Discrete Fourier Analysis and Chebyshev Polynomials with G₂ Group
The discrete Fourier analysis on the 30°-60°-90° triangle is deduced from the corresponding results on the regular hexagon by considering functions invariant under the group G₂, which leads to the definition of four families generalized Chebyshev polynomials. The study of these polynomials leads to...
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| citation_txt | Discrete Fourier Analysis and Chebyshev Polynomials with G₂ Group / H. Li, J. Sun, Y. Xu // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 19 назв. — англ. |
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| description | The discrete Fourier analysis on the 30°-60°-90° triangle is deduced from the corresponding results on the regular hexagon by considering functions invariant under the group G₂, which leads to the definition of four families generalized Chebyshev polynomials. The study of these polynomials leads to a Sturm-Liouville eigenvalue problem that contains two parameters, whose solutions are analogues of the Jacobi polynomials. Under a concept of m-degree and by introducing a new ordering among monomials, these polynomials are shown to share properties of the ordinary orthogonal polynomials. In particular, their common zeros generate cubature rules of Gauss type.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 8 (2012), 067, 29 pages
Discrete Fourier Analysis and Chebyshev Polynomials
with G2 Group
Huiyuan LI :, Jiachang SUN : and Yuan XU ;
: Institute of Software, Chinese Academy of Sciences, Beijing 100190, China
E-mail: huiyuan@iscas.ac.cn, sun@mail.rdcps.ac.cn
; Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222, USA
E-mail: yuan@uoregon.edu
URL: http://uoregon.edu/~yuan/
Received May 04, 2012, in final form September 06, 2012; Published online October 03, 2012
http://dx.doi.org/10.3842/SIGMA.2012.067
Abstract. The discrete Fourier analysis on the 300–600–900 triangle is deduced from the
corresponding results on the regular hexagon by considering functions invariant under the
group G2, which leads to the definition of four families generalized Chebyshev polynomials.
The study of these polynomials leads to a Sturm–Liouville eigenvalue problem that contains
two parameters, whose solutions are analogues of the Jacobi polynomials. Under a concept
of m-degree and by introducing a new ordering among monomials, these polynomials are
shown to share properties of the ordinary orthogonal polynomials. In particular, their
common zeros generate cubature rules of Gauss type.
Key words: discrete Fourier series; trigonometric; group G2; PDE; orthogonal polynomials
2010 Mathematics Subject Classification: 41A05; 41A10
1 Introduction
In our recent works [9, 10, 11] we studied discrete Fourier analysis associated with translation
lattices. In the case of two dimension, our results include discrete Fourier analysis of expo-
nential functions on the regular hexagon and, by restricting to symmetric and antisymmetric
exponentials on the hexagon under the reflection group A2 (the group of symmetry of the regu-
lar hexagon), the generalized cosine and sine functions on the equilateral triangle, which can
also be transformed into the generalized Chebyshev polynomials on a domain bounded by the
hypocycloid. These polynomials possess maximal number of common zeros, which implies the
existence of Gaussian cubature rules, a rarity that is only the second example ever found. The
first example of Gaussian cubature rules is connected with the trigonometric functions on the
450–450–900 triangle. The richness of these results prompts us to look into similar results on
the 300–600–900 triangle in the present work. This case is also considered recently in [13] as
an example under a general framework of cubature rules and orthogonal polynomials for the
compact simple Lie groups, for which the group is G2.
It turns out that much of the discrete Fourier analysis on the 300–600–900 triangle can be
obtained, perhaps not surprisingly, though symmetry from our results on the hexagonal domain.
The most direct way of deduction, however, is not through our results on the equilateral triangle.
The reason lies in the underline group G2, which is a composition of A2 and its dual A˚2 , the
symmetric group of the regular hexagon and its rotation. Our framework of discrete Fourier
analysis incorporates two lattices, one determines the domain and the other determines the space
of exponentials. Our results on the equilateral triangle are obtained from the situation when
both lattices are taken to be the same hexagonal lattices [9]. Another choice is to take one lattice
mailto:huiyuan@iscas.ac.cn
mailto:sun@mail.rdcps.ac.cn
mailto:yuan@uoregon.edu
http://uoregon.edu/~yuan/
http://dx.doi.org/10.3842/SIGMA.2012.067
2 H. Li, J. Sun and Y. Xu
as the hexagonal lattice and the other as the rotation of the same lattice by 900 degree [10], with
the symmetric groups A2 and A˚2 , respectively. As we shall see, it is from this set up that our
results on the 300–600–900 triangle can be deduced directly via symmetry. The results include
cubature rules and orthogonal trigonometric functions that are analogues of cosine and sine
functions. There are four families of such functions and they have also been studied recently
in [13, 18]. While the results in these two papers concern mainly with orthogonal polynomials,
our emphasis is on the discrete Fourier analysis and cubature rules, and on the connection to
the results in the hexagonal domain.
The generalized cosine and sine functions on the 300–600–900 triangle are also eigenfunctions
of the Laplace operator with suitable boundary conditions. There are four families of such
functions. Under proper change of variables, they become orthogonal polynomials on a domain
bounded by two curves. However, unlike the equilateral triangle, these polynomials do not form
a complete orthogonal basis in the usual sense of total order of monomials. To understand
the structure of these polynomials, we consider the Sturm–Liouville problem for a general pair
of parameters α, β, with the four families that correspond to the generalized cosine and sine
functions as α “ ˘1
2 , β “ ˘1
2 . The differential operator of this eigenvalue problem has the form
Lα,β :“ ´A11px, yqB
2
x ´ 2A12px, yqBxBy ´A22px, yqB
2
y `B1px, yqBx `B2px, yqBy.
Such operators have long been studied in association with orthogonal polynomials in two va-
riables; see for example [6, 7, 8, 16], as well as [1] and the references therein. Our operator Lα,β,
however, is different in the sense that the coefficient functions Ai,j are usually assumed to be
of quadratic polynomials to ensure that the operator has n ` 1 polynomials of degree n as
eigenfunctions, whereas A2,2 in our Lα,β is a polynomial of degree 3 for which it is no longer
obvious that a full set of eigenfunctions exists. Nevertheless, we shall prove that the eigenvalue
problem Lα,βu “ λu has a complete set of polynomial solutions, which are also orthogonal
polynomials, analogue of the Jacobi polynomials. Upon introducing a new ordering among
monomials, these polynomials can be shown to be uniquely determined by their highest term
in the new ordering. As a matter of fact, this ordering defines the region of influence and
dependence in the polynomial space for each solution. Furthermore, it preserves the m-degree
of polynomials, a concept introduced in [13], rather than the total degree. In the case of α “ ˘1
2
and β “ ˘1
2 , the common zeros of these polynomials determine the Gauss, Gauss–Lobatto and
Gauss–Radau cubature rules, respectively, all in the sense of m-degree. It is known that the
cubature rule of degree 2n´ 1 exists if and only if its nodes form a variety of an ideal generated
by certain orthogonal polynomials. It is somewhat surprising that this relation is preserved
when the m-degree is used in place of the ordinary degree.
The paper is organized as follows. The following section contains what we need from the
discrete Fourier analysis on the hexagonal domain. The results on the 300–600–900 triangle is
developed in Section 3, which are translated into generalized Chebyshev polynomials in Section 4.
The Sturm–Liouville problem is defined and studied in Section 5 and the cubature rules are
presented in Section 6.
2 Discrete Fourier analysis on hexagonal domain
Before stating the results on the hexagonal domain, we give a short narrative of the neces-
sary background on the discrete Fourial analysis with lattice as developed in [9, 11]. We refer
to [2, 3, 12, 14] for some applications of discrete Fourier analysis in several variables.
A lattice L in Rd is a discrete subgroup L “ LA :“ AZd, where A, called a generator matrix,
is nonsingular. A bounded set Ω of Rd, called the fundamental domain of L, is said to tile Rd
Discrete Fourier Analysis and Chebyshev Polynomials with G2 Group 3
with the lattice L if Ω` L “ Rd, that is,
ÿ
αPL
χΩpx` αq “ 1, for almost all x P Rd,
where χΩ denotes the characteristic function of Ω. For a given lattice LA, the dual lattice LKA is
given by LKA “ A´trZd. A result of Fuglede [5] states that a bounded open set Ω tiles Rd with
the lattice L if, and only if, te2πiα¨x : α P LKu is an orthonormal basis with respect to the inner
product
xf, gyΩ “
1
|Ω|
ż
Ω
fpxqgpxqdx. (2.1)
Since LKA “ A´trZd, we can write α “ A´trk for α P LKA and k P Zd, so that e2πiα¨x “ e2πiktrA´1x.
For our discrete Fourier analysis, the boundary of Ω matters. We shall fix an Ω such that
0 P Ω and Ω`AZd “ Rd holds pointwisely and without overlapping.
Definition 2.1. Let ΩA and ΩB be the fundamental domains of AZd and BZd, respectively.
Assume all entries of the matrix N :“ BtrA are integers. Define
ΛN :“
k P Zd : B´trk P ΩA
(
and Λ:N :“
k P Zd : A´trk P ΩB
(
.
Furthermore, define the finite-dimensional subspace of exponential functions
VN :“ span
e2πi ktrA´1x, k P Λ:N
(
.
A function f defined on Rd is called a periodic function with respect to the lattice AZd if
fpx`Akq “ fpxq for all k P Zd.
The function x ÞÑ e2πiktrA´1x is periodic with respect to the lattice AZd and VN is a space
of periodic exponential functions. We can now state the central result in the discrete Fourier
analysis.
Theorem 2.2. Let A, B and N be as in Definition 2.1. Define
xf, gyN “
1
| detpNq|
ÿ
jPΛN
fpB´trjqgpB´trjq
for f , g in CpΩAq, the space of continuous functions on ΩA. Then
xf, gyΩA “ xf, gyN , f, g P VN . (2.2)
It follows readily that (2.2) gives a cubature formula exact for functions in VN . Furthermore,
it implies an explicit Lagrange interpolation by exponential functions, which we shall not state
since it will not be needed in the present work.
In the following, we shall call the lattice LA as the lattice for the physical space, as it
determines the domain on which our analysis lies, and the lattice LB as the lattice for the
frequency space, as it determines the points that defines the inner product.
The classical discrete Fourier analysis of two variables is the tensor product of the results
in one variable, which corresponds to A “ B “ I, the identity matrix. We are interested in
choosing A as the generating matrix H of the hexagonal domain,
H “
ˆ?
3 0
´1 2
˙
with ΩH “
!
x P R2 : ´1 ď x2,
?
3x1
2 ˘ x2
2 ă 1
)
.
4 H. Li, J. Sun and Y. Xu
If we choose B “ n
2H, so that N “ BtrA has all integer entries, we are back to the situation
studied in [9], which is the one that leads to the discrete Fourier analysis on the equilateral
triangle. The other choices are considered in [10].
For the case that we are interested in, we choose A “ H, the matrix for the hexagonal lattice
in the physical space, and B “ nH´tr with n P Z, the matrix for the hexagonal lattice in the
frequency space. Then N “ BtrA “ nI has all integer entries. This case was studied in [10],
which will be used to deduce the case that we are interested in by an additional symmetry. As
shown in [9, 17], it is more convenient to use homogeneous coordinates pt1, t2, t3q defined by
¨
˝
t1
t2
t3
˛
‚“
¨
˚
˝
?
3
2 ´1
2
0 1
´
?
3
2 ´1
2
˛
‹
‚
ˆ
x1
x2
˙
:“ Ex, (2.3)
which satisfy t1` t2` t3 “ 0. We adopt the convention of using bold letters, such as t to denote
points in homogeneous coordinates. We define by
R3
H :“
t “ pt1, t2, t3q P R3 : t1 ` t2 ` t3 “ 0
(
and H: :“ Z3 X R3
H
the spaces of points and integers in homogeneous coordinates, respectively. In such coordinates,
the fundamental domains of the lattices LA and LB are then given by
Ω :“ ΩA “
t P R3
H : ´1 ă t1, t2,´t3 ď 1
(
,
ΩB “
t P R3
H : ´n ă t1 ´ t2, t1 ´ t3, t2 ´ t3 ď n
(
,
where ΩA can be viewed as the intersection of the plane t1 ` t2 ` t3 “ 0 with the cube r´1, 1s3.
