Acta Polytechnica


doi:10.14311/AP.2016.56.0202
Acta Polytechnica 56(3):202–213, 2016 © Czech Technical University in Prague, 2016

available online at http://ojs.cvut.cz/ojs/index.php/ap

ON CUBATURE RULES ASSOCIATED TO WEYL
GROUP ORBIT FUNCTIONS

Lenka Hákováa, ∗, Jiří Hrivnákb, Lenka Motlochováb

a Department of Mathematics, Faculty of Chemical Engineering, University of Chemistry and Technology,
Prague, Technická 5, CZ-166 28 Prague, Czech Republic

b Department of Physics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University
in Prague, Břehová 7, CZ-115 19 Prague, Czech Republic

∗ corresponding author: lenka.hakova@vscht.cz

Abstract. The aim of this article is to describe several cubature formulas related to the Weyl
group orbit functions, i.e. to the special cases of the Jacobi polynomials associated to root systems.
The diagram containing the relations among the special functions associated to the Weyl group orbit
functions is presented and the link between the Weyl group orbit functions and the Jacobi polynomials
is explicitly derived in full generality. The four cubature rules corresponding to these polynomials are
summarized for all simple Lie algebras and their properties simultaneously tested on model functions.
The Clenshaw-Curtis method is used to obtain additional formulas connected with the simple Lie
algebra C2.

Keywords: Weyl group orbit functions; Jacobi polynomials; cubature formulas.

1. Introduction
The purpose of this paper is to explicitly overview in
full generality the link between the Weyl group orbit
functions and the Jacobi and Macdonald polynomials
and further examine and compare related methods of
numerical integration. These methods of numerical
integration, known as cubature rules, emerged recently
for all four cases of the Weyl group orbit functions.

The four families of the Weyl group orbit functions
[10, 18, 19, 26, 30] are connected to four families of
orthogonal polynomials via similar relations between
Chebyshev polynomials of the first and second kinds
and the ordinary cosine and sine functions. A full set
of four families of orbit functions, called C-, S-, Ss-
and Sl-functions, arises from root systems of simple
Lie algebras with two different lengths of roots. These
four families of orthogonal polynomials are in fact
special cases of the multivariate Jacobi polynomials
[11, 12]. The Jacobi polynomials associated to root
systems are in turn limiting cases of the Macdonald
polynomials [24]. This connection between the four
cases of the Jacobi polynomials and the underlying
orbit functions allows to formulate the corresponding
methods for numerical integration in terms of the
Jacobi polynomials.

Among methods for numerical integration, the
quadrature and cubature formulas related to polyno-
mials of a bounded degree hold a prominent place [1, 6–
8, 35, 38]. Such formulas estimate a given weighted
integral over a fixed domain in Euclidean space. This
estimation holds exactly for all polynomials up to a
certain degree. A significant effort put into develop-
ment of various types of cubature formulas results
in multitude of types of integration domains with

varying efficiencies. The shapes of the integration
domains and the nodes for cubature formulas corre-
sponding to the orthogonal polynomials of the Weyl
group orbit functions are determined by the symme-
tries of the affine Weyl groups and a certain transform
[15, 22, 23, 26, 27]. This transform is generated by the
transform which induces the given set of orthogonal
polynomials. Moreover, a specific notion of the modi-
fied degree of multivariate polynomials is essential for
establishing the final cubature formulas.

One of the specific methods of deriving quadrature
formulas, known as Clenshaw-Curtis method [5], is
classically related to Chebyshev polynomials of one
variable [9]. Its two-dimensional version related to two-
variable Chebyshev polynomials of the root system A2
is also developed [31]. The importance of this method
lies e.g. in its utilization for practical optimization of
the shapes of integration domains. The shapes of the
integration domains are determined by the underlying
Lie algebra [15] and are, however, of non-standard
form. In case of simple Lie algebras related to two-
variable functions, one of the possible optimizations of
these shapes is, similarly to [31], inscribing a triangle
into the original fundamental domain. The focus of
the present article is on simple Lie algebra C2 and its
corresponding cubature rules.

The integration domain in the case of C2 is a region
bounded by two lines and a parabola depicted in Fig. 4.
Except from a general perspective in [15, 26, 27], inte-
gration over this region is studied in [32]. Similarly to
[31] for A2, the non-standard shape of this integration
domain motivates further exploration of the Clenshaw-
Curtis method. This method crucially depends on the
choice of the weight function and the inscribed integra-
tion region and has not yet been studied in detail for

202

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vol. 56 no. 3/2016 On Cubature Rules Associated to Weyl Group Orbit Functions

the case of C2. In this case, the domain inscribed in
the original integration region is considered either the
original domain itself or the triangle depicted in Fig.
6. Two fundamental choices of the weight functions
are detailed. Prior to practical implementation, exact
values of certain integrals are also needed. One of the
goals of this article is to provide all data necessary
for practical implementation of these cubature formu-
las. This is achieved by tabulating and calculating
all needed stabilizer coefficients and exact integral
values for each choice it the inscribed integration do-
main and weight function. To demonstrate usefulness
and viability of the presented methods, the numerical
tests on model functions, including multidimensional
step-functions, are also performed.

The development of novel cubature formulas is mo-
tivated by their widespread use in applied numerical
simulations and engineering problems. Among direct
numerical applications of the cubature formulas is the
induced method of polynomial approximation. The
cubature formulas are ubiquitous in the modern the-
ory of electromagnetism, especially in its branches
of electromagnetic wave propagation [33], magneto-
static modeling [39] and micromagnetic simulations [4].
Other fields include fluid flows simulations, laser optics
and stochastic dynamics.

In Section 2 are reviewed the notions necessary for
definition of the Weyl group orbit functions. The
relation between the orbit functions and the Jacobi
polynomials is detailed. In Section 3, the cubature
formulas from [15, 26, 27] are summarized, Clenshaw-
Curtis method is described and used to derive addi-
tional cubature formulas. Furthermore, numerical test
results are presented.

2. Special functions associated
to root systems

2.1. Basic definitions
This section reviews the basic concepts and notation
from the theory of root systems, Weyl groups and
Weyl group orbit functions. It is consistent with the
notation used in recent papers regarding the topic,
such as [10, 13–15, 26] and others. We consider sim-
ple Lie algebras, i.e. four infinite families An(n ≥ 1),
Bn(n ≥ 3), Cn(n ≥ 2) and Dn(n ≥ 4) and five excep-
tional algebras E6, E7, E8, F4 and G2 (for the clas-
sification see [2, 16]). In particular, we focus on the
algebra C2 as the simplest non-trivial example. Each
simple Lie algebra is completely described by its set
of simple roots ∆ = {α1, . . . ,αn} which forms a non-
orthogonal basis of the Euclidean space Rn equipped
with a scalar product denoted by 〈·, ·〉. Simple roots
are either of the same length or of two different lengths,
in the latter case we distinguish so-called short and
long roots and write ∆ = ∆s ∪ ∆l.
The set of dual roots is denoted by

∆∨ = {α∨1 , . . . ,α
∨
n} ,

where α∨i =
2αi
〈αi,αi〉

. In addition to the bases of simple
roots and dual roots we introduce the weight basis
ω1, . . . ,ωn and the dual weight basis ω∨1 , . . . ,ω∨n where
〈α∨i , ωj〉 = 〈αi, ω

∨
j 〉 = δij. The Cartan matrix C

is defined as Cij =
2〈αi,αj〉
〈αj,αj〉

and its determinant is
denoted by c.

