Cubature formulas of multivariate polynomials arising from symmetric orbit functions
Abstract.
The paper develops applications of symmetric orbit functions, known from irreducible representations of simple Lie groups, in numerical analysis. It is shown that these functions have remarkable properties which yield to cubature formulas, approximating a weighted integral of any function by a weighted finite sum of function values, in connection with any simple Lie group. The cubature formulas are specialized for simple Lie groups of rank two. An optimal approximation of any function by multivariate polynomials arising from symmetric orbit functions is discussed.
Department of Physics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Břehová 7, CZ115 19 Prague, Czech Republic
Centre de recherches mathématiques, Université de Montréal, C. P. 6128 – Centre ville, Montréal, H3C 3J7, Québec, Canada
Département de mathématiques et de statistique, Université de Montréal, Québec, Canada
MIND Research Institute, 111 Academy Drive, Irvine, CA 92617
Email: jiri.hrivnak@fjfi.cvut.cz, lenka.motlochova@fjfi.cvut.cz, patera@crm.umontreal.ca
Keywords: cubature formulas, symmetric orbit functions, simple Lie groups, Weyl groups
MSC: 65D32, 33C52, 41A10, 22E46, 20F55, 17B22
1. Introduction
The purpose of this paper is to extend the results of [25, 24], where cubature formulas for numerical integration connected with three types of multivariate Chebyshevlike polynomials arising from Weyl group orbit functions are developed. The specific goal of this article is to derive the cubature rules and corresponding approximation methods for the family of the polynomials arising from symmetric exponential Weyl group orbit sums [2, 15] and detail specializations of the general results for the twovariable polynomials.
The family of polynomials induced by the symmetric Weyl group orbit functions (functions) forms one of the most natural generalizations of the classical Chebyshev polynomials of one variable — indeed, the lowest symmetric orbit function arising from the Weyl group of coincides with the common cosine function of one variable and thus induces the family of Chebyshev polynomials of the first kind [27]. The continuous and discrete orthogonality of the sets of cosine functions generalize to the families of multivariate functions [13, 15, 26]. This generalization serves as an essential starting point for deriving the cubature formulas and approximation methods.
Cubature formulas for numerical integration constitute multivariate generalizations of classical quadrature formulas for functions of one variable. A weighted integral over some domain inside of any given function is estimated by a finite weighted sum of values of the same function on a specific set of points (nodes). A standard requirement is imposed: the cubature formula has to hold as an exact equality for polynomials up to a certain degree. Numerous types of cubature formulas with diverse shapes of the integration domains and various efficiencies exist [8]. The efficiency of a given cubature formula reflects how the achieved maximal degree of the polynomials relates to the number of the necessary nodes. Optimal cubature formulas of the highest possible efficiency (Gaussian formulas) are for multivariate functions obtained for instance in [23, 25, 28].
The sequence of Gaussian cubature formulas derived in [23] arises from the antisymmetric orbit functions (functions) of the Weyl groups of type , . Generalization of these cubature formulas from [23] to polynomials of the functions of Weyl groups of any type and rank is achieved in [25]. A crucial concept, which allows the generalization of the formulas, is a novel definition of a degree of the underlying polynomials. This generalized degree (degree) is based on invariants of the Weyl groups and their corresponding root systems. Besides the polynomials corresponding to the and functions, two additional families of multivariate polynomials arise from Weyl group orbit functions of mixed symmetries [24]. These hybrid orbit functions ( and functions) exist only for root systems of Weyl groups with two different lengths of roots — , , and . The cubature formulas related to the polynomials of these and functions are developed in [24]. Deduction of the remaining cubature formulas, which correspond to the polynomials of the functions, completes in this paper the results of [23, 25, 24]. The integration domains and nodes of these cubature formulas are constructed in a similar way as onedimensional GaussChebyshev formulas and their Chebyshev nodes.
Instead of a classical onedimensional interval, the multivariate functions are considered in the fundamental domain of the affine Weyl group — a simplex . The discrete orthogonality relations of functions are performed over a finite fragment of a grid , with the parameter controlling the density of inside . The simplex together with the set of points have to be transformed via a transform which induces the corresponding family of polynomials (transform). This process results in the integration domain of nonstandard shape and the set of nodes , with specifically distributed points inside . The last ingredient, needed for successful practical implementation, is the explicit form of the weight polynomial . For practical purposes, the explicit construction of all twovariable cases is presented.
Except for direct numerical integration, one of the most immediate applications of the developed cubature formulas is related multivariate polynomial approximation [4]. The Hilbert basis of the orthogonal multivariate polynomials induced by the functions guarantees that any function from the corresponding Hilbert space is expressed as a series involving these polynomials. A specific truncated sum of this expansion provides the best approximation of the function by the polynomials. Among other potential applications of the developed cubature formulas are calculations in fluid flows [9], laser optics [5], stochastic dynamics [33], magnetostatic modeling [34], micromagnetic simulations [6], electromagnetic wave propagation [29], liquid crystal colloids [31] and quantum dynamics [20].
The paper is organized as follows. In Section 2, notation and pertinent properties of Weyl groups, affine Weyl groups and functions are reviewed. In Section 3, the cubature formulas related to functions are deduced. In Section 4, the explicit cubature formulas of the rank two cases , , are constructed. In Section 5, polynomial approximation methods are developed.
2. Root systems and polynomials
2.1. Pertinent properties of root systems and weight lattices
The notation, established in [13], is used. Recall that, to the Lie algebra of the compact, connected, simply connected simple Lie group of rank , corresponds the set of simple roots [2, 1, 14, 32]. The set spans the Euclidean space , with the scalar product denoted by . The following standard objects related to the set of simple roots are used:

The marks of the highest root .

The Coxeter number of .

The Cartan matrix and its determinant
(1) 
The root lattice .

The dual lattice to ,
with the vectors given by

The dual root lattice , where .

The dual marks of the highest dual root . The marks and the dual marks are summarized in Table 1 in [13]. The highest dual root satisfies for all
(2) 
The dual weight lattice to
with the vectors given by For the following notation is used,
(3) 
The partial ordering on is given: for it holds that if and only if with for all .

The half of the sum of the positive roots

The cone of positive weights and the cone of strictly positive weights

reflections , in dimensional ‘mirrors’ orthogonal to simple roots intersecting at the origin denoted by
Following [25], we define so called degree of as the scalar product of with the highest dual root , i.e. by the relation
Let us denote a finite subset of the cone of the positive weights consisting of the weights of the degree not exceeding by , i.e.
Recall also the separation lemma which asserts for , and any that
(4) 
Note that this lemma is proved in [25] for only — the proof, however, can be repeated verbatim with any .
For two dominant weights for which we have for their degrees
(5) 
Taking into account equation (2), we have the following proposition.
Proposition 2.1.
For two dominant weights with it holds that .
2.2. Affine Weyl groups
The Weyl group is generated by reflections and its order can be calculated using the formula
(6) 
The affine Weyl group is the semidirect product of the Abelian group of translations and of the Weyl group ,
(7) 
The fundamental domain of , which consists of precisely one point of each orbit, is the convex hull of the points . Considering real parameters , we have
(8) 
The volumes vol of the simplices are calculated in [13].
Considering the standard action of on , we denote for the isotropy group and its order by
and denote the orbit by
Then the orbitstabilizer theorem gives for the orders
(9) 
Considering the standard action of on the torus , we denote for the order of its orbit by , i.e.
(10) 
For an arbitrary , the grid is given as cosets from the invariant group with a representative element in the fundamental domain
The representative points of can be explicitly written as
(11) 
The numbers of elements of , denoted by , are also calculated in [13] for all simple Lie algebras.
2.3. Orbit functions
Symmetric orbit functions [16] are defined as complex functions with the labels ,
(12) 
Note that in [13] the results for functions are formulated for the normalized functions which are related to the orbit sums (12) as
Due to the symmetries with respect to the Weyl group as well as with respect to the shifts from
(13) 
it is sufficient to consider functions restricted to the fundamental domain of the affine Weyl group . Moreover, the functions are continuously orthogonal on ,
(14) 
and form a Hilbert basis of the space [16], i.e. any function can be expanded into the series of functions
(15) 
Special case of the orthogonality relations (14) is when one of the weights is equal to zero,
(16) 
For any , the functions from a certain subset of are also discretely orthogonal on and form a basis of the space of discretized functions of dimension [13]; special case of these orthogonality relations is when one of the weights is equal to zero modulo the lattice ,
(17) 
The key point in developing the cubature formulas is comparison of formulas (16) and (17) in the following proposition.
Proposition 2.2.
For any and it holds that
(18) 
Proof.
Let us denote for convenience the functions corresponding to the basic dominant weights by , i.e.
Recall from [2], Ch. VI, §4 that any invariant sum of the exponential functions can be expresssed as a linear combination of some functions with . Also for any a function of the monomial type can be expressed as the sum of functions by less or equal dominant weights than , i.e.
(19) 
Conversely, any function , can be expressed as a polynomial in variables , i.e. there exist a multivariate polynomials such that
(20) 
Antisymmetric orbit functions [16] are defined as complex functions with the labels ,
(21) 
The antisymmetry with respect to the Weyl group and the symmetry with respect to the shifts from holds
Recall that Proposition 9 in [16] states that for the lowest function it holds that
(22)  
(23) 
Since the square of the absolute value is a invariant sum of exponentials it can be expressed as a linear combinations of functions. Each function in this combination is moreover a polynomial of the form (20). Thus there exist a unique polynomial such that
(24) 
3. Cubature formulas
3.1. The transform
The key component in the development of the cubature formulas is the integration by substitution. The transform transforms the fundamental domain to the domain on which are the cubature rules defined. In order to obtain a real valued transform we first need to examine the values of the functions.
The functions of the algebras
(25) 
are realvalued [15]. Using the notation (3), for the remaining cases it holds that
(26)  
Specializing the relations (3.1) for the functions corresponding to the basic dominant weights , we obtain that the functions are real valued, except for the following cases for which it holds that
(27)  
Taking into account (3.1), we introduce the realvalued functions as follows. For the cases (25) we set
