Discrete Fourier Analysis and Chebyshev Polynomials with $G_2$ Group

The discrete Fourier analysis on the $30^{\degree}$-$60^{\degree}$-$90^{\degree}$ triangle is deduced from the corresponding results on the regular hexagon by considering functions invariant under the group $G_2$, 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.


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 exponential functions on the regular hexagon and, by restricting to symmetric and antisymmetric exponentials on the hexagon under the reflection group A 2 (the group of symmetry of the regular 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 45 0 -45 0 -90 0 triangle. The richness of these results prompts us to look into similar results on the 30 0 -60 0 -90 0 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 G 2 .
It turns out that much of the discrete Fourier analysis on the 30 0 -60 0 -90 0 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 G 2 , which is a composition of A 2 and its dual A2, 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 as the hexagonal lattice and the other as the rotation of the same lattice by 90 0 degree [10], with the symmetric groups A 2 and A2, respectively. As we shall see, it is from this set up that our results on the 30 0 -60 0 -90 0 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 30 0 -60 0 -90 0 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 L α,β :"´A 11 px, yqB 2 x´2 A 12 px, yqB x B y´A22 px, yqB 2 y`B 1 px, yqB x`B2 px, yqB y .
Such operators have long been studied in association with orthogonal polynomials in two variables; see for example [6,7,8,16], as well as [1] and the references therein. Our operator L α,β , however, is different in the sense that the coefficient functions A i,j are usually assumed to be of quadratic polynomials to ensure that the operator has n`1 polynomials of degree n as eigenfunctions, whereas A 2,2 in our L α,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 L α,β u " 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 30 0 -60 0 -90 0 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.
with the lattice L if Ω`L " R d , that is, where χ Ω denotes the characteristic function of Ω. For a given lattice L A , the dual lattice L K A is given by L K A " A´t r Z d . A result of Fuglede [5] states that a bounded open set Ω tiles R d with the lattice L if, and only if, te 2πiα¨x : α P L K u is an orthonormal basis with respect to the inner product Since L K A " A´t r Z d , we can write α " A´t r k for α P L K A and k P Z d , so that e 2πiα¨x " e 2πik tr A´1x . For our discrete Fourier analysis, the boundary of Ω matters. We shall fix an Ω such that 0 P Ω and Ω`AZ d " R d holds pointwisely and without overlapping.
Let Ω A and Ω B be the fundamental domains of AZ d and BZ d , respectively. Assume all entries of the matrix N :" B tr A are integers. Define Furthermore, define the finite-dimensional subspace of exponential functions The function x Þ Ñ e 2πik tr A´1x is periodic with respect to the lattice AZ d and V N is a space of periodic exponential functions. We can now state the central result in the discrete Fourier analysis.
It follows readily that (2.2) gives a cubature formula exact for functions in V N . 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 L A as the lattice for the physical space, as it determines the domain on which our analysis lies, and the lattice L B 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,  .
Theorem 2.4 ([10]). The following cubature rule holds for any f P H : 2n´1 , where H 0 n , H v n and H e 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, H 0 n " tj P H :´n ă j 1 , j 2 , j 3 ă nu, H v n " tpn, 0,´nqσ P H : σ P A 2 u and H e n " H n zpH 0 n Y H v n q " tpj, n´j,´nqσ P H : 1 ď j ď n´1u. In particular, let Q n f denote the right hand side of (2.4); then for any k P H : , Q n φ k " 1 ifk " 0 pmod 3nq and Q n φ k " 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 30 0 -60 0 -90 0 triangle In this section we deduce a discrete Fourier analysis on the 30 0 -60 0 -90 0 triangle from the analysis on the hexagon by working with invariant functions.
Because of the relations σ 3 " σ 1 σ 2 σ 1 " σ 2 σ 1 σ 2 , the group is given by The group A2 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 A 2 by a rotation of 90 0 and is exactly the permutation group of three elements. To describe the elements in A2, we define the reflection´σ for any σ P A 2 by tp´σq :"´tσ, @ t P R 3 H .
Let G denote the group of A 2 or A2 or G 2 . For a function f in homogeneous coordinates, the action of the group G on f is defined by σf ptq " f ptσq, 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,˘σ 1 σ2,˘σ 2 σ 1 , and p´1q |σ| "´1 if σ "˘σ 1 ,˘σ 2 ,˘σ 3 . The following proposition is easy to verify (see [6]). rf ptq`f ptσ 1 σ 2 q`f ptσ 2 σ 1 q˘f ptσ 1 q˘f ptσ 2 q˘f ptσ 3 qs . rf ptq`f ptσ 1 σ 2 q`f ptσ 2 σ 1 q˘f p´tσ 1 q˘f p´tσ 2 q˘f p´tσ 3 qs .  (1,0,-1) (1,0,-1) (0, n 2 ,-n 2 ) For σ P G 2 , the number of inversion |σ| satisfies |´σ| " |σ|. The following lemma can be easily verified (writing down the table of σσ˚for σ P A 2 and σ˚P A2 if necessary). For φ k ptq " e 2πik¨t 3 , the action of P`and P´on φ k are called the generalized cosine and generalized sine functions in [9], which are trigonometric functions given by Because of the symmetry, we only need to consider these functions on the fundamental domain of the group A 2 , 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 G 2 . Since the fundamental domain of the group G 2 is the 30 0 -60 0 -90 0 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: 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 In particular, it follows from (3.6)-(3.8) that CS k ptq " SS k ptq " 0 whenever k contains zero component and SC k ptq " SS k ptq " 0 whenever k contains equal elements. Similar formulas can be derived from the permutations of t 1 , t 2 , t 3 . In fact, the functions CC k and SS k are invariant and anti-invariant under G 2 , respectively, whereas the functions CS k and SC k are of the mixed type, with the first one invariant under A 2 and anti-invariant under A2 and the second one invariant under A2 and anti-invariant under A 2 . More precisely, these invariant properties lead to the following identities: In particular, it follows from (3.6)-(3.8) that CS k ptq " SS k ptq " 0 whenever k contains zero component and SC k ptq " SS k ptq " 0 whenever k contains equal elements. Moreover, for any k P H : , CS k ptq " SS k ptq " 0 whenever t contains zero component and SC k ptq " SS k ptq " 0 whenever t contains equal elements. Because of their invariant properties, we only need to consider these functions on one of the twelve 30 0 -60 0 -90 0 triangles in the hexagon Ω. We shall choose the triangle as 14) The region and its relative position in the hexagon are depicted in Figs. 3.3 and 3.1.
(1,0,-1) When CC k , SC k , CS k , SS k are restricted to the triangle , 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 respectively, where the notation is self-explanatory; for example, Γ cc is the index set for CC k .
We define an inner product on by xf, gy :" 1 If fḡ is invariant under the group G 2 , then it is easy to see that xf, gy Ω " xf, gy . Consequently, we can deduce the orthogonality of CC k , SC k , CS k , SS k from that of φ k on Ω.
where kG 2 " tkσ : σ P G 2 u denotes the orbit of k under G 2 .