Each simple root αi relates to a reflection ri defined
for every a ∈ Rn as

ria = a−
2〈a, αi〉
〈αi, αi〉

αi.

The set of reflections {r1, . . . ,rn} generates a finite
group W called the Weyl group. By the action of W on
the set of simple roots we obtain the root system Π =
W ∆. Analogously, we define Π∨ = W ∆∨, Πs = W ∆s
and Πl = W ∆l. Every element of Π can be written as
a combination of simple roots with only non-negative
(positive roots) or non-positive integer coefficients
(negative roots). The set of positive roots is denoted
by Π+. We define a partial ordering � of roots, µ � λ
if µ − λ is a sum of simple roots with non-negative
integer coefficients. There is a unique highest root ξ
with respect to this ordering, its coordinates in the
basis of simple roots are called marks and denoted by
m1, . . . ,mn. Dual root system Π∨ contains the highest
root η = m∨1 α∨1 + · · · + m∨nα∨n with the coefficients
m∨i called dual marks.
An infinite extension of the Weyl group W is the

affine Weyl group W aff which is obtained by adding
to the set of generators of W the affine reflection r0,

r0a = rξa +
2ξ
〈ξ, ξ〉

, rξa = a−
2〈a, ξ〉
〈ξ, ξ〉

ξ.

It can also be written as a semidirect product of W and
a set of shifts by integer combinations of dual roots [13].
We denote by ψ the retraction homomorphism W aff →
W [14]. The fundamental domain F - a set containing
exactly one point from each W aff orbit - can be chosen
as

F =
{
b1ω
∨
1 + · · · + bnω

∨
n |

bi ∈ R≥0, b0 + b1m1 + · · · + bnmn = 1
}
. (1)

Analogously we define dual affine Weyl group as a
semidirect product of W and shifts by integer combi-
nations of simple roots.
We introduce three lattices P,P + and P∨ as

P = Zω1 + · · · + Zωn,
P + = Z≥0ω1 + · · · + Z≥0ωn,
P∨ = Zω∨1 + · · · + Zω

∨
n.

Note that the root system Π is contained in P , there-
fore, the partial ordering � can be extended to the
lattice P.
A function k : α ∈ Π → kα ∈ R≥0 such that

kα = kw(α) for all w ∈ W

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L. Háková, J. Hrivnák, L. Motlochová Acta Polytechnica

is known as a multiplicity function on Π. The trivial
example is to take kα = const for all α ∈ Π which
we denote by kconst. For simple Lie algebras with
two different root lengths, it is natural to distinguish
between short and long roots by defining

ksα ≡

{
1 if α ∈ Πs,
0 if α ∈ Πl,

klα ≡

{
0 if α ∈ Πs,
1 if α ∈ Πl.

The notion of multiplicity function allows us to
define sums of positive roots %(k) and numbers h(k),

%(k) ≡
1
2
∑
α∈Π+

kαα, (2)

h(k) = kξ +
n∑
i=1

mikαi.

In particular, with the choice of kt , where t is one of
the symbols {0, 1,s, l}, we have

%0 ≡ %(k0) = 0,

%1 ≡ %(k1) =
n∑
i=1

ωi,

%s ≡ %(ks) =
∑
αi∈∆s

ωi,

%l ≡ %(kl) =
∑
αi∈∆l

ωi (3)

and

h0 ≡ h(k0) = 0,

h1 ≡ h(k1) = 1 +
n∑
i=1

mi,

hs ≡ h(ks) =
∑
αi∈∆s

mi,

hl ≡ h(kl) = 1 +
∑
αi∈∆l

mi. (4)

The number h ≡ h1 is called the Coxeter number,
analogously, we call hs and hl short and long Coxeter
number.

The set of simple roots ∆ = {α1,α2} of the algebra
C2 decomposes into the set of the short simple roots
∆s = {α1} and the set of the long simple roots ∆l =
{α2}. The highest root is of the form ξ = 2α1 + α2
and the dual highest root is η = α∨1 + 2α∨2 . Thus,
the sets of mark and dual marks are (m1,m2) = (2, 1)
and (m∨1 ,m∨2 ) = (1, 2). The vectors %t are of the
form (%1,%s,%l) = (ω1 + ω2,ω1,ω2) and the Coxeter
numbers are (h1,hs,hl) = (4, 2, 2). The roots and
dual roots, the weights and dual weights, together
with the fundamental domain F and the vectors %t
are depicted in Figure 1.

2.2. Weyl group orbit functions
The definition of Weyl group orbit function uses the
notion of a sign homomorphisms σt : W 7→±1, where

Figure 1. Root system of C2. The white circles
denote the roots, black dots depict the dual roots.
The triangle denotes the fundamental domain F . The
lines denoted r1, r2 and r0 depict reflecting mirrors
which realize the corresponding reflections.

t ∈{0, 1,s, l}. These can be defined by the values on
the reflections corresponding to the simple roots ri,
namely

σ0(ri) = 1, αi ∈ ∆,
σ1(ri) = −1, αi ∈ ∆,

σs(ri) =
{

1 if αi ∈ ∆l,
−1 if αi ∈ ∆s,

σl(ri) =
{

1 if αi ∈ ∆s,
−1 if αi ∈ ∆l.

Several families of special functions are connected
with each Weyl group W. They are labelled by
vectors λ ∈ P + and defined as weighed sums over
the corresponding Weyl group orbit, i.e. the set
Wλ = {wλ | w ∈ W }. For every x ∈ Rn and
t ∈{0, 1,s, l} we define

Stλ+%t(x) =
∑

µ∈W(λ+%t)

σt(µ)e2πi〈µ,x〉, (5)

where %t is given by (3) and σt(µ) ≡ σt(w) for w such
that µ = w(λ + %t). Functions corresponding to the
choice of t = 0 and t = 1 are usually called C- and S-
functions respectively, in the formulas in next sections
we use the notation S0 and S1 for a simplicity.

Families of C-, S-, Ss- and Sl-functions are com-
plex multivariate functions with remarkable properties
such as (anti-)invariance with respect to the action
of the affine Weyl group, continuous and discrete or-
thogonality. They were studied in many papers, see
for example [18, 19, 26] for the general properties
and [13, 14, 28] for their discretization.

