Acta Polytechnica


doi:10.14311/AP.2016.56.0156
Acta Polytechnica 56(3):156–165, 2016 © Czech Technical University in Prague, 2016

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

THREE-VARIABLE ALTERNATING TRIGONOMETRIC
FUNCTIONS AND CORRESPONDING FOURIER TRANSFORMS

Agata Bezubika, Severin Poštab,∗

a Institute of Mathematics, University of Białystok, Akademicka 2, 15-267 Białystok, Poland
b Department of mathematics, Faculty of nuclear sciences and physical engineering, Czech Technical University

in Prague, Trojanova 13, 120 00 Prague, Czech Republic
∗ corresponding author: severin.posta@fjfi.cvut.cz

Abstract. The common trigonometric functions admit generalizations to any higher dimension.
In this paper, we restrict ourselves to three dimensional generalization only, focusing on alternating
case in detail. Many specific properties of this new class of special functions are studied, such as the
orthogonalities, both the continuous one and the discrete one on the 3D lattice of any density, discrete
and continuous Fourier transforms, and others. Rapidly increasing precision of the interpolation with
increasing density of the 3D lattice is shown in an example.
Keywords: trigonometric functions; orthogonal polynomials.

1. Introduction
There are many applications of functions which are symmetric or antisymmetric with respect to the symmetry
group Sn in mathematics. They appear for example in quantum theory or in theory of integrable systems.
In [1] (anti)symmetric multivariate exponential functions were defined and corresponding Fourier transforms
were given. These functions were examined in detail in dimension two and three in [2]. The natural question
which arises is the restriction of the symmetry to the subgroup An of Sn consisting of transformations w with
detw = 1. Such functions were examined in [3] for An involving the general number n, and in [4] in greater
detail concerning the smallest possible n, namely n = 3.

The exponential functions are not the only functions which admit symmetric, antisymmetric and alternating
generalizations to higher dimensions. Their odd and even counterparts, sines and cosines, admit similar
generalizations and fulfil many analogical properties as pure exponentials. Symmetric and antisymmetric case in
general was covered in [5], alternating case was covered in [6]. Two-dimensional symmetric and antisymmetric
generalizations were studied in detail in [7, 8].

The aim of this paper is to study in detail the alternating generalization of sine and cosine functions in
dimension three.

(Anti)symmetric multivariate sine and cosine functions, considered in [5, 7, 8], as well as alternating
multivariate sine and cosine functions, studied in [6] and in this paper, are closely related to symmetric and
antisymmetric orbit functions studied in [9–11]. Because the definition of the considered functions and orbit
functions are similar (roughly speaking, the exponentials are replaced by sines and cosines), they satisfy the
same or similar properties, namely symmetry relations, reductions to the dominant or semidominant forms,
periodicity etc. The discrete Fourier transforms of sine and cosine functions can be derived with the help of
relations valid for exponential functions.[8] The restriction of the functions to three variables allows us to be
more specific about the details of their properties in the following sections. This is most notable when describing
their discretization and orthogonality relations.

The paper consists of several parts. After the first part, devoted to the definition of three dimensional
alternating sine and cosine functions and their basic properties, come the parts describing their continuous
orthogonality relations, their decomposition rules and the two parts treating four kinds of discrete cosine and
sine transforms.

2. Definition
Three dimensional alternating sine functions sin(λ,µ,ν) : R3 → R have the following explicit form [6]:

sin(λ,µ,ν)(x, y, z) = 1
2

∣∣∣∣∣∣
sin 2πλx sin 2πλy sin 2πλz
sin 2πµx sin 2πµy sin 2πµz
sin 2πνx sin 2πνy sin 2πνz

∣∣∣∣∣∣+ 1
2

∣∣∣∣∣∣
sin 2πλx sin 2πλy sin 2πλz
sin 2πµx sin 2πµy sin 2πµz
sin 2πνx sin 2πνy sin 2πνz

∣∣∣∣∣∣
+

= sin 2πλx sin 2πµy sin 2πνz + sin 2πλz sin 2πµx sin 2πνy + sin 2πλy sin 2πµz sin 2πνx,
x, y, z, λ, µ, ν ∈ R, (1)

156

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vol. 56 no. 3/2016 Three-Variable Alternating Trigonometric Functions

where the second determinant with superscipt + stands for permanent [12], which is symmetric with respect to
permutations of its rows and columns.

