DISCRETE AND CONTINUOUS COSINE TRANSFORM GENERALIZED TO …appmath/Abstracts2005/Patera.pdf ·...

DISCRETE AND CONTINUOUS COSINE TRANSFORM GENERALIZED TOLIE GROUPS SU(2)× SU(2) AND O(5)

J. PATERA, A. ZARATSYAN

Abstract. We develop and describe continuous and discrete transforms of class functions on com-pact semisimple Lie group G as their expansions into series of uncommon special functions, calledhere C-functions in recognition of the fact that the functions generalize cosine to any dimensionn < ∞. A uniform discretization of the problem on lattices of any density is described. Continuousand discrete orthogonality of C-functions is shown. Discrete transform is known in case n = 1 asthe cosine transform. Continuous extension of the discrete transform is described.

In general, C-functions are the contributions to irreducible characters from just one orbit of theWeyl group of G. Their products are fully decomposable, so are they reductions to the sums ofC-functions of subgroups of the Lie group. They are eigenfunctions of Laplace operator, satisfyingNeumann conditions at the boundary of the fundamental region of G, etc...

A ready-to-use presentation is made of two of the four variants of the 2-dimensional transforms.Both variants have in common exploitation of square lattices for the discrete version of the trans-forms. They are based on the compact Lie groups SU(2)×SU(2) and O(5), or, equivalently, Sp(4).Remaining two groups, SU(3) and G(2), involve triangular lattices. They are considered separately[1]. Processing digital data, sampled on square lattices, is our motivating application.

[Accepted for May 2005 publication in the Journal of Mathematical Physics.]

1. Introduction

This is the first in the series of three papers [1, 2] dealing with two families of special functions,called here C- and S-functions, whose many properties, very useful for applications, apparentlywent mostly unnoticed, although the functions have been known in Lie theory for decades. TheC-functions were called “orbit functions” or “orbit sums” since [3], while the S-functions appear inWeyl’s character formula under the name “characteristic functions”.

The aim of the series is twofold: (i) to formulate, derive and bring forward practically usefulproperties of these functions and their discretization in general, and (ii) to work out all the detailsof all the four variants in two dimensions.

In this paper we consider two of the four variants of C-functions, since both of them involverectangular lattices. Two other cases, which involve triangular lattices, are given in [1]. TheS-functions for all four cases are described in [2].

Present approach can be viewed as originating from three sources. First it is the traditional theoryof compact semisimple Lie groups and their finite dimensional representations. Indeed, from therewe take the definition of C- and S-functions (though under different names and to be used for avery different purpose), second is the general discretization of the Fourier-like transforms involvingC-functions [3], and finally it is the observation, made in [7], that the continuous extensions ofdiscrete C-expansions interpolates smoothly between points where digital data are given.

Two and three dimensional digital data is often collected in physical experiments, the largest and,probably, the most costly of them being the elaborate batteries of particle counters in high-energyexperimental astrophysics and particle physics. In the recent years digital images have also been

Date: February 24, 2005.

1

2 J. PATERA, A. ZARATSYAN

generated in innumerable applications. Quite often processing such data involves Fourier analysisand discrete transforms [4].

In this paper we describe a new versatile approach to the treatment of such data, which hasbeen mostly unexplored so far. One of our objectives is to make the approach as ready-to-use aspossible. Its 1-dimensional (discrete) version was discovered some 30 years ago and is extensivelyused ever since under the name of the cosine transform [5, 6]. Its straightforward generalizationto 2 dimensions is, in our notation, the (discrete) case of A1×A1 (equivalently SU(2) × SU(2)).Apparently, the first exploitation of the case A2 (equivalently SU(3)) is quite recent [7, 8, 15],although the problem of processing the digital data sampled on triangular lattices is not new [9].

The important difference between the presented method and the traditional decomposition intoFourier series is not in that here we consider other compact semisimple Lie groups of rank two thanthe traditional SU(2)× SU(2). This is merely a technicality; the reason for the new approach lieselsewhere. Traditionally, in any dimension 1 ≤ n ≤ ∞, one uses the periodicity of the functions(the functions to be decomposed as well as the expansion functions). The underlying group in thatcase is the U(1) of our example (2.2), or a product of n of them. Thus, the involved group is thegroup of discrete translations. In our method a larger group is involved: the affine Weyl group,which contains the translations as its subgroup. It requires the use of more complicated functions(the C-functions or S-functions), but, in comparison, it involves lower harmonics in the expansions,which in turn results in much smoother interpolation between digital points during the continuousextension of the method [7], etc...

As a quantitative measure of the difference between the two methods (the number of “harmonics”for example), we can take the comparison of the area of the fundamental region for the affineWeyl group and for its translation subgroup, assuming we have the grids of comparable densities.The fundamental regions F of the affine groups are shown on Figure 1, the fundamental regionof the translation subgroup is the Voronoi domain [10] (also called the proximimty cell) of thecorresponding root lattice. For A1 the ratio is 1 : 2, for A1×A1, A2, C2, and G2 the ratio is 1 : 4,1 : 6, 1 : 8, and 1 : 12 respectively.

There are three compact simple Lie groups of rank two, namely the following: SU(3), O(5) orSp(4), and G(2). There is only one non-simple but semisimple compact Lie group: SU(2)× SU(2).In this paper, we use the notation more familiar in Lie algebra theory:

A1 ←→ SU(2),and

A1×A1 ←→ SU(2)× SU(2),A2 ←→ SU(3), (1.1)C2 ←→ O(5) or Sp(4),G2 ←→ G(2).

In principle, one could also consider the compact Lie groups of rank two U(1)× U(1) and U(1)×SU(2), which are not semisimple. In view of (2.2), that would lead to the traditional Fourierdecompositions. Here we disregard those cases.

In Section 2, our general goals are compared with the traditional approach, namely, with thedecomposition of a class function f into series irreducible characters. It is pointed out that ourexpansion functions are different, and that we are considering in parallel continuous f as well asf sampled on a discrete lattice of any density. Section 3 deals with the 1-dimensional case, bothcontinuous and discrete. It serves as a didactic illustration and an introduction to higher rank cases.General continuous case of any semisimple compact Lie group is presented in an uniform way inSection 4. Properties of C-functions are our main target. Sections 5 and 6 contain respectively all

DISCRETE AND CONTINUOUS COSINE TRANSFORM GENERALIZED TO SU(2)× SU(2) AND O(5) 3

the details for exploitation of the method for A1×A1 and C2. Discretization of 2-dimensional casesin general, as well as the specialization to A1×A1 and C2 is the subject of Section 7. Examplesare shown. Concluding remarks and related problems are brought forward in Section 8.

Let us underline some notations used throughout the paper. The symbols R, Z, and N denotethe real numbers, integers, and positive integers respectively. The scalar product of a, b ∈ R2 in areal Euclidean space Rn of dimension n, is denoted by 〈a | b〉. The same notation is used for theHermitian product of the class functions in the functional space spanned by irreducible characters(or C-functions).