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vol. 56 no. 3/2016 On Cubature Rules Associated to Weyl Group Orbit Functions

Fundamental domains Ft are defined as subsets of
F such that we omit the part of the boundary of F
which is stabilized by certain generating reflections
r ∈ R = {r0,r1 . . . ,rn}. More precisely, with the
notation

Rt = {r ∈ R | σt ◦ψ(r) = −1},
Ht = {a ∈ F | ∃r ∈ Rt, ra = a},

we define Ft = F\Ht. The explicit forms are obtained
from (1) and can be found in [14].

The C-, S-, Ss- and Sl-functions can be viewed as
functional forms of elements from the algebra C[P]
containing all complex linear combinations of formal
exponentials ea,a ∈ P, with multiplication defined
by ea · eb = ea+b, the inverse given by (ea)−1 = e−a
and the identity e0 = 1. The connection is based
on the exponential mapping from Lie algebra to the
corresponding Lie group [2, 16, 27].

2.3. Jacobi polynomials
We assume that the multiplicity function k satisfies
kα ≥ 0. The Jacobi polynomial P (λ,k) [11, 12] associ-
ated to the root system Π with highest weight λ ∈ P +
and multiplicity function k as parameter is defined by
the following formulas.

P(λ,k) ≡
∑
µ∈P+
µ�λ

cλµ(k)Cµ, Cµ =
∑

µ′∈Wµ
eµ

′
, (6)

where the coefficients cλµ(k) are recursively given by(
〈λ + %(k),λ + %(k)〉−〈µ + %(k),µ + %(k)〉

)
cλµ(k)

= 2
∑
α∈Π+

kα

∞∑
j=1
〈µ + jα,α〉cλ,µ+jα(k)

along with the initial value cλλ = 1 and the assump-
tion cλµ = cλ,w(µ) for all w ∈ W. Recall that %(k) is
defined by (2).
By setting kα = 0 for all α ∈ Π, the Jacobi poly-

nomials lead trivially to C-functions. In the case
k = k1, the formula for the calculation of the co-
efficients becomes the Freudenthal’s recurrence for-
mula [16]. Therefore, each P(λ,k1) specializes to a
character χλ of an irreducible representation of the
simple Lie algebra of the highest weight λ, i.e.,

P(λ,k1) = χλ =
Sλ+%1

S%1
.

In addition, we show that the Jacobi polynomials are
related to the Ss- and Sl-functions in the following
way.

P(λ,ks) =
Ssλ+%s

Ss%s
and P(λ,kl) =

Sl
λ+%l

Sl
%l

.

We first observe that Ssλ+%s/S
s
%s are Weyl group

invariant elements of C[P ] (see Proposition 4.2 of [26])

with well-known basis formed by C-functions [2].
Therefore, each Ssλ+%s/S

s
%s can be expressed as a linear

combination of C-functions. By the definition of the
Weyl group, the weights of exponentials in Ssλ+%s are
of the form λ+%s−g(α1, . . . ,αn), where g(α1, . . . ,αn)
denotes a sum of simple roots with non-negative in-
teger coefficients, and the unique maximal weight is
λ + %s. Similarly, the unique maximal weight of Ss%s
is %s. Therefore, we have

Ssλ+%s

Ss%s
=
∑
µ∈P+
µ�λ

bµCµ, bλ = 1.

To prove bµ = cλµ(ks), we proceed by using an equiv-
alent definition of Jacobi polynomials with the mul-
tiplicity function satisfying kα ∈ Z≥0 [12]. For any
f =

∑
λ aλe

λ, we define

f ≡
∑
λ

aλe
−λ and CT(f) ≡ a0.

If we introduce the scalar product (·, ·) on C[P ] by

(f,g) ≡ CT
(
fgδ(k)

1
2 δ(k)

1
2
)
, f,g ∈ C[P ],

δ(k)
1
2 ≡

∏
α∈Π+

(
e

1
2 α −e−

1
2 α
)kα

,

then the Jacobi polynomials P(λ,k) are the unique
polynomials of the form (6) satisfying the requirement(

P(λ,k),P(µ,k)
)

= 0

for all µ ∈ P + such that µ � λ and λ 6= µ assuming
cλλ = 1.

Using Proposition 4.1 of [26], i.e. δ(ks)
1
2 = Ss%s, we

obtain(
Ssλ+%s

Ss%s
,
Ssµ+%s

Ss%s

)
= CT(Ssλ+%sSsµ+%s)

= CT
( ∑
λ′∈W(λ+%s)

∑
µ′∈W(µ+%s)

σs(λ′)σs(µ′)eλ
′−µ′

)
.

Clearly λ′ = µ′ if and only if there exists w ∈ W such
that λ + %s = w(µ + %s). For we consider λ ∈ P +
different from µ ∈ P +, it is not possible to have
λ′ = µ′. This implies that(

Ssλ+%s

Ss%s
,
Ssµ+%s

Ss%s

)
= 0 and P(λ,ks) =

Ssλ+%s

Ss%s
.

The proof for the long root case is similar.
Finally, note that the Jacobi polynomials can be

viewed as the limiting case of the Macdonald poly-
nomials Pλ(q,tα) when tα = qkα with kα fixed and
q → 1. See [24] for more details. The relations among
several special functions associated with the Weyl
groups, which are summarized in [29], are depicted in
Figure 2.

205



L. Háková, J. Hrivnák, L. Motlochová Acta Polytechnica

Macdonald polynomials [24]
Pλ(q,tα), tα = qkα

Jacobi polynomials associated to root systems [11]
P(λ,k) (Section 2.3)

C-functions and S-functions [18, 19]
Cλ = P(λ,k0)

Sλ+%1 = P(λ,k
1)S%1

Ss-functions and Sl-functions [26]
Ssλ+%s = P(λ,k

s)Ss%s
Sl
λ+%l = P(λ,k

l)Sl
%l

CCk and SSk [21]? cos+ and sin− [17] SCk and CSk [21] cos− and sin+ [17]

Chebyshev polynomials [9]
Tm,Um,Vm and Wm

2-D Jacobi polynomials pα,β,γk1,k2
with α,β,γ ∈

{
±12
}
[20]

2-D Jacobi polynomials
on Steiner’s hypocycloid [20]

Jacobi polynomials [36]
P

(α,β)
m

q → 1, kα fixed

k ∈{ks,kl}k ∈{k0,k1}

G2An Bn and Cn G2 Bn and Cn

A1

A2 C2 C2

α,β ∈
{
±12
}

Figure 2. The diagram of relations among several special functions associated with Weyl groups.