Three dimensional alternating cosine functions cos(λ,µ,ν) : R3 → R are defined similarly:

cos(λ,µ,ν)(x, y, z) = 1
2

∣∣∣∣∣∣
cos 2πλx cos 2πλy cos 2πλz
cos 2πµx cos 2πµy cos 2πµz
cos 2πνx cos 2πνy cos 2πνz

∣∣∣∣∣∣+ 1
2

∣∣∣∣∣∣
cos 2πλx cos 2πλy cos 2πλz
cos 2πµx cos 2πµy cos 2πµz
cos 2πνx cos 2πνy cos 2πνz

∣∣∣∣∣∣
+

= cos 2πλx cos 2πµy cos 2πνz + cos 2πλz cos 2πµx cos 2πνy + cos 2πλy cos 2πµz cos 2πνx,
x, y, z, λ, µ, ν ∈ R. (2)

From (1) we immediately see the (rotational) symmetry of sin(λ,µ,ν)(x, y, z) with respect to cyclic permutations
of variables (x, y, z) and (λ, µ, ν):

sin(λ,µ,ν)(x, y, z) = sin(λ,µ,ν)(z, x, y) = sin(λ,µ,ν)(y, z, x) = sin(ν,λ,µ)(x, y, z) = sin(µ,ν,λ)(x, y, z).

From (2) we have similar symmetries for cos(λ,µ,ν)(x, y, z):

cos(λ,µ,ν)(x, y, z) = cos(λ,µ,ν)(z, x, y) = cos(λ,µ,ν)(y, z, x) = cos(ν,λ,µ)(x, y, z) = cos(µ,ν,λ)(x, y, z).

The rotational symmetries above allow us to put any real triple (λ, µ, ν) to the canonical form. For example, we
may restrict ourselves to the functions sin(λ,µ,ν) and cos(λ,µ,ν) with so called semidominant (λ, µ, ν), that is,
triples (λ, µ, ν) where λ ≥ µ ≥ ν or µ > λ > ν holds.

The functions sin(k,l,m) and cos(k,l,m) with k, l,m ∈ Z have additional symmetries induced by the periodicity
of sine and cosine functions:

sin(k,l,m)(x+ r, y + s, z + t) = sin(k,l,m)(x, y, z), r, s, t ∈ Z,
cos(k,l,m)(x+ r, y + s, z + t) = cos(k,l,m)(x, y, z), r, s, t ∈ Z. (3)

Another symmetries of sin(λ,µ,ν) and cos(λ,µ,ν) follow from the well known fact that ordinary sine and cosine
functions of one variable are odd and even functions, respectively. Therefore for all λ, µ, ν and x, y, z we have

sin(λ,µ,ν)(−x, y, z) = sin(λ,µ,ν)(x,−y, z) = sin(λ,µ,ν)(x, y,−z) = − sin(λ,µ,ν)(x, y, z),
cos(λ,µ,ν)(−x, y, z) = cos(λ,µ,ν)(x,−y, z) = cos(λ,µ,ν)(x, y,−z) = cos(λ,µ,ν)(x, y, z). (4)

There is also duality of sin(λ,µ,ν) and cos(λ,µ,ν) functions,

sin(λ,µ,ν)(x, y, z) = sin(x,y,z)(λ, µ, ν), cos(λ,µ,ν)(x, y, z) = cos(x,y,z)(λ, µ, ν). (5)

Moreover,
sin(k,l,m)(x, y, z) = 0 when k = 0 or l = 0 or m = 0, (6)

therefore we can exclude sin(k,l,m) with at least one index being zero from our considerations.
The relations (3)–(6) imply that it is sufficient to restrict to the following families of functions:

sin(k,l,m), k, l,m ∈ N, k ≥ l ≥ m or l > k > m,

and
cos(k,l,m), k, l,m ∈ Z, k ≥ l ≥ m ≥ 0 or l > k > m ≥ 0,

and to consider them on the closure of the fundamental domain F (Ãaff
3 ) [6] only. This fundamental domain of

the dual affine group of A3 can be chosen in 3D to be equal to the part of the cube shown in Fig. 1.

F (Ãaff
3 ) = {(x, y, z) ∈ (0, 1

2 )× (0, 1
2 )× (0, 1

2 )|x > z, y > z}.