2. General goals

The standard general problem of harmonic analysis on a compact Lie group G, is to considerthe functions depending on conjugacy classes of elements of the group, i.e. such that

f(g′) = f(g0g g−10 ), for all g, g0 ∈ G,

and to find their expansions, along with their inversions

f(g) =∑

λ

dλ χλ(g), dλ =∫

Ff(g) χλ(g) dg, (2.1)

in terms of irreducible characters χλ. The inversion is possible due to the orthogonality of thecharacters when integrated over the fundamental region F of the group.

The simplest example of (2.1) is the case when G is the 1-parametric compact Lie group U(1).In this case the irreducible characters are the exponential functions

χm(θ) = e2πimθ, where m ∈ Z and θ ∈ R. (2.2)We take a similar problem, but it differs from (2.1) in two important ways:(i) Our expansions are into series of C-functions (4.7), rather than irreducible characters of

compact semisimple Lie groups. It offers a considerable practical advantage, since the C-functions, unlike the characters, do not get more complicated as λ increases. The secondadvantage has yet to be utilized: the C-functions are the eigenfunctions of the correspondingLaplace operator [12] and their eigenvalues are explicitly known.

(ii) We are interested in discrete expansions (7.7) besides the continuous ones. That is, expan-sions of functions given digitally by their values on a discrete grid of points FM in F . Thosefunctions are then expanded into a series in terms of C-functions given on the same grid.Inversion of such expansions is possible because of the discrete orthogonality of C-functions(7.4) established in [11] for any semisimple compact Lie group. Practically useful appearsto be subsequent continuous extensions of discrete expansions (7.9).

Also it is difficult to overestimate the versatility of different choices of the grid FM inF offered by our approach, which are equivalent to choosing, for each M ∈ N, the finiteAbelian subgroup of the maximal torus of the Lie group generated by all elements of orderM .

3. Discretization in the case of A1

Leaving aside the simplest case (2.2), there remains only one other compact Lie group of rank 1,namely A1. One can recognize a familiar situation in this 1-dimensional setup, without any grouptheory, once the C-functions Ωm(θ) are explicitly written below, in (3.1). In spite of that, it isuseful to go through it using terms and notions that are indispensable in the case of higher ranks.


3.1. The continuous case. The Weyl group of A1 has two elements W = {1, r}, where r is thereflection in the origin. The weight lattice P (A1) consists of all the points x = Zω, where Z standsfor any integer, while the root lattice Q(A1) consists of the even points of P . Symbolically, wewrite it as:

Q(A1) = Zα = 2Zω,P (A1) = Zω = Q(A1) ∪ (ω + Q(A1)).

Here α is called the simple root of A1 and ω is the fundamental weight. Hence the relative lengthsof the basis vectors of Q and P are fixed: α = 2ω. We fix also their absolute length by choosingthe value of the scalar product 〈α | α〉 = 4〈ω | ω〉 = 2. The root system, ∆(A1) = {±α} has justtwo roots.

Given a point λ = mω ∈ P , its Weyl group orbit Wλ is the following:

Wλ ≡ Wm ={{0}, if m = 0,{mω,−mω}, if 0 6= m ∈ Z.

The A1 C-function Ωλ(z) ≡ Ωm(θ) is defined for θ ∈ R as:

Ωm(θ)def=

∑

µ∈Wλe2πi〈µ|z〉 =

{1, for m = 0,2 cos πmθ, for m ∈ N. (3.1)

Here we have used 〈µ | z〉 = ±mθ〈ω | ω〉 = ±12mθ. Sometimes it is convenient to introduce adifferent normalization of the C-functions, namely the following one:

Φm(θ) = 2 cosπmθ, for all m ∈ Z≥0. (3.2)The fundamental region F (A1) is the closed segment with the end points 0 and ω. Its length is

|ω| = 1√2. Hence 0 ≤ θ ≤ 1 within F (A1).

It is straightforward to verify the decomposition of the products:

Φm(θ)Φm′(θ) = Φm+m′(θ) + Φm−m′(θ), for m,m′ ∈ Z≥0.Any two functions Ωm(θ), Ωm′(θ), where m 6= m′, are orthogonal, i.e.:

∫ 10

Ωm(θ)Ωm′(θ)dθ =

0, if m 6= m′,1, if m = m′ = 0,2, if m = m′ > 0.

(3.3)

It is useful to notice the special case of (3.3) arising for m > 0 and m′ = 0,∫ 1

0Ωm(θ)dθ = 0, for any m ∈ N.

The orthogonality can be used to invert the Fourier series (2.1) on F (A1) in the traditional way.Every element of the group A1 is conjugate to an element of its torus. Elements of the torus are

parameterized by points of a circle. Every point of F stands for one conjugacy class of the elementsof A1.

3.2. Discretization of A1. The objective of this subsection is the description of the A1 versionof the discrete orthogonality (7.4) of the C-functions.

Here we are interested in the elements sω ∈ F with rational values of s. They are specified bytwo non-negative integers s0 and s1. It is convenient to set it up as follows,

F 3 sω = s1M

ω, where s0 + s1 = M > 0, s0, s1 ∈ Z≥0.


Fixing M determines an equidistant grid of M + 1 points sω ∈ F . The set FM of their coordinatesis

FMdef= {0, 1M , 2M , 3M , . . . , M−1M , 1}.

3.3. Scalar product on the grid FM . The points of F are in one-to-one correspondence withconjugacy classes of elements of A1. They represent the conjugacy classes of elements of ordersequal to M , as well as all the divisors of M .

In general, one introduces a scalar product in the space of functions defined digitally on FM :

〈f | h〉M def=∑

s∈TMf(s)h(s) =

∑

s∈FMcsf(s)h(s). (3.4)

Here TM stands for the Abelian subgroup of the maximal torus of A1, which is generated by allthe elements of order M . The coefficients cs in the sum over FM count the number of points in thetorus that are conjugate to s. One has

c0 = c1 = 1, and c 1M

= c 2M

= · · · = cM−1M

= 2. (3.5)

The C-functions take the following values on the points of the grid:

Ω0(s) = 1, Ωm(s) = 2 cosπms, for s ∈ FM and m ∈ N. (3.6)In order to reduce the number of calculations, it is more convenient to utilize in the discretization

the normalized form of the C-functions given in (3.2).

Φm(s) = 2 cosπms, for s ∈ FM and all m ∈ Z≥0. (3.7)The crucial discrete orthogonality property of the normalized C-functions over FM is the follow-

ing:

〈Φm | Φm′〉M =

8M, if m = m′ = 0 mod M ,4M, if m = m′ 6= 0 mod M ,0, if m 6= m′ mod M .

(3.8)

Note that the orthogonality (3.8) is valid not only when m and m′ are constrained to the range{0, 1, . . . ,M}.

Examples of several A1 normalized C-functions on the grid F3 are given in the Table 1.

s 0 1323 1

Φ0(s) 2 2 2 2Φ1(s) 2 1 −1 −2Φ2(s) 2 −1 −1 2Φ3(s) 2 −2 2 −2Φ4(s) 2 −1 −1 2Φ5(s) 2 1 −1 −2Φ6(s) 2 2 2 2

cs 1 2 2 1

Table 1. Values of several normalized C-functions of A1 on the points of the gridFM with M = 3. Note that only the first four C-functions are pairwise orthogonal.The higher ones repeat the values of the lowest four. cs are the coefficients from(3.5).