3. Cubature formulas
3.1. General form of cubature formulas
Analogously to Chebyshev polynomials, we identify
polynomial variables y1, . . . ,yn with real-valued func-
tions in the following way. Let zj ≡ Cωj, then

A2k : yj = <(zj), y2k−j+1 = =(zj), j = 1, . . . ,k,
A2k+1 : yj = <(zj), yk+1 = zk+1, y2k−j+2 = =(zj),

j = 1, . . . ,k,
D2k+1 : yj = zj, y2k = <(z2k),

y2k+1 = =(z2k), j = 1, . . . , 2k − 1,
E6 : y1 = <(z1), y2 = <(z2), y3 = z3,

y4 = =(z2), y5 = =(z1), y6 = z6,

otherwise we put yj = zj. We say that a monomial
yλ11 . . .y

λn
n has an m-degree

degm y
λ1
1 . . .y

λn
n = m

∨
1 λ1 + · · · + m

∨
nλn

and any polynomial p in C[y1, . . . ,yn] has an m-degree
equal to the largest m-degree of the monomials oc-
curring in p. We denote the subspace containing all
polynomials of m-degree at most M by ΠM. For
λ = λ1ω1 + · · · + λnωn, the C-functions Cλ and thus
all Jacobi polynomials P(λ,k) can be rewritten as

orthogonal polynomials in variables y1, . . . ,yn of m-
degree equal to m∨1 λ1 + · · · + m∨nλn [15].

The variables y1, . . . ,yn viewed as functions induce
a map

Ξ : Rn → Rn, Ξ(x) = (y1(x), . . . ,yn(x)).

The map Ξ is used to define the integration region Ω
and the sets of nodes ΩtM, t ∈{0, 1,s, l} by

Ω ≡ Ξ(F 1), ΩtM ≡ Ξ
( 1
M + ht

P∨ ∩Ft
)
.

Let

StabWaff (x) ≡
{
w ∈ W aff

∣∣ wx = x},
ε(x) ≡

|W |
|StabWaff (x)|

,

and define a map ε̃ : Ω0M → N by

ε̃(y) ≡ ε(Ξ−1M y), ΞM = Ξ � 1
M+ht

P ∨∩F0 .

Denoting K(y1, . . . ,yn) ≡
√
S%1S%1, the weight

functions are given by

st(y1, . . . ,yn) ≡ St%tSt%t,

wt(y1, . . . ,yn) ≡
st(y1, . . . ,yn)
K(y1, . . . ,yn)

, t ∈{0, 1,s, l}.

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vol. 56 no. 3/2016 On Cubature Rules Associated to Weyl Group Orbit Functions

Figure 3. The fundamental domain F = F 0 of C2
is depicted as the triangle with dashed boundary Hs
and dot-and-dashed boundary Hl. The black dots
correspond to the points from 110 P

∨∩F . The numbers
1, 2, 4 of the dots are the values of ε(x), the inner dots
have ε(x) = 8.

Since the products St%tS
t
%t
, t ∈ {0, 1,s, l} are W-

invariant sums of exponentials from C[P], they
are expressible as functions in polynomial variables
y1, . . . ,yn.

In [15, 26, 27] is shown that the following cubature
formulas are exact equalities for any M ∈ N and any
polynomial p which satisfies the following constraints,
• degm p ≤ 2M − 1 for t ∈{0, l},
• degm p ≤ 2M + 1 for t ∈{1,s}.
Thus, it holds that∫

Ω
p(y)wt(y) dy

=
κ

c|W |

( 2π
M + ht

)n ∑
y∈Ωt

M

ε̃(y)st(y)p(y), (7)

where

κ =




2−b
n
2 c for An,

1
2 for D2k+1,
1
4 for E6,
1 otherwise.

To numerically compare the efficiency of these cuba-
ture formulas, we may consider an integrable function
f, such that f/wt is well defined on ΩtM, and rewrite
the cubatures (7) in the following form:

I(f) ≡
∫

Ω
f(y) dy ≈ ItM (f), (8)

ItM (f) ≡
κ

c|W|

( 2π
M + ht

)n ∑
y∈Ωt

M

ε̃(y)K(y)f(y).

3.2. Cubature formulas of C2
In this section, the general cubature formulas (8) are
specialized and tested on model examples for the case
of algebra C2. The region Ft of C2 with the points
from 110P

∨ ∩ Ft is depicted in Fig. 3, whereas the
corresponding integration region Ω with the trans-
formed grid points is depicted in Fig. 4. Note that

Figure 4. The integration region Ω of C2 contains
the points of the grid Ω010. The inner points of Ω
corresponds to the grid Ω16, the points not lying on
the dashed boundary corresponds to the grid Ωs8 and
finally, the points not lying on the dot-and-dashed
boundary corresponds to the grid Ωl8. The num-
bers 1, 2, 4 are the values of ε̃(y), the inner dots have
ε̃(y) = 8.

the numbers of points in grids ΩtM, t ∈{0, 1,s, l} are
the same. Fixing the basis x = b1ω∨1 + b2ω∨2 results
in the polynomial variables expressed as

y1 = 2
(
cos π(2b1 + b2) + cos πb2

)
,

y2 = 2
(
cos 2π(b1 + b2) + cos 2πb1

)
.

Formula (8) specializes into

ItM (f) =
π2

4(M + ht)2
∑
y∈Ωt

M

ε̃(y)K(y)f(y), (9)

where:
• M ∈ N is arbitrary;
• h0 = 1,h1 = 4 and hs = hl = 2;
• the integration region Ω, depicted in Fig. 4, is

bounded by two lines y2 = ±y1−4 and the parabola
y2 =

y21
4 ;

• the finite grid ΩtM , depicted for M = 10 in
Fig. 4), consists of points (y1(x),y2(x)) where
x = s

t
1

M+ht ω
∨
1 +

st2
M+ht ω

∨
2 with sti satisfying

s0i ∈ Z
≥0, 2s01 + s

0
2 ≤ M,

s1i ∈ Z
>0, 2s11 + s

1
2 < M + 4,

ss1 ∈ Z
≥0,ss2 ∈ Z

>0, 2ss1 + s
s
2 ≤ M + 2,

sl1 ∈ Z
>0,sl2 ∈ Z

≥0, 2sl1 + s
l
2 < M + 2;

• the weight function K becomes

K(y1,y2) =
√

(y21 − 4y2)((y2 + 4)2 − 4y21 );

• the weight function ε̃ is equal to

ε̃(y) =




1 if (y1,y2) = (±4, 4),
2 if (y1,y2) = (0,−4),
8 if (y1,y2) is an inner point of Ω,
4 otherwise.

207



L. Háková, J. Hrivnák, L. Motlochová Acta Polytechnica

Figure 5. The graphs of error values |1 − ItM (fi)| of the integral I(fi) = 1 and its estimations I
t
M (fi), M =

10, 15, 20, . . . , 195 given by (8). The values for t = 0, 1,s are depicted as circles, “+” signs and diamonds, respectively.

For the purpose of numerical tests and comparison,
we choose

f1(y1,y2) = q1(y201 −y1y2 + y
20
2 ),

f2(y1,y2) = q2
(
e−(y

2
1 +(y2+1.8)

2)/2×0.352),
f3(y1,y2) =

q3
1 + y21 + y22

,

f4(y1,y2) = q4ey1+y2,

f5(y1,y2) =
{
q5 if y21 + (y2 + 1.5)2 ≤ 1,
0 otherwise.

as model functions. Each value of qi ∈ R is set to
satisfy the condition I(fi) = 1. Fig. 5 shows for
M = 10, 15, 20, . . . , 195 and t ∈ {0, 1,s} the graphs
of the absolute value of the difference |1 − ItM (fi)|.
Note that the cases t = s and t = l give the same
results since hs = hl = 2 and K(y)fi(y) vanish on the
boundary of Ω.