Define the index sets in homogeneous coordinates
Hn :“
j P H: : ´n ď j1, j2, j3 ď n, j ” 0 pmod 3q
(
,
H:n :“
k P H: : ´n ď k3 ´ k2, k1 ´ k3, k2 ´ k1 ď n
(
,
where t ” 0 pmod mq means, by definition, t1 ” t2 ” t3 pmod mq. We note that Hn and H:n
serve as the symmetric counterparts of ΛN and Λ:N , respectively, so that Hn determines the
points in the discrete inner product and H:n determines the space of exponentials. Moreover,
the index set Hn can be obtained from a rotation of H:n, as shown in the following proposition.
( 1√
3
,1)(- 1√
3
,1)
(- 2√
3
,0)
(- 1√
3
,-1) ( 1√
3
,-1)
( 2√
3
,0)
O x1
x2
(0,1,-1)(-1,1,0)
(-1,0,1)
(0,-1,1) (1,-1,0)
(1,0,-1)
O
t1
t2
t3
Figure 2.1. ΩA in Cartesian coordinates (left) and homogeneous coordinates (right).
Proposition 2.3 ([10]). For t “ pt1, t2, t3q P R3
H , define pt :“ pt3 ´ t2, t1 ´ t3, t2 ´ t1q. Then
pk
3 P H
:
n if k P Hn and pk P Hn if k P H:n.
Discrete Fourier Analysis and Chebyshev Polynomials with G2 Group 5
(
√
3n
6 , n6 )
(
√
3n
6 ,- n6 )
(0,- n3 )
(-
√
3n
6 , n6 )
(-
√
3n
6 , n6 )
(0, n3 )
O x1
x2
( n3 ,
n
3 ,-
2n
3 )
( 2n3 ,- n3 ,-
n
3 )
( n3 ,-
2n
3 , n3 )
(- n3 ,-
n
3 ,
2n
3 )
(- 2n3 , n3 ,
n
3 )
(- n3 ,
2n
3 ,- n3 )
O
t1
t2
t3
Figure 2.2. ΩB in Cartesian coordinates (left) and homogeneous coordinates (right).
Proposition 2.3 states that Hn “ pH:n :“
pk : k P H:n
(
. Similarly, we can define H “ pH: :“
pk : k P H:
(
“
j P H: : j ” 0 pmod 3q
(
. The set H:n is the index set for the space of
exponentials. Define the finite-dimensional space H:n of exponential functions
H:n :“ span
!
φk “ e
2iπ
3
k¨t : k P H:n
)
.
By induction, it is not difficult to verify that
dimH:n “ |H:n| “ |Hn| “
#
n2 ` n` 1, if n ı 1 pmod 3q,
n2 ` n´ 1, if n ” 1 pmod 3q.
Under the homogeneous coordinates (2.3), x ” y pmod Hq becomes t ” s pmod 3q. We call
(0,n,-n)(-n,n,0)
(-n,0,n)
(0,-n,n) (n,-n,0)
(n,0,-n)
(0,n,-n)(-n,n,0)
(-n,0,n)
(0,-n,n) (n,-n,0)
(n,0,-n)
(0,n,-n)(-n,n,0)
(-n,0,n)
(0,-n,n) (n,-n,0)
(n,0,-n)
Figure 2.3. Hn for n “ 9 (left), n “ 10 (center) and n “ 11 (right).
(a,a,-2a)
(2a,-a,-a)
(a,-2a,a)
(-a,-a,2a)
(-2a,a,a)
(-a,2a,-a)
(a,a,-2a)
(2a,-a,-a)
(a,-2a,a)
(-a,-a,2a)
(-2a,a,a)
(-a,2a,-a)
(a,a,-2a)
(2a,-a,-a)
(a,-2a,a)
(-a,-a,2a)
(-2a,a,a)
(-a,2a,-a)
Figure 2.4. H:n for n “ 9 (left), n “ 10 (center) and n “ 11 (right), where a “ n
3 .
a function f H-periodic if fptq “ fpt ` jq whenever j ” 0 pmod3q. Since j,k P H implies that
2j ¨ k “ pj1 ´ j2qpk1 ´ k2q ` 3j3k3, we see that φj is H-periodic.
6 H. Li, J. Sun and Y. Xu
Theorem 2.4 ([10]). The following cubature rule holds for any f P H:2n´1,
1
|Ω|
ż
Ω
fptqdt “
1
n2
ÿ
jPHn
c
pnq
j f
`
j
n
˘
, c
pnq
j “
$
’
&
’
%
1, j P H0n,
1
2 , j P He
n,
1
3 , j P Hv
n,
(2.4)
where H0n, Hv
n and He
n denote the set of points in interior, set of vertices, and set of points
on the edges but not on the vertices; more precisely, H0n “ tj P H : ´n ă j1, j2, j3 ă nu, Hv
n “
tpn, 0,´nqσ P H : σ P A2u and He
n “ HnzpH0nYHv
nq “ tpj, n´ j,´nqσ P H : 1 ď j ď n´ 1u. In
particular, let Qnf denote the right hand side of (2.4); then for any k P H:, Qnφk “ 1 if k̂ ” 0
pmod 3nq and Qnφk “ 0 otherwise.
Here we state the main result in terms of the cubature rule (2.4), from which the discrete
inner product can be easily deduced. For further results in this regard, including interpolation,
we refer to [10].
3 Discrete Fourier analysis on the 300–600–900 triangle
In this section we deduce a discrete Fourier analysis on the 300–600–900 triangle from the analysis
on the hexagon by working with invariant functions.
3.1 Generalized trigonometric functions
The group A2 is generated by the reflections in the edges of the equilateral triangles inside the
regular hexagon Ω. In homogeneous coordinates, the three reflections σ1, σ2, σ3 are defined by
tσ1 :“ ´pt1, t3, t2q, tσ2 :“ ´pt2, t1, t3q, tσ3 :“ ´pt3, t2, t1q.
Because of the relations σ3 “ σ1σ2σ1 “ σ2σ1σ2, the group is given by
A2 “ t1, σ1, σ2, σ3, σ1σ2, σ2σ1u .
The group A˚2 of isometries of the hexagonal lattice is generated by the reflections in the
median of the equilateral triangles inside it, which can be derived from the reflection group A2
by a rotation of 900 and is exactly the permutation group of three elements. To describe the
elements in A˚2 , we define the reflection ´σ for any σ P A2 by
tp´σq :“ ´tσ, @ t P R3
H .
With this notation, the group A˚2 is given by
A˚2 “ t1,´σ1,´σ2,´σ3, σ1σ2, σ2σ1u ,
in which ´σ1, ´σ2, ´σ3 serve as the three basic reflections. The group A˚2 is the same as the
permutation group S3 with three elements.
The group G2 is exactly the composition of A2 and A˚2 ,
G2 “ tσσ
˚ : σ P A2, σ
˚ P A˚2u “ t˘1,˘σ1,˘σ2,˘σ3,˘σ1σ2,˘σ2σ1u .
Let G denote the group of A2 or A˚2 or G2. For a function f in homogeneous coordinates, the
action of the group G on f is defined by σfptq “ fptσq, σ P G. A function f is called invariant
under G if σf “ f for all σ P G, and called anti-invariant under G if σf “ p´1q|σ|f for all σ P G,
where |σ| denotes the inversion of σ and p´1q|σ| “ 1 if σ “ ˘1,˘σ1σ2,˘σ2σ1, and p´1q|σ| “ ´1
if σ “ ˘σ1,˘σ2,˘σ3. The following proposition is easy to verify (see [6]).
Discrete Fourier Analysis and Chebyshev Polynomials with G2 Group 7
Proposition 3.1. Define the operators P` and P´ acting on fptq by
P˘fptq “ 1
6
rfptq ` fptσ1σ2q ` fptσ2σ1q ˘ fptσ1q ˘ fptσ2q ˘ fptσ3qs . (3.1)
Then the operators P` and P´ are projections from the class of H-periodic functions onto
the class of invariant, respectively anti-invariant, functions under A2. Furthermore, define the
operators P`˚ and P´˚ acting on fptq by
P˘˚ fptq “
1
6
rfptq ` fptσ1σ2q ` fptσ2σ1q ˘ fp´tσ1q ˘ fp´tσ2q ˘ fp´tσ3qs . (3.2)
Then the operators P`˚ and P´˚ are projections from the class of H-periodic functions onto the
class of invariant, respectively anti-invariant functions under A˚2 .
(0,1,-1)(-1,1,0)
(-1,0,1)
(0,-1,1) (1,-1,0)
(1,0,-1)
( 12 ,
1
2 ,-1)
(- 12 ,1,-
1
2 )
(-1, 12 ,
1
2 )
(- 12 ,-
1
2 ,1) (1,- 12 ,-
1
2 )
( 12 ,-1,
1
2 )
( 12 ,
1
2 ,-1)
(- 12 ,1,-
1
2 )
(-1, 12 ,
1
2 )
(- 12 ,-
1
2 ,1) (1,- 12 ,-
1
2 )
( 12 ,-1,
1
2 )
(0,1,-1)(-1,1,0)
(-1,0,1)
(0,-1,1) (1,-1,0)
(1,0,-1)
Figure 3.1. Symmetry under A2 (left), A˚2 (center) and G2 (right) in the physical space. The shaded
area is the fundamental triangle of ΩA under G2.
(0, n2 ,-
n
2 )
( n2 ,0,-
n
2 )
( n2 ,-
n
2 ,0)(0,- n2 ,
n
2 )
(- n2 ,
n
2 ,0)
(- n2 ,0,
n
2 )
( n3 ,
n
3 ,-
2n
3 )
( 2n3 ,- n3 ,-
n
3 )
( n3 ,-
2n
3 , n3 )
(- n3 ,-
n
3 ,
2n
3 )
(- 2n3 , n3 ,
n
3 )
(- n3 ,
2n
3 ,- n3 )
(0, n2 ,-
n
2 )
( n2 ,0,-
n
2 )
( n2 ,-
n
2 ,0)(0,- n2 ,
n
2 )
(- n2 ,
n
2 ,0)
(- n2 ,0,
n
2 )
( n3 ,
n
3 ,-
2n
3 )
( 2n3 ,- n3 ,-
n
3 )
( n3 ,-
2n
3 , n3 )
(- n3 ,-
n
3 ,
2n
3 )
(- 2n3 , n3 ,
n
3 )
(- n3 ,
2n
3 ,- n3 )
Figure 3.2. Symmetry under A2 (left), A˚2 (center) and G2 (right) in the frequency space. The shaded
area is the fundamental triangle of ΩB under G2.
For σ P G2, the number of inversion |σ| satisfies | ´ σ| “ |σ|. The following lemma can be
easily verified (writing down the table of σσ˚ for σ P A2 and σ˚ P A˚2 if necessary).
Lemma 3.2. Let f be a generic H-periodic function. Then
P`˚ P`fptq “ 1
12
ÿ
σPA2
pfptσq ` fp´tσqq “
1
12
ÿ
σPG2
fptσq,
P´˚ P`fptq “ 1
12
ÿ
σPA2
pfptσq ´ fp´tσqq “
1
12
ÿ
σPA˚2
p´1q|σ| pfptσq ´ fp´tσqq ,
P`˚ P´fptq “ 1
12
ÿ
σPA2
p´1q|σ| pfptσq ´ fp´tσqq “
1
12
ÿ
σPA˚2
pfptσq ´ fp´tσqq ,
P´˚ P´fptq “ 1
12
ÿ
σPA2
p´1q|σ| pfptσq ` fp´tσqq “
1
12
ÿ
σPG2
p´1q|σ|fptσq.
8 H. Li, J. Sun and Y. Xu
For φkptq “ e
2πik¨t
3 , the action of P` and P´ on φk are called the generalized cosine and
generalized sine functions in [9], which are trigonometric functions given by
Ckptq :“ P`φkptq “
1
3
”
e
iπ
3
pk1´k3qpt1´t3q cos k2πt2
` e
iπ
3
pk1´k3qpt2´t1q cos k2πt3 ` e
iπ
3
pk1´k3qpt3´t2q cos k2πt1
ı
, (3.3)
Skptq :“
1
i
P´φkptq “
1
3
”
e
iπ
3
pk1´k3qpt1´t3q sin k2πt2
` e
iπ
3
pk1´k3qpt2´t1q sin k2πt3 ` e
iπ
3
pk1´k3qpt3´t2q sin k2πt1
ı
. (3.4)
Because of the symmetry, we only need to consider these functions on the fundamental domain
of the group A2, which is one of the equilateral triangles of the regular hexagon. These functions
form a complete orthogonal basis on the equilateral triangle and they are the analogues of the
cosine and sine functions on the equilateral triangle. These generalized cosine and sine functions
are the building blocks of the discrete Fourier analysis on the equilateral triangle and subsequent
analysis of generalized Chebyshev polynomials in [9].