3. Continuous orthogonality
The alternating trigonometric functions sin(k,l,m) resp. cos(k,l,m) are pairwise orthogonal on F (Ãaff

3 ). Let us
denote

Gklm =
{

3 if k = l = m,

1 otherwise.

157



Agata Bezubik, Severin Pošta Acta Polytechnica

xy

z

0

1

2
1

2

1

2

Figure 1. Fundamental domain F (Ãaff
3 )

Then we have ∫
F (Ãaff

3 )
sin(k,l,m)(x, y, z) sin(k′,l′,m′)(x, y, z) dxdy dz = Gklm

64 δkk′δll′δmm′ ,

where k, l,m, k′, l′,m′ ∈ N, k ≥ l ≥ m or l > k > m, k′ ≥ l′ ≥ m′ or l′ > k′ > m′, and∫
F (Ãaff

3 )
cos(k,l,m)(x, y, z) cos(k′,l′,m′)(x, y, z) dxdy dz = Gklm

64 δkk′δll′δmm′ ,

where k, l,m, k′, l′,m′ ∈ {0, 1, 2, . . . }, k ≥ l ≥ m or l > k > m, k′ ≥ l′ ≥ m′ or l′ > k′ > m′.
Let us have a function f : R3 → R with the following properties: rotational symmetry, that is,

f(x, y, z) = f(y, z, x) = f(z, x, y);
periodic, that is

f(x+ r, y + s, z + t) = f(x, y, z) for all r, s, t ∈ Z,
and odd in each variable, that is,

f(−x, y, z) = f(x,−y, z) = f(x, y,−z) = −f(x, y, z).

Let f be sufficiently smooth. Then it can be expanded in terms of the alternating sine functions, respectively,
using formulas

f(x, y, z) =
∑

k,l,m≥1
k≥l≥m or l>k>m

cklm sin(k,l,m)(x, y, z),

cklm = 64G−1
klm

∫
F (Ãaff

3 )
f(x, y, z) sin(k,l,m)(x, y, z)dxdydz.

Similarly, any function f̃ : R3 → R with the symmetries

f̃(x, y, z) = f(y, z, x) = f(z, x, y),
f̃(x+ r, y + s, z + t) = f(x, y, z) for all r, s, t ∈ Z,
f̃(−x, y, z) = f(x,−y, z) = f(x, y,−z) = f(x, y, z),

and sufficiently smooth can be expanded in terms of the alternating cosine functions. The expansion is done
using the formulas

f̃(x, y, z) =
∑

k,l,m≥0
k≥l≥m or l>k>m

c̃klm cos(k,l,m)(x, y, z),

c̃klm = 64G−1
klm

∫
F (Ãaff

3 )
f̃(x, y, z) cos(k,l,m)(x, y, z)dxdydz.

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vol. 56 no. 3/2016 Three-Variable Alternating Trigonometric Functions

xy

z

0

1

2
1

2

1

2

Figure 2. Finite grid of points in F (Ãaff
3 )

4. Product decomposition
The product of two alternating cosines, cos(λ,µ,ν) cos(λ′,µ′,ν′) (we omit general argument (x, y, z) here) can be
decomposed into the sum of alternating cosines using the following formulas. The decomposition can serve e. g.
for the derivation of recursion relations for the considered functions.

We have

cos(λ,µ,ν) cos(λ′,µ′,ν′) = 1
8
(
cos(λ−λ′,µ−µ′,ν−ν′) + cos(λ−λ′,µ−µ′,ν+ν′) + cos(λ−λ′,µ+µ′,ν−ν′)

+ cos(λ−λ′,µ+µ′,ν+ν′) + cos(λ+λ′,µ−µ′,ν−ν′) + cos(λ+λ′,µ−µ′,ν+ν′) + cos(λ+λ′,µ+µ′,ν−ν′) + cos(λ+λ′,µ+µ′,ν+ν′)

+ cos(λ−µ′,µ−ν′,ν−λ′) + cos(λ−µ′,µ−ν′,λ′+ν) + cos(λ−µ′,µ+ν′,ν−λ′) + cos(λ−µ′,µ+ν′,λ′+ν) + cos(λ+µ′,µ−ν′,ν−λ′)

+ cos(λ+µ′,µ−ν′,λ′+ν) + cos(λ+µ′,µ+ν′,ν−λ′) + cos(λ+µ′,µ+ν′,λ′+ν) + cos(λ−ν′,µ−λ′,ν−µ′) + cos(λ−ν′,µ−λ′,µ′+ν)