Our aim in the development of the formalism so far is to use it for the expansion in terms of theC-functions of any function f(s), given by its values on the grid FM .

More precisely, a function f(s), with known real values on FM , can be decomposed as follows,

f(s) =M∑

k=0

dkΦk(s), s ∈ FM . (3.9)

Then we can compute the coefficients dk from

〈f | Φk〉M =∑

s∈FMcsf(s)Φk(s) =

{8Mdk, if k = 0 or k = M ,4Mdk, if k = 1, 2, . . . , M − 1.

(3.10)

After the coefficients dk have been calculated, one can replace s in (3.9) by the continuousvariable θ.

fcont(θ)def=

M∑

k=0

dkΦk(θ), where θ ∈ R. (3.11)

At θ = s ∈ FM , the continuous function fcont(θ) coincides with f(s).The all-important property that distinguishes (3.11) from the standard Fourier transform and

that was apparently established only recently, [7], is the smoothness of the interpolation of fcontbetween the points of the grid FM .

4. Higher rank cases in general

The basic tools that we need, in order to develop the formalism of every case, can be introducedequally simply for all the cases at once and for any rank n < ∞. Thus, we do not always requirethat n = 2. The rank of the Lie group is the dimension of the decomposition problem, i.e. thenumber of variables in a C-function.

4.1. The α- and ω-bases and their dual bases. The root system ∆ ∈ Rn contains k distinctvectors/roots. A suitable basis Π = {α1, . . . , αn} ⊂ ∆ consists of the simple roots (α-basis). Forany simple Lie algebra, its simple roots are of at most of two different lengths.

Relative lengths and angles between simple roots of the basis Π are concisely specified in termsof the elements of the Cartan matrix C:

Cjk =2〈αj | αk〉〈αk | αk〉 = 〈αj | α̂k〉, for j, k = 1, . . . , n, (4.1)

where α̂k is the simple root of the dual root system ∆̂. Adopting the standard convention for thelengths of the long root of Π, namely 〈αl | αl〉 = 2, the elements of the Cartan matrix become thesmallest possible integers. During the dualization,

α ←→ α̂ = 2α〈α | α〉 , of α ∈ ∆,

the long roots do not change, while the short ones become long (see Figure 1).In addition to the α- and α̂-bases, we introduce the basis of the fundamental weights and its

dual basis, called ω- and ω̂-basis respectively. In the matrix form, using (4.1), we have:

α = Cω, ω = C−1α, α̂ = CT ω̂, ω̂ = CT−1α̂. (4.2)

Given a specific Cartan matrix, one can verify the following frequently used multiplication rules,

〈α̂j | ωk〉 = 〈αj | ω̂k〉 = 2〈αj | ωk〉〈αj | αj〉 = δjk. (4.3)


4.2. Reflections of the finite Weyl group. For each ξ ∈ ∆ there are reflections rξ, which actin Rn according to:

rξx = x− 2〈x | ξ〉〈ξ | ξ〉 ξ, where ξ ∈ ∆, x ∈ Rn. (4.4)

Those with ξ ∈ Π generate the finite Weyl group W .Observe that rξ does depend on the direction of ξ, but not on its length or orientation along

that direction. In particular, it is easy to check that rξξ = −ξ and also that

rξrξx = rξ

(x− 2〈x | ξ〉〈ξ | ξ〉 ξ

)= rξx− 2〈x | ξ〉〈ξ | ξ〉 rξξ = x.

The application of W to the basis Π yields the root system, WΠ = ∆. Consequently, one hasthe W -invariance: ∆ = W∆. Thus, a root is transformed into a root of the same system by anyelement of W . Each root appears exactly once in ∆. It is known that for simple Lie groups theroot system is obtained by applying W to either one, or at most two roots of the basis Π.

4.3. The affine Weyl group. The group W contains the reflections rξ in mirrors, which areorthogonal to any ξ ∈ ∆ and pass through the origin. The affine Weyl group W aff contains all thereflections RNξ in mirrors, which are orthogonal to ξ and displaced from the origin by any Nξ,where N ∈ Z. Hence W aff is of infinite order, and W ⊂ W aff .

One has the affine reflections RNξ

RNξxdef= Nξ + rξx, where x ∈ Rn, N ∈ Z and ξ ∈ ∆. (4.5)

Note that R0ξ = rξ and that RNξ 6= R−Nξ. In particular, RNξ0 = Nξ is the reflection of theorigin in the midpoint 12Nξ, and RNξ(Nξ) = 0. In general, one has

RNξRNξx = RNξ(Nξ + rξx) = Nξ + rξ(Nξ + rξx)

= N(ξ + rξξ) + rξrξx = x.

It is useful to note the presence of a translation subgroup T ⊂ W aff . It is an Abelian subgroupof W aff , whose elements are all the translations tNξ, N ∈ Z.

tNξxdef= RNξrξx = rξR−Nξx

= (Nξ + r2ξx) = x + Nξ, where N ∈ Z, ξ ∈ ∆ and x ∈ Rn. (4.6)As an example, one may verify that t−Nξx = rξRNξx.In general, W aff can be defined as the semidirect product W n T .

4.4. Root and weight lattice. The root lattice Q consists of all the elements that can be writtensymbolically as

Zα1 + · · ·+ Zαn ∈ Q, where α1, . . . , αn ∈ Π,in which Z stands for any integer chosen independently in each term of the sum. In particular,∆ ⊂ Q.

We will mainly consider the weight lattice P and its positive chamber P+:

Zω1 + · · ·+ Zωn ∈ P, and Z≥0ω1 + · · ·+ Z≥0ωn ∈ P+.In all cases Q ⊆ P .

In addition to the root and weight lattices Q and P , we have also their dual latices: Q̂ and P̂ ,each along with its basis, namely the α-, ω-, α̂-, ω̂-basis respectively.


4.5. Weyl group orbits. Let λ ∈ P ⊂ Rn. The Weyl group orbit Wλ of λ is the set of distinctelements of P obtained by all possible applications of W to λ. We write Wλ = Wλ. In a similarway, one defines W affλ = W

affλ. The orbit Wλ is always finite, while W affλ is infinite. The size |Wλ|of Wλ is the number of distinct points which are generated from λ by W . The maximal value of|Wλ| equals to the order |W | of the Weyl group. Other possible values of |Wλ| are some of thedivisors of |W |.

In particular, we are interested in the following properties:

W0 = 0 and WΠ = ∆.

In general, P is a union of several W aff -orbits.Each λ ∈ P is contained in precisely one W -orbit Wλ. Elements of Wλ are called weights.

Each W -orbit contains a unique element belonging to P+, called the dominant weight of Wλ. Thedominant weight is easy to recognize since its coordinates in ω-basis are non-negative integers. Itis then usually taken as the λ to be used in the symbol Wλ.