We now define the analogue of such functions on G2. Since the fundamental domain of the
group G2 is the 300–600–900 triangle, which is half of the equilateral triangle, we can relate the
new functions to the generalized cosine and sine functions on the latter domain. There are,
however, four families of such functions, defined as follows:
CCkptq :“ P`˚ P`φkptq “
1
12
ÿ
σPA2
pφkσptq ` φ´kσptqq “
1
2
`
Ckptq ` C´kptq
˘
,
SCkptq :“
1
i
P´˚ P`φkptq “
1
12i
ÿ
σPA2
pφkσptq ´ φ´kσptqq “
1
2i
`
Ckptq ´ C´kptq
˘
,
CSkptq :“
1
i
P`˚ P´φkptq “
1
12i
ÿ
σPA2
p´1q|σ| pφkσptq ´ φ´kσptqq “
1
2
`
Skptq ´ S´kptq
˘
,
SSkptq :“ ´P´˚ P´φkptq “ ´
1
12
ÿ
σPA2
p´1q|σ| pφkσptq ` φ´kσptqq “
1
2i
`
Skptq ` S´kptq
˘
,
where the second and the third equalities follow directly from the definition. We call these
functions generalized trigonometric functions. As their names indicate, they are of the mixed
type of cosine and sine functions.
From (3.3) and (3.4), we can derive explicit formulas for these functions, which are
CCkptq “
1
3
”
cos πpk1´k3qpt1´t3q3 cosπk2t2 ` cos πpk1´k3qpt2´t1q3 cosπk2t3
` cos πpk1´k3qpt3´t2q3 cosπk2t1
ı
, (3.5)
SCkptq “
1
3
”
sin πpk1´k3qpt1´t3q
3 cosπk2t2 ` sin πpk1´k3qpt2´t1q
3 cosπk2t3
` sin πpk1´k3qpt3´t2q
3 cosπk2t1
ı
, (3.6)
CSkptq “
1
3
”
cos πpk1´k3qpt1´t3q3 sinπk2t2 ` cos πpk1´k3qpt2´t1q3 sinπk2t3
` cos πpk1´k3qpt3´t2q3 sinπk2t1
ı
, (3.7)
SSkptq “
1
3
”
sin πpk1´k3qpt1´t3q
3 sinπk2t2 ` sin πpk1´k3qpt2´t1q
3 sinπk2t3
` sin πpk1´k3qpt3´t2q
3 sinπk2t1
ı
. (3.8)
Discrete Fourier Analysis and Chebyshev Polynomials with G2 Group 9
In particular, it follows from (3.6)–(3.8) that CSkptq ” SSkptq ” 0 whenever k contains zero
component and SCkptq ” SSkptq ” 0 whenever k contains equal elements. Similar formulas can
be derived from the permutations of t1, t2, t3. In fact, the functions CCk and SSk are invariant
and anti-invariant under G2, respectively, whereas the functions CSk and SCk are of the mixed
type, with the first one invariant under A2 and anti-invariant under A˚2 and the second one
invariant under A˚2 and anti-invariant under A2. More precisely, these invariant properties lead
to the following identities:
CCkptσq “ CCkptq, SSkptσq “ p´1q|σ|SSkptq, σ P G2, (3.9)
SCkptσq “ ´SCkp´tσq “ SCkptq, σ P A2, (3.10)
CSkptσq “ ´CSkp´tσq “ p´1q|σ|CSkptq, σ P A2, (3.11)
SCkptσq “ ´SCkp´tσq “ p´1q|σ|SCkptq, σ P A˚2 , (3.12)
CSkptσq “ ´CSkp´tσq “ CSkptq, σ P A˚2 . (3.13)
In particular, it follows from (3.6)–(3.8) that CSkptq ” SSkptq ” 0 whenever k contains zero
component and SCkptq ” SSkptq ” 0 whenever k contains equal elements. Moreover, for any
k P H:, CSkptq “ SSkptq “ 0 whenever t contains zero component and SCkptq “ SSkptq “ 0
whenever t contains equal elements.
Because of their invariant properties, we only need to consider these functions on one of the
twelve 300–600–900 triangles in the hexagon Ω. We shall choose the triangle as
4 :“ tt P R3
H : 0 ď t2 ď t1 ď ´t3 ď 1u. (3.14)
The region 4 and its relative position in the hexagon are depicted in Figs. 3.3 and 3.1.
(1,0,-1)
( 12 ,
1
2 ,-1)
(0,0,0)
( n2 ,0,-
n
2 )
( n3 ,
n
3 ,-
2n
3 )
(0,0,0)
Figure 3.3. The fundamental triangles in ΩA (left) and ΩB (right).
When CCk, SCk, CSk, SSk are restricted to the triangle 4, we only need to consider a subset
of k P H: as can be seen by the relations in (3.9)–(3.13). Indeed, we can restrict k to the index
sets
Γ “ Γcc :“
k P H: : 0 ď k2 ď k1
(
, Γsc :“
k P H: : 0 ď k2 ă k1
(
, (3.15)
Γcs :“
k P H: : 0 ă k2 ď k1
(
, Γss :“
k P H: : 0 ă k2 ă k1
(
, (3.16)
respectively, where the notation is self-explanatory; for example, Γcc is the index set for CCk.
We define an inner product on 4 by
xf, gy4 :“
1
|4|
ż
4
fptqgptqdt “ 4
ż 1
2
0
dt2
ż 1´t2
t2
fptqgptqdt1.
If fḡ is invariant under the group G2, then it is easy to see that xf, gyΩ “ xf, gy4. Consequently,
we can deduce the orthogonality of CCk, SCk, CSk, SSk from that of φk on Ω.
10 H. Li, J. Sun and Y. Xu
Proposition 3.3. It holds that
xCCk,CCjy4 “
4k,j
|kG2|
“ 4k,j
$
’
&
’
%
1, k “ 0,
1
6 , k2pk1 ´ k2q “ 0, k1 ą 0,
1
12 , k1 ą k2 ą 0,
j,k P Γcc, (3.17)
xSCk,SCjy4 “
4k,j
|kG2|
“ 4k,j
#
1
6 , k2 “ 0,
1
12 , k1 ą k2 ą 0,
j,k P Γsc, (3.18)
xCSk,CSjy4 “
4k,j
|kG2|
“ 4k,j
#
1
6 , k1 “ k2 ą 0,
1
12 , k1 ą k2 ą 0,
j,k P Γcs, (3.19)
xSSk,SSjy4 “
4k,j
|kG2|
“ 1
124k,j, j,k P Γss, (3.20)
where kG2 “ tkσ : σ P G2u denotes the orbit of k under G2.
3.2 Discrete Fourier analysis on the 300–600–900 triangle
Using the fact that CCk, SCk and CSk, SSk are invariant and anti-invariant under A2 and that
CCk, CSk and SCk, SSk are invariant and anti-invariant under A˚2 , we can deduce a discrete
orthogonality for the generalized trignometric functions. Again, we state the main result in
terms of cubature rules. The index set for the nodes of the cubature rule is given by
Υn :“ tj P H : 0 ď j2 ď j1 ď ´j3 ď nu ,
which are located inside n4 as seen by (3.14). The space of invariant functions being integrated
exactly by the cubature rule are indexed by
Γn “ Γcc
n :“ ΓYH:n “
k P H: : 0 ď k2 ď k1 ď k3 ` n
(
,
Γsc
n :“ Γsc YH:n “
k P H: : 0 ď k2 ă k1 ă k3 ` n
(
,
Γcs
n :“ Γcs YH:n “
k P H: : 0 ă k2 ď k1 ď k3 ` n
(
,
Γss
n :“ Γss YH:n “
k P H: : 0 ă k2 ă k1 ă k3 ` n
(
.
Correspondingly, we define the following subspaces of H:n,
Hcc
n :“ spantCCk : k P Γcc
n u, Hsc
n :“ spantSCk : k P Γsc
n u,
Hcs
n :“ spantCSk : k P Γcs
n u, Hss
n :“ spantSSk : k P Γss
n u.
It is easy to verify that
dimHcc
n “ |Γ
cc
n | “
1
2
`
3tn3 u´ 2n
˘`
tn3 u` 1
˘
´
`
tn2 u´ n´ 1
˘`
tn2 u` 1
˘
,
dimHss
n “ |Γ
ss
n | “ |Γn´6|, dimHsc
n “ |Γ
sc
n | “ dimHcs
n “ |Γ
cs
n | “ |Γn´3|. (3.21)
Theorem 3.4. The following cubature is exact for all f P Hcc
2n´1
1
|4|
ż
4
fptqdt “
1
n2
ÿ
jPΥn
ω
pnq
j f
ˆ
j
n
˙
, (3.22)
where
ω
pnq
j :“ c
pnq
j |jG2| “
$
’
’
’
’
’
&
’
’
’
’
’
%
12, j P Υ0n, pinteriorq,
1, j “ 0, p300-vertexq,
2, j “ pn, 0,´nq, p600-vertexq,
3, j “ pn2 ,
n
2 ,´nq, p900-vertexq,
6, otherwise, pboundariesq.
Discrete Fourier Analysis and Chebyshev Polynomials with G2 Group 11
(n,0,-n)
( n2 ,
n
2 ,-n)
(0,0,0) (n,0,-n)
( n2 ,
n
2 ,-n)
(0,0,0) (n,0,-n)
( n2 ,
n
2 ,-n)
(0,0,0)
Figure 3.4. The index set Υn. n ” 0 pmod 3q (left), n ” 1 pmod 3q (center) and n ” 2 pmod 3q (right).
( n2 ,0,-
n
2 )
( n3 ,
n
3 ,-
2n
3 )
(0,0,0) ( n2 ,0,-
n
2 )
( n3 ,
n
3 ,-
2n
3 )
(0,0,0)
(cc) (sc)
( n2 ,0,-
n
2 )
( n3 ,
n
3 ,-
2n
3 )
(0,0,0) ( n2 ,0,-
n
2 )
( n3 ,
n
3 ,-
2n
3 )
(0,0,0)
(cs) (ss)
Figure 3.5. The index set Γn.
Moreover, if we define the discrete inner product xf, gy4,n “
1
n2
ř
jPΥn
ω
pnq
j fp jnqgp
j
nq, then
xCCj,CCky4,n “
4j,k
c
pnq
pk
|kG2|
“
4j,k
ω
pnq
pk
, j,k P Γn,
xSCj, SCky4,n “
4j,k
c
pnq
pk
|kG2|
“
4j,k
ω
pnq
pk
, j,k P Γsc
n ,
xCSj,CSky4,n “
4j,k
c
pnq
pk
|kG2|
“
4j,k
ω
pnq
pk
, j,k P Γcs
n ,
xSSj,SSky4,n “
4j,k
c
pnq
pk
|kG2|
“
4j,k
12
, j,k P Γss
n ,
where pk “ pk3 ´ k2, k1 ´ k3, k2 ´ k1q.
The formula (3.22) is derived from (2.4) by using the invariance of the functions in Hcc
2n´1 and
upon writing Ω “
`
Ť
σPG2
ttσ : t P 40u
˘
Ť
`
Ť
σPG2
ttσ : t P B4u
˘
. The reason that pk appears
goes back to Proposition 2.3. As the proof is similar to that in [9], we shall omit the details.
One may note that the formulation of the result resembles a Gaussian quadrature. The
connection will be discussed in Section 6.
12 H. Li, J. Sun and Y. Xu
3.3 Sturm–Liouville eigenvalue problem for the Laplace operator
Recall the relation (2.3) between the coordinates px1, x2q and the homogeneous coordinates
pt1, t2, t3q. A quick calculation gives the expression of the Laplace operator in homogeneous
coordinates,
∆ :“
B2
Bx2
1
`
B2
Bx2
2
“
1
2
«
ˆ
B
Bt1
´
B
Bt2
˙2
`
ˆ
B
Bt2
´
B
Bt3
˙2
`
ˆ
B
Bt3
´
B
Bt1
˙2
ff
.