+ cos(λ−ν′,λ′+µ,ν−µ′) + cos(λ−ν′,λ′+µ,µ′+ν) + cos(λ+ν′,µ−λ′,ν−µ′) + cos(λ+ν′,−λ′+µ,µ′+ν) + cos(λ+ν′,λ′+µ,ν−µ′) +
cos(λ+ν′,λ′+µ,µ′+ν)

)
. (7)

Similarly, the product of two alternating sines, sin(λ,µ,ν) sin(λ′,µ′,ν′) can be decomposed into the sum of alternating
cosines using the following formula:

sin(λ,µ,ν) sin(λ′,µ′,ν′) = 1
8
(
cos(λ−λ′,µ−µ′,ν−ν′)− cos(λ−λ′,µ−µ′,ν+ν′)− cos(λ−λ′,µ+µ′,ν−ν′)

+ cos(λ−λ′,µ+µ′,ν+ν′)− cos(λ+λ′,µ−µ′,ν−ν′) + cos(λ+λ′,µ−µ′,ν+ν′) + cos(λ+λ′,µ+µ′,ν−ν′)− cos(λ+λ′,µ+µ′,ν+ν′)

+ cos(λ−µ′,µ−ν′,ν−λ′)− cos(λ−µ′,µ−ν′,λ′+ν)− cos(λ−µ′,µ+ν′,ν−λ′) + cos(λ−µ′,µ+ν′,λ′+ν)− cos(λ+µ′,µ−ν′,ν−λ′)

+ cos(λ+µ′,µ−ν′,λ′+ν) + cos(λ+µ′,µ+ν′,ν−λ′)− cos(λ+µ′,µ+ν′,λ′+ν) + cos(λ−ν′,µ−λ′,ν−µ′)− cos(λ−ν′,µ−λ′,µ′+ν)

− cos(λ−ν′,λ′+µ,ν−µ′) + cos(λ−ν′,λ′+µ,µ′+ν)− cos(λ+ν′,µ−λ′,ν−µ′) + cos(λ+ν′,µ−λ′,µ′+ν) + cos(λ+ν′,λ′+µ,ν−µ′)

− cos(λ+ν′,λ′+µ,µ′+ν)
)
. (8)

For the sake of completeness, one can decompose also the product sin(λ,µ,ν) cos(λ′,µ′,ν′). This can be however
easily obtained from the decomposition (7) by substitution cos→ sin on the right hand side.

5. Discrete cosine transforms
To each of standard discrete cosine transforms, denoted in the literature often as DCT-1, DCT-2, DCT-3 and
DCT-4 [13], there corresponds an alternating three-dimensional discrete cosine transforms. We denote these
corresponding transforms as AMDCT-1, AMDCT-2, AMDCT-3, AMDCT-4.

5.1. AMDCT-1
For given N ∈ N let us define a lattice

FN =
{

( r
2N ,

s
2N ,

t
2N )

∣∣ 0 ≤ r, s, t ≤ N, r ≥ s ≥ t or s > r > t
}
. (9)

159



Agata Bezubik, Severin Pošta Acta Polytechnica

Figure 3. The function (10) used in the interpolation example.

N
∫
F (Ãaff

3 ) |f − fN |
2

1 0.0486711
2 0.0391407
3 0.0096493
4 0.0029516

Table 1. Integral error estimates
for the approximations in Fig. 4.

Figure 4. Alternating cosines approximations of (10) for N = 1, 2, 3, 4.

This lattice is chosen such way that it fulfills the whole space when symmetries (3) and (4) are applied to it.
Evidently, we have a partial freedom which parts of boundary to include in the lattice. The lattice (9) contains
1
3 (N3 + 3N2 + 5N + 3) points. For example,

F2 =
{

(0, 0, 0), ( 1
4 , 0, 0), ( 1

4 ,
1
4 , 0), ( 1

4 ,
1
4 ,

1
4 ), ( 1

4 ,
1
2 , 0), ( 1

2 , 0, 0), ( 1
2 ,

1
4 , 0), ( 1

2 ,
1
4 ,

1
4 ), ( 1

2 ,
1
2 , 0), ( 1

2 ,
1
2 ,

1
4 ), ( 1

2 ,
1
2 ,

1
2 )
}
.