4.6. C-functions. The definition of a C-function Ωλ(z) involves λ ∈ P+ and z ∈ Rn. It requiresalso the W -orbit Wλ of λ. The compact semisimple Lie group figures only through its Weyl group.One has,

Ωλ(z)def=

∑

µ∈Wλe2πi〈µ|z〉, where λ ∈ P+ and z ∈ Rn. (4.7)

The number of summands in (4.7) is the number |Wλ| of weights in Wλ:

|Wλ| = |W ||StabW (λ)| ,

where StabW (λ) is the stabilizer of λ in W : the subgroup of W generated by reflections which donot move λ.

A different normalization of C-functions than (4.7) may occasionally be more convenient. Wewill use the following one

Φλ(z)def= |StabW (λ)|Ωλ(z) (4.8)

All C-functions, renormalized in such way, take the same value at the origin,

Φλ(0) = Φλ′(0), for all λ, λ′ ∈ P+.In order to make explicit the dependencies of C-functions on λ and z, one needs to fix a particular

semisimple Lie group, or, equivalently, a Weyl group, choose the weight λ, and provide more detailson how to calculate the products 〈µ | z〉. That is, in which bases one has z and λ. In the rest ofthis paper it is done explicitly for the cases of interest, where n = 2. Most often λ is given relativeto either ω- or α-basis, while z is taken relative to ω̂-basis.

4.7. Other properties of C-functions. Important symmetry properties of C-functions are thefollowing:

Ω(a,b)(z) = Ω(a,b)(wz),

Ω(a,b)(z) = Ω(a,b)(RNγz) = Ω(a,b)(rγz + Nγ) = Ω(a,b)(rγz), (4.9)

Ω(a,b)(z) = Ω(a,b)(rγRNγz) = Ω(a,b)(RNγrγz) = Ω(a,b)(z ±Nγ),where w ∈ W , γ ∈ ∆̂ and N ∈ Z>0.

In view of (4.3), 〈λ | γ〉 for all γ ∈ Q̂, λ ∈ P take integer values. Hence, they do not changethe value of a C-function (4.7). The first property in (4.9) follows from 〈η | z〉 = 〈wη | wz〉 for allw ∈ W and η ∈ Wλ.


Another useful property of C-functions is the complete decomposability of their products into alinear combination of C-functions with positive integer coefficients:

ΩλΩλ′ = Ωλ+λ′ + · · · . (4.10)The problem of finding the remaining terms of the sum and their multiplicities is a question ofcomputation. Many examples are found in [16] and [17].

4.8. The fundamental region. The fundamental region F of a group, in general, is a finiteregion, where every conjugacy class of the elements of the group is represented precisely by onepoint. For a compact simple Lie group of rank n, F is a simplex in Rn. When n = 2, it forms atriangle. For a non-simple group, it is a Cartesian product of corresponding simplexes. Thus, forA1, F is a segment, for A1×A1 – a square.

The definition (4.7) of the C-function clearly allows one to consider z in the entire space Rn.However, due to the symmetries (4.9), we are mainly interested in the C-functions with z in thefundamental region of the corresponding semisimple Lie group.

The fundamental region of a simple Lie group consists of the points x such that 0 ≤ 〈x | ξh〉 ≤ 1,where ξh is the highest root of ∆. Suppose,

ξh = q1α1 + q2α2 + · · ·+ qnαn, (4.11)where q1, . . . , qn are well known positive integers for each root system. Then the vertices of thesimplex F are the following:

F : {0, 1q1 ω̂1, 1q2 ω̂2, . . . , 1qn ω̂n}. (4.12)4.9. C-functions and irreducible characters. The character χλ of a finite-dimensional irre-ducible representation of a group G is a linear combination of C-functions,

χλ(z) =∑

µ

mλµΩµ(z), (4.13)

where the summation extends over the set of distinct dominant weights in the weight system Vλof the representation labeled by λ. Coefficients mλµ are the multiplicities of the dominant weightsin Vλ. It is a well-known computational problem in Lie theory to find the multiplicities mλµ for agiven λ, see, for example, the tables [14] and references therein. The matrix (mλµ) is non-singular.A suitable ordering of the dominant weights makes it triangular. Hence, it can be inverted, so thatone has:

Ωλ(z) =∑

µ

nλµχµ(z), (4.14)

where nλµ are the (integer) matrix elements of the inverse matrix of dominant weight multiplicities.The summation in (4.14) ranges over the same finite set of dominant weights as in (4.13).

Consequently, C-functions form another basis in the space spanned by irreducible characters ofG.

4.10. Orthogonality of C-functions. The orthogonality property of C-functions,∫

FΩλ(z)Ωλ′(z)dF = 0, where λ, λ′ ∈ P+ and λ 6= λ′, (4.15)

is a consequence of the orthogonality of characters of irreducible representations and the decom-posability of products (4.10) of C-functions. Indeed, let the zero weight be denoted 0 for any rank.Because W0 = 0, one has Ω0(z) = 1 for every group and all z ∈ Rn. Putting λ′ = 0 in (4.15), wehave: ∫

FΩλ(z)dF = 0, for any 0 6= λ ∈ P+. (4.16)


Complex conjugate Ωλ is a C-function with another (contragredient) dominant weight, say λ′.Therefore Ωλ(z)Ωλ′(z) decomposes according to (4.10). The decomposition contains Ω0 preciselyif λ′ = λ. Therefore, only in that case, the integral (4.15) is not zero.

4.11. Eigenfunctions of the Laplace operator. Consider the differential operator

L = (α1∂1 + α2∂2 + · · ·+ αn∂n)2. (4.17)Since the matrix of scalar products of simple roots is positive definite, by a suitable choice of

basis, the operator can be brought to the sum of second derivatives with positive coefficients. Hence,one is justified in calling L the Laplace operator.

Subsequently, we will verify the validity of the eigenvalue equation [13]:

LΩλ(z) = −4π〈λ | λ〉Ωλ(z), for all λ ∈ P+ and z ∈ Rn. (4.18)From (4.7) it is clear that C-functions are continuous functions with continuous derivatives of

all orders in Rn. Then it follows from (4.9) that their derivatives, normal to the boundary of thefundamental region, must be equal to zero (Newmann boundary value condition).

5. The case A1×A1This is a simple concatenation of two cases of A1 described in Section 3.

5.1. Roots and weights. Relative length and angles of the simple roots are given by the scalarproducts

〈α1 | α2〉 = 0, 〈α1 | α1〉 = 〈α2 | α2〉 = 2.The Cartan matrix and its inverse are the following

C =(

2 00 2

)and C−1 =

(12 00 12

).

Consequently, α1 = 2ω1 and α2 = 2ω2. Their duals α̂k and ω̂j coincide with αk and ωj . The rootsystem ∆ = {±α1,±α2} geometrically represents the vertices of a square of a side length 2

√2.

5.2. Weyl group orbits. Suppose λ = aω1 + bω2 ∈ P . Then the Weyl group orbit Wλ is given by

Wλ ≡ W(a,b) =

{(0, 0)}, if a, b = 0,{±(a, 0)}, if a 6= 0 and b = 0,{±(0, b)}, if a = 0 and b 6= 0,{±(a, b),±(a,−b)}, if a, b 6= 0.