A further computation shows that φkptq “ e
2πi
3
k¨t are the eigenfunctions of the Laplace operator:
for k P H,
∆φk “ ´λkφk, λk :“
2π2
9
“
pk1 ´ k2q
2 ` pk2 ´ k3q
2 ` pk3 ´ k1q
2
‰
. (3.23)
As a consequence, our generalized trigonometric functions are the solutions of the Sturm–
Liouville eigenvalue problem for the Laplace operator with certain boundary conditions on the
300–600–900 triangle. To be more precise, we denote the three linear segments that are the
boundary of this triangle by B1, B2, B3,
B1 :“ tt P 4 : t3 “ ´1u, B2 :“ tt P 4 : t2 “ 0u, B3 :“ tt P 4 : t1 “ t2u.
Let B
Bn denote the partial derivative in the direction of the exterior norm of 4. Then
B
Bn
ˇ
ˇ
ˇ
B1
“ ´
B
Bt3
,
B
Bn
ˇ
ˇ
ˇ
B2
“ ´
B
Bt2
,
B
Bn
ˇ
ˇ
ˇ
B1
“
B
Bt2
´
B
Bt1
.
Theorem 3.5. The generalized trigonometric functions CCk, SCk, CSk, SSk are the eigenfunc-
tions of the Laplace operator, ∆u “ ´λku, that satisfy the boundary conditions:
CCk :
Bu
Bn
ˇ
ˇ
ˇ
B1YB2YB3
“ 0, SCk :
Bu
Bn
ˇ
ˇ
ˇ
B1YB2
“ 0, u|B3 “ 0,
CSk :
Bu
Bn
ˇ
ˇ
ˇ
B3
“ 0, u|B1YB2 “ 0, SSk : u|B1YB2YB3 “ 0.
Proof. Since λk is invariant under G2, that is, λk “ λkσ, @σ P G2, that these functions satisfy
∆u “ ´λku follows directly from their definitions. The boundary conditions can be verified
directly via the equations (3.5), (3.6), (3.7) and (3.8). �
In particular, CCk satisfies the Neumann boundary conditions and SSk satisfies the Dirichlet
type boundary conditions.
3.4 Product formulas for the generalized trigonometric functions
Below we give a list of identities on the product of the generalized trigonometric functions, which
will be needed in the following section.
Lemma 3.6. The generalized trigonometric functions satisfy the relations,
CCjCCk “
1
12
ÿ
σPG2
CCk`jσ “
1
12
ÿ
σPG2
CCj`kσ, (3.24)
CCjSCk “
1
12
ÿ
σPG2
SCk`jσ “
1
12
ÿ
τPA˚2
p´1qτ
`
SCj`kτ ´ SCj´kτ
˘
, (3.25)
Discrete Fourier Analysis and Chebyshev Polynomials with G2 Group 13
CCjCSk “
1
12
ÿ
σPG2
CSk`jσ “
1
12
ÿ
τPA˚2
`
CSj`kτ ´ CSj´kτ
˘
, (3.26)
CCjSSk “
1
12
ÿ
σPG2
SSk`jσ “
1
12
ÿ
σPG2
p´1q|τ |SSj`kσ, (3.27)
SCjSCk “ ´
1
12
ÿ
τPA˚2
p´1q|τ |
`
CCk`jτ ´ CCk´jτ
˘
“ ´
1
12
ÿ
τPA˚2
p´1q|τ |
`
CCj`kτ ´ CCj´kτ
˘
, (3.28)
SCjCSk “
1
12
ÿ
τPA˚2
p´1q|τ |
`
SSk`jτ ´ SSk´jτ
˘
“
1
12
ÿ
τPA˚2
`
SSj`kτ ´ SSj´kτ
˘
, (3.29)
CSjCSk “ ´
1
12
ÿ
τPA˚2
`
CCk`jτ ´ CCk´jτ
˘
“ ´
1
12
ÿ
τPA˚2
`
CCj`kτ ´ CCj´kτ
˘
, (3.30)
SSjSSk “
1
12
ÿ
σPG2
p´1q|σ|CCk`jσ “
1
12
ÿ
σPG2
p´1q|σ|CCj`kσ. (3.31)
Furthermore, the following formulas hold:
3SC1,0,´1ptqCS1,1,´2ptq “ SS2,1,´3ptq, (3.32)
rSC1,0,´1ptqs
2 “
1
3
“
1` 2CC1,1,´2
‰
´ rCC1,0,´1s
2, (3.33)
rCS1,1,´2s
2 ` rCC1,1,´2s
2 “
1
3
“
1` 2CC3,0,´3
‰
, (3.34)
rCC1,0,´1s
3 “
1
36
CC3,0,´3 `
1
4
CC1,0,´1 `
1
6
CC1,1,´2 `
1
18
`
1
2
CC1,1,´2CC1,0,´1. (3.35)
Proof. For (3.24)–(3.31), we only prove (3.29). Other identities can be proved similarly. By
the definition of the generalized trigonometric functions,
SCjCSk “
1
12i
ÿ
σPA˚2
p´1q|σ|
`
φjσ ´ φ´jσ
˘
ˆ
1
12i
ÿ
τPA˚2
`
φkτ ´ φ´kτ
˘
“ ´
1
122
ÿ
τPA˚2
p´1q|τ |
ÿ
σPA˚2
p´1q|στ
´1|
“
φpk`jστ´1qτ ` φ´pk`jστ´1qτ
´ φpk´jστ´1qτ ´ φ´pk´jστ´1qτ
‰
upon using the relation p´1q|τ |`|στ
´1| “ p´1q|σ|, consequently,
SCjCSk “ ´
1
122
ÿ
σPA˚2
p´1q|σ|
ÿ
τPA˚2
p´1q|τ |
“
φpk`jσqτ ` φ´pk`jσqτ ´ φpk´jσqτ ´ φ´pk´jσqτ
‰
“
1
12
ÿ
σPA˚2
p´1q|σ|
`
SSk`jσ ´ SSk´jσ
˘
,
proving the first equality in (3.29). Further by (3.9),
1
12
ÿ
σPA˚2
p´1q|σ|
`
SSk`jσ ´ SSk´jσ
˘
“
1
12
ÿ
σPA˚2
`
SSkσ´1`j ´ SSkσ´1´j
˘
“
1
12
ÿ
σPA˚2
`
SSj`kσ ´ SSkσ´j
˘
“
1
12
ÿ
σPA˚2
`
SSj`kσ ´ SSj´kσ
˘
,
since SSj “ SS´j by (3.5). This completes the proof of (3.29).
14 H. Li, J. Sun and Y. Xu
We now prove the relations (3.32)–(3.35). By (3.29),
CS1,1,´2ptqSC1,0,´1ptq “
1
6
”
`
SS2,1,´3ptq ´ SS0,´1,1ptq
˘
`
`
SS´1,1,0ptq ´ SS3,´1,´2ptq
˘
`
`
SS2,´2,0ptq ´ SS0,2,´2ptq
˘
ı
“
1
3
SS2,1,´3ptq,
which proves (3.32). By (3.28) and (3.24), we have
rSC1,0,´1s
2 ` rCC1,0,´1s
2 “ ´
1
6
“
CC2,0,´2 ´ 1` 2CC1,´1,0 ´ 2CC1,1,´2
‰
`
1
6
“
CC2,0,´2 ` 1` 2CC1,´1,0 ` 2CC1,1,´2
‰
“
1
3
“
1` 2CC1,1,´2
‰
,
which is (3.33). Next, from (3.30) and (3.24) we deduce that
rCS1,1,´2s
2 ` rCC1,1,´2s
2 “ ´
1
6
“
CC2,2,´4 ´ 1` 2CC1,1,´2 ´ 2CC3,0,´3
‰
`
1
6
“
CC2,2,´4 ` 1` 2CC1,1,´2 ` 2CC3,0,´3
‰
“
1
3
“
1` 2CC3,0,´3
‰
,
which is (3.34). Finally, the identity (3.35) follows from a successive use of (3.24). The proof is
completed. �
4 Generalized Chebyshev polynomials
In [9], the generalized cosine and sine functions Ck and Sk are shown to be polynomials under
a change of variables, which are analogues of Chebyshev polynomials of the first and the second
kind, respectively, in two variables. These polynomials, first studied in [6, 7], are orthogonal
polynomials on the region bounded by the hypocycloid and they enjoy a remarkable property
on its common zeros, which yields a rare example of the Gaussian cubature rule.
In this section, we consider analogous polynomials related to our new generalized trigono-
metric functions, which has a structure different from those related to Ck and Sk.
The classical Chebyshev polynomials, Tnpxq, are obtained from the trigonometric functions
cosnθ by setting x “ cos θ, the lowest degree nontrivial trigonometric function. In analogy, we
make a change of variables based on the first two nontrivial generalized cosine functions:
x “ xptq :“ CC1,0,´1ptq “
1
3
´
cos 2πpt1´t2q
3 ` cos 2πp2t1`t2q
3 ` cos 2πp2t2`t1q
3
¯
,
y “ yptq :“ CC1,1,´2ptq “
1
3
pcos 2πt1 ` cos 2πt2 ` cos 2πpt1 ` t2qq . (4.1)
If we change variables pt1, t2q ÞÑ px, yq, then the region 4 is mapped onto the region 4˚ bounded
by two hypocycloids,
4˚ “
px, yq :
`
1` 2y ´ 3x2
˘`
24x3 ´ y2 ´ 12xy ´ 6x´ 4y ´ 1
˘
ě 0
(
. (4.2)
The curve that defined the boundary of the domain ∆˚ satisfies the following relation:
Lemma 4.1. Let F px, yq :“ p1`2y´3x2qp24x3´y2´12xy´6x´4y´1q. Then, in homogeneous
coordinates,
F px, yq “ 3 rSC1,0,´1ptqs
2
rCS1,1,´2ptqs
2
“
1
3
rSS2,1,´3ptqs
2 . (4.3)
Discrete Fourier Analysis and Chebyshev Polynomials with G2 Group 15
(1,0,-1)
( 12 ,
1
2 ,-1)
(0,0,0) ( 12 ,0,-
1
2 )
( 13 ,
1
3 ,-
2
3 )
(1,1)(- 12 ,1)
(- 13 ,-
1
3 )
(0,- 12 )
( 16 ,-
1
3 )
Figure 4.1. The region ∆˚ (right) bounded by two hypocycloids, which is mapped from the triangle ∆
(left).
Furthermore, let Jpx, yq be the Jacobian of the changing of variable (4.1); then
Jpx, yq “
64π2
27
sinπt1 sinπt2 sinπpt1 ` t2q sin
πpt1 ´ t2q
3
sin
πpt1 ` 2t2q
3
sin
πp2t1 ` t2q
3
“
4π2
3
SC1,0,´1ptqCS1,1,´2ptq. (4.4)
Proof. Under the change of variables (4.1), by (3.33), (3.34) and (3.35), it follows that
rSC1,0,´1ptqs
2
“
1
3
`
1` 2y ´ 3x2
˘
,
rCS1,1,´2ptqs
2
“ 24x3 ´ y2 ´ 12xy ´ 6x´ 4y ´ 1, (4.5)
from which the first equality in (4.3) follows, whereas the second one follows from (3.32).
Taking derivatives and simplifying, we derive the formula of Jpx, yq in terms of the product
of sine functions. Furthermore, under the change of variables (4.1), it is not hard to verify that
24x3 ´ y2 ´ 12xy ´ 6x´ 4y ´ 1 “
16
9
sin2 πt1 sin2 πt2 sin2 πpt1 ` t2q,
1` 2y ´ 3x2 “
16
3
sin2 πpt1 ´ t2q
3
sin2 πpt1 ` 2t2q
3
sin2 πp2t1 ` t2q
3
,
from which the second equality of (4.4) follows readily. �
Definition 4.2. Under the change of variables (4.1), define for k1, k2 ě 0,
P
´ 1
2
,´ 1
2
k1,k2
px, yq :“ CCk1`k2,k2,´k1´2k2ptq,
P
1
2
,´ 1
2
k1,k2
px, yq :“
SCk1`k2`1,k2,´k1´2k2´1ptq
SC1,0,´1ptq
,
P
´ 1
2
, 1
2
k1,k2
px, yq :“
CSk1`k2`1,k2`1,´k1´2k2´2ptq
CS1,1,´2ptq
,
P
1
2
, 1
2
k1,k2
px, yq :“
SSk1`k2`2,k2`1,´k1´2k2´3ptq
SS2,1,´3ptq
.
We call these functions generalized Chebyshev polynomials and, in particular, call P
´ 1
2
,´ 1
2
k px, yq
and P
1
2
, 1
2
k px, yq the first kind and the second kind, respectively.