Another example, for N = 5, is shown in Fig. 2.
The alternating cosines cos(k,l,m) are pairwise orthogonal on this lattice, namely we have∑

0≤r,s,t≤N,
r≥s≥t or s>r>t

G−1
rstcrcsct cos(k,l,m)( r

2N ,
s

2N ,
t

2N ) cos(k′,l′,m′)( r
2N ,

s
2N ,

t
2N ) = N3c̃k c̃lc̃mGklmδ(k,l,m),(k′,l′,m′),

where

cα =
{

1
2 if α = 0 or α = N,

1 otherwise,
c̃α =

{
1 if α = 0 or α = N,
1
2 otherwise,

and 0 ≤ k, l,m ≤ N , k ≥ l ≥ m or l > k > m, and 0 ≤ k′, l′,m′ ≤ N , k′ ≥ l′ ≥ m′ or l′ > k′ > m′.
Let f be a real function defined on FN . Then it can be expanded into a sum of alternating cosines,

f(x, y, z) =
∑

0≤k,l,m≤N,
k≥l≥m or l>k>m

a(k,l,m) cos(k,l,m)(x, y, z),

where the coefficients a(k,l,m) are given by the formula

a(k,l,m) = 1
N3ckclcmGklm

∑
0≤r,s,t≤N,

r≥s≥t or s>r>t

c̃r c̃sc̃t
Grst

f( r
2N ,

s
2N ,

t
2N ) cos(k,l,m)( r

2N ,
s

2N ,
t

2N ).

As an interpolation example, we take

f(x, y, z) = cos 2πx cos 2πy cos 2πz cos 10(x+ y + z), (10)

and compute four interpolations fN for N = 1, 2, 3, 4 using alternating cosines. The graph of the function (10)
is shown in Fig. 3 and four approximations in Fig. 4.

By visual inspection, approximations fN get closer to f as N increases. This is verified by computing integral
error estimates shown in Table 1.

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vol. 56 no. 3/2016 Three-Variable Alternating Trigonometric Functions

5.2. AMDCT-2
Alternating discrete cosine transform of second kind uses a lattice

F̃N =
{ ( r+1/2

2N , s+1/2
2N , t+1/2

2N
) ∣∣ 0 ≤ r, s, t ≤ N − 1, r ≥ s ≥ t or s > r > t

}
. (11)

This lattice contains 1
3N(N2 + 2) points. For example,

F̃2 =
{

( 1
8 ,

1
8 ,

1
8 ), ( 3

8 ,
1
8 ,

1
8 ), ( 3

8 ,
3
8 ,

1
8 ), ( 3

8 ,
3
8 ,

3
8 )
}
.

The alternating cosines cos(k,l,m) are pairwise orthogonal on this lattice, namely we have∑
0≤r,s,t≤N−1,
r≥s≥t or s>r>t

G−1
rstcrcsct cos(k,l,m)

( r+1/2
2N , s+1/2

2N , t+1/2
2N

)
cos(k′,l′,m′)

( r+1/2
2N , s+1/2

2N , t+1/2
2N

)
= N3c̃k c̃lc̃mGklmδ(k,l,m),(k′,l′,m′),

where 0 ≤ k, l,m ≤ N − 1, k ≥ l ≥ m or l > k > m, and 0 ≤ k′, l′,m′ ≤ N − 1, k′ ≥ l′ ≥ m′ or l′ > k′ > m′.
Let f̃ be a real function defined on F̃N . Then it can be expanded into a sum of alternating cosines,

f̃(x, y, z) =
∑

0≤k,l,m≤N−1,
k≥l≥m or l>k>m

b(k,l,m) cos(k,l,m)(x, y, z),

where the coefficients b(k,l,m) are given by the formula

b(k,l,m) = 1
N3ckclcmGklm

∑
0≤r,s,t≤N−1,
r≥s≥t or s>r>t

c̃r c̃sc̃t
Grst

f̃
( r+1/2

2N , s+1/2
2N , t+1/2

2N
)

cos(k,l,m)
( r+1/2

2N , s+1/2
2N , t+1/2

2N
)
.

5.3. AMDCT-3
Alternating discrete cosine transform of the third kind uses the lattice

F ′N =
{

( r
2N ,

s
2N ,

t
2N )

∣∣ 0 ≤ r, s, t ≤ N − 1, r ≥ s ≥ t or s > r > t
}
.