5.3. C-functions. The C-functions of A1×A1, with λ = aω1+bω2 and z = xω1+yω2, are productsof two A1 C-functions

Ωa.b(x, y) = Ωa(x)Ωb(y).They are the following:

Ω(0,0)(x, y) = 1,

Ω(a,0)(x, y) = 2 cos(πax),

Ω(0,b)(x, y) = 2 cos(πby), (5.1)

Ω(a,b)(x, y) = 4 cos(πax) cos(πby).

The C-functions, normalized as in (4.8), are written for all a, b ∈ Z≥0 in one expression:Φ(a,b)(x, y) = 4 cos(πax) cos(πby).


r0

r2

α1

α2 ω2

ω1

C2

α1

^

ω1

^

α2

^ ω2

^

F

α1

ω1

ξh

F

α2ω2

r2

r1

A1× A1

r0,2

r0,1

ξh

Figure 1. The simple roots, the fundamental weights, along with their duals, andthe fundamental region for the cases A1×A1 and C2.

5.4. Orthogonality of C-functions. The fundamental region of A1×A1 here is:F (A1×A1) = {xω1 + yω2 | 0 ≤ x, y ≤ 1},

its vertices being 0, ω1, ω2 and ω1 +ω2 (see Figure 1). Hence its sides have length |ω1| = |ω2| = 1√2 .Orthogonality of the functions is readily verified directly:

∫

FΩ(a,b)(x, y)Ω(c,d)(x, y)dF = 12

∫ 10

dx

∫ 10

Ω(a,b)(x, y)Ω(c,d)(x, y)dy

=

0, if a 6= c and b 6= d,12 , if a = b = c = d = 0,1, if a = c > 0 and b = d = 0,

or a = c = 0 and b = d > 0,2, if a = b > 0 and c = d > 0.

(5.2)

5.5. Laplace operator. The Laplace operator in this case is

L = 2∂xx + 2∂yy.

The C-functions are its eigenfunctions, and

LΩλ = −4π2〈λ | λ〉Ωλ = −2π2(a2 + b2)Ωλ,where the scalar product for λ = aω1 + bω2 is computed using the inverse Cartan matrix C−1:

〈λ | λ〉 = (a b) C−1(

ab

)= 12(a

2 + b2).

6. The case C2

6.1. Roots and weights. Relative length and angles of the simple roots of C2 are given by:

〈α1 | α2〉 = −1, 〈α1 | α1〉 = 1, 〈α2 | α2〉 = 2.The Cartan matrix and its inverse are:

C =(

2 −1−2 2

)and C−1 =

(1 121 1

).


Consequently,

α1 = 2ω1 − ω2, ω1 = α1 + 12α2, α̂1 = 2α1, ω̂1 = α̂1 + α̂2,α2 = −2ω2 + 2ω2, ω2 = α1 + α2, α̂2 = α2, ω̂2 = 12 α̂1 + α̂2.

The root system ∆ = {±α1,±α2,±(α1 + α2),±(2α1 + α2)} geometrically represents the verticesand midpoints of a square. The highest root is ξh = 2α1 + α2.

6.2. Weyl group orbits. Let λ = aω1 + bω2 ∈ P+. Then the Weyl group orbit Wλ ≡ W(a,b)contains 1, 4, or 8 points. More precisely,

W(a,b) =

{(0, 0)} if a = b = 0,{±(a, 0),±(−a, a)} if a 6= 0 and b = 0,{±(0, b),±(2b,−b)} if a = 0 and b 6= 0,{±(a, b),±(−a, a + b),±(a + 2b,−b),±(a + 2b,−a− b)} if a, b 6= 0.

In particular, ∆ = W(2,0) ∪W(0,1).6.3. C-functions. The C-functions of C2, with λ = aω1+bω2 and z = xω̂1+yω̂2, are the following:

Ω(0,0)(x, y) = 1,

Ω(a,0)(x, y) = 2 cos(πay) + 2 cos(πa(2x + y)),

Ω(0,b)(x, y) = 2 cos(2πbx) + 2 cos(2πb(x + y)), (6.1)

Ω(a,b)(x, y) = 2 cos(π(2bx + (a + 2b)y)) + 2 cos(π((2a + 2b)x + (a + 2b)y))

+ 2 cos(π(ay + (2a + 2b)x)) + 2 cos(π(2bx− ay)), where a, b > 0.C-functions, normalized as in (4.8), are written for all a, b ∈ Z≥0 in one expression:

Φ(a,b)(x, y) = 2 cos(π(2bx + (a + 2b)y)) + 2 cos(π((2a + 2b)x + (a + 2b)y))

+ 2 cos(π(ay + (2a + 2b)x)) + 2 cos(π(2bx− ay)).6.4. Decomposition of products of C-functions. Products of the C-functions decompose intosums of C-functions (4.10). For example, one has:

Ω(0,a)Ω(0,b) = Ω(0,a+b) + Ω(2a,b−a) + Ω(0,b−a), when a < b,Ω(0,a)Ω(b,0) = Ω(b,a) + Ω(2a−b,b−a), when a < b < 2a,Ω(a,0)Ω(0,b) = Ω(a,b) + Ω(a,b−2a), when b > 2a.

It is possible to build up recursively the higher C-functions, starting from the lowest three,namely Ω(0,0), Ω(1,0) and Ω(0,1):

Ω(1,1) = Ω(1,0)Ω(0,1) − 2Ω(1,0),Ω(2,0) = Ω(1,0)Ω(1,0) − 4Ω(0,0) − 2Ω(0,1),Ω(0,2) = Ω(0,1)Ω(0,1) − 4Ω(0,0) − 2Ω(2,0),Ω(2,1) = Ω(0,1)Ω(2,0) − 2Ω(0,1),Ω(3,0) = Ω(1,0)Ω(2,0) − Ω(1,0) − Ω(1,1),Ω(1,2) = Ω(0,1)Ω(1,1) − 2Ω(1,0) − Ω(1,1) − 2Ω(3,0),

...


6.5. Orthogonality of C-functions. The fundamental region F (C2) is defined as follows:

F (C2) = {xω̂1 + yω̂2 | where x, y ≥ 0 and 2x + y ≤ 1}.Therefore, its vertices are 0, ω̂12 and ω̂2. Geometrically it is a triangle with angles

π2 ,

π4 and

π4 (see

Figure 1).Orthogonality of C-functions of C2 can be verified, if somewhat laboriously, by using (6.1) in

(4.15):∫

FΩ(a,b)(x, y)Ω(c,d)(x, y)dF =

∫ 12

0dx

∫ 1−2x0

Ω(a,b)(x, y)Ω(c,d)(x, y)dy

=

0, if a 6= c and b 6= d,14 , if a = b = c = d = 0,1, if a = c > 0 and b = d = 0,

or a = c = 0 and b = d > 0,2, if a = c > 0 and b = d > 0.

(6.2)

In particular, we have for any (a, b) 6= (0, 0) and (c, d) = (0, 0)∫

FΩ(a,b)(x, y)dF = 0.

6.6. Laplace operator. The Laplace operator (4.17) specializes to C2 as follows:

L = (α1∂x + α2∂y)2 = ∂xx − 2∂xy + 2∂yy.