Discrete
Fourier analysis on the 30 0 -60 0 -90 0 triangle Using the fact that CC k , SC k and CS k , SS k are invariant and anti-invariant under A 2 and that CC k , CS k and SC k , SS k are invariant and anti-invariant under A2, 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 which are located inside n as seen by (3.14). The space of invariant functions being integrated exactly by the cubature rule are indexed by Correspondingly, we define the following subspaces of H : n , H cc n :" spantCC k : k P Γ cc n u, H sc n :" spantSC k : k P Γ sc n u, H cs n :" spantCS k : k P Γ cs n u, H ss n :" spantSS k : k P Γ ss n u. It is easy to verify that Theorem 3.4. The following cubature is exact for all f P H cc pboundariesq. (n,0,-n) The index set Υ n . n " 0 pmod 3q (left), n " 1 pmod 3q (center) and n " 2 pmod 3q (right). Moreover, if we define the discrete inner product xf, gy ,n " 1 where p k " pk 3´k2 , k 1´k3 , k 2´k1 q.
The formula (3.22) is derived from (2.4) by using the invariance of the functions in H cc 2n´1 and upon writing Ω "`Ť σPG 2 ttσ : t P 0 u˘Ť`Ť σPG 2 ttσ : t P B u˘. The reason that p k 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.

Sturm-Liouville eigenvalue problem for the Laplace operator
Recall the relation (2.3) between the coordinates px 1 , x 2 q and the homogeneous coordinates pt 1 , t 2 , t 3 q. A quick calculation gives the expression of the Laplace operator in homogeneous coordinates, ff .
A further computation shows that φ k ptq " e 2πi 3 k¨t are the eigenfunctions of the Laplace operator: for k P H, 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 30 0 -60 0 -90 0 triangle. To be more precise, we denote the three linear segments that are the boundary of this triangle by B 1 , B 2 , B 3 , Let B Bn denote the partial derivative in the direction of the exterior norm of . Then Theorem 3.5. The generalized trigonometric functions CC k , SC k , CS k , SS k are the eigenfunctions of the Laplace operator, ∆u "´λ k u, that satisfy the boundary conditions: Proof . Since λ k is invariant under G 2 , that is, λ k " λ kσ , @ σ P G 2 , that these functions satisfy ∆u "´λ k u 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, CC k satisfies the Neumann boundary conditions and SS k satisfies the Dirichlet type boundary conditions.