That these functions are indeed algebraic polynomials in x and y variables can be seen from
the following recursive relations, which can be derived from (3.24)–(3.27).
16 H. Li, J. Sun and Y. Xu
Proposition 4.3. For α, β “ ˘1
2 , Pα,βk1,k2
satisfy the recursion relation
Pα,βk1`1,k2
px, yq “ 6xPα,βk1,k2
px, yq ´ Pα,βk1`2,k2´1px, yq ´ P
α,β
k1´1,k2`1px, yq
´ Pα,βk1`1,k2´1px, yq ´ P
α,β
k1´2,k2`1px, yq ´ P
α,β
k1´1,k2
px, yq, (4.6)
Pα,βk1,k2`1px, yq “ 6yPα,βk1,k2
px, yq ´ Pα,βk1`3,k2´2px, yq ´ P
α,β
k1`3,k2´1px, yq
Pα,βk1,k2`1px, yq “ ´ P
α,β
k1´3,k2`1px, yq ´ P
α,β
k1´3,k2`2px, yq ´ P
α,β
k1,k2´1px, yq (4.7)
for k1, k2 ě 0. Furthermore, the following symmetric relations hold,
P
α,´ 1
2
µ,´ν px, yq “ P
α,´ 1
2
µ´3ν,νpx, yq, P
α, 1
2
µ,´ν´1px, yq “ ´P
α, 1
2
µ´3ν,ν´1px, yq, µ ě 3ν ě 0, (4.8)
P
´ 1
2
,β
´µ,ν px, yq “ P
´ 1
2
,β
µ,ν´µpx, yq, P
1
2
,β
´µ´1,νpx, yq “ ´P
1
2
,β
µ´1,ν´µpx, yq, ν ě µ ě 0. (4.9)
Proof. The recursive relations (4.6) and (4.7) follow directly from (3.24) and (3.27). As for (4.8)
and (4.9), we resort to the following identities of the trigonometric functions,
CCµ´ν,´ν,2ν´µpx, yq “ CCpµ´3νq`ν,ν,ν´µpx, yq,
SCµ´ν`1,´ν,2ν´µ´1px, yq “ SCpµ´3νq`ν`1,ν,ν´µ´1px, yq,
CSµ´pν`1q`1,´pν`1q`1,2ν´µpx, yq “ ´CSpµ´3νq`pν´1q`1,pν´1q`1,ν´µpx, yq,
SSµ´pν`1q`2,´pν`1q`1,2ν´µ´1px, yq “ ´SSpµ´3νq`pν´1q`2,pν´1q`1,ν´µ´1px, yq,
CC´µ`ν,ν,µ´2νpx, yq “ CCµ`pν´µq,ν´µ,µ´2νpx, yq,
CS´µ`ν`1,ν`1,µ´2ν´2px, yq “ CSµ`pν´µq`1,pν´µq`1,µ´2ν´2px, yq,
SC´pµ`1q`ν`1,ν,µ´2νpx, yq “ ´SCpµ´1q`pν´µq`1,ν´µ,µ´2νpx, yq,
SS´pµ`1q`ν`2,ν`1,µ´2ν´2px, yq “ ´SSpµ´1q`pν´µq`2,pν´µq`1,µ´2ν´2px, yq,
which are derived from (3.9)–(3.13). �
The recursive relations (4.6) and (4.7) can be used to generate all polynomials Pα,βk1,k2
recur-
sively. The task, however, is non-trivial. Below we describe an algorithm for the recursion. Our
starting point is
P
´ 1
2
,´ 1
2
0,0 px, yq “ 1, P
´ 1
2
,´ 1
2
1,0 px, yq “ x, P
´ 1
2
,´ 1
2
0,1 px, yq “ y,
P
1
2
,´ 1
2
0,0 px, yq “ 1, P
1
2
,´ 1
2
1,0 px, yq “ 6x` 2, P
1
2
,´ 1
2
0,1 px, yq “ 6x` 3y ` 1,
P
´ 1
2
, 1
2
0,0 px, yq “ 1, P
´ 1
2
, 1
2
1,0 px, yq “ 3x, P
´ 1
2
, 1
2
0,1 px, yq “ 6y ` 2,
P
1
2
, 1
2
0,0 px, yq “ 1, P
1
2
, 1
2
1,0 px, yq “ 6x` 1, P
1
2
, 1
2
0,1 px, yq “ 6x` 6y ` 2.
The first few cases are complicated as the right side of the (4.6) and (4.7) involve negative
indexes, for which we need to use (4.8) and (4.9). We give these cases explicitly below
P
´ 1
2
,´ 1
2
2,0 px, yq “ 6x2 ´ 2x´ 2y ´ 1, P
´ 1
2
,´ 1
2
1,1 px, yq “ 3xy ´ 6x2 ` x` 2y ` 1,
P
´ 1
2
, 1
2
2,0 px, yq “ 18x2 ´ 3x´ 6y ´ 3, P
´ 1
2
, 1
2
1,1 px, yq “ 18xy ` 6x´ 18x2 ` 6y ` 3,
P
1
2
,´ 1
2
2,0 px, yq “ 36x2 ´ 6y ´ 3, P
1
2
,´ 1
2
1,1 px, yq “ 18xy ` 6x` 9y ` 2,
P
1
2
, 1
2
2,0 px, yq “ 36x2 ´ 6y ´ 3; P
1
2
, 1
2
1,1 px, yq “ 36xy ` 12x` 12y ` 4;
Discrete Fourier Analysis and Chebyshev Polynomials with G2 Group 17
P
´ 1
2
,´ 1
2
3,0 px, yq “ 36x3 ´ 18xy ´ 9x´ 6y ´ 2,
P
´ 1
2
, 1
2
3,0 px, yq “ 108x3 ´ 54xy ´ 27x´ 12y ´ 5,
P
1
2
,´ 1
2
3,0 px, yq “ 216x3 ´ 72xy ´ 48x´ 24y ´ 8,
P
1
2
, 1
2
3,0 px, yq “ 216x3 ´ 72xy ´ 42x´ 18y ´ 7;
P
´ 1
2
,´ 1
2
0,2 px, yq “ 6y2 ` 10y ´ 72x3 ` 36xy ` 18x` 3,
P
´ 1
2
, 1
2
0,2 px, yq “ 36y2 ` 36y ´ 216x3 ` 108xy ` 54x` 9,
P
1
2
,´ 1
2
0,2 px, yq “ 126xy ` 18y2 ` 36y ` 54x` 10´ 216x3,
P
1
2
, 1
2
0,2 px, yq “ 144xy ` 36y2 ` 42y ´ 216x3 ` 60x` 11.
The above formulas are derived from the recursive relations in the order of p2, 0q, p1, 1q, p3, 0q,
p0, 2q, that is, we need to deduce p3, 0q before proceeding to p0, 2q. It should be pointed out
that our polynomial Pα,β0,2 is of degree 3, rather than degree 2, which shows that our polynomials
do not satisfy the property of spantPα,βk1,k2
: k1 ` k2 ď nu “ Π2
n. In particular, they cannot be
ordered naturally in the graded lexicographical order.
We shall show in the following section that our polynomials are best ordered in another
graded order for which the order is defined by 2k1`3k2 “ n. We have displayed the polynomials
Pα,βk1,k2
px, yq for all 2k1 ` 3k2 ď 6. In Algorithm 1 below we give an algorithm for the evaluation
of all Pα,βk1,k2
px, yq with 2k1 ` 3k2 “ n and n ě 7.
The polynomials P
˘ 1
2
,˘ 1
2
k defined in the Definition 4.2 satisfy an orthogonality relation. Let
us define a weight function wα,β on the domain 4˚,
wα,βpx, yq :“
p4π2qα`β
32α`β
`
1` 2y ´ 3x2
˘α `
24x3 ´ y2 ´ 12xy ´ 6x´ 4y ´ 1
˘β
“
ˆ
4π2
3
˙α`β
pSC1,0,´1ptqq
2α
pCS1,1,´2ptqq
2β
where the second equality follows from (4.5). This weight function is closely related to the
Jacobian of the changing variables (4.1), as seen in Lemma 4.1. With respect to this weight
function, we define
xf, gywα,β :“ cα,β
ż
∆˚
fpx, yqgpx, yqwα,βpx, yqdxdy,
where cα,β :“ 1{
ş
4˚ wα,βpx, yqdxdy is a normalization constant; in particular, c´ 1
2
,´ 1
2
“ 4,
c 1
2
,´ 1
2
“ c´ 1
2
, 1
2
“ 18{π2 and c 1
2
, 1
2
“ 243{π4. Since the change of variables (4.1) implies immedi-
ately that
cα,β
ż
4˚
fpx, yqwα,βpx, yqdxdy “
1
|4|
ż
4
fptq
`
SC1,0,1ptq
˘2α`1`
CS1,1,´2ptq
˘2β`1
dt, (4.10)
we can translate the orthogonality of CCj, SCj, CSj and SSj to that of Pα,βk1,k2
for α, β “ ˘1
2 .
Indeed, from Proposition 3.3 we can deduce the following theorem.
Theorem 4.4. For α “ ˘1
2 , β “ ˘
1
2 ,
xPα,βk1,k2
, Pα,βj1,j2
ywα,β “ dα,βk1,k2δk1,j1δk2,j2 , (4.11)
18 H. Li, J. Sun and Y. Xu
Algorithm 1. A recursive algorithm for the evaluation of Pα,βk1,k2
px, yq.
Step 1 if n “ 2m
Pα,βm,0px, yq “ 6xPα,βm´1,0px, yq ´ cβP
α,β
m´2,1px, yq ´ P
α,β
m´2,0px, yq ´ cβP
α,β
m´3,1px, yq,
where cβ “ 2 if β “ ´1
2 , and cβ “ 1 if β “ 1
2 ;
Step 2 for k2 from 2´ modpn, 2q with increment 2 up to tn3 u´ 2 do
k1 “
n´3k2
2 ,
Pα,βk1,k2
px, yq “ 6xPα,βk1´1,k2
px, yq ´ Pα,βk1`1,k2´1px, yq ´ P
α,β
k1´2,k2`1px, yq
´ Pα,βk1,k2´1px, yq ´ P
α,β
k1´3,k2`1px, yq ´ P
α,β
k1´2,k2
px, yq;
Step 3 if n “ 3m
Pα,β0,mpx, yq “ 6yPα,β0,m´1px, yq ´ P
α,β
3,m´3px, yq ´ P
α,β
3,m´2px, yq ´ P
α,β
0,m´2px, yq
`
#
´Pα,β3,m´3px, yq ´ P
α,β
3,m´2px, yq, α “ ´1
2 ,
Pα,β1,m´2px, yq ` P
α,β
1,m´1px, yq, α “ 1
2 ;
if n “ 3m` 1
Pα,β2,m´1px, yq “ 6xPα,β1,m´1px, yq ´ P
α,β
3,m´2px, yq ´ P
α,β
0,mpx, yq ´ P
α,β
2,m´2px, yq
´ Pα,β0,m´1px, yq ´
#
Pα,β1,m´1px, yq, α “ ´1
2 ,
0, α “ 1
2 ;
if n “ 3m` 2
Pα,β1,mpx, yq “
#
3xPα,β0,mpx, yq ´ P
α,β
2,m´1px, yq ´ P
α,β
1,m´1px, yq, α “ ´1
2 ,
p6x` 1qPα,β0,mpx, yq ´ P
α,β
2,m´1px, yq ´ P
α,β
1,m´1px, yq, α “ 1
2 .
where
d
´ 1
2
,´ 1
2
k1,k2
:“
$
’
&
’
%
1, k1 “ k2 “ 0,
1
6 , k1k2 “ 0, k1 ` k2 ą 0,
1
12 , k1 ą 0, k2 ą 0,
d
1
2
,´ 1
2
k1,k2
:“
#
1
6 , k1 ě 0, k2 “ 0,
1
12 , k1 ě 0, k2 ą 0,
d
´ 1
2
, 1
2
k1,k2
:“
#
1
6 , k1 “ 0, k2 ě 0,
1
12 , k1 ą 0, k2 ě 0,
d
1
2
, 1
2
k1,k2
:“
1
12
, k1 ě 0, k2 ě 0.