Applying L to C-functions, we see that they are its eigenfunctions:

LΩλ = −4π2〈λ | λ〉Ωλ = −2π2(a2 + 2ab + 2b2)Ωλ,where the scalar product 〈λ | λ〉 is computed for λ = (aω1 + bω2) ≡

(a b

), using the quadrature

matrix Q.

〈λ | λ〉 = λQλT = 12

(a b

)(1 11 2

)(ab

)= 12a

2 + ab + b2.

6.7. The branching rules for C-functions of C2 to the subgroup A1×A1. The problemconsidered here is to calculate the “branching rules” for C-functions. There are two rather differentways how a semisimple Lie group of rank 2 is related to C2. In both cases the group is of typeA1×A1. In order to distinguish the two cases we use the notations A1×A1⊂ and A1×A1


and [Ω(a,b)

]C2

=[Ω(a+2b,a) + (1− δb,0)Ω(a,a+2b)

]A1×A1< (6.4)

C2 A1×A1⊂ A1×A1<Ω(0,0) Ω(0,0) Ω(0,0)Ω(1,0) Ω(1,0) + Ω(0,1) Ω(1,1)Ω(0,1) Ω(1,1) Ω(2,0) + Ω(0,2)Ω(2,0) Ω(2,0) + Ω(0,2) Ω(2,2)Ω(1,1) Ω(2,1) + Ω(1,2) Ω(3,1) + Ω(1,3)Ω(0,2) Ω(2,2) Ω(4,0) + Ω(0,4)Ω(3,0) Ω(3,0) + Ω(0,3) Ω(3,3)Ω(2,1) Ω(3,1) + Ω(1,3) Ω(4,2) + Ω(2,4)Ω(1,2) Ω(3,2) + Ω(2,3) Ω(5,1) + Ω(1,5)Ω(0,3) Ω(3,3) Ω(6,0) + Ω(0,6)Ω(4,0) Ω(4,0) + Ω(0,4) Ω(4,4)

Table 2. Examples of the decomposition of C-functions of C2 (left column) intoC-functions of the maximal subgroup A1×A1⊂ (middle column) and of the maximalsubjoint group A1×A1< (right column).

7. Discretization of 2-dimensional transforms

In this section we describe all the necessary tools, which allow one to decompose a function f(s),given by its values on points s of certain 2-dimensional grids FM , M = 1, 2, · · · < ∞, in terms offinite series of C-functions. Such decomposition is possible, due to the discrete orthogonality ofC-functions on the points of FM , i.e., its coefficients can be computed.

7.1. Equidistant grids of points in the fundamental region. The fundamental region F canbe used to tile the entire plane by its copies, W affF = R2. Our next task is to describe grids FMof discrete points in F of any density, characterized by a positive integer M , which extends into alattice with the tiling of R2 by copies of F .

The points s of FM are conveniently described in barycentric coordinates. That is, by threenon-negative integers, [s0, s1, s2]. A point s belongs the grid FM , provided

s = s1M ω̂1 +s2M ω̂2, where s0, s1, s2 ∈ Z≥0 and M = s0 + q1s1 + q2s2 > 0. (7.1)

Here q1 and q2 are positive integers, specific for each Lie group. They are the coefficients of thehighest root of ∆ in α-basis. Equivalently, we write

FMdef=

{(s1M ,

s2M

) | s0, s1, s2 ∈ Z≥0, s0 + q1s1 + q2s2 = M > 0}

. (7.2)

7.2. Bilinear form on FM . Given the set of points FM in the fundamental region, and twofunctions f(s) and h(s), given by their values at the points s ∈ FM , one defines a hermitian formas follows [3]:

〈f | h〉M def=∑

s∈FMcsf(s)h(s). (7.3)

The line over h(s) stands for complex conjugation. The coefficients cs are positive integer numbersfor each Lie group. They are given below for the groups of rank 2, for a general case, see [3].


7.3. Discrete orthogonality of C-functions. For a fixed value of M ∈ N, the discrete orthogo-nality of C-functions on FM ⊂ F is crucial for this study:

〈Φ(a,b) | Φ(a′,b′)〉M =∑

s∈FMcsΦ(a,b)(s)Φ(a′,b′)(s) = δa,a′δb,b′〈Φ(a,b) | Φ(a,b)〉M . (7.4)

The orthogonality (7.4) holds for Φ(a,b), Φ(a′,b′) from a finite subset SM of C-functions.In general, for all cases of rank 2, we can construct (infinitely many) such subsets SM by using

the following:

SMdef= {Φ(a,b) | ( bM , aM ) = r

(s1M ,

s2M

), where

(s1M ,

s2M

) ∈ FM and r ∈ W aff}. (7.5)As an easy example, one can obtain the set of the lowest Φ(a,b) ∈ SM , by taking r = 1 and a, b

that satisfy the inequality:

aq2 + bq1 ≤ M =⇒ Φ(a,b) ∈ SM , (7.6)where q1, q2 are the same as in (4.11).

From the definition of SM it is easy to see that

|SM | = |FM |.The numbers 〈Φ(a,b) | Φ(a,b)〉M in (7.4) take only a few integer values for each M . Subsequently,

we will provide them for all 0 ≤ a, b < ∞, for all the cases considered in this paper.

7.4. Decomposition into C-functions and continuous extension. A function f(s), withknown values on points of the grid FM , can be decomposed as follows:

f(s) =∑

Φ(a,b)∈SMd(a,b)Φ(a,b)(s), s ∈ FM , (7.7)

Orthogonality (7.4) makes it possible to compute the coefficients d(a,b) from

d(a,b) =〈f | Φ(a,b)〉M

〈Φ(a,b) | Φ(a,b)〉Mwhere 〈f | Φ(a,b)〉M =

∑

s∈FMcsf(s)Φ(a,b)(s) (7.8)

After d(a,b) have been calculated, one can replace s in (7.7) by the continuous variables x and y:

fcont(x, y)def=

∑

Φ(a,b)∈SMd(a,b)Φ(a,b)(x, y), x, y ∈ R, (7.9)

The function fcont(x, y) is the continuous extension of the decomposition (7.7). Both functionscoincide at the points (x, y) = ( s1M ,

s2M ) = s ∈ FM .

7.5. Discretization in the case of A1×A1. The fundamental region in the case of A1×A1 is aCartesian product of two fundamental regions of A1. It forms a square described by the following:

F = {xω1 + yω2 | 0 ≤ x, y ≤ 1}.A square lattice FM of order M is a Cartesian product of two lattices FM (A1), built on F thefollowing way:

FMdef=

{(s1M ,

s2M

) | s1, s2 ∈ Z≥0, s1, s2 ≤ M ∈ N}

.


Coefficients cs are in this case:

cs ≡ c( s1M , s2M ) =

1, if s1 = s2 = 0,or s1 = s2 = M ,or s1 = 0 and s2 = M ,or s1 = M and s2 = 0,

2, if s1 = 0 and 0 < s2 < M ,or s2 = 0 and 0 < s1 < M ,or s1 = M and 0 < s2 < M ,or s2 = M and 0 < s1 < M ,

4, if 0 < s1, s2 < M .