4. Clenshaw-Curtis cubature
formulas

4.1. Clenshaw-Curtis method
Assuming that we have an interpolation of a function
f in terms of P(λ,kt) ∈ ΠM in points ΩtM, i.e.

f ≈
∑
λ∈P+
〈λ,η〉≤M

btλP(λ,k
t),

f(y) =
∑
λ∈P+
〈λ,η〉≤M

btλP(λ,k
t; y), y ∈ ΩtM,

we estimate a weighted integral of f with a weight
function w over a domain D ⊂ Ω by∑

λ∈P+
〈λ,η〉≤M

btλ

∫
D

P(λ,kt; y)w(y) dy.

Such construction of the Clenshaw-Curtis cubature
rule implies the exact equality for any polynomial f
of m-degree at most M. Denoting

atλ(w) ≡
∫
D

P(λ,kt; y)w(y) dy,

the Clenshaw-Curtis cubature is thus given by∫
D

f(y)wt(y) dy ≈
∑
λ∈P+
〈λ,η〉≤M

btλa
t
λ(w),

where the coefficients btλ and a
t
λ(w) need to be de-

termined. The coefficients btλ are readily obtained
using the discrete orthogonality relations of the orbit
functions from [13, 14]. Denoting the order of the sta-
bilizer of λ+%

t

M+ht with respect to the dual affine Weyl
group by h∨λ+%t, it holds that

btλ =
|StabW (λ + %t)|2

c|W |(M + ht)nh∨
λ+%t

×
∑
y∈ΩtM

ε̃(y)st(y)f(y)P(λ,kt; y). (10)

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vol. 56 no. 3/2016 On Cubature Rules Associated to Weyl Group Orbit Functions

A0(λ1,λ2)

(λ1,λ2) = (2i, 2j)
∑

(µ1,µ2)∈M(2i,2j)
64(4µ22+4µ1µ2−3)

|StabW (2i,2j)|(µ21−1)(4µ22−1)(4µ22−9)
(λ1,λ2) = (2i, 2j + 1)

∑
(µ1,µ2)∈M(2i,2j+1)

−64(4µ22+4µ1µ2+3)
|StabW (2i,2j+1)|(µ21−4)(4µ22−1)(4µ22−9)

Otherwise 0

A1(λ1,λ2)

(λ1,λ2) = (2i, 2j)
32(i+j+1)

(2i+4j+3)(2j+1)(2i+1)
(λ1,λ2) = (2i, 2j + 1)

32(j+1)
(2i+4j+5)(2i+2j+3)(2i+1)

Otherwise 0

As(λ1,λ2)

(λ1,λ2) = (2i, 2j)
∑

(µ1,µ2)∈Ms1(2i+1,2j)
8(µ1+µ2)

|StabW (2i+1,2j)|µ2(µ21−1)(µ22−1)
(λ1,λ2) = (2i, 2j + 1)

∑
(µ1,µ2)∈Ms2(2i+1,2j+1)

−8(µ1+µ2)
µ2(µ21−1)(µ22−1)

Otherwise 0

Al(λ1,λ2)

(λ1,λ2) = (2i, 2j)
(∑

(µ1,µ2)∈Ml1(2i,2j+1)
−
∑

(µ1,µ2)∈Ml2(2i,2j+1)

)
32µ2

|StabW (2i,2j+1)|µ1(4µ22−1)
(λ1,λ2) = (2i, 2j + 1)

(∑
(µ1,µ2)∈Ml2(2i,2j+2)

−
∑

(µ1,µ2)∈Ml1(2i,2j+2)

)
16(2µ1µ2+1)

|StabW (2i,2j+2)|(µ21−1)(4µ22−1)
Otherwise 0

M(λ1,λ2)
{

(λ1 + λ2, λ12 + λ2), (λ2,
λ1
2 + λ2), (λ1 + λ2,

λ1
2 ), (λ2,−

λ1
2 )
}

Ms1(λ1,λ2)
{

(λ2, λ12 + λ2), (λ2,−
λ1
2 )
}

Ms2(λ1,λ2)
{

(λ1 + λ2, λ12 + λ2), (λ1 + λ2,
λ1
2 )
}

Ml1(λ1,λ2)
{

(λ1 + λ2, λ12 + λ2), (λ2,
λ1
2 + λ2)

}
Ml2(λ1,λ2)

{
(λ1 + λ2, λ12 ), (λ2,−

λ1
2 )
}

Table 2. Values of Atλ (11) for t ∈{0, 1,s, l}, λ = λ1ω1 + λ2ω2 and i,j are non-negative integers.

[λ0,λ1,λ2] |StabW (λ + %t)| h∨λ+%t
(?,?,?) 1 1
(0,?,?) 1 2
(?, 0,?) 2 2
(?,?, 0) 2 2
(0, 0,?) 2 4
(0,?, 0) 2 8
(?, 0, 0) 8 8

Table 1. The values of |StabW (λ + %t)| and h∨λ+%t of
C2, where λ+%t = λ1ω1 +λ2ω2 and λ0 ≡ M +ht−λ1−
2λ2. Asterisks denote non-zero positive integers.

It remains to evaluate the integrals atλ(w) which de-
pend on the chosen weight and the integration domain
D ⊂ Ω.

Since the Jacobi polynomials have several properties
connected to the domain Ω (e.g. continuous and
discrete orthogonality), we firstly take D = Ω. The
cubature rules with the choice w = wt coincide for
any simple Lie algebra with the formulas (7). The
difference lies in the fact that Clenshaw-Curtis method
guarantees the exact equality only for polynomials up
to m-degree M.

4.2. Integration domain Ω of C2
In this section, the Clenshaw-Curtis integration
method is applied to the algebra C2. The values
of |StabW (λ + %t)| and h∨λ+%t, needed in (10), are
tabulated in Tab. 1.

Since the choice of w = wt gives standard cubature
formulas, the next natural choice of the weight func-
tion is to set w = 1. In this case are the coefficients
atλ(1), denoted by A

t
λ, expressed as the following inte-

grals:

Atλ = 2π
2



∫
F
Cλ(x)S%1 (x) dx if t = 0,∫

F
Sλ+%1 (x) dx if t = 1,∫

F
Ssλ+%s(x)S

l
%l

(x) dx if t = s,∫
F
Sl
λ+%l(x)S

s
%s(x) dx if t = l.

(11)

The exact values of Atλ are explicitly calculated in
Tab. 2.

4.3. Triangular domain of C2
The next choice of the domain D, for which we derive
the Clenshaw-Curtis cubature rules, is the triangle
T ⊂ Ω depicted on Fig. 6 and given explicitly by

T ≡
{

(y1,y2)
∣∣∣ y2 ≤ 0, −y22 − 2 ≤ y1 ≤ y22 + 2

}
.

