: ::: ::: ::: 0 B0 3 B0 2 B0 1
B10 0 ... ... ... ... ... 0 B1 2 B1 1
B20 B21 0 ... ... ... ... ... 0 B2 1
B30 B31 B32 0 ... ... ... ... ... 0
0 B41 B42 B43 0 ... ... ... ... 0
0 0 ... ... ... ... ... ... ... 0
... 0 0 ... ... ... ... ... ... ...
0 ::: ::: ::: Bn 3n 6 Bn 3n 5 Bn 3n 4 0 0 0
0 ::: ::: ::: ::: Bn 2n 4 Bn 2n 3 0 0
0 ::: ::: ::: ::: ::: Bn 1n 3 Bn 1n 2 0
1
CC
CC
CC
CC
CC
CC
CC
CC
CC
CC
CC
CC
CC
CA
(3.9)
The matrix B has as its entries B-splines evaluated at the nodal points where Bji
represents Bi;m(xj); i = m+ 1; m+ 2;:::;n 1 and j = 0;1;:::;n 1: We have
27
the zero entries in the above matrix because the B-spline Bi;m(x) vanishes outside its
local support [xi;xi+m]:
The vector of all coe cients is represented as C and the vector of all function
values at the nodal points is represented as F, then
C =
0
BB
BB
BB
BB
BB
BB
BB
BB
BB
BB
BB
BB
BB
B@
C0
C1
C2
C3
C4
C5
...
Cn 3
Cn 2
Cn 1
1
CC
CC
CC
CC
CC
CC
CC
CC
CC
CC
CC
CC
CC
CA
and F =
0
BB
BB
BB
BB
BB
BB
BB
BB
BB
BB
BB
BB
BB
B@
f0
f1
f2
f3
f4
f5
...
fn 3
fn 2
fn 1
1
CC
CC
CC
CC
CC
CC
CC
CC
CC
CC
CC
CC
CC
CA
(3.10)
The problem of representing the cubic spline as a linear combination of the basis
functions is numerically equivalent to the problem of nding the B-spline coe cients,
i.e., the vector C: Since the matrix B and the vector F is completely known we can
easily solve for C on a digital computer.
Table 3.4: The following table shows the error and the observed rate of convergence
for various step sizes for the function f( ) = sin( 4 ).
Step Size Error Order of Convergence
1:000000000000000e+ 00 3:412826951770349e 03
5:000000000000000e 01 9:534612985020590e 05 5:161649073886882e+ 00
2:500000000000000e 01 4:520615188600750e 06 4:398583359486799e+ 00
1:250000000000000e 01 2:618848335825963e 07 4:109514698295172e+ 00
6:250000000000000e 02 1:605166782410018e 08 4:028137400685653e+ 00
3:125000000000000e 02 9:983146507216004e 10 4:007084797322440e+ 00
28
As we are approximating a function de ned on a square of edge length 2, the
approximating spline will have a periodicity of b a which in our case is 8: Hence the
factor of 4 in the function f( ) = sin 4 :
0 1 2 3 4 5 6 7 8
!1
0
1
Number of Points
Function f
0 1 2 3 4 5 6 7 8
!1
0
1
Number of Points
Spline
0 1 2 3 4 5 6 7 8
!5
0
5
x 10
!7
Number of Points
Function f
!
spline
Figure 3.9: Approximation of a function
f( ) = sin 4 ;h = 6:25 10 2
0 1 2 3 4 5 6 7 8
!1
0
1
Number of Points
Function f
0 1 2 3 4 5 6 7 8
!1
0
1
Number of Points
Spline
0 1 2 3 4 5 6 7 8
!2
0
2
x 10
!9
Number of Points
Function f
!
spline
Figure 3.10: Approximation of a function
f( ) = sin 4 ;h = 1:56 10 2
Table 3.5: The following table shows the error and the observed rate of convergence
for various step sizes for the function f =2C1[a;b]
Step Size Error Order of Convergence
1:000000000000000e+ 00 7:968190728250351e 03
5:000000000000000e 01 7:968190728250389e 03 7:047538308392802e 15
2:500000000000000e 01 2:576965563447752e 03 1:628578922185994e+ 00
1:250000000000000e 01 8:877169929217066e 04 1:537501539184818e+ 00
6:250000000000000e 02 3:137585531721901e 04 1:500445025990139e+ 00
3:125000000000000e 02 1:109295998711029e 04 1:500010409963090e+ 00
1:562500000000000e 02 3:921497851505102e 05 1:500167662913683e+ 00
3.3 Tensor Product Splines
In this section we will construct the tensor product spline rst on a square patch
and then extend this notion to the cube treating each face of the cube as a square
29
patch. This will give us a method to approximate any function posed on the sphere
with the help of the mapping P and P 1 from the surface of the cube to the surface
of the sphere and back respectively.