The C-functions are orthogonal:

〈Φ(a,b) | Φ(a′,b′)〉M = 0, if a 6= a′ and b 6= b′,

otherwise, for the set of the lowest pairwise orthogonal normalized C-functions:

〈Φ(a,b) | Φ(a,b)〉M = 16M2 ×

1, if 0 < a, b < M ,2, if 0 < a < M and b = 0,

or a = 0 and 0 < b < M ,or 0 < a < M and b = M ,or a = M and 0 < b < M ,

4, if a = b = 0,or a = 0 and b = M ,or a = M and b = 0,or a = b = M ,

with the higher C-functions repeating the values of the lowest ones.For example, F2 consists of the nine points given as [s1, s2] =

(s12 ,

s22

):

[0, 0] = (0, 0), [0, 1] = (0, 12), [0, 2] = (0, 1),

[1, 0] = (12 , 0), [1, 1] = (12 ,

12), [1, 2] = (

12 , 1),

[2, 0] = (1, 0), [2, 1] = (1, 12), [2, 2] = (1, 1), (7.10)

while F1 has only four points:

[0, 0] = (0, 0), [0, 1] = (0, 1), [1, 0] = (1, 0), [1, 1] = (1, 1).

7.6. Discretization in the case of C2. The highest root of C2 is 2α1 + α2. Therefore:

FMdef=

{(s1M ,

s2M

) | s0, s1, s2 ∈ Z≥0, s0 + 2s1 + s2 = M > 0}

.

Vertices of F are (0, 0), (0, 1), (12 , 0), relative to ω̂-basis.


Φ(0,0) Φ(0,1) Φ(0,2)

Φ(1,0) Φ(1,1) Φ(1,2)

Φ(2,0) Φ(2,1) Φ(2,2)

Figure 2. The set of nine lowest pairwise orthogonal normalized C-functions ofA1×A1 for the grid F2.

F F A1C2 A1

F2

F4

F3

Figure 3. The lattice points of F2, F3 and F4 in the fundamental region F for thecases C2 and A1×A1.

Coefficients cs are in this case:

cs ≡ c( s1M , s2M ) =

1, if s1 = 0 and s2 = 0,or s1 = 0 and s2 = M ,

2, if s1 = 0 and s2 = M2 ,4, if s1 = 0 and 0 < s2 < M ,

or s2 = 0 and 0 < s1 < M ,or s1, s2 > 0 and 2s1 + s2 = M ,

8, if s1, s2 > 0 and 2s1 + s2 < M .

(7.11)


Φ(0,0) Φ(0,1) Φ(0,2)

Φ(1,0) Φ(1,1) Φ(2,0)

Φ(2,1) Φ(3,0) Φ(4,0)

Figure 4. The set of nine lowest pairwise orthogonal normalized C-functions of C2for the grid F4.

The discrete orthogonality:

〈Φ(a,b) | Φ(a′,b′)〉M = 0, if a 6= a′ and b 6= b′,otherwise, for the set of the lowest pairwise orthogonal normalized C-functions:

〈Φ(a,b) | Φ(a,b)〉M = 16M2 ×

1, if 0 < a, b and a + 2b < M ,2, if 0 < a < M and b = 0,

or a = 0 and 0 < 2b < M ,or 0 < a, b and a + 2b = M ,

4, if a = 0 and 2b = M ,8, if a = b = 0,

or a = M and b = 0,

with the higher C-functions repeating the values of the lowest ones.For example, F3 consists of the six points given as [s0, s1, s2] =

(s1M ,

s2M

):

[1, 0, 0] = (0, 0), [0, 0, 1] = (0, 1), [1, 1, 0] = (13 , 0),

[0, 1, 1] = (13 ,13), [1, 0, 2] = (0,

23), [2, 0, 1] = (0,

13), (7.12)

while F2 has only four points:

[1, 0, 0] = (0, 0), [0, 0, 1] = (0, 1), [0, 1, 0] = (12 , 0), [1, 0, 1] = (0,12).

As an example, the values of C2 normalized orbit functions on the grid F3 are given in Table 3.


s (0, 0)(0, 13

) (0, 23

) (13 , 0

) (13 ,

13

)(0, 1)

Φ(0,0)(s) 8 8 8 8 8 8Φ(0,1)(s) 8 2 2 −4 −4 8Φ(1,0)(s) 8 4 −4 2 −2 −8Φ(1,1)(s) 8 −2 2 −4 4 −8Φ(2,0)(s) 8 −4 −4 2 2 8Φ(3,0)(s) 8 −8 8 8 −8 −8Φ(2,1)(s) 8 −4 −4 2 2 8Φ(3,1)(s) 8 −2 2 −4 4 −8Φ(1,2)(s) 8 4 −4 2 −2 −8Φ(0,2)(s) 8 2 2 −4 −4 8Φ(0,3)(s) 8 8 8 8 8 8

cs 1 4 4 4 4 1

Table 3. Values of several normalized C-functions of C2 at the points of the gridF3. Note that only the first six C-functions are pairwise orthogonal. The higherones repeat the values of the lowest six. The cs are the coefficients from (7.11).

8. Motivating examples

Comparison of expansions of into series of both groups considered here, as well as the twoadditional groups of [1], will require further study. Related questions are illustrated by the followingexamples.

There are two examples shown in this section, involving decomposition of the same functionf(x, y) into series of C-functions of A1×A1 and C2. Goal of the examples is (i) to illustrate discretedecomposition of a given function followed by the continuous extension, and (ii) to compare thecontinuous extensions in both cases.

We choose for f(x, y) the square step function with sharp edges,

f(x, y) =

{1 for 0.30 < x < 0.45, and 0.05 < y < 0.20,0 elsewhere in F .

(8.1)

In order to make the comparison, we set up the vertices of the two fundamental regions as follows(relative to an orthonormal basis),

F (A1×A1) = {(0, 0), (12 , 0), (0, 12), (12 , 12)} (8.2)F (C2) = {(0, 0), (12 , 0), (12 , 12)} (8.3)

Thus F (C2) is exactly half of F (A1×A1) with three vertices in common. In order to have the gridof the same density in both regions, one has to make sure that there is the same number of pointsalong the edges of sides of F adjacent to angle π2 .

Suppose M is fixed. Then each side of F (A1×A1) contains M + 1 points each. For a given M ′,the F (C2)-edge (0, 0), (12 , 0) contains [

M ′2 ] points, where [

M ′2 ] stands for the integer part of [

M ′2 ].

Consequently, to have the same density of the grid in both cases, we have to have M + 1 = [M′

2 ].Figures 5 and 6 contain results of our two examples. The same function f(x, y) of (8.1) is

placed into the fundamental regions F of the two groups, their values are sampled at the points sof the grids FM (7.2) and taken as our digital data f(s). Then the functions are expanded (7.7)into C-functions of A1×A1 and of C2 on the corresponding grid FM , i.e. expansion coefficients are


M=8

M=16 M=32

M=4

Figure 5. Decomposition and continuous extension of a square step function placedin the fundamental region of A1×A1 on the grids of orders M = 4, 8, 16 and 32.

calculated (7.8). After that, continuous extensions of the discrete expansions of f(s) are made (7.9)by replacing the C-functions of the discrete argument s in the expansions by the same functions ofthe continuous argument, while keeping the expansion coefficients unchanged. Each figure shows thefunction fcont(x, y) resulting from the continuous extension of discrete expansions. More precisely,four different continuous extensions are shown on each figure. They differ by the densities of thegrid FM , namely M = 4, 8, 16, and 32, from which the continuous extension is made. For the samevalue of M , the densities of grids of A1×A1 and C2 are the same. The points of the grids are notshown on the figures.