209



L. Háková, J. Hrivnák, L. Motlochová Acta Polytechnica

A0(λ1,λ2)
(λ1,λ2) = (0, 0) π

2

4
(λ1,λ2) = (2i, 2j + 1) (−1)i+1 8|StabW (2i,2j+1)|(2i+2j+1)(2j+1)
Otherwise 0

A1(λ1,λ2)
(λ1,λ2) = (0, 0) 2π2 + 1289
(λ1,λ2) = (2i, 2j), i + j 6= 0 (−1)i+1

128(i+j+1)
(2i+2j+1)(2j−1)(2i+2j+3)(2j+3)

(λ1,λ2) = (2i, 2j + 1) (−1)i+1
128(j+1)

(2j+1)(2i+2j+1)(2j+3)(2i+2j+5)
Otherwise 0

As(λ1,λ2)
(λ1,λ2) = (0, 0) π2 + 8
(λ1,λ2) = (2i, 2j), i + j 6= 0 (−1)i+1 16|StabW (2i+1,2j)|(2j+1)(2i+2j+1)(2j−1)
(λ1,λ2) = (2i, 2j + 1) (−1)i+1 16(2j+1)(2i+2j+3)(2i+2j+1)
Otherwise 0

Al(λ1,λ2)
(λ1,λ2) = (0, 0) π2

(λ1,λ2) = (2i, 2j + 1) (−1)i+1
128(i+j+1)(j+1)

|StabW (2i,2j+2)|(2i+2j+3)(2j+3)(2i+2j+1)(2j+1)
Otherwise 0

Table 3. Values of Atλ, given by (12), for t ∈ {0, 1,s, l} and λ = λ1ω1 + λ2ω2. The indices i,j are non-negative
integers.

Figure 6. The domain bounded by the two lines and
the parabola with the inscribed triangle T corresponds
to the integration region Ω of C2.

Choosing the weight function w = wt, the integrals
atλ(w

t) are calculated by a change of variables induced
by the map Ξ. Denoting Atλ ≡ a

t
λ(w

t), it holds that

Atλ = 2π
2
∫
P

Stλ+%t(x)St%t(x) dx, (12)

where P is the pre-image of the triangle T under
the map Ξ. This pre-image P, depicted as a square
in Fig. 7, contains the points b1ω∨1 + b2ω∨2 satisfy-
ing 2b1 + b2 ≥ 1/2, 2b1 + b2 ≤ 1, b2 ≥ 0 and
b2 ≤ 1/2. The exact values of Atλ are tabulated
in Tab. 3.

Finally, choosing w = 1, we calculate the coefficients

Figure 7. The fundamental domain F corresponding
to C2 is depicted as the triangle containing the square
P with the boundaries α,β,γ and δ.

Btλ ≡ a
t
λ(1) on T as the following integrals

Btλ = 2π
2



∫
P
Cλ(x)S%1 (x) dx if t = 0,∫

P
Sλ+%1 (x) dx if t = 1,∫

P
Ssλ+%s(x)S

l
%l

(x) dx if t = s,∫
P
Sl
λ+%l(x)S

s
%s(x) dx if t = l.

(13)

The exact values of Btλ are tabulated in Tab. 4.
We choose the following functions as model func-

tions for numerical tests,

g1(y1,y2) = r1(y201 −y1y2 + y
20
2 ),

g2(y1,y2) = r2
(
e−(y

2
1 +(y2+1.8)

2)/2×0.352),
g3(y1,y2) =

r3
1 + y21 + y22

,

g4(y1,y2) = r4ex+y,

210



vol. 56 no. 3/2016 On Cubature Rules Associated to Weyl Group Orbit Functions

B0(λ1,λ2)
(λ1,λ2) = (0, 1) −323
(λ1,λ2) = (2, 0) −163

(λ1,λ2) = (2i, 2), i 6= 0
16(1+(−1)i+1)

3i(i+1)

(λ1,λ2) = (2i, 1), i 6= 0 16
[

2
(2i+3)(2i−1) +

1+(2i+1)(−1)i+1
3i(i+1)

]
(λ1,λ2) = (2i, 2j + 1),j 6= 0 64|StabW (2i,2j+1)|

[
−1+(2j+1)(−1)j

((2i+2j+1)2−4)((2j+1)2−1) +
−1+(2i+2j+1)(−1)i+j

((2i+2j+1)2−1)((2j+1)2−4)

]
(λ1,λ2) = (2i, 2j),j 6= 1, i + j 6= 1 16|StabW (2i,2j)|

[
1+(−1)i+j

((i+j)2−1)(4j2−1) +
1+(−1)j

(4(i+j)2−1)(j2−1)

]
Otherwise 0

B1(λ1,λ2)

(λ1,λ2) = (2i, 2j)
4(1+(−1)i+j)

(i+j+1)(2j+1)

(λ1,λ2) = (2i, 2j + 1)
−4(1+(−1)j)

(2i+2j+3)(j+1)

Otherwise 0

Bs(λ1,λ2)
(λ1,λ2) = (0, 0) 8
(λ1,λ2) = (2i, 1) 16(2i+3)(2i+1)

(λ1,λ2) = (2i, 2j), i + j 6= 0
8(1+(2i+2j+1)(−1)i+j+1)

|StabW (2i+1,2j)|(i+j+1)(i+j)(4j2−1)

(λ1,λ2) = (2i, 2j + 1),j 6= 0
8(−1+(2j+1)(−1)j)

(2i+2j+3)(2i+2j+1)j(j+1)

Otherwise 0

Bl(λ1,λ2)
(λ1,λ2) = (0, 0) 8

(λ1,λ2) = (2i, 0), i 6= 0 8
[

2i+1+(−1)i+1
2i(i+1) +

1
2i+1

]
(λ1,λ2) = (2i, 2j),j 6= 0 4|StabW (2i,2j+1)|

[
2i+2j+1+(−1)i+j+1
(i+j+1)(2j+1)(i+j) +

2j+1+(−1)j+1
(2i+2j+1)(j+1)j

]
(λ1,λ2) = (2i, 2j + 1), 16|StabW (2i,2j+2)|

[
(−1+(−1)j+1)(i+j+1)

(2i+2j+3)(j+1)(2i+2j+1) +
(−1+(−1)i+j+1)(j+1)
(i+j+1)(2j+3)(2j+1)

]
Otherwise 0

Table 4. Values of Btλ, given by (13), for t ∈ {0, 1,s, l} and λ = λ1ω1 + λ2ω2. The indices i,j are non-negative
integers.

g5(y1,y2) =
{
r5 if y21 + (y2 + 1.5)2 ≤ 1,
0 otherwise.