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,

Generalized Chebyshev polynomials
In [9], the generalized cosine and sine functions C k and S k 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 trigonometric functions, which has a structure different from those related to C k and S k .
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 P α,β 0,2 is of degree 3, rather than degree 2, which shows that our polynomials do not satisfy the property of spantP α,β k 1 ,k 2 : k 1`k2 ď nu " Π 2 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 2k 1`3 k 2 " n. We have displayed the polynomials P α,β k 1 ,k 2 px, yq for all 2k 1`3 k 2 ď 6. In Algorithm 1 below we give an algorithm for the evaluation of all P α,β k 1 ,k 2 px, yq with 2k 1`3 k 2 " 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 w α,β on the domain ˚, w α,β px, yq :" 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, gy w α,β :" c α,β ż ∆˚f px, yqgpx, yqw α,β px, yqdxdy, where c α,β :" 1{ ş ˚wα,β px, yqdxdy is a normalization constant; in particular, c´1 " 18{π 2 and c 1 2 , 1 2 " 243{π 4 . Since the change of variables (4.1) implies immediately that we can translate the orthogonality of CC j , SC j , CS j and SS j to that of P α,β k 1 ,k 2 for α, β "˘1 2 . Indeed, from Proposition 3.3 we can deduce the following theorem.
Although the polynomials P˘1 2 ,˘1 2 k 1 ,k 2 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 is much more complicated. To understand their structure, we study them as solutions of the corresponding Sturm-Liouville problem in the following section.

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).
A tedious but straightforward computation shows that where we define A 22 :"´18y 2`1 08x 3´5 4xy´27x´9y.  for the polynomials P´1 2 ,´1 2 k 1 ,k 2 px, yq. It is easy to verify that the operator can be rewritten as where in the second line we have used ∇ :" pB x , B y q tr and Λ :" It is not difficult to verify that the matrix Λ is positive definite in the interior of the domain ˚. Indeed, det Λ " 3F px, yq, where F is defined in Lemma 4.1, and A 1,1 px, yq " 3px´yq`2p12 y´3x 2 q 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 A 1,1 ą 0 in the interior of ˚. The expression of L´1 2 ,´1 2 prompts the following definition.
Proof . By Green's formula, α,β f rpB x gqpA 11 dy´A 21 dxq´pB y gqpA 22 dx´A 12 dyqs , where B ˚d enotes the boundary of the triangle. Recall that B ˚i s defined by F px, yq " 0, where F is defined in Lemma 4.1. It follows then On the other other hand, a quick computation shows that on B ˚. Solving (5.3) and (5.4) shows that A 11 dy´A 21 dx " 0, whereas solving (5.3) and (5.5) shows that A 22 dx´A 12 dy " 0 on B ˚. Consequently, the integral over B ˚i s zero and we conclude that ĳ ˚f L α,β gw α,β dxdy " ĳ ˚p ∇f q tr Λp∇gqw α,β dxdy "´ĳ ˚g L α,β f w α,β dxdy, which shows that L α,β is self-adjoint and positive definite.
We consider polynomial solutions for the eigenvalue problem L α,β u " λu. Differential operators 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 A i,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 n, since such choices ensure that the differential operator preserves the degree of polynomials. In our case, however, the coefficient A 2,2 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 Π 2 n to Π 2 n , the space of polynomials of degree at most n in two variables.
Definition 5.3. For k 1 , k 2 ě 0, the m-degree of the monomial x k 1 y k 2 is defined as |k|˚:" 2k 1`3 k 2 . 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 N 0 , let Πn denote the space of polynomials of m-degree at most n; that is, Πn :" span x k 1 y k 2 : 0 ď k 1 , k 2 ; 2k 1`3 k 2 ď n ( . The dimension of the space Πn is the same as that of H cc n , by (3.21), Here is a list of the dimension for small n: n 1 2 3 4 5 6 7 8 9 10 11 12 dim Πn 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 G 2 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 N 2 0 , we define an order ă by k ă j if 2j 1`3 j 2 ą 2k 1`3 k 2 or 2pk 1´j1 q " 3pj 2´k2 q ą 0, and k ĺ j if k ă j or k " j. We call ă the˚-order. If ppx, yq " ř pk 1 ,k 2 qĺpm,nq c k 1 ,k 2 x k 1 y k 2 with c m,n ‰ 0, we call c m,n x m y n the leading term of p in the -order.
It is easy to see that Πn " Π2 n´3t 2n 3 u,2t 2n 3 u´n . The˚-order is well-defined. The following lemma justifies our definitions.
From this computation, it follows readily that L α,β maps Πm ,n into Πm ,n . Furthermore, with respect to the˚-order, it is easy to see that a m,n 0,0 x m y n is the leading term of L α,β by (5.7), which shows that L α,β maps Πm ,n onto Πm ,n .
The identity (5.7) also shows that L α,β has a complete set of eigenfunctions in Πm ,n .
Theorem 5.6. For α, β ě´1{2 and k 1 , k 2 ě 0, there exists a polynomial P α,β k 1 ,k 2 P Πk 1 ,k 2 with the leading term x k 1 y k 2 such that Furthermore, if we require all the polynomials are orthogonal to each other with respect to the inner product x¨,¨y w α,β , then P α,β k 1 ,k 2 is uniquely determined by its leading term in the˚-order.
This set of dependence of the polynomial solution is determined by the˚-ordering. Indeed, it is easy to see that Γ m,n " Γm ,n Y Γḿ ,n , Γm ,n :" pm´2p`3q, n`p´2qqq P Z 2 : 0 ď q ď t p`n 2 u, 0 ď p ď 2m`3n ( , ) .
Consequently, up to a constant multiple, we see that Q α,β k 1 ,k 2 coincides with the Jacobi polynomials.
In particular, this shows that the Chebyshev polynomials are elements in Π| k|˚a nd they are determined, as eigenfunctions of L α,β , uniquely by the leading term in the˚-order.