From equation (1.24) we have the unique tensor product representation of a spline
de ned over a rectangle [a;b] [c;d] with respect to the extended partition 1 and 2
given by De nition 1.5 in the horizontal and vertical directions respectively. Keeping
in mind that we do not want periodicity on a single patch but rather across the four
faces of the cube we do not implement the periodic conditions for the single patch
which we create. The cube in question is the box Bd as de ned in Chapter 2. It is
centered at the origin with d = 1:
It is clear from the above setup that that the total number of coe cients that
need to be determined are (n+3) (p+3) since m = l = 4: However we only have the
function values at the nodal points (xi;yj); i = 0;1;:::;n; j = 0;1;:::;p: Hence we
still need 2n+ 2p+ 8 conditions to have a unique spline representation on the patch.
These extra conditions are given as restrictions on the derivatives of the spline at the
boundary and the corner points of the grid formed by the partitions 1 and 2:
We give here a brief list of the boundary conditions generally associated with the
tensor product cubic splines.
Boundary conditions of the rst type
@s
@x(xi;yj) = f
x
ij; i = 0;n; j = 0;1;:::;p
@s
@y(xi;yj) = f
y
ij; i = 0;1;:::;n; j = 0;p
@2s
@x@y(xi;yj) = f
xy
ij ; i = 0;n; j = 0;p (3.11)
The total number of rst boundary conditions here is 2n+ 2p+ 8:
Boundary conditions of the second type
30
@2s
@x2(xi;yj) = f
x
ij; i = 0;n; j = 0;1;:::;p
@2s
@y2(xi;yj) = f
y
ij; i = 0;1;:::;n; j = 0;p
@4s
@x2@y2(xi;yj) = f
xy
ij ; i = 0;n; j = 0;p (3.12)
The total number of second boundary conditions is again 2n+ 2p+ 8:
Boundary conditions of the third type
Boundary conditions of the third type are called periodic boundary conditions. Peri-
odicity with respect to the horizontal variable must be Px = b a and with respect
to the vertical variable must be Py = d c: For our particular case we must have
Px = Py = 8
s(x0;yj) = s(xn;yj); j = 0;1;:::;p
s(xi;y0) = s(xi;yp); i = 0;1;:::;n
@ks
@xk(x0;yj) =
@ks
@xk(xn;yj); j = 0;1;:::;p; k = 1;2
@ls
@yl(xi;y0) =
@ls
@yl(xi;yp); i = 0;1;:::;n; l = 1;2
@2ks
@xk@yk(x0;yj) =
@ks
@xk@yk(xn;yj); j = 0;1;:::;p; k = 1;2
@2ks
@xk@yk(xi;y0) =
@2ks
@xk@yk(xi;yp); i = 0;1;:::;n; k = 1;2 (3.13)
Applying the boundary conditions of the rst type in order to determine the
coe cients we end up with a system of linear equations which in matrix form may be
represented as
MCN = F
C = M 1FN 1 (3.14)
31
where M(p+3) (p+3) is the matrix of B-splines in the vertical direction, N(n+3) (n+3)
is the matrix of B-splines in the horizontal direction, C(p+3) (n+3) is the coe cient
matrix and F(p+3) (n+3) is the matrix of all function values at the nodal points in the
grid along with its derivatives at the boundary and corner points.
Table 3.6: The following table shows the error and the observed rate of convergence
for various step sizes for the function f(x;y) = x6y6
Step Size Error Order of Convergence
1:000000000000000e+ 00 7:060256060377759e+ 00
5:000000000000000e 01 9:550664958381095e 01 2:886047419513821e+ 00
2:500000000000000e 01 5:683782321068801e 02 4:070677975228285e+ 00
1:250000000000000e 01 3:408501271897639e 03 4:059641876435491e+ 00
6:250000000000000e 02 2:095067852178469e 04 4:024068647573170e+ 00
!1
!0.5
0
0.5
1
!1
!0.5
0
0.5
1
0
0.2
0.4
0.6
0.8
1
X ! Axis
Graph of the Function values at Intermediate Points
Y !Axis
Function Values
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 3.11: Function f(x;y) = x6y6
!1
!0.5
0
0.5
1
!1
!0.5
0
0.5
1
!0.4
!0.2
0
0.2
0.4
0.6
0.8
1
X ! Axis
Graph of the Spline Function values at intermediate Points
Y !Axis
Function Values
!0.2
0
0.2
0.4
0.6
0.8
Figure 3.12: Approximation of a function
f(x;y) = x6y6;h = 1
To obtain the the tensor product spline representation on the cube we note that
spline function must be periodic with periodicity along x-direction Px, y-direction Py
and z-direction Pz and Px = Py = Pz = 8. Even though the tensor products are
de ned on individual patches, the spline function must be periodic across the four
adjacent faces of the cube. Let us consider here the tensor product de ned on a
32
!1
!0.5
0
0.5
1
!1
!0.5
0
0.5
1
0
0.2
0.4
0.6
0.8
1
X ! Axis
Graph of the Spline Function values at intermediate Points
Y !Axis
Function Values
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 3.13: Approximation of a function f(x;y) = x6y6;h = 1:25 10 1
single patch. Then to obtain periodicity along the horizontal we need to wrap the
ctitious nodes u i;i = m+1;:::; 1 and un+i;i = 1;:::;m 1 along the horizontal
so that they coincide with the nodal points on the adjacent sides. For periodicity
along the vertical we do the same, i.e., wrap the nodes v i;i = m + 1;:::; 1 and
vp+i;i = 1;:::;m 1 along the vertical so that they coincide with nodal points on
the adjacent sides. To do this we must have the following
u i = ui hw; i = 1;2;3
un+i = un +hw; i = 1;2;3 (3.15)
and
v i = vi hw; i = 1;2;3
vp+i = vp +hw; i = 1;2;3 (3.16)
where ui denotes the nodal points along the horizontal, vj denotes the nodal points
along the vertical and hw is the step size of the adjacent side.