Instead of alternating cosines cos(k,l,m), shifted cosines cos(k+1/2,l+1/2,m+1/2) are used. They have similar
properties as normal alternating cosines with integer arguments. For scalar product we obtain∑

0≤r,s,t≤N−1,
r≥s≥t or s>r>t

G−1
rstcrcsct cos(k+1/2,l+1/2,m+1/2)( r

2N ,
s

2N ,
t

2N ) cos(k′+1/2,l′+1/2,m′+1/2)( r
2N ,

s
2N ,

t
2N )

= N3c̃k c̃lc̃mGklmδ(k,l,m),(k′,l′,m′),

where 0 ≤ k, l,m ≤ N − 1, k ≥ l ≥ m or l > k > m, and 0 ≤ k′, l′,m′ ≤ N − 1, k′ ≥ l′ ≥ m′ or l′ > k′ > m′.
Let g be a real function defined on F ′N . Then it can be expanded into a sum of shifted alternating cosines,

g(x, y, z) =
∑

0≤k,l,m≤N−1,
k≥l≥m or l>k>m

c(k,l,m) cos(k+1/2,l+1/2,m+1/2)(x, y, z),

where the coefficients c(k,l,m) are given by the formula

c(k,l,m) = 1
N3ckclcmGklm

∑
0≤r,s,t≤N−1,
r≥s≥t or s>r>t

c̃r c̃sc̃t
Grst

g
(
r

2N ,
s

2N ,
t

2N
)

cos(k+1/2,l+1/2,m+1/2)
(
r

2N ,
s

2N ,
t

2N
)
.

5.4. AMDCT-4
Alternating discrete cosine transform of the fourth kind uses the lattice F̃N and shifted cosines as in the third
case. We have∑

0≤r,s,t≤N−1,
r≥s≥t or s>r>t

G−1
rstcrcsct cos(k+1/2,l+1/2,m+1/2)

( r+1/2
2N , s+1/2

2N , t+1/2
2N

)
× cos(k′+1/2,l′+1/2,m′+1/2)

( r+1/2
2N , s+1/2

2N , t+1/2
2N

)
= N3c̃k c̃lc̃mGklmδ(k,l,m),(k′,l′,m′),

where 0 ≤ k, l,m ≤ N − 1, k ≥ l ≥ m or l > k > m, and 0 ≤ k′, l′,m′ ≤ N − 1, k′ ≥ l′ ≥ m′ or l′ > k′ > m′.

161



Agata Bezubik, Severin Pošta Acta Polytechnica

Let f̃ be a real function defined on F̃N . Then it can be expanded into a sum

f̃(x, y, z) =
∑

0≤k,l,m≤N−1,
k≥l≥m or l>k>m

d(k,l,m) cos(k+1/2,l+1/2,m+1/2)(x, y, z),

where the coefficients d(k,l,m) are given by the formula

d(k,l,m) = 1
N3ckclcmGklm

∑
0≤r,s,t≤N−1,
r≥s≥t or s>r>t

c̃r c̃sc̃t
Grst

f̃
( r+1/2

2N , s+1/2
2N , t+1/2

2N
)

cos(k+1/2,l+1/2,m+1/2)
( r+1/2

2N , s+1/2
2N , t+1/2

2N
)
.

6. Discrete sine transforms
Analogical formulas for discrete sine transforms 1–4 can be derived. To each of standard discrete sine transforms,
denoted in the literature often as DST-1, DST-2, DST-3 and DST-4, there corresponds an alternating three-
dimensional discrete sine transforms. We denote these corresponding transforms as AMDST-1, AMDST-2,
AMDST-3, AMDST-4.

6.1. AMDST-1
For given N ∈ N let us define a lattice

F
(S)
N =

{
( r

2N ,
s

2N ,
t

2N )
∣∣ 1 ≤ r, s, t ≤ N − 1, r ≥ s ≥ t or s > r > t

}
.

This lattice contains 1
3 (N3 − 3N2 + 5N − 3) points. For example,

F
(S)
3 =

{
( 1

6 ,
1
6 ,

1
6 ), ( 1

3 ,
1
6 ,

1
6 ), ( 1

3 ,
1
3 ,

1
6 ), ( 1

3 ,
1
3 ,

1
3 )
}
.