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 R 2 . A cubature rule of degree 2n´1 for the integral with respect to w is a sum of point evaluations that satisfies for every f P Π 2 2n´1 . It is well-known that a cubature rule of degree 2n´1 exists only if N ě dim Π 2 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.

Gaussian cubature rule of m-degree
We first consider the case of w 1 Theorem 6.2. The set Y 0 n`5 is the variety of the polynomial ideal @ P 1 2 , 1 2 k 1 ,k 2 pxq : 2k 1`3 k 2 " n D .
Proof . By the definition of P 1 2 , 1 2 k 1 ,k 2 , it suffices to show that SS k´j n`5¯" 0 for j P Υ, k P Γ and k 1´k3 " n`5.
In [13], the existence of the Gaussian cubature rule in the sense of m-degree and the connection to orthogonal polynomials were established in the context of compact simple Lie groups. The case of the group G 2 was used as an example, where a numerical example was given. The domain ˚a nd 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 explanation for the result. Chebyshev-Guass-Radau II  holds for f P Π2 n´1 .
only have three common zeros on ˚, px, yq "´? 2 7`1 q,´1 ? 7`1¯, µ " 0, 1, 2, whereas dim Π5 " 5. For cubature rules in the ordinary sense, that is, with Π 2 n in place of Πn, the nodes of a cubature rule of degree 2n´1 with dim Π 2 n nodes must be the variety of a polynomial ideal generated by dim Πn`1 linearly independent polynomials of degree n`1, and these polynomials are necessarily quasi-orthogonal in the sense that they are orthogonal to all polynomials 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˚" pα 1´1 , α 2 q, a 1 ą a 2 , and α˚" pα 1 , α 1´1 q if α 1 " α 2 . Then Y n is the variety of the polynomial ideal xT α pxq´T α˚p xq : |α|˚" n`1y . (6.5) Furthermore, the polynomial T α pxq´T α˚p xq is of m-degree n`1 and orthogonal to all polynomials in Πn´2 with respect to w´1 Proof . A direct computation shows that, for any k P Γ with k 1´k3 " n`1, CC k 1´1 ,k 2 ,k 3`1 ptq´CC k ptq " where we have used the definition of CC k for the first equality sign. Hence, for any j P Υ n , CC k 1´1 ,k 2 ,k 3`1´j n¯´C C k´j n¯" 2 3 " sin πpj 1´j3 q 3 sin πpj 1´j3 q 3n cos πk 2 j 2 n