Proof. All four cases follow from Proposition 3.3. For α “ β “ ´1
2 , this is immediate. For the
other three cases, we observe that the weight function cancels the denominator in the definition
of Pα,βk1,k2
Pα,βj1,j2
(see Definition 4.2), which requires (3.32) in the case of α “ β “ 1
2 . �
Although the polynomials P
˘ 1
2
,˘ 1
2
k1,k2
are mutually orthogonal, they are not quite the usual
orthogonal polynomials as we have seen from the recursive relations. In fact, there are only
two such polynomials with the total degree 2, which is one less than the number of monomials
of degree 2. As we have seen from the recursive relations, the structure of these polynomials
Discrete Fourier Analysis and Chebyshev Polynomials with G2 Group 19
is much more complicated. To understand their structure, we study them as solutions of the
corresponding Sturm–Liouville problem in the following section.
5 Sturm–Liouville eigenvalue problem
and generalized Jacobi polynomials
Recall that our generalized trigonometric polynomials are solutions of the Sturm–Liouville
eigenvalue problems with corresponding boundary conditions. The Laplace operator becomes
a second-order linear differential operator in x, y variables under the change of variables (4.1).
Using the fact that t3 “ ´t1 ´ t2, we rewrite the change of variables (4.1) as
x “
1
3
´
cos 2πpt1´t2q
3 ` cos 2πpt2´t3q
3 ` cos 2πpt3´t1q
3
¯
,
y “
1
3
pcos 2πt1 ` cos 2πt2 ` cos 2πt3q.
A tedious but straightforward computation shows that
pBt1 ´ Bt2q
2 ` pBt2 ´ Bt3q
2 ` pBt3 ´ Bt1q
2
“
4π2
9
“
A1,1px, yqB
2
x ` 2A1,2px, yqBxBy `A2,2px, yqB
2
y ` 6xBx ` 18yBy
‰
“:
4π2
9
L´ 1
2
,´ 1
2
,
where we define
A11 :“ ´6x2 ` y ` 3x` 2, A12 “ A21 :“ ´9xy ` 18x2 ´ 6y ´ 3,
A22 :“ ´18y2 ` 108x3 ´ 54xy ´ 27x´ 9y. (5.1)
Consequently, we can translate the Laplace equation satisfied by CCk into the equation in
L´ 1
2
,´ 1
2
for the polynomials P
´ 1
2
,´ 1
2
k1,k2
px, yq. It is easy to verify that the operator can be rewritten
as
L´ 1
2
,´ 1
2
“ ´w 1
2
, 1
2
“
Bxwα,β
`
A11Bx `A12By
˘
` Byω
´ 1
2
,´ 1
2
`
A21Bx `A22By
˘‰
“ ´w 1
2
, 1
2
∇trw´ 1
2
,´ 1
2
Λ∇,
where in the second line we have used
∇ :“ pBx, Byq
tr and Λ :“
ˆ
A11 A12
A21 A22
˙
.
It is not difficult to verify that the matrix Λ is positive definite in the interior of the domain 4˚.
Indeed, det Λ “ 3F px, yq, where F is defined in Lemma 4.1, and A1,1px, yq “ 3px ´ yq ` 2p1 `
2y ´ 3x2q is positive if x ą y and it attains its minimal on the left most boundary, as seen by
taking partial derivatives, in the rest of the domain, from which it is easy to verify that A1,1 ą 0
in the interior of 4˚. The expression of L´ 1
2
,´ 1
2
prompts the following definition.
Definition 5.1. For α, β ą ´1, define a second-order differential operator
Lα,β :“ ´w´α,´β∇trwα,βΛ∇
“ ´w´α,´β
“
Bxwα,β
`
A11Bx `A12By
˘
` Bywα,β
`
A21Bx `A22By
˘‰
.
20 H. Li, J. Sun and Y. Xu
The explicit formula of this differential operator is given by
Lα,β “ ´A11B
2
x ´ 2A12BxBy ´A22B
2
y `B1Bx `B2By. (5.2)
where we define
B1px, yq “ 21x` 12αx` 18βx` 6α` 3,
B2px, yq “ 18x` 36αx` 18β ` 45y ` 36βy ` 18αy ` 9.
Theorem 5.2. Let α, β ą ´1. Then, the differential operator
Lα,β “ ´w´α,´β∇trwα,βΛ∇
is self-adjoint and positive definite with respect to the inner product x¨, ¨ywα,β .
Proof. By Green’s formula,
ij
4˚
fLα,βgwα,βdxdy “
ij
4˚
f∇trwα,βΛ∇gdxdy “ ´
ij
4˚
p∇fqtrΛp∇gqwα,βdxdy
`
¿
B4˚
wα,βf rpA11Bxg `A12Bygqdy ´ fpA22Byg `A21Bxgqdxs
“ ´
ij
4˚
p∇fqtrΛp∇gqwα,βdxdy
`
¿
B4˚
wα,βf rpBxgqpA11dy ´A21dxq ´ pBygqpA22dx´A12dyqs ,
where B4˚ denotes the boundary of the triangle. Recall that B4˚ is defined by F px, yq “ 0,
where F is defined in Lemma 4.1. It follows then
dF “ F1dx` F2dy “ 0, where F1 “
BF
Bx
, F2 “
BF
By
. (5.3)
On the other other hand, a quick computation shows that
F1A11 ` F2A21 “ ´6p5x` 1qF px, yq “ 0, (5.4)
F1A12 ` F2A22 “ ´6p3y ` 2x` 1qF px, yq “ 0 (5.5)
on B4˚. Solving (5.3) and (5.4) shows that A11dy´A21dx “ 0, whereas solving (5.3) and (5.5)
shows that A22dx ´ A12dy “ 0 on B4˚. Consequently, the integral over B4˚ is zero and we
conclude that
´
ij
4˚
fLα,βgwα,βdxdy “
ij
4˚
p∇fqtrΛp∇gqwα,βdxdy “ ´
ij
4˚
gLα,βfwα,βdxdy,
which shows that Lα,β is self-adjoint and positive definite. �
We consider polynomial solutions for the eigenvalue problem Lα,βu “ λu. Differential opera-
tors in the form of (5.2) have long been associated with orthogonal polynomials of two variables
(see, for example, [8, 16]). However, in most of the studies, the coefficients Ai,j are chosen to
be polynomials of degree 2, which is necessary if, for each positive integer n, the solution of the
eigenvalue problem is required to consist of n ` 1 linearly independent polynomials of degree
Discrete Fourier Analysis and Chebyshev Polynomials with G2 Group 21
n, since such choices ensure that the differential operator preserves the degree of polynomials.
In our case, however, the coefficient A2,2 in (5.1) is of degree 3, which causes a number of
complications. In particular, our differential operator does not preserve the polynomial degree;
in other words, it does not map Π2
n to Π2
n, the space of polynomials of degree at most n in two
variables.
Definition 5.3. For k1, k2 ě 0, the m-degree of the monomial xk1yk2 is defined as |k|˚ :“
2k1 ` 3k2. A polynomial p in two variables is said to have m-degree n if one monomial in p has
m-degree of exactly n and all other monomials in p have m-degree at most n. For n P N0, let Π˚n
denote the space of polynomials of m-degree at most n; that is,
Π˚n :“ span
xk1yk2 : 0 ď k1, k2; 2k1 ` 3k2 ď n
(
.
The dimension of the space Π˚n is the same as that of Hcc
n , by (3.21),
dim Π˚n “
1
2
`
3tn3 u´ 2n
˘`
tn3 u` 1
˘
´
`
tn2 u´ n´ 1
˘`
tn2 u` 1
˘
. (5.6)
Here is a list of the dimension for small n:
n 1 2 3 4 5 6 7 8 9 10 11 12
dim Π˚n 1 2 3 4 5 7 8 10 12 14 16 19
The name m-degree is coined in [13] after the marks, or co-marks, in the root system for
the simple compact Lie group, where the case of the group G2 is used as an example. For
polynomials graded by the m-degree, we introduce an ordering among monomials.
Definition 5.4. For any k, j P N2
0, we define an order ă by k ă j if 2j1 ` 3j2 ą 2k1 ` 3k2
or 2pk1 ´ j1q “ 3pj2 ´ k2q ą 0, and k ĺ j if k ă j or k “ j. We call ă the ˚-order. If
ppx, yq “
ř
pk1,k2qĺpm,nq
ck1,k2x
k1yk2 with cm,n ‰ 0, we call cm,nx
myn the leading term of p in the
˚-order.
For m,n ě 0, define
Π˚m,n “ span
xjyk : pj, kq ĺ pm,nq
(
.
It is easy to see that Π˚n “ Π˚
2n´3t 2n
3
u,2t 2n
3
u´n
. The ˚-order is well-defined. The following lemma
justifies our definitions.
Lemma 5.5. For m,n ě 0, the operator Lα,β maps Π˚m,n onto Π˚m,n.
Proof. We apply the operator Lα,β on the monomial xjyk. The result is
Lα,βxjyk “ ´A11B
2
xx
jyk ´A22B
2
yx
jyk ´ 2A12BxByx
jyk `B1Bxx
jyk `B2Byx
jyk
“
“
6pj2 ` 3k2 ` 3jkq ` 3p5` 4α` 6βqj ` 3p9` 6α` 12βqk
‰
xjyk
´ 108kpk ´ 1qxj`3yk´2 ´ jpj ´ 1qxj´2yk`1 ` 18kp3k ´ 2´ 2j ` 2αqxj`1yk´1
` 3jp´j ` 2` 4k ` 2αqxj´1yk ` 9kpk ` 2βqxjyk´1
´ 2jpj ´ 1qxj´2yk ` 27kpk ´ 1qxj`1yk´2 ` 6jkxj´1yk´1.
Introducing the notation
Υ “ tp0, 0q, p0, 1q, p1, 0q, p1, 1q, p2, 1q, p3, 2q, p4, 2q, p4, 3q, p5, 3qu ,
22 H. Li, J. Sun and Y. Xu
we write the expression as
Lα,βxjyk “
ÿ
pµ,νqPΥ
aj,kµ,νx
j´2µ`3νyk`µ´2ν , (5.7)
where
aj,k0,0 “ 6
`
j2 ` 3k2 ` 3jk
˘
` 3p5` 4α` 6βqj ` 3p9` 6α` 12βqk,
aj,k0,1 “ ´108kpk ´ 1q, aj,k1,0 “ ´jpj ´ 1q, aj,k1,1 “ 18kp3k ´ 2´ 2j ` 2αq,
aj,k2,1 “ 3jp´j ` 2` 4k ` 2αq, aj,k3,2 “ 9kpk ` 2βq,
aj,k4,2 “ ´2jpj ´ 1q, aj,k4,3 “ 27kpk ´ 1q, aj,k5,3 “ 6jk.
From this computation, it follows readily that Lα,β maps Π˚m,n into Π˚m,n. Furthermore, with
respect to the ˚-order, it is easy to see that am,n0,0 x
myn is the leading term of Lα,β by (5.7), which
shows that Lα,β maps Π˚m,n onto Π˚m,n. �
The identity (5.7) also shows that Lα,β has a complete set of eigenfunctions in Π˚m,n.
Theorem 5.6. For α, β ě ´1{2 and k1, k2 ě 0, there exists a polynomial Pα,βk1,k2
P Π˚k1,k2 with
the leading term xk1yk2 such that
Lα,βPα,βk1,k2
“ λα,βk1,k2P
α,β
k1,k2
, (5.8)
where
λα,βk1,k2 :“
3
2
|k|˚p|k|˚ ` 5` 4α` 6βq `
9
2
k2pk2 ` 1` 2βq. (5.9)
Furthermore, if we require all the polynomials are orthogonal to each other with respect to the
inner product x¨, ¨ywα,β , then Pα,βk1,k2
is uniquely determined by its leading term in the ˚-order.
Proof. We first apply the Gram–Schmidt orthogonality process to monomials
xk1yk2
(
in the
˚-order, which uniquely determines a complete system of orthogonal polynomials with leading
term xk1yk2 with respect to x¨, ¨ywα,β ; that is, Pα,β0,0 px, yq “ 1 and
Pα,βk1,k2
px, yq “ xk1yk2 ´
ÿ
pj1,j2qăpk1,k2q
xxk1yk2 , Pα,βj1,j2
ywα,β
xPα,βj1,j2
, Pα,βj1,j2
ywα,β
Pα,βj1,j2
px, yq, p0, 0q ă pk1, k2q.