The alternating sines sin(k,l,m) are pairwise orthogonal on this lattice, namely we have

∑
1≤r,s,t≤N−1,
r≥s≥t or s>r>t

G−1
rst sin(k,l,m)( r

2N ,
s

2N ,
t

2N ) sin(k′,l′,m′)( r
2N ,

s
2N ,

t
2N ) =

(N
2

)3
Gklmδ(k,l,m),(k′,l′,m′),

where 1 ≤ k, l,m ≤ N − 1, k ≥ l ≥ m or l > k > m, and 1 ≤ k′, l′,m′ ≤ N − 1, k′ ≥ l′ ≥ m′ or l′ > k′ > m′.
Let f (S) be a real function defined on F (S)

N . Then it can be expanded into a sum of alternating sines,

f (S)(x, y, z) =
∑

1≤k,l,m≤N−1,
k≥l≥m or l>k>m

a(k,l,m) sin(k,l,m)(x, y, z),

where the coefficients a(S)
(k,l,m) are given by the formula

a
(S)
(k,l,m) = 8

N3Gklm

∑
1≤r,s,t≤N−1,
r≥s≥t or s>r>t

1
Grst

f (S)( r
2N ,

s
2N ,

t
2N ) sin(k,l,m)( r

2N ,
s

2N ,
t

2N ).

As an interpolation example, we take

f (S)(x, y, z) = sin 2πx sin 2πy sin 2πz sin 10(x+ y + z), (12)

and compute four interpolations f (S)
N for N = 1, 2, 3, 4 using alternating sines. Picture of (12) is shown in Fig. 5

and four approximations in Fig. 6.
By visual inspection, approximations f (S)

N get closer to (12) as N increases. This is verified by computing
integral error estimates shown in Table 2.

6.2. AMDST-2
Alternating discrete sine transform of second kind uses the same lattice as AMDCT-2, that is

F̃
(S)
N = F̃N , (13)

where F̃N is given by (11).

162



vol. 56 no. 3/2016 Three-Variable Alternating Trigonometric Functions

Figure 5. The function (12) used in interpolation example.

N
∫
F (Ãaff

3 ) |f
(S) − f (S)

N |2

1 0.0482575
2 0.0282054
3 0.0078227
4 0.0006646

Table 2. Integral error estimates
for the approximations in Fig. 6.

Figure 6. The alternating sines approximations of (12) for N = 1, 2, 3, 4.

The alternating sines sin(k,l,m) are pairwise orthogonal on this lattice, namely we have∑
0≤r,s,t≤N−1,
r≥s≥t or s>r>t

G−1
rst sin(k,l,m)

( r+1/2
2N , s+1/2

2N , t+1/2
2N

)
sin(k′,l′,m′)

( r+1/2
2N , s+1/2

2N , t+1/2
2N

)
=
(N

2

)3
ckclcmGklmδ(k,l,m),(k′,l′,m′),

where cp = 1
2 for p = N and cp = 1 otherwise and 1 ≤ k, l,m ≤ N , k ≥ l ≥ m or l > k > m, and

1 ≤ k′, l′,m′ ≤ N , k′ ≥ l′ ≥ m′ or l′ > k′ > m′.
Let f̃ (S) be a real function defined on F̃ (S)

N . Then it can be expanded into a sum of alternating sines,

f̃ (S)(x, y, z) =
∑

1≤k,l,m≤N,
k≥l≥m or l>k>m

b
(S)
(k,l,m) sin(k,l,m)(x, y, z),

where the coefficients b(k,l,m) are given by the formula

b
(S)
(k,l,m) = 8

N3ckclcmGklm

∑
0≤r,s,t≤N−1,
r≥s≥t or s>r>t

1
Grst

f̃ (S)( r+1/2
2N , s+1/2

2N , t+1/2
2N

)
sin(k,l,m)

( r+1/2
2N , s+1/2

2N , t+1/2
2N

)
.

6.3. AMDST-3
Alternating discrete sine transform of the third kind uses the lattice

F ′N
(S) =

{
( r

2N ,
s

2N ,
t

2N )
∣∣ 1 ≤ r, s, t ≤ N, r ≥ s ≥ t or s > r > t

}
.