Each value of ri ∈ R is set to satisfy the normal-
ization condition

∫
T
gi(y) dy = 1. We compute the

approximations

ItM (gi) =
∑
λ∈P+
〈λ,η〉≤M

btλB
t
λ (14)

of
∫
T
gi(y) dy = 1 with the formula for Btλ given

by (13). Figs. 8 and 9 show for t ∈ {0, 1,s, l} the

graphs of of the absolute value of the difference
|1 −ItM (gi)|.

5. Concluding Remarks
(1.) Establishing the explicit connection between the
Jacobi and Macdonald polynomials and the Weyl
group orbit functions in Section 2.3 forms a crucial
step for generalizing known cubature formulas to
the entire class of the Jacobi polynomials.

(2.) Numerical tests results in Figs. 5 and 8 indicate
in general excellent convergence rates of the devel-
oped cubature rules including their Clenshaw-Curtis

211



L. Háková, J. Hrivnák, L. Motlochová Acta Polytechnica

Figure 8. The graphs of error values |1−ItM (gi)| of the integral
∫
T
gi(y) dy = 1, i = 1, . . . , 4 and its approximations

ItM (gi), M = 10, 11, . . . , 50 given by (14). The values for t = 0, 1,s, l are depicted as circles, “+”, diamonds and “×”,
respectively.

Figure 9. The graphs of error values |1 −ItM (g5)|
of the integral

∫
T
g5(y) dy = 1 and its approximations

ItM (g5), M = 10, 15, 20, . . . , 170 given by (14). The
values for t = 0, 1,s, l are depicted as circles, “+”,
diamonds and “×”, respectively.

versions for the case C2. The convergence rate of
the multidimensional step-functions, even though
less uniform, still appears to be very good. Devel-
oping similar methods for the two-variable case G2
and extending the rules to higher dimensions poses
an open problem.

(3.) The hyperinterpolation methods [3, 25, 34, 35] are
among the tools which directly use cubature rules.
For the standard cubature rules of the Weyl group
orbit functions, several tests with very good results
are also performed in [15]. Developing and test-
ing hyperinterpolation methods for the presented
cubature rules merits further study.

(4.) The present work demonstrates wide variety of
possibilities of constructing the cubature rules in
the orbit functions setting. Comparison of the de-
veloped methods is necessary for establishing range
of their viable applications. Especially, comparison

of the Gauss and Clenshaw-Curtis cubature meth-
ods, similar to [37], regarding their efficiency, speed,
model function and integration domain dependence
merits further research.

6. Acknowledgments
LM and JH gratefully acknowledge the support of this
work by RVO68407700.

References
[1] H. Berens, H. J. Schmid, Y. Xu, Multivariate
Gaussian cubature formulae, Arch. Math. (Basel) 64
(1995), no. 1, 26–32, doi:10.1007/BF01193547.

[2] N. Bourbaki, Groupes et algèbres de Lie, Chapitres
IV, V, VI, Hermann, Paris, 1968.

[3] M. Caliari, S. De Marchi, M. Vianello,
Hyperinterpolation in the cube, Comput. & Math. with
Appl. 55 (2008), 2490–2497,
doi:10.1016/j.camwa.2007.10.003.

[4] D. Chernyshenko, H. Fangohr, Computing the
demagnetizing tensor for finite difference micromagnetic
simulations via numerical integration, J. Magn. Magn.
Mat. 381 (2015) 440–445,
doi:10.1016/j.jmmm.2015.01.013.

[5] C.W. Clenshaw, A.R. Curtis, A method for numerical
integration on an automatic computer, Numer. Math. 2
(1960), 197–205, doi:10.1007/BF01386223.

[6] R. Cools, An encyclopaedia of cubature formulas,
Journal of Complexity 19 (2003), 445–453,
doi:10.1016/S0885-064X(03)00011-6.

[7] R. Cools, I. P. Mysovskikh, H. J. Schmid, Cubature
formulae and orthogonal polynomials, J. Comput. Appl.
Math. 127 (2001), no. 1-2, 121–152,
doi:10.1016/S0377-0427(00)00495-7.

212

http://dx.doi.org/10.1007/BF01193547
http://dx.doi.org/10.1016/j.camwa.2007.10.003
http://dx.doi.org/10.1016/j.jmmm.2015.01.013
http://dx.doi.org/10.1007/BF01386223
http://dx.doi.org/10.1016/S0885-064X(03)00011-6
http://dx.doi.org/10.1016/S0377-0427(00)00495-7


vol. 56 no. 3/2016 On Cubature Rules Associated to Weyl Group Orbit Functions

[8] P. de la Harpe, C. Pache, B. Venkov, Construction of
spherical cubature formulas using lattices, Algebra i
Analiz 18 (2006), no. 1, 162–186; reprinted in St.
Petersburg Math. J. 18 (2007), no. 1, 119–139,
doi:10.1090/S1061-0022-07-00946-6.

[9] D. C. Handscomb, J. C. Mason, Chebyshev
polynomials, Chapman&Hall/CRC, USA, 2003,
doi:10.1201/9781420036114.

[10] L. Háková, J. Hrivnák, J. Patera, Four families of
Weyl group orbit functions of B3 and C3, J. Math. Phys.
54 (2013), 083501, 19, doi:10.1063/1.4817340.

[11] G. Heckman, H. Schlichtkrull, Harmonic Analysis
and Special Functions on Symmetric Spaces, Academic
Press Inc., San Diego, 1994.

[12] G. Heckman, E. M. Opdam, Root systems and
hypergeometric functions. I, II, Composition Math. 64
(1987), 329-373.

[13] J. Hrivnák, J. Patera, On discretization of tori of
compact simple Lie groups, J. Phys. A: Math. Theor. 42
(2009) 385208, doi:10.1088/1751-8113/42/38/385208.

[14] J. Hrivnák, L. Motlochová, J. Patera, On
discretization of tori of compact simple Lie groups II., J.
Phys. A 45 (2012), 255201, 18,
doi:10.1088/1751-8113/45/25/255201.

[15] J. Hrivnák, L. Motlochová, J. Patera, Cubature
formulas of multivariate polynomials arising from
symmetric orbit functions, arXiv:1512.01710.

[16] J. E. Humphreys, Introduction to Lie algebras and
representation theory, Spinger-Verlag, New York, 1978,
doi:10.1007/978-1-4612-6398-2.

[17] A. Klimyk, J. Patera, (Anti)symmetric multivariate
trigonometric functions and corresponding Fourier
transforms, J. Math. Phys. 48 (2007), 093504, 24,
doi:10.1063/1.2779768.

[18] A. U. Klimyk, J. Patera, Orbit functions, SIGMA 2
(2006), 006, 60 pages, doi:10.3842/SIGMA.2006.006.

[19] A. U. Klimyk, J. Patera, Antisymmetric orbit
functions, SIGMA 3 (2007), paper 023, 83 pages,
doi:10.3842/SIGMA.2007.023.

[20] T. H. Koornwinder, Two-variable analogues of the
classical orthogonal polynomials, Theory and
Application of Special Functions, edited by R. A. Askey,
Academic Press, New York (1975) 435–495,
doi:10.1016/B978-0-12-064850-4.50015-X.