The Gram–Schmidt orthogonality and Lemma 5.5 show that
Lα,βPα,βk1,k2
px, yq P span
Pα,βj1,j2
px, yq : pj1, j2q ĺ pk1, k2q
(
“ Π˚k1,k2 . (5.10)
Evidently, Lα,βPα,β0,0 “ 0 “ a0,0
0,0P
α,β
0,0 . We apply induction. Assume that
Lα,βPα,βj1,j2
“ a0,0
j1,j2
Pα,βj1,j2
, pj1, j2q ă pk1, k2q.
It then follows from Theorem 5.2 and the orthogonality of Pα,βk1,k2
that
xLα,βPα,βk1,k2
, Pα,βj1,j2
ywα,β “ xP
α,β
k1,k2
,Lα,βPα,βj1,j2
ywα,β “ aj1,j20,0 xPα,βk1,k2
, Pα,βj1,j2
ywα,β “ 0,
so that, as a consequence of (5.10),
Lα,βPα,βk1,k2
“ cPα,βk1,k2
. (5.11)
Discrete Fourier Analysis and Chebyshev Polynomials with G2 Group 23
Comparing the leading term of the above identity, we obtain from (5.7) that
ak1,k20,0 xk1,k2 “ cxk1,k2 ,
which gives c “ ak1,k20,0 . Ultimately, this inductive process shows that
Lα,βPα,βk1,k2
“ λα,βk1,k2P
α,β
k1,k2
, with λα,βk1,k2 “ ak1,k20,0 .
As shown in the proof of Lemma 5.5,
λα,βk1,k2 “ 6
`
k2
1 ` 3k2
2 ` 3k1k2
˘
` 3p5` 4α` 6βqk1 ` 3p9` 6α` 12βqk2
“
3
2
p2k1 ` 3k2qpp2k1 ` 3k2q ` 4` 4α` 6βq `
9
2
k2pk2 ` 2` 2βq ` 3k1,
which is (5.9) since |k|˚ “ 2k1 ` 3k2 by definition.
Moreover, suppose rPα,βk1,k2
px, yq P Π˚k1,k2 is another polynomial with the leading term xk1yk2
such that
Lα,β rPα,βk1,k2
px, yq “ λ rPα,βk1,k2
px, yq,
x rPα,βk1,k2
, pywα,β “ 0, @ p P span
xj1yj2 : pj1, j2q ă pk1, k2q
(
.
Using the same argument that determines c in (5.11), we see that λ “ λα,βk1,k2 “ ak1,k20,0 . Moreover,
it is easy to see that
Pα,βk1,k2
´ rPα,βk1,k2
P span
xj1yj2 : pj1, j2q ă pk1, k2q
(
,
@
Pα,βk1,k2
´ rPα,βk1,k2
, Pα,βj1,j2
D
wα,β
“ 0, @ pj1, j2q ă pk1, k2q.
This finally leads to Pα,βk1,k2
´ rPα,βk1,k2
“ 0, which shows that Pα,βk1,k2
is uniquely determined by its
leading term in the ˚-order and the orthogonality xPα,βk1,k2
, xj1yj2ywα,β “ 0 for all pj1, j2q ă pk1, k2q.
This completes the proof. �
Let Pα,βk1,k2
be orthogonal to each other with respect to the inner product x¨, ¨ywα,β . The first
few polynomials and the eigenvalues can be readily checked to be
Pα,β0,0 px, yq “ 1, λα,β0,0 “ 0;
Pα,β1,0 px, yq “ x`
1` 2α
7` 4α` 6β
, λα,β1,0 “ 3p7` 4α` 6βq;
Pα,β0,1 px, yq “ y `
3p1` 2αq
4` α` 3β
x`
5` 5α` 11β ` 2αβ ` 6β2 ` 4α2
p4` α` 3βqp5` 2α` 4βq
,
λα,β0,1 “ 9p5` 2α` 4βq;
Pα,β2,0 px, yq “ x2 ´
2y
3p3` 2αq
y `
4pα` 1qp2α´ 1q
p3` 2αqp4α` 11` 6βq
x
`
´105´ 86α´ 120β ´ 36β2 ´ 48βα` 8α2 ` 24α3
3p3` 2αqp4α` 11` 6βqp4α` 6β ` 9q
,
λα,β2,0 “ 6p9` 4β ` 6αq,
Pα,β1,1 px, yq “ xy `
3p2α´ 1q
5` α` 3β
x2 `
6βα` 11α` 15β ` 2α2 ` 27
p5` α` 3βqp4α` 6β ` 13q
y
`
119`229β`40α3`36β3`111α`80βα2`156β2`132βα`140α2`36β2α
p2α` 7` 4βqp4α` 6β ` 13qp5` α` 3βq
x
24 H. Li, J. Sun and Y. Xu
`
8α3 ` 6α2 ` 4βα2 ` 12β2α` 9α` 28βα` 70` 93β ` 30β2
p2α` 7` 4βqp4α` 6β ` 13qp5` α` 3βq
,
λα,β1,1 “ 6p14` 9β ` 5αq.
For each Pα,βm,n, (5.10) shows that the Lα,βPα,βm,n involves only Pα,βj1,j2
with pj1, j2q in
Γm,n :“
pj1, j2q P N2 : pj1, j2q ĺ pm,nq
(
.
This set of dependence of the polynomial solution is determined by the ˚-ordering. Indeed, it is
easy to see that
Γm,n “ Γ`m,n Y Γ´m,n,
Γ`m,n :“
pm´ 2p` 3q, n` p´ 2qqq P Z2 : 0 ď q ď t
p`n
2 u, 0 ď p ď 2m` 3n
(
,
Γ´m,n :“
!
pm´ 2p` 3q, n` p´ 2qq P Z2 : r
2p´m
3 s ď q ď ´1, 1 ď p ď tm´3
2 u
)
.
For p, q as in Γk1,k2 but not both 0, we have that for α, β ě ´1
2 ,
λα,βk1,k2 ´ λ
α,β
k1´2p`3q,k2`p´2q “ 3p2k1 ´ 2p` 3q ` 2α` 1qp` 9p2k2 ` p´ 2q ` 2β ` 1qq ą 0,
which shows that λα,βk1,k2 ‰ λα,βj1,j2 for any pj1, j2q P Γ`k1,k2 . This implies that polynomial solutions
of the same m-degree below to different eigenvalues. Moreover, if λα,βk1,k2 “ λα,βj1,j2 for pj1, j2q ă
pk1, k2q, then pj1, j2q P Γ´k1,k2 .
In the case of pα, βq “ p´1
2 ,´
1
2q, our polynomials Pα,βk1,k2
agree with the generalized Chebyshev
polynomial that we defined in the last section. For the other three cases of pα, βq “ p˘1
2 ,˘
1
2q,
this requires proof. Let us denote the Chebyshev polynomials temporarily by Qα,βk1,k2 , pα, βq “
p´1
2 ,´
1
2q. It is not hard to see, from Algorithm 1, that the leading term of Qα,βk1,k2 is cxk1yk2
with certain c ą 0, which implies that span
Qα,βj1,j2px, yq : pj1, j2q ĺ pk1, k2q
(
“ Π˚k1,k2 . Thus, we
can write
Lα,βQα,βk1,k2px, yq “ λα,βk1,k2Q
α,β
k1,k2
px, yq `
ÿ
pj1,j2qăpk1,k2q
ck1,k2j1,j2
Qα,βj1,j2px, yq. (5.12)
On the other hand, by the orthogonality and the self-adjointness of Lα,β, for any pl1, l2q ă
pk1, k2q,
`
Lα,βQα,βk1,k2 , Q
α,β
l1,l2
˘
wα,β
“
`
Qα,βk1,k2 ,Lα,βQ
α,β
l1,l2
˘
wα,β
“
¨
˝Qα,βk1,k2 , λ
α,β
l1,l2
Qα,βl1,l2 `
ÿ
pj1,j2qăpl1,l2q
cl1,l2j1,j2
Qα,βj1,j2
˛
‚
wα,β
“ 0.
As a result, we deduce from (5.12) that
Lα,βQα,βk1,k2px, yq “ λα,βk1,k2Q
α,β
k1,k2
px, yq.
Consequently, up to a constant multiple, we see that Qα,βk1,k2 coincides with the Jacobi polyno-
mials.
Corollary 5.7. The Chebyshev polynomials defined in Definition 4.2 satisfy the equation (5.8).
In particular, this shows that the Chebyshev polynomials are elements in Π˚
|k|˚
and they are
determined, as eigenfunctions of Lα,β, uniquely by the leading term in the ˚-order.
Discrete Fourier Analysis and Chebyshev Polynomials with G2 Group 25
6 Cubature rules for polynomials
In the case of the equilateral triangle, the cubature rules for the trigonometric functions are
transformed into cubature rules of high quality for polynomials on the region bounded by the
Steiner’s hypocycloid. In this section we discuss analogous results for the cubature rules in the
Section 3. To put the results in perspective, let us first recall the relevant background.
Let w be a nonnegative weight function defined on a compact set Ω in R2. A cubature rule
of degree 2n´ 1 for the integral with respect to w is a sum of point evaluations that satisfies
ż
Ω
fpxqwpxqdx “
N
ÿ
j“1
λjfpxjq, λj P R
for every f P Π2
2n´1. It is well-known that a cubature rule of degree 2n ´ 1 exists only if N ě
dim Π2
n´1 “ npn` 1q{2. A cubature that attains such a lower bound is called Gaussian. Unlike
one variable, the Gaussian cubature rule exists rarely and it exists if and only if the corresponding
orthogonal polynomials of degree n, all n ` 1 linearly independent ones, have npn ` 1q{2 real
distinct common zeros. We refer to [4, 15] for these results and further discussions. At the
moment there are only two regions with weight functions that admit the Gaussian cubature
rule. One is the region bounded by the Steiner’s hypocycloid and the Gaussian cubature rule is
obtained by transformation from one cubature rule for trigonometric functions on the equilateral
triangle.
6.1 Gaussian cubature rule of m-degree
We first consider the case of w 1
2
, 1
2
, which turns out to admit the Gaussian cubature rule in the
sense of m-degree.
Theorem 6.1. For w 1
2
, 1
2
on 4˚, the cubature rule
c 1
2
, 1
2
ij
∆˚
fpx, yqw
1
2
, 1
2 px, yqdxdy “
12
pn` 5q2
ÿ
jPΥ0n`5
ˇ
ˇSS2,1,´3
`
j
n`5
˘ˇ
ˇ
2
f
`
x
`
j
n`5
˘
, y
`
j
n`5
˘˘
, (6.1)
is exact for all polynomials f P P ˚2n´1.
Proof. Using (4.10) with α “ β “ 1
2 and (3.32), we see that
c 1
2
, 1
2
ż
4˚
fpx, yqw 1
2
, 1
2
px, yqdxdy “
1
|4|
ż
4
fpxptq, yptqq rSS2,1,´3ptqs
2 dt. (6.2)
By (4.3) and (4.5), rSS2,1,´3ptqs
2 has m-degree 10, so that fpxptq, yptqq rSS2,1,´3ptqs
2
P Hcc
2n`9 if
f P Π˚2n´1. Since SS2,1,´3ptq vanishes on the boundary of 4, applying the cubature rule (3.22)
of degree 2n` 9 to the right hand side of (6.2) gives the stated result. �
What makes this result interesting is the fact that, by (3.21),
|Υ0n`5| “ |Γ
ss
n`5| “ |Γ
cc
n´1| “ dim Π˚n´1,
which shows that the cubature rule (6.1) resembles the Gaussian cubature rule under the m-
degree. Furthermore, it turns out that it is again characterized by the common zeros of ortho-
gonal polynomials. Let Y 0n be the image of
j
n : j P Υss
n
(
under the mapping t ÞÑ x,
Y 0n :“
`
x
`
j
n
˘
, y
`
j
n
˘˘
: j P Υ0n
(
,
which is the set of nodes for (6.1). Then all polynomials P
1
2
, 1
2
k1,k2
with m-degree n vanish on Y 0n .
26 H. Li, J. Sun and Y. Xu
Theorem 6.2. The set Y 0n`5 is the variety of the polynomial ideal
@
P
1
2
, 1
2
k1,k2
pxq : 2k1 ` 3k2 “ n
D
.