33
Table 3.7: The following table shows the error and the observed rate of convergence
for various step sizes for the function f(x;y;z) = sin(xyz)
Step Size Error Order of Convergence
1:000000000000000e+ 00 1:670920218139921e 01
5:000000000000000e 01 7:961222685614932e 03 4:391509023007923e+ 00
2:500000000000000e 01 4:648371361131741e 04 4:098192780859205e+ 00
1:250000000000000e 01 2:836252157206563e 05 4:034667624804363e+ 00
6:250000000000000e 02 1:759114473732576e 06 4:011064527357291e+ 00
3:125000000000000e 02 1:097374360545641e 07 4:002721689943840e+ 00
!1
!0.5
0
0.5
1
!0.5
0
0.5
!0.8
!0.6
!0.4
!0.2
0
0.2
0.4
0.6
0.8
x
y
z
Figure 3.14: Mesh on the sphere Sr
!1
!0.5
0
0.5
1
!1
!0.5
0
0.5
1
!1
!0.5
0
0.5
1
x
y
z
Figure 3.15: Mesh on the cube Bd
34
Chapter 4
Conclusion
We developed and analyzed a method to approximate functions posed on the
sphere. The method described here is based on tensor product splines and because
of the tensor nature of the resulting space many algebraic properties of univariate
polynomial splines can easily be carried over. Although the tensor product splines
have a restricted use they are computationally advantageous due to their easy imple-
mentation on a digital computer.
35
Bibliography
[1] Carl de Boor, \A Practical Guide to Splines," Applied Mathematical Sciences,
Vol. 27. Springer, New York, 1978.
[2] Paul Dierckx, \Curve and Surface Fitting with Splines," Oxford Science Publi-
cations, 1993.
[3] Larry L. Schumaker, \Spline Functions:Basic theory," Wiley, New York, 1981.
[4] Necibe Tuncer, \A Novel Finite Element Discretization of Domains with
Spheroidal Geometry," Ph.D Thesis, Auburn University, AL, May 2007.
[5] Al o Quarteroni, Ricardo Sacco, Fausto Saleri, \Numerical Mathematics,"
Springer, 2000.
[6] Martin H. Schultz, \Spline Analysis," Prentice-Hall, Inc., 1973.
[7] M. G. Cox, \The Numerical evaluation of B-Splines," Journal of the Institute of
Mathematical Applications, 10 (1972), 134-149.
[8] A. Ralston, \A rst course in Numerical Analysis," Mc-Graw-Hill, New York,
1965.
[9] C. Ronchi, R. Iaconco, P. S. Paolucci \The \cubed-sphere": A New Method for
the Solution of Partial Di erential Equations in Spherical Geometry," J. Comput.
Phy., 124 (1996), 93-114.
[10] C. R. Trass, \Smooth Approximation of Data on the Sphere," Computing, 38
(1987), 177-184.
[11] Amnon J. Meir, Necibe Tuncer, \Radially Projected Finite Elements," SIAM J.
Sci. Comput., 31 (2007), 2368-2385.
36
Appendices
37
Appendix A
Notations
A.0.1 One dimensional case
I [a;b] fxja6x6bg:
For each nonnegative integer t and for each p; 1 6 p 61; PCpt(I) will denote the
set of all real valued functions f(x) such that
1. f(x) is t 1 times continuously di erentiable,
2. there exists xi; 0 6i6n 1; with
a = x0