Inspecting and comparing the two figures, one readily observes the following:

- Increasing density of the grid, i.e. increasing the value of M , makes the continuous extensionto match more closely the given model function f(x, y) of (8.1).

- Quality of the extension, i.e. the match between the continuous extension fcont(x, y) and theoriginal function f(x, y), is comparable for the same density of the grid in both cases, thoughC2 expansion may be slightly superior, as noticeable by comparing the two at M = 32.

The observations have important consequences:The number of points of FM (A1×A1) and FM (C2) are approximately in the ratio 2:1, due to

the ration of the areas of the fundamental regions (see Figure 1) and to the equal density of thepoints in both cases.

The number of terms in the expansions equals the number of points in the corresponding FM .Hence the C2 expansions are half as long as those of A1×A1 for the same quality of continuousexpansion.


M=8

M=16 M=32

M=4

Figure 6. Decomposition and continuous extension of a square step function placedin the fundamental region of C2 on the grids of orders M = 4, 8, 16 and 32.

Consequently, as much as one can draw a conclusion from an example of using only one functionf(x, y), it appears that the C2 expansions are considerably more efficient than the A1×A1 ones.

In a way of objection to the conclusion just made, one may point out that C-functions of C2are formed as sums of twice as many exponential functions (up to eight), in comparison with C-functions of A1×A1 (sums of up to four exponentials). Consequently, computing the values ofmore complicated C-functions of C2 on half as many points of its grid, may require comparablecomputational efforts to those for the simpler C-functions of A1×A1 on larger number of points.Although such an objection is undoubtedly true, it is hardly pertinent. Indeed, in any extensivecomputations of the expansions, the values of C-functions at the grid points would be calculated inadvance, and used as an look-up table during the actual expansions. Size of such look-up table isgiven by the number of points in FM , and by, what is the same, the number of discretely orthogonalC-functions in the set SM . Moreover, the same table should be used for expansion of any functionon the same grid.

9. Concluding remarks

1. In general terms, the families of C- and S-functions, based on any compact semisimple Liegroup, have all the properties of traditional special functions, and more [12]. Indeed, discretization,as demonstrated in this paper and generally in [3], is not a standard feature of traditional specialfunctions.

2. The application, which mostly motivated our interest in rank 2 group transforms, like thetwo considered in this paper, is the decomposition of functions sampled on 2D latices into finite


sums of discretized C-functions. The remaining two cases: the groups A2 and G2, are consideredin a similar manner in [1].

3. Recently it was recognized [7] that the continuous extension of the decompositions on thegrid FM is particularly useful. Once coefficients of a discrete expansion are found, one replaces inthe expansion the C-functions, sampled on FM , by their continuous versions. Unlike similar exten-sion of conventional Fourier expansions, continuous extensions of C-function expansions smoothlyinterpolate between grid points of FM . This property is likely to turn out extremely useful forvarious methods of image enhancement and data compression [7, 8].

4. In parallel with cosine transform, there exists a sine transform (see for example [20]). Sim-ilarly, as C-function transforms generalize cosine transform to any compact semisimple Lie group,the sine transforms generalize as S-function transforms [2] (continuous and discrete). Most of theproperties of C-transforms carry over as properties of S-transforms, one of the noticeable differencebeing their behaviors at the boundary of F .

5. In our opinion, independent interest represents the construction of the lattices in Rn throughthe grids in FM , even without the problem of expansion of functions on F ,. The flexibility anduniformity of its construction for any density specified by just one natural number M , should beexploited in other applications. Let us also single out the fact that the grid FM , for any M , isgroup-theoretically defined. It represents an Abelian subgroup of the maximal torus generated bythe elements of order M . Each point s ∈ FM is a representative of a conjugacy class of elements offinite order in the Lie group.

6. Among problems of interest related to this paper and to [1, 2], one can point out the followingones:

- The fact, that the C- and S-functions are the eigenfunctions of the Laplace operator withknown eigenvalues and known value at the boundary of F , should find a number of usefulapplications in physics. That property, along with their relative simplicity, distinguishesthem from the irreducible characters. Equally useful should prove to be the fact that theyform bases for lattice problems.

- Also, we are interested in the question of identification of the types of functions on F ,which are most efficiently decomposed into C- and/or S-function series for each of the foursemisimple Lie groups of rank 2. Examples of comparison are given in Section 8 and in [1],but more conclusive and more definite results would be interesting.

- Practical processing of 2D digital data often involves grids with millions of points. Thequestion of computational efficiency in large scale applications needs to be investigated.Furthermore, rational coordinates of points s ∈ FM make C-functions into linear combina-tions of M -th roots of unity. It was shown in [3] that some, even very large, decompositionproblems can be reformulated and done entirely in integers, for example [16].

- There exists a similarity to Fast Fourier Transform, which merits further investigation.Indeed, one has the freedom to work with series of gradually refining grids, for exampleF2 ⊂ F22 ⊂ · · · ⊂ F2k ⊂ · · · .

7. Every C- or S-function is a sum of a finite number of exponential functions. In rank 2, thatis:

e2πi〈λ|z〉 = e2πiA|C| θ1+2πi

B|C| θ2 ,

where A and B are integers and |C| is the determinant of the corresponding Cartan matrix. Thesubstitution

e2πi

θ1|C| −→ x, e2πi

θ2|C| −→ y

transforms any C- and S-function into a polynomial in x and y. Indeed, one gets e2πi〈λ|z〉 = xAyB.


Thus, C- and S-functions are families of orthogonal polynomials, each related to a particularsemisimple Lie group and to a particular W -orbit, in as many variables as is the rank of the group.

10. Acknowledgements

We are grateful for the support from the National Science and Engineering Research Councilof Canada, Laboratoires Universitaires Bell, MITACS, and to Lockheed Martin Canada. We arealso grateful to A. Atoyan, J.P. Gazeau and A. Klimyk for their helpful comments. One of us(A.Z.) would also like to acknowledge the scholarship for doctorate studies from the Departmentof Mathematics and Statistics of University of Montreal.

References

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[19] R.V. Moody, A. Pianzola, Lie algebras with triangular decompositions, Wiley, New York 1995[20] M. Frigo, S. G. Johnson, FFTW, A free (GPL) C library that can compute fast DSTs (types I-IV) in one or

more dimensions, of arbitrary size. http://www.fftw.org/

Centre de recherches mathématiques, Université de Montréal, C.P. 6128 succ. Centre-Ville,Montréal, Québec H3C3J7, Canada.

E-mail address: [email protected], [email protected]

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