[21] H. Li, J. Sun, Y. Xu, Discrete Fourier Analysis and
Chebyshev Polynomials with G2 Group, SIGMA 8,
Paper 067, 29, 2012, doi:10.3842/SIGMA.2012.067.

[22] H. Li, J. Sun, Y. Xu, Discrete Fourier analysis,
cubature and interpolation on a hexagon and a triangle,
SIAM J. Numer. Anal. 46 (2008), 1653–1681,
doi:10.1137/060671851.

[23] H. Li, Y. Xu, Discrete Fourier analysis on
fundamental domain and simplex of Ad lattice in
d-variables, J. Fourier Anal. Appl. 16, 383–433, (2010),
doi:10.1007/s00041-009-9106-9.

[24] I. G. Macdonald, Orthogonal polynomials associated
with root systems, Sém. Lothar. Combin. 45 (2000/01),
Art. B45a, 40, doi:10.1007/978-94-009-0501-6_14.

[25] S. De Marchi, M. Vianello, Y. Xu, New cubature
formulae and hyperinterpolation in three variables, BIT.
Numerical Mathematics 49 (2009), Number 1, 55–73,
doi:10.1007/s10543-009-0210-7.

[26] R. V. Moody, L. Motlochová, J. Patera, Gaussian
cubature arising from hybrid characters of simple Lie
groups , J. Fourier Anal. Appl. 20 (2014), Issue 6,
1257–1290, doi:10.1007/s00041-014-9355-0.

[27] R. V. Moody, J. Patera, Cubature formulae for
orthogonal polynomials in terms of elements of finite
order of compact simple Lie groups, Adv. in Appl. Math.
47 (2011) 509–535, doi:10.1016/j.aam.2010.11.005.

[28] R. V. Moody and J. Patera, Orthogonality within the
families of C-, S-, and E-functions of any compact
semisimple Lie group, SIGMA 2 (2006) 076, 14 pages,
doi:10.3842/SIGMA.2006.076.

[29] L. Motlochová, Special functions of Weyl groups and
their continuous and discrete orthogonality, Ph.D.
Thesis, Université de Montréal (2014),
http://hdl.handle.net/1866/11153

[30] H. Z. Munthe-Kaas, M. Nome, B. N. Ryland, Through
the kaleidoscope: symmetries, groups and Chebyshev-
approximations from a computational point of view,
Foundations of computational mathematics, Budapest
2011, 188–229, London Math. Soc. Lecture Note Ser.,
403, Cambridge Univ. Press, Cambridge, 2013.

[31] B. N. Ryland, H. Z. Munthe-Kaas, On multivariate
Chebyshev polynomials and spectral approximation on
triangles, Spectral and High Order Methods for Partial
Differential Equations, Lecture Notes in computational
science and engineering, Springer, 2011,
doi:10.1007/978-3-642-15337-2_2.

[32] H. J. Schmid, Y. Xu, On bivariate Gaussian cubature
formulae, Proc. Amer. Math. Soc., 122 (1994), 833–841,
doi:10.2307/2160762.

[33] I. Sfevanovic, F. Merli, P. Crespo-Valero, W. Simon,
S. Holzwarth, M. Mattes, J. R. Mosig, Integral Equation
Modeling of Waveguide-Fed Planar Antennas, IEEE
Antenn. Propag. M. 51 (2009), 82–92,
doi:10.1109/MAP.2009.5433099.

[34] I. H. Sloan, Polynomial interpolation and
hyperinterpolation over general regions, J. Approx.
Theory 83 (1995), no. 2, 238–254,
doi:10.1006/jath.1995.1119.

[35] A. Sommariva, M. Vianello, R. Zanovello,
Nontensorial Clenshaw-Curtis cubature, Numer.
Algorithms 49 (2008), Number 1-4, 409–427,
doi:10.1007/s11075-008-9203-x.

[36] G. Szegő, Orthogonal polynomials, American
Mathematical Society, Providence, R.I., 1975.

[37] L. N. Trefethen, Is Gauss quadrature better than
Clenshaw-Curtis? SIAM Rev. 50 (2008) 67–87,
doi:10.1137/060659831.

[38] J. Waldvogel, Fast construction of the Fejér and
Clenshaw-Curtis quadrature rules, BIT 46 (2006), no. 1,
195–202, doi:10.1007/s10543-006-0045-4.

[39] J. C. Young, S. D. Gedney, R. J. Adams, Quasi-
Mixed-Order Prism Basis Functions for Nyström-Based
Volume Integral Equations, IEEE Trans. Magn. 48
(2012), 2560–2566, doi:10.1109/TMAG.2012.2197634.

213

http://dx.doi.org/10.1090/S1061-0022-07-00946-6
http://dx.doi.org/10.1201/9781420036114
http://dx.doi.org/10.1063/1.4817340
http://dx.doi.org/10.1088/1751-8113/42/38/385208
http://dx.doi.org/10.1088/1751-8113/45/25/255201
http://dx.doi.org/10.1007/978-1-4612-6398-2
http://dx.doi.org/10.1063/1.2779768
http://dx.doi.org/10.3842/SIGMA.2006.006
http://dx.doi.org/10.3842/SIGMA.2007.023
http://dx.doi.org/10.1016/B978-0-12-064850-4.50015-X
http://dx.doi.org/10.3842/SIGMA.2012.067
http://dx.doi.org/10.1137/060671851
http://dx.doi.org/10.1007/s00041-009-9106-9
http://dx.doi.org/10.1007/978-94-009-0501-6_14
http://dx.doi.org/10.1007/s10543-009-0210-7
http://dx.doi.org/10.1007/s00041-014-9355-0
http://dx.doi.org/10.1016/j.aam.2010.11.005
http://dx.doi.org/10.3842/SIGMA.2006.076
http://hdl.handle.net/1866/11153
http://dx.doi.org/10.1007/978-3-642-15337-2_2
http://dx.doi.org/10.2307/2160762
http://dx.doi.org/10.1109/MAP.2009.5433099
http://dx.doi.org/10.1006/jath.1995.1119
http://dx.doi.org/10.1007/s11075-008-9203-x
http://dx.doi.org/10.1137/060659831
http://dx.doi.org/10.1007/s10543-006-0045-4
http://dx.doi.org/10.1109/TMAG.2012.2197634

	Acta Polytechnica 56(3):202–213, 2016
	1 Introduction
	2 Special functions associated to root systems
	2.1 Basic definitions
	2.2 Weyl group orbit functions
	2.3 Jacobi polynomials

	3 Cubature formulas
	3.1 General form of cubature formulas
	3.2 Cubature formulas of C2

	4 Clenshaw-Curtis cubature formulas
	4.1 Clenshaw-Curtis method
	4.2 Integration domain Omega of C2
	4.3 Triangular domain of C2

	5 Concluding Remarks
	6 Acknowledgments
	References