Instead of alternating sines sin(k,l,m), shifted sines sin(k+1/2,l+1/2,m+1/2) are used. They have similar properties
as normal alternating sines with integer arguments. For scalar product we obtain∑

1≤r,s,t≤N,
r≥s≥t or s>r>t

G−1
rstcrcsct sin(k+1/2,l+1/2,m+1/2)( r

2N ,
s

2N ,
t

2N ) sin(k′+1/2,l′+1/2,m′+1/2)( r
2N ,

s
2N ,

t
2N )

=
(N

2

)3
Gklmδ(k,l,m),(k′,l′,m′),

where cp = 1
2 for p = N and cp = 1 otherwise and 0 ≤ k, l,m ≤ N − 1, k ≥ l ≥ m or l > k > m, and

0 ≤ k′, l′,m′ ≤ N − 1, k′ ≥ l′ ≥ m′ or l′ > k′ > m′.

163



Agata Bezubik, Severin Pošta Acta Polytechnica

Let g(S) be a real function defined on F ′N
(S). Then it can be expanded into a sum of shifted alternating sines,

g(S)(x, y, z) =
∑

0≤k,l,m≤N−1,
k≥l≥m or l>k>m

c
(S)
(k,l,m) sin(k+1/2,l+1/2,m+1/2)(x, y, z),

where the coefficients c(S)
(k,l,m) are given by the formula

c
(S)
(k,l,m) = 8

N3Gklm

∑
1≤r,s,t≤N,

r≥s≥t or s>r>t

crcsct
Grst

g(S)( r
2N ,

s
2N ,

t
2N ) sin(k+1/2,l+1/2,m+1/2)( r

2N ,
s

2N ,
t

2N ).

6.4. AMDST-4
Alternating discrete sine transform of the fourth kind uses the lattice F̃ (S)

N given by (13) and shifted sines as in
the third case. We have∑

0≤r,s,t≤N−1,
r≥s≥t or s>r>t

G−1
rst sin(k+1/2,l+1/2,m+1/2)

( r+1/2
2N , s+1/2

2N , t+1/2
2N

)
sin(k′+1/2,l′+1/2,m′+1/2)

( r+1/2
2N , s+1/2

2N , t+1/2
2N

)
=
(N

2

)3
Gklmδ(k,l,m),(k′,l′,m′),

where 0 ≤ k, l,m ≤ N − 1, k ≥ l ≥ m or l > k > m, and 0 ≤ k′, l′,m′ ≤ N − 1, k′ ≥ l′ ≥ m′ or l′ > k′ > m′.
Let f̃ (S) be a real function defined on F̃ (S)

N . Then it can be expanded into a sum

f̃ (S)(x, y, z) =
∑

0≤k,l,m≤N−1,
k≥l≥m or l>k>m

d
(S)
(k,l,m) sin(k+1/2,l+1/2,m+1/2)(x, y, z),

where the coefficients d(S)
(k,l,m) are given by the formula

d
(S)
(k,l,m) = 8

N3Gklm

∑
0≤r,s,t≤N−1,
r≥s≥t or s>r>t

1
Grst

f̃
( r+1/2

2N , s+1/2
2N , t+1/2

2N
)

sin(k+1/2,l+1/2,m+1/2)
( r+1/2

2N , s+1/2
2N , t+1/2

2N
)
.

7. Conclusion
We have presented a detailed study of alternating three dimensional trigonometric functions and their properties
including product decomposition, continuous and discrete orthogonality and interpolation problem. Practical
computational aspects of the formalism presented in this paper need further investigation, namely a fast Fourier
transform analog and comparing to the usual discrete Fourier transform in three dimensions.

Acknowledgements
We gratefully acknowledge the support of this work by the Doppler Institute of the Czech Technical University in Prague.
AB is grateful for the hospitality extended to her at Department of mathematics FNSPE CTU. SP acknowledges the
support of SGS15/215/OHK4/3T/14, project of the Czech Technical University in Prague.

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	Acta Polytechnica 56(3):156–165, 2016
	1 Introduction
	2 Definition
	3 Continuous orthogonality
	4 Product decomposition
	5 Discrete cosine transforms
	5.1 AMDCT-1
	5.2 AMDCT-2
	5.3 AMDCT-3
	5.4 AMDCT-4

	6 Discrete sine transforms
	6.1 AMDST-1
	6.2 AMDST-2
	6.3 AMDST-3
	6.4 AMDST-4

	7 Conclusion
	Acknowledgements
	References