Proof. By the definition of P
1
2
, 1
2
k1,k2
, it suffices to show that
SSk
´
j
n`5
¯
“ 0 for j P Υ, k P Γ and k1 ´ k3 “ n` 5. (6.3)
Directly form its definition,
SSk
`
j
n`5
˘
“
1
3
”
sin πpk1´k3qpj1´j3q
3pn`5q sin πk2j2
n`5 ` sin πpk1´k3qpj2´j1q
3pn`5q sin πk2j3
n`5
` sin πpk1´k3qpj3´j2q
3pn`5q sin πk2j1
n`5
ı
“
1
3
”
sin πpj1´j3q
3 sin πk2j2
n`5 ` sin πpj2´j1q
3 sin πk2j3
n`5 ` sin πpj3´j2q
3 sin πk2j1
n`5
ı
,
Since j1 ” j2 ” j3 pmod3q, we conclude then SSk
`
j
n`5
˘
“ 0. The proof is completed. �
In [13], the existence of the Gaussian cubature rule in the sense ofm-degree and the connection
to orthogonal polynomials were established in the context of compact simple Lie groups. The
case of the group G2 was used as an example, where a numerical example was given. The
domain 4˚ and the one in [13] differ by an affine change of variables.
Our results give explicit nodes and weights of the cubature rule and provide further expla-
nation for the result.
(1,1)(- 12 ,1)
(- 13 ,-
1
3 )
Chebyshev–Guass
(1,1)(- 12 ,1)
(- 13 ,-
1
3 )
Chebyshev–Guass–Lobatto
(1,1)(- 12 ,1)
(- 13 ,-
1
3 )
Chebyshev–Guass–Radau I
(1,1)(- 12 ,1)
(- 13 ,-
1
3 )
Chebyshev–Guass–Radau II
Figure 6.1. The cubature nodes on the region ∆˚.
6.2 Gauss–Lobatto cubature and Chebyshev polynomials of the first kind
In the case of w´ 1
2
,´ 1
2
, the change of variables t ÞÑ x shows that (3.22) leads to a cubature of
m-degree 2n´ 1 based on the nodes of Yn.
Discrete Fourier Analysis and Chebyshev Polynomials with G2 Group 27
Theorem 6.3. For the weight function w´ 1
2
,´ 1
2
on 4˚ the cubature rule
c´ 1
2
,´ 1
2
ij
4˚
fpx, yqw´ 1
2
,´ 1
2
px, yqdxdy “
1
n2
ÿ
jPΥn
ω
pnq
j f
`
x
`
j
n
˘
, y
`
j
n
˘˘
(6.4)
holds for f P Π˚2n´1.
The set Yn includes points on the boundary of 4˚, hence, the cubature rule in (6.4) is an
analogue of the Gauss–Lobatto type cubature for w´ 1
2
,´ 1
2
on 4˚. The number of nodes of this
cubature is dim Π˚n, instead of dim Π˚n´1. In this case, the corresponding orthogonal polynomials
are the generalized Chebyshev polynomials of the first kind, Tk1,k2px, yq :“ P
´ 1
2
,´ 1
2
n px, yq. The
polynomials in tTα : |α|˚ “ nu do not have enough common zeros in general. In fact, the two
orthogonal polynomials of m-degree 6,
T3,0px, yq “ 36x3 ´ 18xy ´ 9x´ 6y ´ 2,
T2,2px, yq “ 6y2 ` 10y ´ 72x3 ` 36xy ` 18x` 3.
only have three common zeros on 4˚,
px, yq “
´ ?
2?
7`1
cosp2πµ
3 ` 1
3 arccos 3
?
2
2
?
7`1
q,´ 1?
7`1
¯
, µ “ 0, 1, 2,
whereas dim Π˚5 “ 5. For cubature rules in the ordinary sense, that is, with Π2
n in place of Π˚n,
the nodes of a cubature rule of degree 2n´ 1 with dim Π2
n nodes must be the variety of a poly-
nomial ideal generated by dim Π˚n`1 linearly independent polynomials of degree n`1, and these
polynomials are necessarily quasi-orthogonal in the sense that they are orthogonal to all polyno-
mials of degree n´2 [19]. Our next theorem shows that this characterization of such a cubature
carries over to the case of m-degree.
Theorem 6.4. Denote α˚ “ pα1 ´ 1, α2q, a1 ą a2, and α˚ “ pα1, α1 ´ 1q if α1 “ α2. Then Yn
is the variety of the polynomial ideal
xTαpxq ´ Tα˚pxq : |α|˚ “ n` 1y . (6.5)
Furthermore, the polynomial Tαpxq´Tα˚pxq is of m-degree n`1 and orthogonal to all polynomials
in Π˚n´2 with respect to w´ 1
2
,´ 1
2
.
Proof. A direct computation shows that, for any k P Γ with k1 ´ k3 “ n` 1,
CCk1´1,k2,k3`1ptq ´ CCkptq “
1
3
”
cos πpn´1qpt1´t3q
3 cosπk2t2 ` cos πpn´1qpt2´t1q
3 cosπk2t3
` cos πpn´1qpt3´t2q
3 cosπk2t1
ı
´
1
3
”
cos πpn`1qpt1´t3q
3 cosπk2t2
` cos πpn`1qpt2´t1q
3 cosπk2t3 ` cos πpn`1qpt3´t2q
3 cosπk2t1
ı
“
2
3
”
sin πnpt1´t3q
3 sin πpt1´t3q
3 cosπk2t2 ` sin πnpt2´t1q
3 sin πpt1´t3q
3 cosπk2t3
` sin πnpt3´t2q
3 sin πpt1´t3q
3 cosπk2t1
ı
,
where we have used the definition of CCk for the first equality sign. Hence, for any j P Υn,
CCk1´1,k2,k3`1
´
j
n
¯
´ CCk
´
j
n
¯
“
2
3
”
sin πpj1´j3q
3 sin πpj1´j3q
3n cos πk2j2n
28 H. Li, J. Sun and Y. Xu
` sin πpj2´j1q
3 sin πpj2´j1q
3n cos πk2j3n ` sin πpj3´j2q
3 sin πpj3´j2q
3n cos πk2j1n
ı
“ 0,
where the last equality sign uses the fact j1 ” j2 ” j3 pmod 3q. With α1 “ k1 ` k2, this shows
that Tα ´ Tα˚ vanishes on Yn. Finally, we note that |α˚|˚ “ |α|˚ ´ 2 or |α|˚ ´ 1, so that Tα˚ is
a Chebyshev polynomial of degree at least n´ 1 and Tα ´ Tα˚ is orthogonal to all polynomials
in Π˚n´2. �
6.3 Gauss–Radau cubature and Chebyshev polynomials of mixed kinds
Under the change of variables t ÞÑ x defined in (4.1), we can also transform (3.22) into cubature
rules with respect to w´ 1
2
, 1
2
and w´ 1
2
, 1
2
, which have nodes on part of the boundary and are
analogue of Gauss–Radau cubature rule. They are associated with Chebyshev polynomials of
the mixed types. We state the result without proof.
Theorem 6.5. The following cubature rules hold,
c´ 1
2
, 1
2
ij
∆˚
fpx, yqw´ 1
2
, 1
2
px, yqdxdy
“
4π2
9pn` 2q2
ÿ
jPΥn`2
ω
pn`2q
j
ˇ
ˇ
ˇ
SC1,0,´1
`
j
n`2
˘
ˇ
ˇ
ˇ
2
f
`
x
`
j
n`2
˘
, y
`
j
n`2
˘˘
, @ f P Π˚2n´1, (6.6)
c 1
2
,´ 1
2
ij
∆˚
fpx, yqw 1
2
,´ 1
2
px, yqdxdy
“
4π2
9pn` 3q2
ÿ
jPΥn`3
ω
pn`3q
j
ˇ
ˇ
ˇ
CS1,1,´2
`
j
n`3
˘
ˇ
ˇ
ˇ
2
f
`
x
`
j
n`3
˘
, y
`
j
n`3
˘˘
, @ f P Π˚2n´1. (6.7)
Since by (4.5), SC1,0,´1 and CS1,1,´2 vanish on part of the boundary of 4, the summation is
not over the entire Υn`2 or Υn`3 but over a subset that exclude points on the respective boun-
dary. Let Y sc
n`1 and Y cs
n`3 denote the set of nodes for the above two cubature rules, respectively.
Theorem 6.6. Y sc
n`2 is the variety of the polynomial ideal
@
P
´ 1
2
, 1
2
α pxq : |α|˚ “ n
D
. (6.8)
And Y cs
n`3 is the variety of the polynomial ideal
@
P
1
2
,´ 1
2
α pxq ´ P
1
2
,´ 1
2
α˚ pxq : |α|˚ “ n` 1
D
. (6.9)
It is of some interests to notice that, in terms of the number of nodes vs the degree, (6.6) is
an analogue of the Gauss cubature rule in m-degree.
Acknowledgements
The work of the first author was partially supported by NSFC Grants 10971212 and 91130014.
The work of the second author was partially supported by NSFC Grant 60970089. The work
of the third author was supported in part by NSF Grant DMS-110 6113 and a grant from the
Simons Foundation (# 209057 to Yuan Xu).
Discrete Fourier Analysis and Chebyshev Polynomials with G2 Group 29
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http://dx.doi.org/10.2307/2001804
http://dx.doi.org/10.1017/CBO9780511565717
http://dx.doi.org/10.1017/CBO9780511565717
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http://dx.doi.org/10.1080/10652469.2011.598265
http://arxiv.org/abs/1101.2502
http://dx.doi.org/10.1023/A:1018989707569
1 Introduction
2 Discrete Fourier analysis on hexagonal domain
3 Discrete Fourier analysis on the 30–60–90 triangle
3.1 Generalized trigonometric functions
3.2 Discrete Fourier analysis on the 30–60–90 triangle
3.3 Sturm–Liouville eigenvalue problem for the Laplace operator
3.4 Product formulas for the generalized trigonometric functions
4 Generalized Chebyshev polynomials
5 Sturm–Liouville eigenvalue problem and generalized Jacobi polynomials
6 Cubature rules for polynomials
6.1 Gaussian cubature rule of m-degree
6.2 Gauss-Lobatto cubature and Chebyshev polynomials of the first kind
6.3 Gauss-Radau cubature and Chebyshev polynomials of mixed kinds
References
|
| id | nasplib_isofts_kiev_ua-123456789-148448 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T18:33:49Z |
| publishDate | 2012 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Li, H. Sun, J. Xu, Y. 2019-02-18T12:42:41Z 2019-02-18T12:42:41Z 2012 Discrete Fourier Analysis and Chebyshev Polynomials with G₂ Group / H. Li, J. Sun, Y. Xu // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 19 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 41A05; 41A10 DOI: http://dx.doi.org/10.3842/SIGMA.2012.067 https://nasplib.isofts.kiev.ua/handle/123456789/148448 The discrete Fourier analysis on the 30°-60°-90° triangle is deduced from the corresponding results on the regular hexagon by considering functions invariant under the group G₂, which leads to the definition of four families generalized Chebyshev polynomials. The study of these polynomials leads to a Sturm-Liouville eigenvalue problem that contains two parameters, whose solutions are analogues of the Jacobi polynomials. Under a concept of m-degree and by introducing a new ordering among monomials, these polynomials are shown to share properties of the ordinary orthogonal polynomials. In particular, their common zeros generate cubature rules of Gauss type. The work of the first author was partially supported by NSFC Grants 10971212 and 91130014.The work of the second author was partially supported by NSFC Grant 60970089. The work of the third author was supported in part by NSF Grant DMS-110 6113 and a grant from the Simons Foundation (# 209057 to Yuan Xu). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Discrete Fourier Analysis and Chebyshev Polynomials with G₂ Group Article published earlier |
| spellingShingle | Discrete Fourier Analysis and Chebyshev Polynomials with G₂ Group Li, H. Sun, J. Xu, Y. |
| title | Discrete Fourier Analysis and Chebyshev Polynomials with G₂ Group |
| title_full | Discrete Fourier Analysis and Chebyshev Polynomials with G₂ Group |
| title_fullStr | Discrete Fourier Analysis and Chebyshev Polynomials with G₂ Group |
| title_full_unstemmed | Discrete Fourier Analysis and Chebyshev Polynomials with G₂ Group |
| title_short | Discrete Fourier Analysis and Chebyshev Polynomials with G₂ Group |
| title_sort | discrete fourier analysis and chebyshev polynomials with g₂ group |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/148448 |
| work_keys_str_mv | AT lih discretefourieranalysisandchebyshevpolynomialswithg2group AT sunj discretefourieranalysisandchebyshevpolynomialswithg2group AT xuy discretefourieranalysisandchebyshevpolynomialswithg2group |