(28) 
and for the remaining cases (3.1) we define
(29)  
Thus, we obtain a crucial mapping given by
(30) 
The image of the fundamental domain under the mapping forms the integration domain on which the cubature rules will be formulated, i.e.
(31) 
In order to use the mapping for an integration by substitution we need to know that it is onetoone except for possibly some set of zero measure. Since the image of the set of points under the mapping forms the set of nodes for the cubature rules, i.e.
(32) 
a discretized version of the onetoone correspondence of the restriced mapping of to , i.e.
(33) 
is also essential. Note that due to the periodicity of functions (13), the restriction (33) is welldefined for the cosets from .
Proposition 3.1.
Proof.
Let us assume that there exists a set of nonzero measure such that with . Since the transforms (28), (29) are as regular linear mappings onetoone correspondences, this fact implies that with . Then from the polynomial expression (20) we obtain for all that it holds that . Since the functions form a Hilbert basis of the space we conclude that for any is valid that , which is contradiction.
Retracing the steps of the continuous case above, let us assume that there exist two distinct points , such that . Since the transforms (28), (29) are as regular linear mappings onetoone correspondences, this fact again implies that . Then from the polynomial expression (20) we obtain for all that it holds that . The same equality has hold for those functions which form a basis of the space . We conclude that for any is valid that , which is contradiction. ∎
The absolute value of the determinant of the Jacobian matrix of the transform (30) is essential for construction of the cubature formulas — its value is determined in the following proposition.
Proposition 3.2.
The absolute value of the Jacobian determinant of the transform (30) is given by
(35) 
where is defined as
(36) 
Proof.
Note that the transform can be composed of the the following two transforms: the transform and the transform via relations (28), (29). To calculate the Jacobian of the transform , let us denote by the matrix of the coordinates (in columns) of the vectors in the standard orthonormal basis of and by the coordinates of a point in basis, i.e. . If denotes the coordinates arranged in a column vector then it holds that . The absolute value of the Jacobian of the mapping is according to equation (32) in [25] given by . Using the chain rule, this implies for the absolute value of the Jacobian of the map that
It can be seen directly from formula (6) and Proposition 2.1 in [13] that
The calculation of the absolute value of the Jacobian determinant is straightforward from definitions (28), (29). ∎
3.2. The cubature formula
We attach to any a monomial and assign to this monomial the degree of . The degree of a polynomials , denoted by , is defined as the largest degree of a monomial occurring in . For instance we observe from Proposition 2.1 and (20) that the degree of the polynomials coincides with the degree of ,
(37) 
The subspace is formed by the polynomials of degree at most , i.e.
(38) 
In order to investigate how the degree of a polynomial changes under the substitution of the type (29) we formulate the following proposition.
Proposition 3.3.
Let be two distinct indices such that and two polynomials such that
holds. Then .
Proof.
Since any polynomial is a linear combination of monomials with , it is sufficient to prove for all monomials of degree at most . If is a monomial with , then
Using the binomial expansion, we obtain
Therefore, the degree of the polynomial is given by
Since we assume that , we conclude that . ∎
Having the transform (33), it is possible to transfer uniquely the values (10) of to the points of , i.e. by the relation
(39) 
Taking the inverse transforms of (28), (29) and substituting them into the polynomials (24), (20) we obtain the polynomials such that
(40) 
Theorem 3.4 (Cubature formula).
For any and any it holds that
(41) 
Proof.
Proposition 3.1 guarantees that the transform is onetoone except for some set of measure zero and Proposition 3.2 together with (23) gives that the Jacobian determinant is nonzero except for the boundary of . Thus using the integration by substitution we obtain
The onetoone correspondence for the points and from Proposition 3.1 enables us to rewrite the finite sum in (41) as
(42) 
Succesively applying Proposition 3.3 to perform the substitutions (29) in we conclude that there exist a polynomial such that Due to (19) we obtain for the polynomial that
and therefore it holds that
(43) 
and
(44) 
Since Proposition 2.1 states that for all it holds that , we connect equations (43) and (44) by Proposition 2.2. ∎
Note that for practical purposes it may be more convenient to use the cubature formula (41) in its less developed form resulting from (42),
(45) 
This form may be more practical since the explicit inverse transform to is usually not available and, on the contrary, the calculation of the coefficients and the points is straightforward.
4. Cubature formulas of rank two
In this section we specialize the cubature formula for the irreducible root systems of rank two. Let us firstly recall some basic facts about root systems of rank , i.e., and . They are characterized by two simple roots which satisfy
(46)  